ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Pretriangulated.inv_rot_of_distTriang
theorem inv_rot_of_distTriang (T : Triangle C) (H : T ∈ distTriang C) :
    T.invRotate ∈ distTriang C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (H : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: T.invRotate ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Pretriangulated.comp_distTriang_mor_zero₁₂
@[reassoc]
theorem comp_distTriang_mor_zero₁₂ (T) (H : T ∈ (distTriang C)) : T.mor₁ ≫ T.mor₂ = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (H : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: T.mor₁ ≫ T.mor₂ = 0
-- CategoryTheory.Pretriangulated.comp_distTriang_mor_zero₂₃
@[reassoc]
theorem comp_distTriang_mor_zero₂₃ (T : Triangle C) (H : T ∈ distTriang C) :
    T.mor₂ ≫ T.mor₃ = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (H : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: T.mor₂ ≫ T.mor₃ = 0
-- CategoryTheory.Pretriangulated.comp_distTriang_mor_zero₃₁
@[reassoc]
theorem comp_distTriang_mor_zero₃₁ (T : Triangle C) (H : T ∈ distTriang C) :
    T.mor₃ ≫ T.mor₁⟦1⟧' = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (H : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: T.mor₃ ≫ (CategoryTheory.shiftFunctor C 1).map T.mor₁ = 0
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangle_X₁
@[simps]
def shortComplexOfDistTriangle (T : Triangle C) (hT : T ∈ distTriang C) : ShortComplex C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangle T hT).X₁ = T.obj₁
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangle_f
@[simps]
def shortComplexOfDistTriangle (T : Triangle C) (hT : T ∈ distTriang C) : ShortComplex C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangle T hT).f = T.mor₁
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangle_X₂
@[simps]
def shortComplexOfDistTriangle (T : Triangle C) (hT : T ∈ distTriang C) : ShortComplex C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangle T hT).X₂ = T.obj₂
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangle_g
@[simps]
def shortComplexOfDistTriangle (T : Triangle C) (hT : T ∈ distTriang C) : ShortComplex C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangle T hT).g = T.mor₂
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangle_X₃
@[simps]
def shortComplexOfDistTriangle (T : Triangle C) (hT : T ∈ distTriang C) : ShortComplex C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangle T hT).X₃ = T.obj₃
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_inv_τ₁
@[simps!]
def shortComplexOfDistTriangleIsoOfIso {T T' : Triangle C} (e : T ≅ T') (hT : T ∈ distTriang C) :
    shortComplexOfDistTriangle T hT ≅ shortComplexOfDistTriangle T'
      (isomorphic_distinguished _ hT _ e.symm) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] {T T' : CategoryTheory.Pretriangulated.Triangle C} (e : T ≅ T') (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso e hT).inv.τ₁ = e.inv.hom₁
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_inv_τ₂
@[simps!]
def shortComplexOfDistTriangleIsoOfIso {T T' : Triangle C} (e : T ≅ T') (hT : T ∈ distTriang C) :
    shortComplexOfDistTriangle T hT ≅ shortComplexOfDistTriangle T'
      (isomorphic_distinguished _ hT _ e.symm) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] {T T' : CategoryTheory.Pretriangulated.Triangle C} (e : T ≅ T') (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso e hT).inv.τ₂ = e.inv.hom₂
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_inv_τ₃
@[simps!]
def shortComplexOfDistTriangleIsoOfIso {T T' : Triangle C} (e : T ≅ T') (hT : T ∈ distTriang C) :
    shortComplexOfDistTriangle T hT ≅ shortComplexOfDistTriangle T'
      (isomorphic_distinguished _ hT _ e.symm) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] {T T' : CategoryTheory.Pretriangulated.Triangle C} (e : T ≅ T') (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso e hT).inv.τ₃ = e.inv.hom₃
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_hom_τ₂
@[simps!]
def shortComplexOfDistTriangleIsoOfIso {T T' : Triangle C} (e : T ≅ T') (hT : T ∈ distTriang C) :
    shortComplexOfDistTriangle T hT ≅ shortComplexOfDistTriangle T'
      (isomorphic_distinguished _ hT _ e.symm) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] {T T' : CategoryTheory.Pretriangulated.Triangle C} (e : T ≅ T') (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso e hT).hom.τ₂ = e.hom.hom₂
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_hom_τ₁
@[simps!]
def shortComplexOfDistTriangleIsoOfIso {T T' : Triangle C} (e : T ≅ T') (hT : T ∈ distTriang C) :
    shortComplexOfDistTriangle T hT ≅ shortComplexOfDistTriangle T'
      (isomorphic_distinguished _ hT _ e.symm) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] {T T' : CategoryTheory.Pretriangulated.Triangle C} (e : T ≅ T') (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso e hT).hom.τ₁ = e.hom.hom₁
-- CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_hom_τ₃
@[simps!]
def shortComplexOfDistTriangleIsoOfIso {T T' : Triangle C} (e : T ≅ T') (hT : T ∈ distTriang C) :
    shortComplexOfDistTriangle T hT ≅ shortComplexOfDistTriangle T'
      (isomorphic_distinguished _ hT _ e.symm) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] {T T' : CategoryTheory.Pretriangulated.Triangle C} (e : T ≅ T') (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso e hT).hom.τ₃ = e.hom.hom₃
-- CategoryTheory.Pretriangulated.distinguished_cocone_triangle₁
lemma distinguished_cocone_triangle₁ {Y Z : C} (g : Y ⟶ Z) :
    ∃ (X : C) (f : X ⟶ Y) (h : Z ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] {Y Z : C} (g : Y ⟶ Z)
Goal: ∃ X f h, CategoryTheory.Pretriangulated.Triangle.mk f g h ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Pretriangulated.distinguished_cocone_triangle₂
lemma distinguished_cocone_triangle₂ {Z X : C} (h : Z ⟶ X⟦(1 : ℤ)⟧) :
    ∃ (Y : C) (f : X ⟶ Y) (g : Y ⟶ Z), Triangle.mk f g h ∈ distTriang C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] {Z X : C} (h : Z ⟶ (CategoryTheory.shiftFunctor C 1).obj X)
Goal: ∃ Y f g, CategoryTheory.Pretriangulated.Triangle.mk f g h ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Pretriangulated.complete_distinguished_triangle_morphism₁
lemma complete_distinguished_triangle_morphism₁ (T₁ T₂ : Triangle C)
    (hT₁ : T₁ ∈ distTriang C) (hT₂ : T₂ ∈ distTriang C) (b : T₁.obj₂ ⟶ T₂.obj₂)
    (c : T₁.obj₃ ⟶ T₂.obj₃) (comm : T₁.mor₂ ≫ c = b ≫ T₂.mor₂) :
    ∃ (a : T₁.obj₁ ⟶ T₂.obj₁), T₁.mor₁ ≫ b = a ≫ T₂.mor₁ ∧
      T₁.mor₃ ≫ a⟦(1 : ℤ)⟧' = c ≫ T₂.mor₃ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T₁ T₂ : CategoryTheory.Pretriangulated.Triangle C) (hT₁ : T₁ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (hT₂ : T₂ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (b : T₁.obj₂ ⟶ T₂.obj₂) (c : T₁.obj₃ ⟶ T₂.obj₃) (comm : T₁.mor₂ ≫ c = b ≫ T₂.mor₂)
Goal: ∃ a, T₁.mor₁ ≫ b = a ≫ T₂.mor₁ ∧ T₁.mor₃ ≫ (CategoryTheory.shiftFunctor C 1).map a = c ≫ T₂.mor₃
-- CategoryTheory.Pretriangulated.complete_distinguished_triangle_morphism₂
lemma complete_distinguished_triangle_morphism₂ (T₁ T₂ : Triangle C)
    (hT₁ : T₁ ∈ distTriang C) (hT₂ : T₂ ∈ distTriang C) (a : T₁.obj₁ ⟶ T₂.obj₁)
    (c : T₁.obj₃ ⟶ T₂.obj₃) (comm : T₁.mor₃ ≫ a⟦(1 : ℤ)⟧' = c ≫ T₂.mor₃) :
    ∃ (b : T₁.obj₂ ⟶ T₂.obj₂), T₁.mor₁ ≫ b = a ≫ T₂.mor₁ ∧ T₁.mor₂ ≫ c = b ≫ T₂.mor₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T₁ T₂ : CategoryTheory.Pretriangulated.Triangle C) (hT₁ : T₁ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (hT₂ : T₂ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (a : T₁.obj₁ ⟶ T₂.obj₁) (c : T₁.obj₃ ⟶ T₂.obj₃) (comm : T₁.mor₃ ≫ (CategoryTheory.shiftFunctor C 1).map a = c ≫ T₂.mor₃)
Goal: ∃ b, T₁.mor₁ ≫ b = a ≫ T₂.mor₁ ∧ T₁.mor₂ ≫ c = b ≫ T₂.mor₂
-- CategoryTheory.Pretriangulated.contractible_distinguished₁
lemma contractible_distinguished₁ (X : C) :
    Triangle.mk (0 : 0 ⟶ X) (𝟙 X) 0 ∈ distTriang C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (X : C)
Goal: CategoryTheory.Pretriangulated.Triangle.mk 0 (𝟙 X) 0 ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Pretriangulated.contractible_distinguished₂
lemma contractible_distinguished₂ (X : C) :
    Triangle.mk (0 : X ⟶ 0) 0 (𝟙 (X⟦1⟧)) ∈ distTriang C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (X : C)
Goal: CategoryTheory.Pretriangulated.Triangle.mk 0 0 (𝟙 ((CategoryTheory.shiftFunctor C 1).obj X)) ∈
  CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Pretriangulated.Triangle.yoneda_exact₂
lemma yoneda_exact₂ {X : C} (f : T.obj₂ ⟶ X) (hf : T.mor₁ ≫ f = 0) :
    ∃ (g : T.obj₃ ⟶ X), f = T.mor₂ ≫ g 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) {X : C} (f : T.obj₂ ⟶ X) (hf : T.mor₁ ≫ f = 0)
Goal: ∃ g, f = T.mor₂ ≫ g
-- CategoryTheory.Pretriangulated.Triangle.yoneda_exact₃
lemma yoneda_exact₃ {X : C} (f : T.obj₃ ⟶ X) (hf : T.mor₂ ≫ f = 0) :
    ∃ (g : T.obj₁⟦(1 : ℤ)⟧ ⟶ X), f = T.mor₃ ≫ g 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) {X : C} (f : T.obj₃ ⟶ X) (hf : T.mor₂ ≫ f = 0)
Goal: ∃ g, f = T.mor₃ ≫ g
-- CategoryTheory.Pretriangulated.Triangle.coyoneda_exact₂
lemma coyoneda_exact₂ {X : C} (f : X ⟶ T.obj₂) (hf : f ≫ T.mor₂ = 0) :
    ∃ (g : X ⟶ T.obj₁), f = g ≫ T.mor₁ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) {X : C} (f : X ⟶ T.obj₂) (hf : f ≫ T.mor₂ = 0)
Goal: ∃ g, f = g ≫ T.mor₁
-- CategoryTheory.Pretriangulated.Triangle.coyoneda_exact₁
lemma coyoneda_exact₁ {X : C} (f : X ⟶ T.obj₁⟦(1 : ℤ)⟧) (hf : f ≫ T.mor₁⟦1⟧' = 0) :
    ∃ (g : X ⟶ T.obj₃), f = g ≫ T.mor₃ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) {X : C} (f : X ⟶ (CategoryTheory.shiftFunctor C 1).obj T.obj₁) (hf : f ≫ (CategoryTheory.shiftFunctor C 1).map T.mor₁ = 0)
Goal: ∃ g, f = g ≫ T.mor₃
-- CategoryTheory.Pretriangulated.Triangle.coyoneda_exact₃
lemma coyoneda_exact₃ {X : C} (f : X ⟶ T.obj₃) (hf : f ≫ T.mor₃ = 0) :
    ∃ (g : X ⟶ T.obj₂), f = g ≫ T.mor₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) {X : C} (f : X ⟶ T.obj₃) (hf : f ≫ T.mor₃ = 0)
Goal: ∃ g, f = g ≫ T.mor₂
-- CategoryTheory.Pretriangulated.Triangle.mor₃_eq_zero_iff_epi₂
lemma mor₃_eq_zero_iff_epi₂ : T.mor₃ = 0 ↔ Epi T.mor₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: T.mor₃ = 0 ↔ CategoryTheory.Epi T.mor₂
-- CategoryTheory.Pretriangulated.Triangle.mor₂_eq_zero_iff_epi₁
lemma mor₂_eq_zero_iff_epi₁ : T.mor₂ = 0 ↔ Epi T.mor₁ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: T.mor₂ = 0 ↔ CategoryTheory.Epi T.mor₁
-- CategoryTheory.Pretriangulated.Triangle.mor₁_eq_zero_iff_epi₃
lemma mor₁_eq_zero_iff_epi₃ : T.mor₁ = 0 ↔ Epi T.mor₃ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: T.mor₁ = 0 ↔ CategoryTheory.Epi T.mor₃
-- CategoryTheory.Pretriangulated.Triangle.mor₃_eq_zero_of_epi₂
lemma mor₃_eq_zero_of_epi₂ (h : Epi T.mor₂) : T.mor₃ = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (h : CategoryTheory.Epi T.mor₂)
Goal: T.mor₃ = 0
-- CategoryTheory.Pretriangulated.Triangle.mor₂_eq_zero_of_epi₁
lemma mor₂_eq_zero_of_epi₁ (h : Epi T.mor₁) : T.mor₂ = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (h : CategoryTheory.Epi T.mor₁)
Goal: T.mor₂ = 0
-- CategoryTheory.Pretriangulated.Triangle.mor₁_eq_zero_of_epi₃
lemma mor₁_eq_zero_of_epi₃ (h : Epi T.mor₃) : T.mor₁ = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (h : CategoryTheory.Epi T.mor₃)
Goal: T.mor₁ = 0
-- CategoryTheory.Pretriangulated.Triangle.epi₂
lemma epi₂ (h : T.mor₃ = 0) : Epi T.mor₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (h : T.mor₃ = 0)
Goal: CategoryTheory.Epi T.mor₂
-- CategoryTheory.Pretriangulated.Triangle.epi₁
lemma epi₁ (h : T.mor₂ = 0) : Epi T.mor₁ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (h : T.mor₂ = 0)
Goal: CategoryTheory.Epi T.mor₁
-- CategoryTheory.Pretriangulated.Triangle.epi₃
lemma epi₃ (h : T.mor₁ = 0) : Epi T.mor₃ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles) (h : T.mor₁ = 0)
Goal: CategoryTheory.Epi T.mor₃
-- CategoryTheory.Pretriangulated.Triangle.mor₁_eq_zero_iff_mono₂
lemma mor₁_eq_zero_iff_mono₂ : T.mor₁ = 0 ↔ Mono T.mor₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: T.mor₁ = 0 ↔ CategoryTheory.Mono T.mor₂
-- CategoryTheory.Pretriangulated.Triangle.mor₂_eq_zero_iff_mono₃
lemma mor₂_eq_zero_iff_mono₃ : T.mor₂ = 0 ↔ Mono T.mor₃ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: T.mor₂ = 0 ↔ CategoryTheory.Mono T.mor₃
-- CategoryTheory.Pretriangulated.Triangle.mor₃_eq_zero_iff_mono₁
lemma mor₃_eq_zero_iff_mono₁ : T.mor₃ = 0 ↔ Mono T.mor₁ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: T.mor₃ = 0 ↔ CategoryTheory.Mono T.mor₁