ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Functor.mapTriangleInvRotateIso_inv_app_hom₃
@[simps!]
noncomputable def mapTriangleInvRotateIso [F.Additive] :
    F.mapTriangle ⋙ Pretriangulated.invRotate D ≅
      Pretriangulated.invRotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleInvRotateIso.inv.app X).hom₃ = CategoryTheory.CategoryStruct.id (F.obj X.obj₂)
-- CategoryTheory.Functor.mapTriangleInvRotateIso_hom_app_hom₂
@[simps!]
noncomputable def mapTriangleInvRotateIso [F.Additive] :
    F.mapTriangle ⋙ Pretriangulated.invRotate D ≅
      Pretriangulated.invRotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleInvRotateIso.hom.app X).hom₂ = CategoryTheory.CategoryStruct.id (F.obj X.obj₁)
-- CategoryTheory.Functor.mapTriangleInvRotateIso_inv_app_hom₂
@[simps!]
noncomputable def mapTriangleInvRotateIso [F.Additive] :
    F.mapTriangle ⋙ Pretriangulated.invRotate D ≅
      Pretriangulated.invRotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleInvRotateIso.inv.app X).hom₂ = CategoryTheory.CategoryStruct.id (F.obj X.obj₁)
-- CategoryTheory.Functor.mapTriangleInvRotateIso_hom_app_hom₃
@[simps!]
noncomputable def mapTriangleInvRotateIso [F.Additive] :
    F.mapTriangle ⋙ Pretriangulated.invRotate D ≅
      Pretriangulated.invRotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleInvRotateIso.hom.app X).hom₃ = CategoryTheory.CategoryStruct.id (F.obj X.obj₂)
-- CategoryTheory.Functor.mapTriangleInvRotateIso_inv_app_hom₁
@[simps!]
noncomputable def mapTriangleInvRotateIso [F.Additive] :
    F.mapTriangle ⋙ Pretriangulated.invRotate D ≅
      Pretriangulated.invRotate C ⋙ F.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [F.Additive] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: (F.mapTriangleInvRotateIso.inv.app X).hom₁ = (F.commShiftIso (-1)).hom.app X.obj₃
-- CategoryTheory.Functor.mapTriangleIdIso_hom_app_hom₂
@[simps!]
def mapTriangleIdIso : (𝟭 C).mapTriangle ≅ 𝟭 _ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIdIso C).hom.app X).hom₂ = CategoryTheory.CategoryStruct.id X.obj₂
-- CategoryTheory.Functor.mapTriangleIdIso_inv_app_hom₁
@[simps!]
def mapTriangleIdIso : (𝟭 C).mapTriangle ≅ 𝟭 _ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIdIso C).inv.app X).hom₁ = CategoryTheory.CategoryStruct.id X.obj₁
-- CategoryTheory.Functor.mapTriangleIdIso_inv_app_hom₂
@[simps!]
def mapTriangleIdIso : (𝟭 C).mapTriangle ≅ 𝟭 _ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIdIso C).inv.app X).hom₂ = CategoryTheory.CategoryStruct.id X.obj₂
-- CategoryTheory.Functor.mapTriangleIdIso_inv_app_hom₃
@[simps!]
def mapTriangleIdIso : (𝟭 C).mapTriangle ≅ 𝟭 _ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIdIso C).inv.app X).hom₃ = CategoryTheory.CategoryStruct.id X.obj₃
-- CategoryTheory.Functor.mapTriangleIdIso_hom_app_hom₃
@[simps!]
def mapTriangleIdIso : (𝟭 C).mapTriangle ≅ 𝟭 _ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIdIso C).hom.app X).hom₃ = CategoryTheory.CategoryStruct.id X.obj₃
-- CategoryTheory.Functor.mapTriangleIdIso_hom_app_hom₁
@[simps!]
def mapTriangleIdIso : (𝟭 C).mapTriangle ≅ 𝟭 _ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIdIso C).hom.app X).hom₁ = CategoryTheory.CategoryStruct.id X.obj₁
-- CategoryTheory.Functor.mapTriangleCompIso_hom_app_hom₁
@[simps!]
def mapTriangleCompIso : (F ⋙ G).mapTriangle ≅ F.mapTriangle ⋙ G.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} {E : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} E] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.HasShift E ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] (G : CategoryTheory.Functor D E) [G.CommShift ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCompIso G).hom.app X).hom₁ = CategoryTheory.CategoryStruct.id (G.obj (F.obj X.obj₁))
-- CategoryTheory.Functor.mapTriangleCompIso_hom_app_hom₂
@[simps!]
def mapTriangleCompIso : (F ⋙ G).mapTriangle ≅ F.mapTriangle ⋙ G.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} {E : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} E] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.HasShift E ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] (G : CategoryTheory.Functor D E) [G.CommShift ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCompIso G).hom.app X).hom₂ = CategoryTheory.CategoryStruct.id (G.obj (F.obj X.obj₂))
-- CategoryTheory.Functor.mapTriangleCompIso_hom_app_hom₃
@[simps!]
def mapTriangleCompIso : (F ⋙ G).mapTriangle ≅ F.mapTriangle ⋙ G.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} {E : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} E] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.HasShift E ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] (G : CategoryTheory.Functor D E) [G.CommShift ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCompIso G).hom.app X).hom₃ = CategoryTheory.CategoryStruct.id (G.obj (F.obj X.obj₃))
-- CategoryTheory.Functor.mapTriangleCompIso_inv_app_hom₃
@[simps!]
def mapTriangleCompIso : (F ⋙ G).mapTriangle ≅ F.mapTriangle ⋙ G.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} {E : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} E] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.HasShift E ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] (G : CategoryTheory.Functor D E) [G.CommShift ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCompIso G).inv.app X).hom₃ = CategoryTheory.CategoryStruct.id (G.obj (F.obj X.obj₃))
-- CategoryTheory.Functor.mapTriangleCompIso_inv_app_hom₂
@[simps!]
def mapTriangleCompIso : (F ⋙ G).mapTriangle ≅ F.mapTriangle ⋙ G.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} {E : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} E] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.HasShift E ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] (G : CategoryTheory.Functor D E) [G.CommShift ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCompIso G).inv.app X).hom₂ = CategoryTheory.CategoryStruct.id (G.obj (F.obj X.obj₂))
-- CategoryTheory.Functor.mapTriangleCompIso_inv_app_hom₁
@[simps!]
def mapTriangleCompIso : (F ⋙ G).mapTriangle ≅ F.mapTriangle ⋙ G.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} {E : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} E] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.HasShift E ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] (G : CategoryTheory.Functor D E) [G.CommShift ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((F.mapTriangleCompIso G).inv.app X).hom₁ = CategoryTheory.CategoryStruct.id (G.obj (F.obj X.obj₁))
-- CategoryTheory.Functor.mapTriangleIso_hom_app_hom₂
@[simps!]
def mapTriangleIso {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ]
    [NatTrans.CommShift e.hom ℤ] : F₁.mapTriangle ≅ F₂.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] {F₁ F₂ : CategoryTheory.Functor C D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ] [CategoryTheory.NatTrans.CommShift e.hom ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIso e).hom.app X).hom₂ = e.hom.app X.obj₂
-- CategoryTheory.Functor.mapTriangleIso_inv_app_hom₁
@[simps!]
def mapTriangleIso {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ]
    [NatTrans.CommShift e.hom ℤ] : F₁.mapTriangle ≅ F₂.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] {F₁ F₂ : CategoryTheory.Functor C D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ] [CategoryTheory.NatTrans.CommShift e.hom ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIso e).inv.app X).hom₁ = e.inv.app X.obj₁
-- CategoryTheory.Functor.mapTriangleIso_inv_app_hom₂
@[simps!]
def mapTriangleIso {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ]
    [NatTrans.CommShift e.hom ℤ] : F₁.mapTriangle ≅ F₂.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] {F₁ F₂ : CategoryTheory.Functor C D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ] [CategoryTheory.NatTrans.CommShift e.hom ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIso e).inv.app X).hom₂ = e.inv.app X.obj₂
-- CategoryTheory.Functor.mapTriangleIso_inv_app_hom₃
@[simps!]
def mapTriangleIso {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ]
    [NatTrans.CommShift e.hom ℤ] : F₁.mapTriangle ≅ F₂.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] {F₁ F₂ : CategoryTheory.Functor C D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ] [CategoryTheory.NatTrans.CommShift e.hom ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIso e).inv.app X).hom₃ = e.inv.app X.obj₃
-- CategoryTheory.Functor.mapTriangleIso_hom_app_hom₃
@[simps!]
def mapTriangleIso {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ]
    [NatTrans.CommShift e.hom ℤ] : F₁.mapTriangle ≅ F₂.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] {F₁ F₂ : CategoryTheory.Functor C D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ] [CategoryTheory.NatTrans.CommShift e.hom ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIso e).hom.app X).hom₃ = e.hom.app X.obj₃
-- CategoryTheory.Functor.mapTriangleIso_hom_app_hom₁
@[simps!]
def mapTriangleIso {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ]
    [NatTrans.CommShift e.hom ℤ] : F₁.mapTriangle ≅ F₂.mapTriangle 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] {F₁ F₂ : CategoryTheory.Functor C D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ] [CategoryTheory.NatTrans.CommShift e.hom ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Functor.mapTriangleIso e).hom.app X).hom₁ = e.hom.app X.obj₁
-- CategoryTheory.Functor.IsTriangulated.map_distinguished
class IsTriangulated : Prop where
  map_distinguished (T : Triangle C) : (T ∈ distTriang C) → F.mapTriangle.obj T ∈ distTriang D
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] {F : CategoryTheory.Functor C D} [F.CommShift ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroObject D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Pretriangulated D] [self : F.IsTriangulated] (T : CategoryTheory.Pretriangulated.Triangle C)
Goal: T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles →
  F.mapTriangle.obj T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Functor.map_distinguished
lemma map_distinguished [F.IsTriangulated] (T : Triangle C) (hT : T ∈ distTriang C) :
    F.mapTriangle.obj T ∈ distTriang D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroObject D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Pretriangulated D] [F.IsTriangulated] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: F.mapTriangle.obj T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Functor.isTriangulated_of_iso
lemma isTriangulated_of_iso {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ]
    [NatTrans.CommShift e.hom ℤ] [F₁.IsTriangulated] : F₂.IsTriangulated 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroObject D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Pretriangulated D] {F₁ F₂ : CategoryTheory.Functor C D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ] [CategoryTheory.NatTrans.CommShift e.hom ℤ] [F₁.IsTriangulated]
Goal: F₂.IsTriangulated
-- CategoryTheory.Functor.isTriangulated_iff_of_iso
lemma isTriangulated_iff_of_iso {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ]
    [NatTrans.CommShift e.hom ℤ] : F₁.IsTriangulated ↔ F₂.IsTriangulated 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroObject D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Pretriangulated D] {F₁ F₂ : CategoryTheory.Functor C D} (e : F₁ ≅ F₂) [F₁.CommShift ℤ] [F₂.CommShift ℤ] [CategoryTheory.NatTrans.CommShift e.hom ℤ]
Goal: F₁.IsTriangulated ↔ F₂.IsTriangulated
-- CategoryTheory.Functor.mem_mapTriangle_essImage_of_distinguished
lemma mem_mapTriangle_essImage_of_distinguished
    [F.IsTriangulated] [F.mapArrow.EssSurj] (T : Triangle D) (hT : T ∈ distTriang D) :
    ∃ (T' : Triangle C) (_ : T' ∈ distTriang C), Nonempty (F.mapTriangle.obj T' ≅ T) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroObject D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Pretriangulated D] [F.IsTriangulated] [F.mapArrow.EssSurj] (T : CategoryTheory.Pretriangulated.Triangle D) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: ∃ T', ∃ (_ : T' ∈ CategoryTheory.Pretriangulated.distinguishedTriangles), Nonempty (F.mapTriangle.obj T' ≅ T)
-- CategoryTheory.Functor.isTriangulated_of_precomp
lemma isTriangulated_of_precomp
    [(F ⋙ G).IsTriangulated] [F.IsTriangulated] [F.mapArrow.EssSurj] :
    G.IsTriangulated 
Arguments: {C : Type u_1} {D : Type u_2} {E : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] [CategoryTheory.Category.{u_5, u_3} E] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.HasShift E ℤ] (F : CategoryTheory.Functor C D) [F.CommShift ℤ] (G : CategoryTheory.Functor D E) [G.CommShift ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroObject D] [CategoryTheory.Limits.HasZeroObject E] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [CategoryTheory.Preadditive E] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor E n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Pretriangulated D] [CategoryTheory.Pretriangulated E] [(F.comp G).IsTriangulated] [F.IsTriangulated] [F.mapArrow.EssSurj]
Goal: G.IsTriangulated
-- CategoryTheory.Functor.isTriangulated_of_precomp_iso
lemma isTriangulated_of_precomp_iso {H : C ⥤ E} (e : F ⋙ G ≅ H) [H.CommShift ℤ]
    [H.IsTriangulated] [F.IsTriangulated] [F.mapArrow.EssSurj] [NatTrans.CommShift e.hom ℤ] :
    G.IsTriangulated 
Arguments: {C : Type u_1} {D : Type u_2} {E : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] [CategoryTheory.Category.{u_5, u_3} E] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.HasShift E ℤ] {F : CategoryTheory.Functor C D} [F.CommShift ℤ] {G : CategoryTheory.Functor D E} [G.CommShift ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroObject D] [CategoryTheory.Limits.HasZeroObject E] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [CategoryTheory.Preadditive E] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor E n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Pretriangulated D] [CategoryTheory.Pretriangulated E] {H : CategoryTheory.Functor C E} (e : F.comp G ≅ H) [H.CommShift ℤ] [H.IsTriangulated] [F.IsTriangulated] [F.mapArrow.EssSurj] [CategoryTheory.NatTrans.CommShift e.hom ℤ]
Goal: G.IsTriangulated
-- CategoryTheory.Triangulated.Octahedron.map_m₃
@[simps]
def map : Octahedron (by dsimp; rw [← F.map_comp, comm])
    (F.map_distinguished _ h₁₂) (F.map_distinguished _ h₂₃) (F.map_distinguished _ h₁₃) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroObject D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Pretriangulated D] {X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C} {u₁₂ : X₁ ⟶ X₂} {u₂₃ : X₂ ⟶ X₃} {u₁₃ : X₁ ⟶ X₃} {comm : CategoryTheory.CategoryStruct.comp u₁₂ u₂₃ = u₁₃} {v₁₂ : X₂ ⟶ Z₁₂} {w₁₂ : Z₁₂ ⟶ (CategoryTheory.shiftFunctor C 1).obj X₁} {h₁₂ : CategoryTheory.Pretriangulated.Triangle.mk u₁₂ v₁₂ w₁₂ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles} {v₂₃ : X₃ ⟶ Z₂₃} {w₂₃ : Z₂₃ ⟶ (CategoryTheory.shiftFunctor C 1).obj X₂} {h₂₃ : CategoryTheory.Pretriangulated.Triangle.mk u₂₃ v₂₃ w₂₃ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles} {v₁₃ : X₃ ⟶ Z₁₃} {w₁₃ : Z₁₃ ⟶ (CategoryTheory.shiftFunctor C 1).obj X₁} {h₁₃ : CategoryTheory.Pretriangulated.Triangle.mk u₁₃ v₁₃ w₁₃ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles} (h : CategoryTheory.Triangulated.Octahedron comm h₁₂ h₂₃ h₁₃) (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [F.IsTriangulated]
Goal: (h.map F).m₃ = F.map h.m₃
-- CategoryTheory.Triangulated.Octahedron.map_m₁
@[simps]
def map : Octahedron (by dsimp; rw [← F.map_comp, comm])
    (F.map_distinguished _ h₁₂) (F.map_distinguished _ h₂₃) (F.map_distinguished _ h₁₃) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.HasShift C ℤ] [CategoryTheory.HasShift D ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroObject D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [CategoryTheory.Pretriangulated C] [CategoryTheory.Pretriangulated D] {X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C} {u₁₂ : X₁ ⟶ X₂} {u₂₃ : X₂ ⟶ X₃} {u₁₃ : X₁ ⟶ X₃} {comm : CategoryTheory.CategoryStruct.comp u₁₂ u₂₃ = u₁₃} {v₁₂ : X₂ ⟶ Z₁₂} {w₁₂ : Z₁₂ ⟶ (CategoryTheory.shiftFunctor C 1).obj X₁} {h₁₂ : CategoryTheory.Pretriangulated.Triangle.mk u₁₂ v₁₂ w₁₂ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles} {v₂₃ : X₃ ⟶ Z₂₃} {w₂₃ : Z₂₃ ⟶ (CategoryTheory.shiftFunctor C 1).obj X₂} {h₂₃ : CategoryTheory.Pretriangulated.Triangle.mk u₂₃ v₂₃ w₂₃ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles} {v₁₃ : X₃ ⟶ Z₁₃} {w₁₃ : Z₁₃ ⟶ (CategoryTheory.shiftFunctor C 1).obj X₁} {h₁₃ : CategoryTheory.Pretriangulated.Triangle.mk u₁₃ v₁₃ w₁₃ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles} (h : CategoryTheory.Triangulated.Octahedron comm h₁₂ h₂₃ h₁₃) (F : CategoryTheory.Functor C D) [F.CommShift ℤ] [F.IsTriangulated]
Goal: (h.map F).m₁ = F.map h.m₁