mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
⊢ Epi (πQ f₂ β)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₁ : B ⟶ X₁
hx₁ : x₁ ≫ f₁ = 0
this : ∃ k', (shiftFunctor C 1).map (ι.map x₁) = k' ≫ α ≫ f₃
⊢ ∃ k, k ≫ ιK f₃ α = x₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.SpectralObject

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b c : EInt
hab : a ≤ b
hbc : b ≤ c
X : C
⊢ (t.triangleω₁δ a b c hab hbc).obj X ≅ (t.eTriangleLTGE.obj b).obj ((t.ω₁.obj (mk₁ (homOfLE ⋯))).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.SpectralObject

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
X : C
D D' : ComposableArrows EInt 2
φ : D ⟶ D'
⊢ (functorArrows EInt 1 2 2 SpectralObject._proof_2 SpectralObject._proof_4 ⋙ t.ω₁ ⋙ (evaluation C C).obj X).map φ ≫
      (t.ω₁δ (D'.obj 0) (D'.obj 1) (D'.obj 2) ⋯ ⋯).app X =
    (t.ω₁δ (D.obj 0) (D.obj 1) (D.obj 2) ⋯ ⋯).app X ≫
      (functorArrows EInt 0 1 2 SpectralObject._proof_6 SpectralObject._proof_2 ⋙
            (t.ω₁ ⋙ (evaluation C C).obj X) ⋙ shiftFunctor C 1).map
        φ

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₁ : B ⟶ X₁
hx₁ : x₁ ≫ f₁ = 0
⊢ (shiftFunctor C 1).map (ι.map x₁) ≫ (shiftFunctor C 1).map (Triangle.mk (ι.map f₁) f₂ f₃).mor₁ = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.SpectralObject

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
X : C
D : ComposableArrows EInt 2
⊢ Triangle.mk
      ((t.ω₁ ⋙ (evaluation C C).obj X).map
        ((mapFunctorArrows EInt 0 1 0 2 2 SpectralObject._proof_6 SpectralObject._proof_8 SpectralObject._proof_10
              SpectralObject._proof_2 SpectralObject._proof_4).app
          D))
      ((t.ω₁ ⋙ (evaluation C C).obj X).map
        ((mapFunctorArrows EInt 0 2 1 2 2 SpectralObject._proof_8 SpectralObject._proof_2 SpectralObject._proof_6
              SpectralObject._proof_4 SpectralObject._proof_4).app
          D))
      ({ app := fun D ↦ (t.ω₁δ (D.obj 0) (D.obj 1) (D.obj 2) ⋯ ⋯).app X, naturality := ⋯ }.app D) ∈
    distinguishedTriangles

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₁ : B ⟶ X₁
hx₁ : x₁ ≫ f₁ = 0
x₃ : (shiftFunctor C 1).obj (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₃
hx₃ : (shiftFunctor C 1).map (ι.map x₁) = x₃ ≫ (Triangle.mk (ι.map f₁) f₂ f₃).mor₃
⊢ 0 < 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.WeakKernels

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
⊢ Nonempty (WeakLimitCone (parallelPair f 0))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.UnivLE

Goal state before:
pp: w : uliftFunctor.{u, v}.EssSurj
α : Type (max u v)
⊢ Small.{v, max u v} α

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₁ : B ⟶ X₁
hx₁ : x₁ ≫ f₁ = 0
x₃ : (shiftFunctor C 1).obj (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₃
hx₃ : (shiftFunctor C 1).map (ι.map x₁) = x₃ ≫ (Triangle.mk (ι.map f₁) f₂ f₃).mor₃
k' : (shiftFunctor C 1).obj (ι.obj B) ⟶ (Triangle.mk α β γ).obj₁
hk' : x₃ = k' ≫ (Triangle.mk α β γ).mor₁
⊢ (shiftFunctor C 1).map (ι.map x₁) = k' ≫ α ≫ f₃

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₁ : B ⟶ X₁
hx₁ : x₁ ≫ f₁ = 0
⊢ ∃ k, k ≫ ιK f₃ α = x₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
W'✝ : A
x₁ : W'✝ ⟶ X₁
hx₁ : x₁ ≫ f₁ = 0
m : W'✝ ⟶ K
hm : m ≫ ιK f₃ α = x₁
⊢ m = ⋯.choose

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
⊢ G.mapTriangle.obj T ∈ distinguishedTriangles

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₂ : X₂ ⟶ B
hx₂ : f₁ ≫ x₂ = 0
⊢ (Triangle.mk (ι.map f₁) f₂ f₃).mor₁ ≫ ι.map x₂ = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : G.IsTriangulated
⊢ F.IsTriangulated

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₂ : X₂ ⟶ B
hx₂ : f₁ ≫ x₂ = 0
x₁ : (Triangle.mk (ι.map f₁) f₂ f₃).obj₃ ⟶ ι.obj B
hx₁ : ι.map x₂ = (Triangle.mk (ι.map f₁) f₂ f₃).mor₂ ≫ x₁
⊢ 0 < 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : Category.{v_2, u_2} D
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : Preadditive C
inst✝⁹ : Preadditive D
inst✝⁸ : HasShift C ℤ
inst✝⁷ : HasShift D ℤ
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Pretriangulated D
E : C ≌ D
inst✝² : E.functor.CommShift ℤ
inst✝¹ : E.inverse.CommShift ℤ
inst✝ : E.CommShift ℤ
⊢ Equivalence.refl.IsTriangulated

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₂ : X₂ ⟶ B
hx₂ : f₁ ≫ x₂ = 0
x₁ : (Triangle.mk (ι.map f₁) f₂ f₃).obj₃ ⟶ ι.obj B
hx₁ : ι.map x₂ = (Triangle.mk (ι.map f₁) f₂ f₃).mor₂ ≫ x₁
k : (Triangle.mk α β γ).obj₃ ⟶ ι.obj B
hk : x₁ = (Triangle.mk α β γ).mor₂ ≫ k
⊢ ι.map (πQ f₂ β ≫ ι.preimage k) = ι.map x₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₂ : X₂ ⟶ B
hx₂ : f₁ ≫ x₂ = 0
⊢ ∃ k, πQ f₂ β ≫ k = x₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝²⁵ : Category.{v_1, u_1} C
inst✝²⁴ : Category.{v_2, u_2} D
inst✝²³ : HasZeroObject C
inst✝²² : HasZeroObject D
inst✝²¹ : Preadditive C
inst✝²⁰ : Preadditive D
inst✝¹⁹ : HasShift C ℤ
inst✝¹⁸ : HasShift D ℤ
inst✝¹⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝¹⁵ : Pretriangulated C
inst✝¹⁴ : Pretriangulated D
E : C ≌ D
inst✝¹³ : E.functor.CommShift ℤ
inst✝¹² : E.inverse.CommShift ℤ
inst✝¹¹ : E.CommShift ℤ
D' : Type u_3
inst✝¹⁰ : Category.{v_3, u_3} D'
inst✝⁹ : HasZeroObject D'
inst✝⁸ : Preadditive D'
inst✝⁷ : HasShift D' ℤ
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D' n).Additive
inst✝⁵ : Pretriangulated D'
E' : D ≌ D'
inst✝⁴ : E'.functor.CommShift ℤ
inst✝³ : E'.inverse.CommShift ℤ
inst✝² : E'.CommShift ℤ
inst✝¹ : E.IsTriangulated
inst✝ : E'.IsTriangulated
⊢ (E.trans E').IsTriangulated

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
Z'✝ : A
x₂ : X₂ ⟶ Z'✝
hx₂ : f₁ ≫ x₂ = 0
m : Q ⟶ Z'✝
hm : πQ f₂ β ≫ m = x₂
⊢ m = ⋯.choose

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝² : Preadditive A
inst✝¹ : ι.Full
inst✝ : ι.Faithful
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
hf₁ : admissibleMorphism ι f₁
⊢ HasKernel f₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝² : Preadditive A
inst✝¹ : ι.Full
inst✝ : ι.Faithful
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
hf₁ : admissibleMorphism ι f₁
⊢ HasCokernel f₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : HasFiniteProducts A
inst✝ : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) (ι.map f₂) f₃ ∈ distinguishedTriangles
⊢ ι.map (f₁ ≫ f₂) = ι.map 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : HasFiniteProducts A
inst✝ : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) (ι.map f₂) f₃ ∈ distinguishedTriangles
⊢ IsZero (Triangle.mk (𝟙 ((shiftFunctor C 1).obj (ι.obj X₁))) 0 0).obj₃

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : HasFiniteProducts A
inst✝ : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) (ι.map f₂) f₃ ∈ distinguishedTriangles
⊢ Triangle.mk (𝟙 ((shiftFunctor C 1).obj (ι.obj X₁))) 0 0 ∈ distinguishedTriangles

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : HasFiniteProducts A
inst✝ : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) (ι.map f₂) f₃ ∈ distinguishedTriangles
hT' : Triangle.mk (𝟙 ((shiftFunctor C 1).obj (ι.obj X₁))) 0 0 ∈ distinguishedTriangles
⊢ (ι ⋙ shiftFunctor C 1).map
      ((-Iso.refl
              (KernelFork.ofι
                  (ιK (-(shiftFunctor C 1).map (Triangle.mk (ι.map f₁) (ι.map f₂) f₃).mor₁)
                    (𝟙 ((shiftFunctor C 1).obj (ι.obj X₁))))
                  ⋯).pt).hom ≫
        Fork.ι (KernelFork.ofι f₁ ⋯)) =
    (ι ⋙ shiftFunctor C 1).map
      (Fork.ι
        (KernelFork.ofι
          (ιK (-(shiftFunctor C 1).map (Triangle.mk (ι.map f₁) (ι.map f₂) f₃).mor₁)
            (𝟙 ((shiftFunctor C 1).obj (ι.obj X₁))))
          ⋯))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : HasFiniteProducts A
inst✝ : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) (ι.map f₂) f₃ ∈ distinguishedTriangles
⊢ IsLimit (KernelFork.ofι f₁ ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : Preadditive C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
inst✝⁴ : Preadditive A
inst✝³ : ι.Full
inst✝² : ι.Faithful
inst✝¹ : HasFiniteProducts A
inst✝ : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) (ι.map f₂) f₃ ∈ distinguishedTriangles
⊢ Triangle.mk 0 (𝟙 (ι.obj X₃)) 0 ∈ distinguishedTriangles

Tactic:
by

Goal state after: