ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.ShortComplex.mapRightHomologyIso_inv_naturality
@[reassoc]
lemma mapRightHomologyIso_inv_naturality [S₁.HasRightHomology] [S₂.HasRightHomology]
    [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂] :
    F.map (rightHomologyMap φ) ≫ (S₂.mapRightHomologyIso F).inv =
      (S₁.mapRightHomologyIso F).inv ≫ rightHomologyMap (F.mapShortComplex.map φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasRightHomology] [S₂.HasRightHomology] [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.ShortComplex.rightHomologyMap φ))
    (S₂.mapRightHomologyIso F).inv =
  CategoryTheory.CategoryStruct.comp (S₁.mapRightHomologyIso F).inv
    (CategoryTheory.ShortComplex.rightHomologyMap (F.mapShortComplex.map φ))
-- CategoryTheory.ShortComplex.mapHomologyIso_hom_naturality
@[reassoc]
lemma mapHomologyIso_hom_naturality [S₁.HasHomology] [S₂.HasHomology]
    [(S₁.map F).HasHomology] [(S₂.map F).HasHomology]
    [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂] :
    @homologyMap _ _ _ (S₁.map F) (S₂.map F) (F.mapShortComplex.map φ) _ _ ≫
      (S₂.mapHomologyIso F).hom = (S₁.mapHomologyIso F).hom ≫ F.map (homologyMap φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasHomology] [S₂.HasHomology] [(S₁.map F).HasHomology] [(S₂.map F).HasHomology] [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap (F.mapShortComplex.map φ))
    (S₂.mapHomologyIso F).hom =
  CategoryTheory.CategoryStruct.comp (S₁.mapHomologyIso F).hom (F.map (CategoryTheory.ShortComplex.homologyMap φ))
-- CategoryTheory.ShortComplex.mapHomologyIso_inv_naturality
@[reassoc]
lemma mapHomologyIso_inv_naturality [S₁.HasHomology] [S₂.HasHomology]
    [(S₁.map F).HasHomology] [(S₂.map F).HasHomology]
    [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂] :
    F.map (homologyMap φ) ≫ (S₂.mapHomologyIso F).inv =
      (S₁.mapHomologyIso F).inv ≫
      @homologyMap _ _ _ (S₁.map F) (S₂.map F) (F.mapShortComplex.map φ) _ _ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasHomology] [S₂.HasHomology] [(S₁.map F).HasHomology] [(S₂.map F).HasHomology] [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.ShortComplex.homologyMap φ)) (S₂.mapHomologyIso F).inv =
  CategoryTheory.CategoryStruct.comp (S₁.mapHomologyIso F).inv
    (CategoryTheory.ShortComplex.homologyMap (F.mapShortComplex.map φ))
-- CategoryTheory.ShortComplex.mapHomologyIso'_hom_naturality
@[reassoc]
lemma mapHomologyIso'_hom_naturality [S₁.HasHomology] [S₂.HasHomology]
    [(S₁.map F).HasHomology] [(S₂.map F).HasHomology]
    [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂] :
    @homologyMap _ _ _ (S₁.map F) (S₂.map F) (F.mapShortComplex.map φ) _ _ ≫
      (S₂.mapHomologyIso' F).hom = (S₁.mapHomologyIso' F).hom ≫ F.map (homologyMap φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasHomology] [S₂.HasHomology] [(S₁.map F).HasHomology] [(S₂.map F).HasHomology] [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap (F.mapShortComplex.map φ))
    (S₂.mapHomologyIso' F).hom =
  CategoryTheory.CategoryStruct.comp (S₁.mapHomologyIso' F).hom (F.map (CategoryTheory.ShortComplex.homologyMap φ))
-- CategoryTheory.ShortComplex.mapHomologyIso'_inv_naturality
@[reassoc]
lemma mapHomologyIso'_inv_naturality [S₁.HasHomology] [S₂.HasHomology]
    [(S₁.map F).HasHomology] [(S₂.map F).HasHomology]
    [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂] :
    F.map (homologyMap φ) ≫ (S₂.mapHomologyIso' F).inv = (S₁.mapHomologyIso' F).inv ≫
      @homologyMap _ _ _ (S₁.map F) (S₂.map F) (F.mapShortComplex.map φ) _ _ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasHomology] [S₂.HasHomology] [(S₁.map F).HasHomology] [(S₂.map F).HasHomology] [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.ShortComplex.homologyMap φ)) (S₂.mapHomologyIso' F).inv =
  CategoryTheory.CategoryStruct.comp (S₁.mapHomologyIso' F).inv
    (CategoryTheory.ShortComplex.homologyMap (F.mapShortComplex.map φ))
-- CategoryTheory.ShortComplex.mapHomologyIso'_eq_mapHomologyIso
lemma mapHomologyIso'_eq_mapHomologyIso [S.HasHomology] [F.PreservesLeftHomologyOf S]
    [F.PreservesRightHomologyOf S] :
    S.mapHomologyIso' F = S.mapHomologyIso F 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (S : CategoryTheory.ShortComplex C) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S.HasHomology] [F.PreservesLeftHomologyOf S] [F.PreservesRightHomologyOf S]
Goal: S.mapHomologyIso' F = S.mapHomologyIso F
-- CategoryTheory.ShortComplex.LeftHomologyMapData.natTransApp_φH
@[simps]
def LeftHomologyMapData.natTransApp (h : LeftHomologyData S) (τ : F ⟶ G) :
    LeftHomologyMapData (S.mapNatTrans τ) (h.map F) (h.map G) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] [F.PreservesLeftHomologyOf S] [G.PreservesLeftHomologyOf S] (h : S.LeftHomologyData) (τ : F ⟶ G)
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.natTransApp h τ).φH = τ.app h.H
-- CategoryTheory.ShortComplex.LeftHomologyMapData.natTransApp_φK
@[simps]
def LeftHomologyMapData.natTransApp (h : LeftHomologyData S) (τ : F ⟶ G) :
    LeftHomologyMapData (S.mapNatTrans τ) (h.map F) (h.map G) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] [F.PreservesLeftHomologyOf S] [G.PreservesLeftHomologyOf S] (h : S.LeftHomologyData) (τ : F ⟶ G)
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.natTransApp h τ).φK = τ.app h.K
-- CategoryTheory.ShortComplex.RightHomologyMapData.natTransApp_φH
@[simps]
def RightHomologyMapData.natTransApp (h : RightHomologyData S) (τ : F ⟶ G) :
    RightHomologyMapData (S.mapNatTrans τ) (h.map F) (h.map G) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] [F.PreservesRightHomologyOf S] [G.PreservesRightHomologyOf S] (h : S.RightHomologyData) (τ : F ⟶ G)
Goal: (CategoryTheory.ShortComplex.RightHomologyMapData.natTransApp h τ).φH = τ.app h.H
-- CategoryTheory.ShortComplex.RightHomologyMapData.natTransApp_φQ
@[simps]
def RightHomologyMapData.natTransApp (h : RightHomologyData S) (τ : F ⟶ G) :
    RightHomologyMapData (S.mapNatTrans τ) (h.map F) (h.map G) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] [F.PreservesRightHomologyOf S] [G.PreservesRightHomologyOf S] (h : S.RightHomologyData) (τ : F ⟶ G)
Goal: (CategoryTheory.ShortComplex.RightHomologyMapData.natTransApp h τ).φQ = τ.app h.Q
-- CategoryTheory.ShortComplex.HomologyMapData.natTransApp_right
@[simps]
def HomologyMapData.natTransApp (h : HomologyData S) (τ : F ⟶ G) :
    HomologyMapData (S.mapNatTrans τ) (h.map F) (h.map G) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] [F.PreservesLeftHomologyOf S] [G.PreservesLeftHomologyOf S] [F.PreservesRightHomologyOf S] [G.PreservesRightHomologyOf S] (h : S.HomologyData) (τ : F ⟶ G)
Goal: (CategoryTheory.ShortComplex.HomologyMapData.natTransApp h τ).right =
  CategoryTheory.ShortComplex.RightHomologyMapData.natTransApp h.right τ
-- CategoryTheory.ShortComplex.HomologyMapData.natTransApp_left
@[simps]
def HomologyMapData.natTransApp (h : HomologyData S) (τ : F ⟶ G) :
    HomologyMapData (S.mapNatTrans τ) (h.map F) (h.map G) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] [F.PreservesLeftHomologyOf S] [G.PreservesLeftHomologyOf S] [F.PreservesRightHomologyOf S] [G.PreservesRightHomologyOf S] (h : S.HomologyData) (τ : F ⟶ G)
Goal: (CategoryTheory.ShortComplex.HomologyMapData.natTransApp h τ).left =
  CategoryTheory.ShortComplex.LeftHomologyMapData.natTransApp h.left τ
-- CategoryTheory.ShortComplex.homologyMap_mapNatTrans
lemma homologyMap_mapNatTrans [S.HasHomology] (τ : F ⟶ G) :
    homologyMap (S.mapNatTrans τ) =
      (S.mapHomologyIso F).hom ≫ τ.app S.homology ≫ (S.mapHomologyIso G).inv 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (S : CategoryTheory.ShortComplex C) {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] [F.PreservesLeftHomologyOf S] [G.PreservesLeftHomologyOf S] [F.PreservesRightHomologyOf S] [G.PreservesRightHomologyOf S] [S.HasHomology] (τ : F ⟶ G)
Goal: CategoryTheory.ShortComplex.homologyMap (S.mapNatTrans τ) =
  CategoryTheory.CategoryStruct.comp (S.mapHomologyIso F).hom
    (CategoryTheory.CategoryStruct.comp (τ.app S.homology) (S.mapHomologyIso G).inv)
-- CategoryTheory.ShortComplex.LeftHomologyMapData.quasiIso_map_iff
lemma LeftHomologyMapData.quasiIso_map_iff
    [(F.mapShortComplex.obj S₁).HasHomology]
    [(F.mapShortComplex.obj S₂).HasHomology]
    [hl₁.IsPreservedBy F] [hl₂.IsPreservedBy F] :
    QuasiIso (F.mapShortComplex.map φ) ↔ IsIso (F.map ψl.φH) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {hl₁ : S₁.LeftHomologyData} {hl₂ : S₂.LeftHomologyData} (ψl : CategoryTheory.ShortComplex.LeftHomologyMapData φ hl₁ hl₂) [(F.mapShortComplex.obj S₁).HasHomology] [(F.mapShortComplex.obj S₂).HasHomology] [hl₁.IsPreservedBy F] [hl₂.IsPreservedBy F]
Goal: CategoryTheory.ShortComplex.QuasiIso (F.mapShortComplex.map φ) ↔ CategoryTheory.IsIso (F.map ψl.φH)
-- CategoryTheory.ShortComplex.RightHomologyMapData.quasiIso_map_iff
lemma RightHomologyMapData.quasiIso_map_iff
    [(F.mapShortComplex.obj S₁).HasHomology]
    [(F.mapShortComplex.obj S₂).HasHomology]
    [hr₁.IsPreservedBy F] [hr₂.IsPreservedBy F] :
    QuasiIso (F.mapShortComplex.map φ) ↔ IsIso (F.map ψr.φH) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {hr₁ : S₁.RightHomologyData} {hr₂ : S₂.RightHomologyData} (ψr : CategoryTheory.ShortComplex.RightHomologyMapData φ hr₁ hr₂) [(F.mapShortComplex.obj S₁).HasHomology] [(F.mapShortComplex.obj S₂).HasHomology] [hr₁.IsPreservedBy F] [hr₂.IsPreservedBy F]
Goal: CategoryTheory.ShortComplex.QuasiIso (F.mapShortComplex.map φ) ↔ CategoryTheory.IsIso (F.map ψr.φH)
-- CategoryTheory.ShortComplex.quasiIso_map_iff_of_preservesLeftHomology
lemma quasiIso_map_iff_of_preservesLeftHomology
    [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂]
    [F.ReflectsIsomorphisms] :
    QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [S₁.HasHomology] [S₂.HasHomology] [(F.mapShortComplex.obj S₁).HasHomology] [(F.mapShortComplex.obj S₂).HasHomology] [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂] [F.ReflectsIsomorphisms]
Goal: CategoryTheory.ShortComplex.QuasiIso (F.mapShortComplex.map φ) ↔ CategoryTheory.ShortComplex.QuasiIso φ
-- CategoryTheory.ShortComplex.quasiIso_map_iff_of_preservesRightHomology
lemma quasiIso_map_iff_of_preservesRightHomology
    [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂]
    [F.ReflectsIsomorphisms] :
    QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [S₁.HasHomology] [S₂.HasHomology] [(F.mapShortComplex.obj S₁).HasHomology] [(F.mapShortComplex.obj S₂).HasHomology] [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂] [F.ReflectsIsomorphisms]
Goal: CategoryTheory.ShortComplex.QuasiIso (F.mapShortComplex.map φ) ↔ CategoryTheory.ShortComplex.QuasiIso φ
-- CategoryTheory.NatTrans.app_homology
lemma NatTrans.app_homology {F G : C ⥤ D} (τ : F ⟶ G)
    (S : ShortComplex C) [S.HasHomology] [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms]
    [F.PreservesLeftHomologyOf S] [G.PreservesLeftHomologyOf S] [F.PreservesRightHomologyOf S]
    [G.PreservesRightHomologyOf S] :
    τ.app S.homology = (S.mapHomologyIso F).inv ≫
      ShortComplex.homologyMap (S.mapNatTrans τ) ≫ (S.mapHomologyIso G).hom 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {F G : CategoryTheory.Functor C D} (τ : F ⟶ G) (S : CategoryTheory.ShortComplex C) [S.HasHomology] [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] [F.PreservesLeftHomologyOf S] [G.PreservesLeftHomologyOf S] [F.PreservesRightHomologyOf S] [G.PreservesRightHomologyOf S]
Goal: τ.app S.homology =
  CategoryTheory.CategoryStruct.comp (S.mapHomologyIso F).inv
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap (S.mapNatTrans τ))
      (S.mapHomologyIso G).hom)
-- CochainComplex.homOfDegreewiseSplit_f
@[simp]
lemma homOfDegreewiseSplit_f (n : ℤ) :
    (homOfDegreewiseSplit S σ).f n =
      (cocycleOfDegreewiseSplit S σ).1.v n (n + 1) rfl 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting) (n : ℤ)
Goal: (CochainComplex.homOfDegreewiseSplit S σ).f n = (↑(CochainComplex.cocycleOfDegreewiseSplit S σ)).v n (n + 1) ⋯
-- CochainComplex.triangleOfDegreewiseSplit_mor₁
@[simps! obj₁ obj₂ obj₃ mor₁ mor₂ mor₃]
def triangleOfDegreewiseSplit : Triangle (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting)
Goal: (CochainComplex.triangleOfDegreewiseSplit S σ).mor₁ = S.f
-- CochainComplex.triangleOfDegreewiseSplit_mor₂
@[simps! obj₁ obj₂ obj₃ mor₁ mor₂ mor₃]
def triangleOfDegreewiseSplit : Triangle (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting)
Goal: (CochainComplex.triangleOfDegreewiseSplit S σ).mor₂ = S.g
-- CochainComplex.triangleOfDegreewiseSplit_mor₃
@[simps! obj₁ obj₂ obj₃ mor₁ mor₂ mor₃]
def triangleOfDegreewiseSplit : Triangle (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting)
Goal: (CochainComplex.triangleOfDegreewiseSplit S σ).mor₃ = CochainComplex.homOfDegreewiseSplit S σ
-- CochainComplex.triangleOfDegreewiseSplit_obj₁
@[simps! obj₁ obj₂ obj₃ mor₁ mor₂ mor₃]
def triangleOfDegreewiseSplit : Triangle (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting)
Goal: (CochainComplex.triangleOfDegreewiseSplit S σ).obj₁ = S.X₁
-- CochainComplex.triangleOfDegreewiseSplit_obj₃
@[simps! obj₁ obj₂ obj₃ mor₁ mor₂ mor₃]
def triangleOfDegreewiseSplit : Triangle (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting)
Goal: (CochainComplex.triangleOfDegreewiseSplit S σ).obj₃ = S.X₃
-- CochainComplex.triangleOfDegreewiseSplit_obj₂
@[simps! obj₁ obj₂ obj₃ mor₁ mor₂ mor₃]
def triangleOfDegreewiseSplit : Triangle (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting)
Goal: (CochainComplex.triangleOfDegreewiseSplit S σ).obj₂ = S.X₂
-- CochainComplex.mappingConeHomOfDegreewiseSplitIso_hom_f
@[simps!]
noncomputable def mappingConeHomOfDegreewiseSplitIso :
    mappingCone (homOfDegreewiseSplit S σ) ≅ S.X₂⟦(1 : ℤ)⟧ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting) [CategoryTheory.Limits.HasBinaryBiproducts C] (i : ℤ)
Goal: (CochainComplex.mappingConeHomOfDegreewiseSplitIso S σ).hom.f i =
  (CochainComplex.mappingConeHomOfDegreewiseSplitXIso S σ i (i + 1) ⋯).hom
-- CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_f
@[simps!]
noncomputable def mappingConeHomOfDegreewiseSplitIso :
    mappingCone (homOfDegreewiseSplit S σ) ≅ S.X₂⟦(1 : ℤ)⟧ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting) [CategoryTheory.Limits.HasBinaryBiproducts C] (i : ℤ)
Goal: (CochainComplex.mappingConeHomOfDegreewiseSplitIso S σ).inv.f i =
  (CochainComplex.mappingConeHomOfDegreewiseSplitXIso S σ i (i + 1) ⋯).inv
-- CochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv
@[reassoc (attr := simp)]
lemma shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv :
    S.f⟦(1 : ℤ)⟧' ≫ (mappingConeHomOfDegreewiseSplitIso S σ).inv = -mappingCone.inr _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting) [CategoryTheory.Limits.HasBinaryBiproducts C]
Goal: (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map S.f ≫
    (CochainComplex.mappingConeHomOfDegreewiseSplitIso S σ).inv =
  -CochainComplex.mappingCone.inr (CochainComplex.homOfDegreewiseSplit S σ)
-- CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor₃
@[reassoc (attr := simp)]
lemma mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor₃ :
    (mappingConeHomOfDegreewiseSplitIso S σ).inv ≫
      (mappingCone.triangle (homOfDegreewiseSplit S σ)).mor₃ = -S.g⟦(1 : ℤ)⟧' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting) [CategoryTheory.Limits.HasBinaryBiproducts C]
Goal: (CochainComplex.mappingConeHomOfDegreewiseSplitIso S σ).inv ≫
    (CochainComplex.mappingCone.triangle (CochainComplex.homOfDegreewiseSplit S σ)).mor₃ =
  -(CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map S.g
-- CochainComplex.mappingCone.triangleRotateShortComplex_X₁
@[simps]
noncomputable def triangleRotateShortComplex : ShortComplex (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C ℤ} (φ : K ⟶ L)
Goal: (CochainComplex.mappingCone.triangleRotateShortComplex φ).X₁ = (CochainComplex.mappingCone.triangle φ).rotate.obj₁
-- CochainComplex.mappingCone.triangleRotateShortComplex_g
@[simps]
noncomputable def triangleRotateShortComplex : ShortComplex (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C ℤ} (φ : K ⟶ L)
Goal: (CochainComplex.mappingCone.triangleRotateShortComplex φ).g = (CochainComplex.mappingCone.triangle φ).rotate.mor₂
-- CochainComplex.mappingCone.triangleRotateShortComplex_X₃
@[simps]
noncomputable def triangleRotateShortComplex : ShortComplex (CochainComplex C ℤ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C ℤ} (φ : K ⟶ L)
Goal: (CochainComplex.mappingCone.triangleRotateShortComplex φ).X₃ = (CochainComplex.mappingCone.triangle φ).rotate.obj₃