ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.ShortComplex.leftHomologyMapIso'_hom
@[simps]
def leftHomologyMapIso' (e : S₁ ≅ S₂) (h₁ : S₁.LeftHomologyData)
    (h₂ : S₂.LeftHomologyData) : h₁.H ≅ h₂.H 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: (CategoryTheory.ShortComplex.leftHomologyMapIso' e h₁ h₂).hom = CategoryTheory.ShortComplex.leftHomologyMap' e.hom h₁ h₂
-- CategoryTheory.ShortComplex.leftHomologyMapIso'_inv
@[simps]
def leftHomologyMapIso' (e : S₁ ≅ S₂) (h₁ : S₁.LeftHomologyData)
    (h₂ : S₂.LeftHomologyData) : h₁.H ≅ h₂.H 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: (CategoryTheory.ShortComplex.leftHomologyMapIso' e h₁ h₂).inv = CategoryTheory.ShortComplex.leftHomologyMap' e.inv h₂ h₁
-- CategoryTheory.ShortComplex.cyclesMapIso'_inv
@[simps]
def cyclesMapIso' (e : S₁ ≅ S₂) (h₁ : S₁.LeftHomologyData)
    (h₂ : S₂.LeftHomologyData) : h₁.K ≅ h₂.K 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: (CategoryTheory.ShortComplex.cyclesMapIso' e h₁ h₂).inv = CategoryTheory.ShortComplex.cyclesMap' e.inv h₂ h₁
-- CategoryTheory.ShortComplex.cyclesMapIso'_hom
@[simps]
def cyclesMapIso' (e : S₁ ≅ S₂) (h₁ : S₁.LeftHomologyData)
    (h₂ : S₂.LeftHomologyData) : h₁.K ≅ h₂.K 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: (CategoryTheory.ShortComplex.cyclesMapIso' e h₁ h₂).hom = CategoryTheory.ShortComplex.cyclesMap' e.hom h₁ h₂
-- CategoryTheory.ShortComplex.leftHomologyMapIso_inv
@[simps]
noncomputable def leftHomologyMapIso (e : S₁ ≅ S₂) [S₁.HasLeftHomology]
    [S₂.HasLeftHomology] : S₁.leftHomology ≅ S₂.leftHomology 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: (CategoryTheory.ShortComplex.leftHomologyMapIso e).inv = CategoryTheory.ShortComplex.leftHomologyMap e.inv
-- CategoryTheory.ShortComplex.leftHomologyMapIso_hom
@[simps]
noncomputable def leftHomologyMapIso (e : S₁ ≅ S₂) [S₁.HasLeftHomology]
    [S₂.HasLeftHomology] : S₁.leftHomology ≅ S₂.leftHomology 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: (CategoryTheory.ShortComplex.leftHomologyMapIso e).hom = CategoryTheory.ShortComplex.leftHomologyMap e.hom
-- CategoryTheory.ShortComplex.cyclesMapIso_hom
@[simps]
noncomputable def cyclesMapIso (e : S₁ ≅ S₂) [S₁.HasLeftHomology]
    [S₂.HasLeftHomology] : S₁.cycles ≅ S₂.cycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: (CategoryTheory.ShortComplex.cyclesMapIso e).hom = CategoryTheory.ShortComplex.cyclesMap e.hom
-- CategoryTheory.ShortComplex.cyclesMapIso_inv
@[simps]
noncomputable def cyclesMapIso (e : S₁ ≅ S₂) [S₁.HasLeftHomology]
    [S₂.HasLeftHomology] : S₁.cycles ≅ S₂.cycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: (CategoryTheory.ShortComplex.cyclesMapIso e).inv = CategoryTheory.ShortComplex.cyclesMap e.inv
-- CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_hom_comp_i
@[reassoc (attr := simp)]
lemma cyclesIso_hom_comp_i : h.cyclesIso.hom ≫ h.i = S.iCycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData) [S.HasLeftHomology]
Goal: CategoryTheory.CategoryStruct.comp h.cyclesIso.hom h.i = S.iCycles
-- CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_inv_comp_iCycles
@[reassoc (attr := simp)]
lemma cyclesIso_inv_comp_iCycles : h.cyclesIso.inv ≫ S.iCycles = h.i 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData) [S.HasLeftHomology]
Goal: CategoryTheory.CategoryStruct.comp h.cyclesIso.inv S.iCycles = h.i
-- CategoryTheory.ShortComplex.LeftHomologyData.leftHomologyπ_comp_leftHomologyIso_hom
@[reassoc (attr := simp)]
lemma leftHomologyπ_comp_leftHomologyIso_hom :
    S.leftHomologyπ ≫ h.leftHomologyIso.hom = h.cyclesIso.hom ≫ h.π 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData) [S.HasLeftHomology]
Goal: CategoryTheory.CategoryStruct.comp S.leftHomologyπ h.leftHomologyIso.hom =
  CategoryTheory.CategoryStruct.comp h.cyclesIso.hom h.π
-- CategoryTheory.ShortComplex.LeftHomologyData.π_comp_leftHomologyIso_inv
@[reassoc (attr := simp)]
lemma π_comp_leftHomologyIso_inv :
    h.π ≫ h.leftHomologyIso.inv = h.cyclesIso.inv ≫ S.leftHomologyπ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData) [S.HasLeftHomology]
Goal: CategoryTheory.CategoryStruct.comp h.π h.leftHomologyIso.inv =
  CategoryTheory.CategoryStruct.comp h.cyclesIso.inv S.leftHomologyπ
-- CategoryTheory.ShortComplex.LeftHomologyMapData.leftHomologyMap_eq
lemma leftHomologyMap_eq [S₁.HasLeftHomology] [S₂.HasLeftHomology] :
    leftHomologyMap φ = h₁.leftHomologyIso.hom ≫ γ.φH ≫ h₂.leftHomologyIso.inv 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.leftHomologyMap φ =
  CategoryTheory.CategoryStruct.comp h₁.leftHomologyIso.hom
    (CategoryTheory.CategoryStruct.comp γ.φH h₂.leftHomologyIso.inv)
-- CategoryTheory.ShortComplex.LeftHomologyMapData.cyclesMap_eq
lemma cyclesMap_eq [S₁.HasLeftHomology] [S₂.HasLeftHomology] :
    cyclesMap φ = h₁.cyclesIso.hom ≫ γ.φK ≫ h₂.cyclesIso.inv 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.cyclesMap φ =
  CategoryTheory.CategoryStruct.comp h₁.cyclesIso.hom (CategoryTheory.CategoryStruct.comp γ.φK h₂.cyclesIso.inv)
-- CategoryTheory.ShortComplex.LeftHomologyMapData.leftHomologyMap_comm
lemma leftHomologyMap_comm [S₁.HasLeftHomology] [S₂.HasLeftHomology] :
    leftHomologyMap φ ≫ h₂.leftHomologyIso.hom = h₁.leftHomologyIso.hom ≫ γ.φH 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftHomologyMap φ) h₂.leftHomologyIso.hom =
  CategoryTheory.CategoryStruct.comp h₁.leftHomologyIso.hom γ.φH
-- CategoryTheory.ShortComplex.LeftHomologyMapData.cyclesMap_comm
lemma cyclesMap_comm [S₁.HasLeftHomology] [S₂.HasLeftHomology] :
    cyclesMap φ ≫ h₂.cyclesIso.hom = h₁.cyclesIso.hom ≫ γ.φK 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.cyclesMap φ) h₂.cyclesIso.hom =
  CategoryTheory.CategoryStruct.comp h₁.cyclesIso.hom γ.φK
-- CategoryTheory.ShortComplex.leftHomologyFunctor_obj
@[simps]
noncomputable def leftHomologyFunctor : ShortComplex C ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.ShortComplex.leftHomologyFunctor C).obj S = S.leftHomology
-- CategoryTheory.ShortComplex.leftHomologyFunctor_map
@[simps]
noncomputable def leftHomologyFunctor : ShortComplex C ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C]
Goal: ∀ {X Y : CategoryTheory.ShortComplex C} (φ : X ⟶ Y),
  (CategoryTheory.ShortComplex.leftHomologyFunctor C).map φ = CategoryTheory.ShortComplex.leftHomologyMap φ
-- CategoryTheory.ShortComplex.cyclesFunctor_obj
@[simps]
noncomputable def cyclesFunctor : ShortComplex C ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.ShortComplex.cyclesFunctor C).obj S = S.cycles
-- CategoryTheory.ShortComplex.cyclesFunctor_map
@[simps]
noncomputable def cyclesFunctor : ShortComplex C ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C]
Goal: ∀ {X Y : CategoryTheory.ShortComplex C} (φ : X ⟶ Y),
  (CategoryTheory.ShortComplex.cyclesFunctor C).map φ = CategoryTheory.ShortComplex.cyclesMap φ
-- CategoryTheory.ShortComplex.leftHomologyπNatTrans_app
@[simps]
noncomputable def leftHomologyπNatTrans : cyclesFunctor C ⟶ leftHomologyFunctor C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.ShortComplex.leftHomologyπNatTrans C).app S = S.leftHomologyπ
-- CategoryTheory.ShortComplex.iCyclesNatTrans_app
@[simps]
noncomputable def iCyclesNatTrans : cyclesFunctor C ⟶ ShortComplex.π₂ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.ShortComplex.iCyclesNatTrans C).app S = S.iCycles
-- CategoryTheory.ShortComplex.toCyclesNatTrans_app
@[simps]
noncomputable def toCyclesNatTrans :
    π₁ ⟶ cyclesFunctor C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.ShortComplex.toCyclesNatTrans C).app S = S.toCycles
-- CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono_H
@[simps]
noncomputable def ofEpiOfIsIsoOfMono (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₁)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₁.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono φ h).H = h.H
-- CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono_π
@[simps]
noncomputable def ofEpiOfIsIsoOfMono (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₁)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₁.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono φ h).π = h.π
-- CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono_K
@[simps]
noncomputable def ofEpiOfIsIsoOfMono (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₁)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₁.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono φ h).K = h.K
-- CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono_i
@[simps]
noncomputable def ofEpiOfIsIsoOfMono (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₁)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₁.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono φ h).i = CategoryTheory.CategoryStruct.comp h.i φ.τ₂
-- CategoryTheory.ShortComplex.LeftHomologyData.τ₁_ofEpiOfIsIsoOfMono_f'
@[simp]
lemma τ₁_ofEpiOfIsIsoOfMono_f' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₁)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : φ.τ₁ ≫ (ofEpiOfIsIsoOfMono φ h).f' = h.f' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₁.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: CategoryTheory.CategoryStruct.comp φ.τ₁ (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono φ h).f' = h.f'
-- CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono'_i
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₂.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono' φ h).i =
  CategoryTheory.CategoryStruct.comp h.i (CategoryTheory.inv φ.τ₂)
-- CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono'_H
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₂.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono' φ h).H = h.H
-- CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono'_π
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₂.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono' φ h).π = h.π
-- CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono'_K
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : LeftHomologyData S₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₂.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono' φ h).K = h.K
-- CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono'_f'
@[simp]
lemma ofEpiOfIsIsoOfMono'_f' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : (ofEpiOfIsIsoOfMono' φ h).f' = φ.τ₁ ≫ h.f' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₂.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono' φ h).f' = CategoryTheory.CategoryStruct.comp φ.τ₁ h.f'
-- CategoryTheory.ShortComplex.hasLeftHomology_of_epi_of_isIso_of_mono
lemma hasLeftHomology_of_epi_of_isIso_of_mono (φ : S₁ ⟶ S₂) [HasLeftHomology S₁]
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : HasLeftHomology S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [S₁.HasLeftHomology] [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: S₂.HasLeftHomology
-- CategoryTheory.ShortComplex.hasLeftHomology_of_epi_of_isIso_of_mono'
lemma hasLeftHomology_of_epi_of_isIso_of_mono' (φ : S₁ ⟶ S₂) [HasLeftHomology S₂]
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : HasLeftHomology S₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [S₂.HasLeftHomology] [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: S₁.HasLeftHomology
-- CategoryTheory.ShortComplex.hasLeftHomology_of_iso
lemma hasLeftHomology_of_iso {S₁ S₂ : ShortComplex C} (e : S₁ ≅ S₂) [HasLeftHomology S₁] :
    HasLeftHomology S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) [S₁.HasLeftHomology]
Goal: S₂.HasLeftHomology
-- CategoryTheory.ShortComplex.LeftHomologyMapData.ofEpiOfIsIsoOfMono_φK
@[simps]
def ofEpiOfIsIsoOfMono (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₁)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] :
    LeftHomologyMapData φ h (LeftHomologyData.ofEpiOfIsIsoOfMono φ h) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₁.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.ofEpiOfIsIsoOfMono φ h).φK = CategoryTheory.CategoryStruct.id h.K
-- CategoryTheory.ShortComplex.LeftHomologyMapData.ofEpiOfIsIsoOfMono_φH
@[simps]
def ofEpiOfIsIsoOfMono (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₁)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] :
    LeftHomologyMapData φ h (LeftHomologyData.ofEpiOfIsIsoOfMono φ h) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₁.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.ofEpiOfIsIsoOfMono φ h).φH = CategoryTheory.CategoryStruct.id h.H
-- CategoryTheory.ShortComplex.LeftHomologyMapData.ofEpiOfIsIsoOfMono'_φH
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] :
    LeftHomologyMapData φ (LeftHomologyData.ofEpiOfIsIsoOfMono' φ h) h 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₂.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.ofEpiOfIsIsoOfMono' φ h).φH =
  CategoryTheory.CategoryStruct.id (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono' φ h).H
-- CategoryTheory.ShortComplex.LeftHomologyMapData.ofEpiOfIsIsoOfMono'_φK
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] :
    LeftHomologyMapData φ (LeftHomologyData.ofEpiOfIsIsoOfMono' φ h) h 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₂.LeftHomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.ofEpiOfIsIsoOfMono' φ h).φK =
  CategoryTheory.CategoryStruct.id (CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono' φ h).K
-- CategoryTheory.ShortComplex.liftCycles_i
@[reassoc (attr := simp)]
lemma liftCycles_i : S.liftCycles k hk ≫ S.iCycles = k 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) {A : C} (k : A ⟶ S.X₂) (hk : CategoryTheory.CategoryStruct.comp k S.g = 0) [S.HasLeftHomology]
Goal: CategoryTheory.CategoryStruct.comp (S.liftCycles k hk) S.iCycles = k
-- CategoryTheory.ShortComplex.comp_liftCycles
@[reassoc]
lemma comp_liftCycles {A' : C} (α : A' ⟶ A) :
    α ≫ S.liftCycles k hk = S.liftCycles (α ≫ k) (by rw [assoc, hk, comp_zero]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) {A : C} (k : A ⟶ S.X₂) (hk : CategoryTheory.CategoryStruct.comp k S.g = 0) [S.HasLeftHomology] {A' : C} (α : A' ⟶ A)
Goal: CategoryTheory.CategoryStruct.comp α (S.liftCycles k hk) = S.liftCycles (CategoryTheory.CategoryStruct.comp α k) ⋯
-- CategoryTheory.ShortComplex.cyclesIsoKernel_hom
@[simps]
noncomputable def cyclesIsoKernel [HasKernel S.g] : S.cycles ≅ kernel S.g 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] [CategoryTheory.Limits.HasKernel S.g]
Goal: S.cyclesIsoKernel.hom = CategoryTheory.Limits.kernel.lift S.g S.iCycles ⋯
-- CategoryTheory.ShortComplex.cyclesIsoKernel_inv
@[simps]
noncomputable def cyclesIsoKernel [HasKernel S.g] : S.cycles ≅ kernel S.g 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] [CategoryTheory.Limits.HasKernel S.g]
Goal: S.cyclesIsoKernel.inv = S.liftCycles (CategoryTheory.Limits.kernel.ι S.g) ⋯
-- CategoryTheory.ShortComplex.liftCycles_leftHomologyπ_eq_zero_of_boundary
@[reassoc]
lemma liftCycles_leftHomologyπ_eq_zero_of_boundary (x : A ⟶ S.X₁) (hx : k = x ≫ S.f) :
    S.liftCycles k (by rw [hx, assoc, S.zero, comp_zero]) ≫ S.leftHomologyπ = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) {A : C} (k : A ⟶ S.X₂) [S.HasLeftHomology] (x : A ⟶ S.X₁) (hx : k = CategoryTheory.CategoryStruct.comp x S.f)
Goal: CategoryTheory.CategoryStruct.comp (S.liftCycles k ⋯) S.leftHomologyπ = 0