ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.ShortComplex.opcycles_ext_iff
lemma opcycles_ext_iff {A : C} (f₁ f₂ : S.opcycles ⟶ A) :
    f₁ = f₂ ↔ S.pOpcycles ≫ f₁ = S.pOpcycles ≫ f₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] {A : C} (f₁ f₂ : S.opcycles ⟶ A)
Goal: f₁ = f₂ ↔ CategoryTheory.CategoryStruct.comp S.pOpcycles f₁ = CategoryTheory.CategoryStruct.comp S.pOpcycles f₂
-- CategoryTheory.ShortComplex.opcycles_ext
@[ext]
lemma opcycles_ext {A : C} (f₁ f₂ : S.opcycles ⟶ A)
    (h : S.pOpcycles ≫ f₁ = S.pOpcycles ≫ f₂) : f₁ = f₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] {A : C} (f₁ f₂ : S.opcycles ⟶ A) (h : CategoryTheory.CategoryStruct.comp S.pOpcycles f₁ = CategoryTheory.CategoryStruct.comp S.pOpcycles f₂)
Goal: f₁ = f₂
-- CategoryTheory.ShortComplex.isIso_pOpcycles
lemma isIso_pOpcycles (hf : S.f = 0) : IsIso S.pOpcycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] (hf : S.f = 0)
Goal: CategoryTheory.IsIso S.pOpcycles
-- CategoryTheory.ShortComplex.opcyclesIsoX₂_inv
@[simps! inv]
noncomputable def opcyclesIsoX₂ (hf : S.f = 0) : S.opcycles ≅ S.X₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] (hf : S.f = 0)
Goal: (S.opcyclesIsoX₂ hf).inv = S.pOpcycles
-- CategoryTheory.ShortComplex.opcyclesIsoX₂_inv_hom_id
@[reassoc (attr := simp)]
lemma opcyclesIsoX₂_inv_hom_id (hf : S.f = 0) :
    S.pOpcycles ≫ (S.opcyclesIsoX₂ hf).hom = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] (hf : S.f = 0)
Goal: CategoryTheory.CategoryStruct.comp S.pOpcycles (S.opcyclesIsoX₂ hf).hom = CategoryTheory.CategoryStruct.id S.X₂
-- CategoryTheory.ShortComplex.opcyclesIsoX₂_hom_inv_id
@[reassoc (attr := simp)]
lemma opcyclesIsoX₂_hom_inv_id (hf : S.f = 0) :
    (S.opcyclesIsoX₂ hf).hom ≫ S.pOpcycles = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] (hf : S.f = 0)
Goal: CategoryTheory.CategoryStruct.comp (S.opcyclesIsoX₂ hf).hom S.pOpcycles = CategoryTheory.CategoryStruct.id S.opcycles
-- CategoryTheory.ShortComplex.isIso_rightHomologyι
lemma isIso_rightHomologyι (hg : S.g = 0) : IsIso S.rightHomologyι 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] (hg : S.g = 0)
Goal: CategoryTheory.IsIso S.rightHomologyι
-- CategoryTheory.ShortComplex.opcyclesIsoRightHomology_inv
@[simps! inv]
noncomputable def opcyclesIsoRightHomology (hg : S.g = 0) : S.opcycles ≅ S.rightHomology 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] (hg : S.g = 0)
Goal: (S.opcyclesIsoRightHomology hg).inv = S.rightHomologyι
-- CategoryTheory.ShortComplex.opcyclesIsoRightHomology_inv_hom_id
@[reassoc (attr := simp)]
lemma opcyclesIsoRightHomology_inv_hom_id (hg : S.g = 0) :
    S.rightHomologyι ≫ (S.opcyclesIsoRightHomology hg).hom = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] (hg : S.g = 0)
Goal: CategoryTheory.CategoryStruct.comp S.rightHomologyι (S.opcyclesIsoRightHomology hg).hom =
  CategoryTheory.CategoryStruct.id S.rightHomology
-- CategoryTheory.ShortComplex.opcyclesIsoRightHomology_hom_inv_id
@[reassoc (attr := simp)]
lemma opcyclesIsoRightHomology_hom_inv_id (hg : S.g = 0) :
    (S.opcyclesIsoRightHomology hg).hom ≫ S.rightHomologyι  = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] (hg : S.g = 0)
Goal: CategoryTheory.CategoryStruct.comp (S.opcyclesIsoRightHomology hg).hom S.rightHomologyι =
  CategoryTheory.CategoryStruct.id S.opcycles
-- CategoryTheory.ShortComplex.p_opcyclesMap'
@[reassoc (attr := simp)]
lemma p_opcyclesMap' : h₁.p ≫ opcyclesMap' φ h₁ h₂ = φ.τ₂ ≫ h₂.p 
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.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.CategoryStruct.comp h₁.p (CategoryTheory.ShortComplex.opcyclesMap' φ h₁ h₂) =
  CategoryTheory.CategoryStruct.comp φ.τ₂ h₂.p
-- CategoryTheory.ShortComplex.opcyclesMap'_g'
@[reassoc (attr := simp)]
lemma opcyclesMap'_g' : opcyclesMap' φ h₁ h₂ ≫ h₂.g' = h₁.g' ≫ φ.τ₃ 
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.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.opcyclesMap' φ h₁ h₂) h₂.g' =
  CategoryTheory.CategoryStruct.comp h₁.g' φ.τ₃
-- CategoryTheory.ShortComplex.rightHomologyι_naturality'
@[reassoc (attr := simp)]
lemma rightHomologyι_naturality' :
    rightHomologyMap' φ h₁ h₂ ≫ h₂.ι = h₁.ι ≫ opcyclesMap' φ h₁ h₂ 
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.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.rightHomologyMap' φ h₁ h₂) h₂.ι =
  CategoryTheory.CategoryStruct.comp h₁.ι (CategoryTheory.ShortComplex.opcyclesMap' φ h₁ h₂)
-- CategoryTheory.ShortComplex.p_opcyclesMap
@[reassoc (attr := simp)]
lemma p_opcyclesMap : S₁.pOpcycles ≫ opcyclesMap φ = φ.τ₂ ≫ S₂.pOpcycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} [S₁.HasRightHomology] [S₂.HasRightHomology] (φ : S₁ ⟶ S₂)
Goal: CategoryTheory.CategoryStruct.comp S₁.pOpcycles (CategoryTheory.ShortComplex.opcyclesMap φ) =
  CategoryTheory.CategoryStruct.comp φ.τ₂ S₂.pOpcycles
-- CategoryTheory.ShortComplex.fromOpcycles_naturality
@[reassoc (attr := simp)]
lemma fromOpcycles_naturality : opcyclesMap φ ≫ S₂.fromOpcycles = S₁.fromOpcycles ≫ φ.τ₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} [S₁.HasRightHomology] [S₂.HasRightHomology] (φ : S₁ ⟶ S₂)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.opcyclesMap φ) S₂.fromOpcycles =
  CategoryTheory.CategoryStruct.comp S₁.fromOpcycles φ.τ₃
-- CategoryTheory.ShortComplex.rightHomologyι_naturality
@[reassoc (attr := simp)]
lemma rightHomologyι_naturality :
    rightHomologyMap φ ≫ S₂.rightHomologyι = S₁.rightHomologyι ≫ opcyclesMap φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} [S₁.HasRightHomology] [S₂.HasRightHomology] (φ : S₁ ⟶ S₂)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.rightHomologyMap φ) S₂.rightHomologyι =
  CategoryTheory.CategoryStruct.comp S₁.rightHomologyι (CategoryTheory.ShortComplex.opcyclesMap φ)
-- CategoryTheory.ShortComplex.RightHomologyMapData.rightHomologyMap'_eq
lemma rightHomologyMap'_eq : rightHomologyMap' φ h₁ h₂ = 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.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂)
Goal: CategoryTheory.ShortComplex.rightHomologyMap' φ h₁ h₂ = γ.φH
-- CategoryTheory.ShortComplex.RightHomologyMapData.opcyclesMap'_eq
lemma opcyclesMap'_eq : opcyclesMap' φ h₁ h₂ = γ.φQ 
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.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂)
Goal: CategoryTheory.ShortComplex.opcyclesMap' φ h₁ h₂ = γ.φQ
-- CategoryTheory.ShortComplex.rightHomologyMap'_id
@[simp]
lemma rightHomologyMap'_id (h : S.RightHomologyData) :
    rightHomologyMap' (𝟙 S) h h = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData)
Goal: CategoryTheory.ShortComplex.rightHomologyMap' (CategoryTheory.CategoryStruct.id S) h h =
  CategoryTheory.CategoryStruct.id h.H
-- CategoryTheory.ShortComplex.opcyclesMap'_id
@[simp]
lemma opcyclesMap'_id (h : S.RightHomologyData) :
    opcyclesMap' (𝟙 S) h h = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData)
Goal: CategoryTheory.ShortComplex.opcyclesMap' (CategoryTheory.CategoryStruct.id S) h h = CategoryTheory.CategoryStruct.id h.Q
-- CategoryTheory.ShortComplex.rightHomologyMap_id
@[simp]
lemma rightHomologyMap_id [HasRightHomology S] :
    rightHomologyMap (𝟙 S) = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology]
Goal: CategoryTheory.ShortComplex.rightHomologyMap (CategoryTheory.CategoryStruct.id S) =
  CategoryTheory.CategoryStruct.id S.rightHomology
-- CategoryTheory.ShortComplex.opcyclesMap_id
@[simp]
lemma opcyclesMap_id [HasRightHomology S] :
    opcyclesMap (𝟙 S) = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology]
Goal: CategoryTheory.ShortComplex.opcyclesMap (CategoryTheory.CategoryStruct.id S) =
  CategoryTheory.CategoryStruct.id S.opcycles
-- CategoryTheory.ShortComplex.rightHomologyMap'_zero
@[simp]
lemma rightHomologyMap'_zero (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) :
    rightHomologyMap' 0 h₁ h₂ = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.ShortComplex.rightHomologyMap' 0 h₁ h₂ = 0
-- CategoryTheory.ShortComplex.opcyclesMap'_zero
@[simp]
lemma opcyclesMap'_zero (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) :
    opcyclesMap' 0 h₁ h₂ = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.ShortComplex.opcyclesMap' 0 h₁ h₂ = 0
-- CategoryTheory.ShortComplex.rightHomologyMap_zero
@[simp]
lemma rightHomologyMap_zero [HasRightHomology S₁] [HasRightHomology S₂] :
    rightHomologyMap (0 : S₁ ⟶ S₂) = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S₁ S₂ : CategoryTheory.ShortComplex C) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.rightHomologyMap 0 = 0
-- CategoryTheory.ShortComplex.opcyclesMap_zero
@[simp]
lemma opcyclesMap_zero [HasRightHomology S₁] [HasRightHomology S₂] :
    opcyclesMap (0 : S₁ ⟶ S₂) = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S₁ S₂ : CategoryTheory.ShortComplex C) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.opcyclesMap 0 = 0
-- CategoryTheory.ShortComplex.rightHomologyMap'_comp
@[reassoc]
lemma rightHomologyMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
    (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) (h₃ : S₃.RightHomologyData) :
    rightHomologyMap' (φ₁ ≫ φ₂) h₁ h₃ = rightHomologyMap' φ₁ h₁ h₂ ≫
      rightHomologyMap' φ₂ h₂ h₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃) (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) (h₃ : S₃.RightHomologyData)
Goal: CategoryTheory.ShortComplex.rightHomologyMap' (CategoryTheory.CategoryStruct.comp φ₁ φ₂) h₁ h₃ =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.rightHomologyMap' φ₁ h₁ h₂)
    (CategoryTheory.ShortComplex.rightHomologyMap' φ₂ h₂ h₃)
-- CategoryTheory.ShortComplex.opcyclesMap'_comp
@[reassoc]
lemma opcyclesMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
    (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) (h₃ : S₃.RightHomologyData) :
    opcyclesMap' (φ₁ ≫ φ₂) h₁ h₃ = opcyclesMap' φ₁ h₁ h₂ ≫ opcyclesMap' φ₂ h₂ h₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃) (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) (h₃ : S₃.RightHomologyData)
Goal: CategoryTheory.ShortComplex.opcyclesMap' (CategoryTheory.CategoryStruct.comp φ₁ φ₂) h₁ h₃ =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.opcyclesMap' φ₁ h₁ h₂)
    (CategoryTheory.ShortComplex.opcyclesMap' φ₂ h₂ h₃)
-- CategoryTheory.ShortComplex.rightHomologyMap_comp
@[simp]
lemma rightHomologyMap_comp [HasRightHomology S₁] [HasRightHomology S₂] [HasRightHomology S₃]
    (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃) :
    rightHomologyMap (φ₁ ≫ φ₂) = rightHomologyMap φ₁ ≫ rightHomologyMap φ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} [S₁.HasRightHomology] [S₂.HasRightHomology] [S₃.HasRightHomology] (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
Goal: CategoryTheory.ShortComplex.rightHomologyMap (CategoryTheory.CategoryStruct.comp φ₁ φ₂) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.rightHomologyMap φ₁)
    (CategoryTheory.ShortComplex.rightHomologyMap φ₂)
-- CategoryTheory.ShortComplex.opcyclesMap_comp
@[simp]
lemma opcyclesMap_comp [HasRightHomology S₁] [HasRightHomology S₂] [HasRightHomology S₃]
    (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃) :
    opcyclesMap (φ₁ ≫ φ₂) = opcyclesMap φ₁ ≫ opcyclesMap φ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} [S₁.HasRightHomology] [S₂.HasRightHomology] [S₃.HasRightHomology] (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
Goal: CategoryTheory.ShortComplex.opcyclesMap (CategoryTheory.CategoryStruct.comp φ₁ φ₂) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.opcyclesMap φ₁)
    (CategoryTheory.ShortComplex.opcyclesMap φ₂)
-- CategoryTheory.ShortComplex.rightHomologyMapIso'_inv
@[simps]
def rightHomologyMapIso' (e : S₁ ≅ S₂) (h₁ : S₁.RightHomologyData)
    (h₂ : S₂.RightHomologyData) : 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.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: (CategoryTheory.ShortComplex.rightHomologyMapIso' e h₁ h₂).inv =
  CategoryTheory.ShortComplex.rightHomologyMap' e.inv h₂ h₁
-- CategoryTheory.ShortComplex.rightHomologyMapIso'_hom
@[simps]
def rightHomologyMapIso' (e : S₁ ≅ S₂) (h₁ : S₁.RightHomologyData)
    (h₂ : S₂.RightHomologyData) : 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.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: (CategoryTheory.ShortComplex.rightHomologyMapIso' e h₁ h₂).hom =
  CategoryTheory.ShortComplex.rightHomologyMap' e.hom h₁ h₂
-- CategoryTheory.ShortComplex.opcyclesMapIso'_inv
@[simps]
def opcyclesMapIso' (e : S₁ ≅ S₂) (h₁ : S₁.RightHomologyData)
    (h₂ : S₂.RightHomologyData) : h₁.Q ≅ h₂.Q 
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.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: (CategoryTheory.ShortComplex.opcyclesMapIso' e h₁ h₂).inv = CategoryTheory.ShortComplex.opcyclesMap' e.inv h₂ h₁
-- CategoryTheory.ShortComplex.opcyclesMapIso'_hom
@[simps]
def opcyclesMapIso' (e : S₁ ≅ S₂) (h₁ : S₁.RightHomologyData)
    (h₂ : S₂.RightHomologyData) : h₁.Q ≅ h₂.Q 
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.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: (CategoryTheory.ShortComplex.opcyclesMapIso' e h₁ h₂).hom = CategoryTheory.ShortComplex.opcyclesMap' e.hom h₁ h₂
-- CategoryTheory.ShortComplex.rightHomologyMapIso_inv
@[simps]
noncomputable def rightHomologyMapIso (e : S₁ ≅ S₂) [S₁.HasRightHomology]
    [S₂.HasRightHomology] : S₁.rightHomology ≅ S₂.rightHomology 
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.HasRightHomology] [S₂.HasRightHomology]
Goal: (CategoryTheory.ShortComplex.rightHomologyMapIso e).inv = CategoryTheory.ShortComplex.rightHomologyMap e.inv
-- CategoryTheory.ShortComplex.rightHomologyMapIso_hom
@[simps]
noncomputable def rightHomologyMapIso (e : S₁ ≅ S₂) [S₁.HasRightHomology]
    [S₂.HasRightHomology] : S₁.rightHomology ≅ S₂.rightHomology 
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.HasRightHomology] [S₂.HasRightHomology]
Goal: (CategoryTheory.ShortComplex.rightHomologyMapIso e).hom = CategoryTheory.ShortComplex.rightHomologyMap e.hom
-- CategoryTheory.ShortComplex.opcyclesMapIso_hom
@[simps]
noncomputable def opcyclesMapIso (e : S₁ ≅ S₂) [S₁.HasRightHomology]
    [S₂.HasRightHomology] : S₁.opcycles ≅ S₂.opcycles 
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.HasRightHomology] [S₂.HasRightHomology]
Goal: (CategoryTheory.ShortComplex.opcyclesMapIso e).hom = CategoryTheory.ShortComplex.opcyclesMap e.hom
-- CategoryTheory.ShortComplex.opcyclesMapIso_inv
@[simps]
noncomputable def opcyclesMapIso (e : S₁ ≅ S₂) [S₁.HasRightHomology]
    [S₂.HasRightHomology] : S₁.opcycles ≅ S₂.opcycles 
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.HasRightHomology] [S₂.HasRightHomology]
Goal: (CategoryTheory.ShortComplex.opcyclesMapIso e).inv = CategoryTheory.ShortComplex.opcyclesMap e.inv
-- CategoryTheory.ShortComplex.RightHomologyData.p_comp_opcyclesIso_inv
@[reassoc (attr := simp)]
lemma p_comp_opcyclesIso_inv : h.p ≫ h.opcyclesIso.inv = S.pOpcycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData) [S.HasRightHomology]
Goal: CategoryTheory.CategoryStruct.comp h.p h.opcyclesIso.inv = S.pOpcycles
-- CategoryTheory.ShortComplex.RightHomologyData.pOpcycles_comp_opcyclesIso_hom
@[reassoc (attr := simp)]
lemma pOpcycles_comp_opcyclesIso_hom : S.pOpcycles ≫ h.opcyclesIso.hom = h.p 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData) [S.HasRightHomology]
Goal: CategoryTheory.CategoryStruct.comp S.pOpcycles h.opcyclesIso.hom = h.p
-- CategoryTheory.ShortComplex.RightHomologyData.rightHomologyIso_inv_comp_rightHomologyι
@[reassoc (attr := simp)]
lemma rightHomologyIso_inv_comp_rightHomologyι :
    h.rightHomologyIso.inv ≫ S.rightHomologyι = h.ι ≫ h.opcyclesIso.inv 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData) [S.HasRightHomology]
Goal: CategoryTheory.CategoryStruct.comp h.rightHomologyIso.inv S.rightHomologyι =
  CategoryTheory.CategoryStruct.comp h.ι h.opcyclesIso.inv
-- CategoryTheory.ShortComplex.RightHomologyData.rightHomologyIso_hom_comp_ι
@[reassoc (attr := simp)]
lemma rightHomologyIso_hom_comp_ι :
    h.rightHomologyIso.hom ≫ h.ι = S.rightHomologyι ≫ h.opcyclesIso.hom 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData) [S.HasRightHomology]
Goal: CategoryTheory.CategoryStruct.comp h.rightHomologyIso.hom h.ι =
  CategoryTheory.CategoryStruct.comp S.rightHomologyι h.opcyclesIso.hom
-- CategoryTheory.ShortComplex.RightHomologyMapData.rightHomologyMap_eq
lemma rightHomologyMap_eq [S₁.HasRightHomology] [S₂.HasRightHomology] :
    rightHomologyMap φ = h₁.rightHomologyIso.hom ≫ γ.φH ≫ h₂.rightHomologyIso.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.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.rightHomologyMap φ =
  CategoryTheory.CategoryStruct.comp h₁.rightHomologyIso.hom
    (CategoryTheory.CategoryStruct.comp γ.φH h₂.rightHomologyIso.inv)
-- CategoryTheory.ShortComplex.RightHomologyMapData.opcyclesMap_eq
lemma opcyclesMap_eq [S₁.HasRightHomology] [S₂.HasRightHomology] :
    opcyclesMap φ = h₁.opcyclesIso.hom ≫ γ.φQ ≫ h₂.opcyclesIso.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.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.opcyclesMap φ =
  CategoryTheory.CategoryStruct.comp h₁.opcyclesIso.hom (CategoryTheory.CategoryStruct.comp γ.φQ h₂.opcyclesIso.inv)
-- CategoryTheory.ShortComplex.RightHomologyMapData.rightHomologyMap_comm
lemma rightHomologyMap_comm [S₁.HasRightHomology] [S₂.HasRightHomology] :
    rightHomologyMap φ ≫ h₂.rightHomologyIso.hom = h₁.rightHomologyIso.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.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.rightHomologyMap φ) h₂.rightHomologyIso.hom =
  CategoryTheory.CategoryStruct.comp h₁.rightHomologyIso.hom γ.φH
-- CategoryTheory.ShortComplex.RightHomologyMapData.opcyclesMap_comm
lemma opcyclesMap_comm [S₁.HasRightHomology] [S₂.HasRightHomology] :
    opcyclesMap φ ≫ h₂.opcyclesIso.hom = h₁.opcyclesIso.hom ≫ γ.φQ 
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.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.opcyclesMap φ) h₂.opcyclesIso.hom =
  CategoryTheory.CategoryStruct.comp h₁.opcyclesIso.hom γ.φQ