ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TotalComplexShapeSymmetry.σ_ε₂
class TotalComplexShapeSymmetry [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] where
  symm (i₁ : I₁) (i₂ : I₂) : ComplexShape.π c₂ c₁ c₁₂ ⟨i₂, i₁⟩ = ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩
    σ (i₁ : I₁) (i₂ : I₂) : ℤˣ
  σ_ε₁ {i₁ i₁' : I₁} (h₁ : c₁.Rel i₁ i₁') (i₂ : I₂) :
    σ i₁ i₂ * ComplexShape.ε₁ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = ComplexShape.ε₂ c₂ c₁ c₁₂ ⟨i₂, i₁⟩ * σ i₁' i₂
  σ_ε₂ (i₁ : I₁) {i₂ i₂' : I₂} (h₂ : c₂.Rel i₂ i₂') :
    σ i₁ i₂ * ComplexShape.ε₂ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = ComplexShape.ε₁ c₂ c₁ c₁₂ ⟨i₂, i₁⟩ * σ i₁ i₂'
Arguments: {I₁ : Type u_1} {I₂ : Type u_2} {I₁₂ : Type u_3} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} {c₁₂ : ComplexShape I₁₂} [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] [self : TotalComplexShapeSymmetry c₁ c₂ c₁₂] (i₁ : I₁) {i₂ i₂' : I₂} (h₂ : c₂.Rel i₂ i₂')
Goal: TotalComplexShapeSymmetry.σ c₁ c₂ c₁₂ i₁ i₂ * c₁.ε₂ c₂ c₁₂ (i₁, i₂) =
  c₂.ε₁ c₁ c₁₂ (i₂, i₁) * TotalComplexShapeSymmetry.σ c₁ c₂ c₁₂ i₁ i₂'
-- TotalComplexShapeSymmetry.σ_ε₁
class TotalComplexShapeSymmetry [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] where
  symm (i₁ : I₁) (i₂ : I₂) : ComplexShape.π c₂ c₁ c₁₂ ⟨i₂, i₁⟩ = ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩
    σ (i₁ : I₁) (i₂ : I₂) : ℤˣ
  σ_ε₁ {i₁ i₁' : I₁} (h₁ : c₁.Rel i₁ i₁') (i₂ : I₂) :
    σ i₁ i₂ * ComplexShape.ε₁ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = ComplexShape.ε₂ c₂ c₁ c₁₂ ⟨i₂, i₁⟩ * σ i₁' i₂
  σ_ε₂ (i₁ : I₁) {i₂ i₂' : I₂} (h₂ : c₂.Rel i₂ i₂') :
    σ i₁ i₂ * ComplexShape.ε₂ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = ComplexShape.ε₁ c₂ c₁ c₁₂ ⟨i₂, i₁⟩ * σ i₁ i₂'
Arguments: {I₁ : Type u_1} {I₂ : Type u_2} {I₁₂ : Type u_3} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} {c₁₂ : ComplexShape I₁₂} [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] [self : TotalComplexShapeSymmetry c₁ c₂ c₁₂] {i₁ i₁' : I₁} (h₁ : c₁.Rel i₁ i₁') (i₂ : I₂)
Goal: TotalComplexShapeSymmetry.σ c₁ c₂ c₁₂ i₁ i₂ * c₁.ε₁ c₂ c₁₂ (i₁, i₂) =
  c₂.ε₂ c₁ c₁₂ (i₂, i₁) * TotalComplexShapeSymmetry.σ c₁ c₂ c₁₂ i₁' i₂
-- TotalComplexShapeSymmetry.symm
class TotalComplexShapeSymmetry [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] where
  symm (i₁ : I₁) (i₂ : I₂) : ComplexShape.π c₂ c₁ c₁₂ ⟨i₂, i₁⟩ = ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩
    σ (i₁ : I₁) (i₂ : I₂) : ℤˣ
  σ_ε₁ {i₁ i₁' : I₁} (h₁ : c₁.Rel i₁ i₁') (i₂ : I₂) :
    σ i₁ i₂ * ComplexShape.ε₁ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = ComplexShape.ε₂ c₂ c₁ c₁₂ ⟨i₂, i₁⟩ * σ i₁' i₂
  σ_ε₂ (i₁ : I₁) {i₂ i₂' : I₂} (h₂ : c₂.Rel i₂ i₂') :
    σ i₁ i₂ * ComplexShape.ε₂ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = ComplexShape.ε₁ c₂ c₁ c₁₂ ⟨i₂, i₁⟩ * σ i₁ i₂'
Arguments: {I₁ : Type u_1} {I₂ : Type u_2} {I₁₂ : Type u_3} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} {c₁₂ : ComplexShape I₁₂} [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] [self : TotalComplexShapeSymmetry c₁ c₂ c₁₂] (i₁ : I₁) (i₂ : I₂)
Goal: c₂.π c₁ c₁₂ (i₂, i₁) = c₁.π c₂ c₁₂ (i₁, i₂)
-- ComplexShape.π_symm
lemma π_symm (i₁ : I₁) (i₂ : I₂) :
    π c₂ c₁ c₁₂ ⟨i₂, i₁⟩ = π c₁ c₂ c₁₂ ⟨i₁, i₂⟩ 
Arguments: {I₁ : Type u_1} {I₂ : Type u_2} {I₁₂ : Type u_3} (c₁ : ComplexShape I₁) (c₂ : ComplexShape I₂) (c₁₂ : ComplexShape I₁₂) [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] [TotalComplexShapeSymmetry c₁ c₂ c₁₂] (i₁ : I₁) (i₂ : I₂)
Goal: c₂.π c₁ c₁₂ (i₂, i₁) = c₁.π c₂ c₁₂ (i₁, i₂)
-- ComplexShape.σ_ε₁
lemma σ_ε₁ {i₁ i₁' : I₁} (h₁ : c₁.Rel i₁ i₁') (i₂ : I₂) :
    σ c₁ c₂ c₁₂ i₁ i₂ * ε₁ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = ε₂ c₂ c₁ c₁₂ ⟨i₂, i₁⟩ * σ c₁ c₂ c₁₂ i₁' i₂ 
Arguments: {I₁ : Type u_1} {I₂ : Type u_2} {I₁₂ : Type u_3} {c₁ : ComplexShape I₁} (c₂ : ComplexShape I₂) (c₁₂ : ComplexShape I₁₂) [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] [TotalComplexShapeSymmetry c₁ c₂ c₁₂] {i₁ i₁' : I₁} (h₁ : c₁.Rel i₁ i₁') (i₂ : I₂)
Goal: c₁.σ c₂ c₁₂ i₁ i₂ * c₁.ε₁ c₂ c₁₂ (i₁, i₂) = c₂.ε₂ c₁ c₁₂ (i₂, i₁) * c₁.σ c₂ c₁₂ i₁' i₂
-- ComplexShape.σ_ε₂
lemma σ_ε₂ (i₁ : I₁) {i₂ i₂' : I₂} (h₂ : c₂.Rel i₂ i₂') :
    σ c₁ c₂ c₁₂ i₁ i₂ * ε₂ c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = ε₁ c₂ c₁ c₁₂ ⟨i₂, i₁⟩ * σ c₁ c₂ c₁₂ i₁ i₂' 
Arguments: {I₁ : Type u_1} {I₂ : Type u_2} {I₁₂ : Type u_3} (c₁ : ComplexShape I₁) {c₂ : ComplexShape I₂} (c₁₂ : ComplexShape I₁₂) [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] [TotalComplexShapeSymmetry c₁ c₂ c₁₂] (i₁ : I₁) {i₂ i₂' : I₂} (h₂ : c₂.Rel i₂ i₂')
Goal: c₁.σ c₂ c₁₂ i₁ i₂ * c₁.ε₂ c₂ c₁₂ (i₁, i₂) = c₂.ε₁ c₁ c₁₂ (i₂, i₁) * c₁.σ c₂ c₁₂ i₁ i₂'
-- ComplexShape.instTotalComplexShapeSymmetryIntUp_σ
@[simps]
instance : TotalComplexShapeSymmetry (up ℤ) (up ℤ) (up ℤ) 
Arguments: (p q : ℤ)
Goal: TotalComplexShapeSymmetry.σ (ComplexShape.up ℤ) (ComplexShape.up ℤ) (ComplexShape.up ℤ) p q = (p * q).negOnePow
-- TotalComplexShapeSymmetrySymmetry.σ_symm
class TotalComplexShapeSymmetrySymmetry [TotalComplexShape c₁ c₂ c₁₂]
    [TotalComplexShape c₂ c₁ c₁₂] [TotalComplexShapeSymmetry c₁ c₂ c₁₂]
    [TotalComplexShapeSymmetry c₂ c₁ c₁₂] : Prop where
  σ_symm i₁ i₂ : ComplexShape.σ c₂ c₁ c₁₂ i₂ i₁ = ComplexShape.σ c₁ c₂ c₁₂ i₁ i₂
Arguments: {I₁ : Type u_1} {I₂ : Type u_2} {I₁₂ : Type u_3} {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} {c₁₂ : ComplexShape I₁₂} [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] [TotalComplexShapeSymmetry c₁ c₂ c₁₂] [TotalComplexShapeSymmetry c₂ c₁ c₁₂] [self : TotalComplexShapeSymmetrySymmetry c₁ c₂ c₁₂] (i₁ : I₁) (i₂ : I₂)
Goal: c₂.σ c₁ c₁₂ i₂ i₁ = c₁.σ c₂ c₁₂ i₁ i₂
-- ComplexShape.σ_symm
lemma σ_symm (i₁ : I₁) (i₂ : I₂) :
    σ c₂ c₁ c₁₂ i₂ i₁ = σ c₁ c₂ c₁₂ i₁ i₂ 
Arguments: {I₁ : Type u_1} {I₂ : Type u_2} {I₁₂ : Type u_3} (c₁ : ComplexShape I₁) (c₂ : ComplexShape I₂) (c₁₂ : ComplexShape I₁₂) [TotalComplexShape c₁ c₂ c₁₂] [TotalComplexShape c₂ c₁ c₁₂] [TotalComplexShapeSymmetry c₁ c₂ c₁₂] [TotalComplexShapeSymmetry c₂ c₁ c₁₂] [TotalComplexShapeSymmetrySymmetry c₁ c₂ c₁₂] (i₁ : I₁) (i₂ : I₂)
Goal: c₂.σ c₁ c₁₂ i₂ i₁ = c₁.σ c₂ c₁₂ i₁ i₂
-- CategoryTheory.ShortComplex.add_τ₁
@[simp] lemma add_τ₁ (φ φ' : S₁ ⟶ S₂) : (φ + φ').τ₁ = φ.τ₁ + φ'.τ₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂)
Goal: (φ + φ').τ₁ = φ.τ₁ + φ'.τ₁
-- CategoryTheory.ShortComplex.add_τ₂
@[simp] lemma add_τ₂ (φ φ' : S₁ ⟶ S₂) : (φ + φ').τ₂ = φ.τ₂ + φ'.τ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂)
Goal: (φ + φ').τ₂ = φ.τ₂ + φ'.τ₂
-- CategoryTheory.ShortComplex.add_τ₃
@[simp] lemma add_τ₃ (φ φ' : S₁ ⟶ S₂) : (φ + φ').τ₃ = φ.τ₃ + φ'.τ₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂)
Goal: (φ + φ').τ₃ = φ.τ₃ + φ'.τ₃
-- CategoryTheory.ShortComplex.sub_τ₁
@[simp] lemma sub_τ₁ (φ φ' : S₁ ⟶ S₂) : (φ - φ').τ₁ = φ.τ₁ - φ'.τ₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂)
Goal: (φ - φ').τ₁ = φ.τ₁ - φ'.τ₁
-- CategoryTheory.ShortComplex.sub_τ₂
@[simp] lemma sub_τ₂ (φ φ' : S₁ ⟶ S₂) : (φ - φ').τ₂ = φ.τ₂ - φ'.τ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂)
Goal: (φ - φ').τ₂ = φ.τ₂ - φ'.τ₂
-- CategoryTheory.ShortComplex.sub_τ₃
@[simp] lemma sub_τ₃ (φ φ' : S₁ ⟶ S₂) : (φ - φ').τ₃ = φ.τ₃ - φ'.τ₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂)
Goal: (φ - φ').τ₃ = φ.τ₃ - φ'.τ₃
-- CategoryTheory.ShortComplex.neg_τ₁
@[simp] lemma neg_τ₁ (φ : S₁ ⟶ S₂) : (-φ).τ₁ = -φ.τ₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂)
Goal: (-φ).τ₁ = -φ.τ₁
-- CategoryTheory.ShortComplex.neg_τ₂
@[simp] lemma neg_τ₂ (φ : S₁ ⟶ S₂) : (-φ).τ₂ = -φ.τ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂)
Goal: (-φ).τ₂ = -φ.τ₂
-- CategoryTheory.ShortComplex.neg_τ₃
@[simp] lemma neg_τ₃ (φ : S₁ ⟶ S₂) : (-φ).τ₃ = -φ.τ₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂)
Goal: (-φ).τ₃ = -φ.τ₃
-- CategoryTheory.ShortComplex.LeftHomologyMapData.neg_φK
@[simps]
def neg : LeftHomologyMapData (-φ) h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂)
Goal: γ.neg.φK = -γ.φK
-- CategoryTheory.ShortComplex.LeftHomologyMapData.neg_φH
@[simps]
def neg : LeftHomologyMapData (-φ) h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂)
Goal: γ.neg.φH = -γ.φH
-- CategoryTheory.ShortComplex.LeftHomologyMapData.add_φK
@[simps]
def add : LeftHomologyMapData (φ + φ') h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) (γ' : CategoryTheory.ShortComplex.LeftHomologyMapData φ' h₁ h₂)
Goal: (γ.add γ').φK = γ.φK + γ'.φK
-- CategoryTheory.ShortComplex.LeftHomologyMapData.add_φH
@[simps]
def add : LeftHomologyMapData (φ + φ') h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) (γ' : CategoryTheory.ShortComplex.LeftHomologyMapData φ' h₁ h₂)
Goal: (γ.add γ').φH = γ.φH + γ'.φH
-- CategoryTheory.ShortComplex.leftHomologyMap'_neg
@[simp]
lemma leftHomologyMap'_neg :
    leftHomologyMap' (-φ) h₁ h₂ = -leftHomologyMap' φ h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.leftHomologyMap' (-φ) h₁ h₂ = -CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂
-- CategoryTheory.ShortComplex.cyclesMap'_neg
@[simp]
lemma cyclesMap'_neg :
    cyclesMap' (-φ) h₁ h₂ = -cyclesMap' φ h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.cyclesMap' (-φ) h₁ h₂ = -CategoryTheory.ShortComplex.cyclesMap' φ h₁ h₂
-- CategoryTheory.ShortComplex.leftHomologyMap'_add
@[simp]
lemma leftHomologyMap'_add :
    leftHomologyMap' (φ + φ') h₁ h₂ = leftHomologyMap' φ h₁ h₂ +
      leftHomologyMap' φ' h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.leftHomologyMap' (φ + φ') h₁ h₂ =
  CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂ + CategoryTheory.ShortComplex.leftHomologyMap' φ' h₁ h₂
-- CategoryTheory.ShortComplex.cyclesMap'_add
@[simp]
lemma cyclesMap'_add :
    cyclesMap' (φ + φ') h₁ h₂ = cyclesMap' φ h₁ h₂ +
      cyclesMap' φ' h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.cyclesMap' (φ + φ') h₁ h₂ =
  CategoryTheory.ShortComplex.cyclesMap' φ h₁ h₂ + CategoryTheory.ShortComplex.cyclesMap' φ' h₁ h₂
-- CategoryTheory.ShortComplex.leftHomologyMap'_sub
@[simp]
lemma leftHomologyMap'_sub :
    leftHomologyMap' (φ - φ') h₁ h₂ = leftHomologyMap' φ h₁ h₂ -
      leftHomologyMap' φ' h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.leftHomologyMap' (φ - φ') h₁ h₂ =
  CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂ - CategoryTheory.ShortComplex.leftHomologyMap' φ' h₁ h₂
-- CategoryTheory.ShortComplex.cyclesMap'_sub
@[simp]
lemma cyclesMap'_sub :
    cyclesMap' (φ - φ') h₁ h₂ = cyclesMap' φ h₁ h₂ -
      cyclesMap' φ' h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.cyclesMap' (φ - φ') h₁ h₂ =
  CategoryTheory.ShortComplex.cyclesMap' φ h₁ h₂ - CategoryTheory.ShortComplex.cyclesMap' φ' h₁ h₂
-- CategoryTheory.ShortComplex.leftHomologyMap_neg
@[simp]
lemma leftHomologyMap_neg : leftHomologyMap (-φ) = -leftHomologyMap φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.leftHomologyMap (-φ) = -CategoryTheory.ShortComplex.leftHomologyMap φ
-- CategoryTheory.ShortComplex.cyclesMap_neg
@[simp]
lemma cyclesMap_neg : cyclesMap (-φ) = -cyclesMap φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.cyclesMap (-φ) = -CategoryTheory.ShortComplex.cyclesMap φ
-- CategoryTheory.ShortComplex.leftHomologyMap_add
@[simp]
lemma leftHomologyMap_add : leftHomologyMap (φ + φ') = leftHomologyMap φ + leftHomologyMap φ' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.leftHomologyMap (φ + φ') =
  CategoryTheory.ShortComplex.leftHomologyMap φ + CategoryTheory.ShortComplex.leftHomologyMap φ'
-- CategoryTheory.ShortComplex.cyclesMap_add
@[simp]
lemma cyclesMap_add : cyclesMap (φ + φ') = cyclesMap φ + cyclesMap φ' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.cyclesMap (φ + φ') =
  CategoryTheory.ShortComplex.cyclesMap φ + CategoryTheory.ShortComplex.cyclesMap φ'
-- CategoryTheory.ShortComplex.leftHomologyMap_sub
@[simp]
lemma leftHomologyMap_sub : leftHomologyMap (φ - φ') = leftHomologyMap φ - leftHomologyMap φ' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.leftHomologyMap (φ - φ') =
  CategoryTheory.ShortComplex.leftHomologyMap φ - CategoryTheory.ShortComplex.leftHomologyMap φ'
-- CategoryTheory.ShortComplex.cyclesMap_sub
@[simp]
lemma cyclesMap_sub : cyclesMap (φ - φ') = cyclesMap φ - cyclesMap φ' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.cyclesMap (φ - φ') =
  CategoryTheory.ShortComplex.cyclesMap φ - CategoryTheory.ShortComplex.cyclesMap φ'
-- CategoryTheory.ShortComplex.RightHomologyMapData.neg_φH
@[simps]
def neg : RightHomologyMapData (-φ) h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂)
Goal: γ.neg.φH = -γ.φH
-- CategoryTheory.ShortComplex.RightHomologyMapData.neg_φQ
@[simps]
def neg : RightHomologyMapData (-φ) h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂)
Goal: γ.neg.φQ = -γ.φQ
-- CategoryTheory.ShortComplex.RightHomologyMapData.add_φQ
@[simps]
def add : RightHomologyMapData (φ + φ') h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (γ' : CategoryTheory.ShortComplex.RightHomologyMapData φ' h₁ h₂)
Goal: (γ.add γ').φQ = γ.φQ + γ'.φQ
-- CategoryTheory.ShortComplex.RightHomologyMapData.add_φH
@[simps]
def add : RightHomologyMapData (φ + φ') h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (γ' : CategoryTheory.ShortComplex.RightHomologyMapData φ' h₁ h₂)
Goal: (γ.add γ').φH = γ.φH + γ'.φH
-- CategoryTheory.ShortComplex.rightHomologyMap'_neg
@[simp]
lemma rightHomologyMap'_neg :
    rightHomologyMap' (-φ) h₁ h₂ = -rightHomologyMap' φ h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.ShortComplex.rightHomologyMap' (-φ) h₁ h₂ = -CategoryTheory.ShortComplex.rightHomologyMap' φ h₁ h₂
-- CategoryTheory.ShortComplex.opcyclesMap'_neg
@[simp]
lemma opcyclesMap'_neg :
    opcyclesMap' (-φ) h₁ h₂ = -opcyclesMap' φ h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.ShortComplex.opcyclesMap' (-φ) h₁ h₂ = -CategoryTheory.ShortComplex.opcyclesMap' φ h₁ h₂
-- CategoryTheory.ShortComplex.rightHomologyMap'_add
@[simp]
lemma rightHomologyMap'_add :
    rightHomologyMap' (φ + φ') h₁ h₂ = rightHomologyMap' φ h₁ h₂ +
      rightHomologyMap' φ' h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.ShortComplex.rightHomologyMap' (φ + φ') h₁ h₂ =
  CategoryTheory.ShortComplex.rightHomologyMap' φ h₁ h₂ + CategoryTheory.ShortComplex.rightHomologyMap' φ' h₁ h₂
-- CategoryTheory.ShortComplex.opcyclesMap'_add
@[simp]
lemma opcyclesMap'_add :
    opcyclesMap' (φ + φ') h₁ h₂ = opcyclesMap' φ h₁ h₂ +
      opcyclesMap' φ' h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.ShortComplex.opcyclesMap' (φ + φ') h₁ h₂ =
  CategoryTheory.ShortComplex.opcyclesMap' φ h₁ h₂ + CategoryTheory.ShortComplex.opcyclesMap' φ' h₁ h₂
-- CategoryTheory.ShortComplex.rightHomologyMap'_sub
@[simp]
lemma rightHomologyMap'_sub :
    rightHomologyMap' (φ - φ') h₁ h₂ = rightHomologyMap' φ h₁ h₂ -
      rightHomologyMap' φ' h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.ShortComplex.rightHomologyMap' (φ - φ') h₁ h₂ =
  CategoryTheory.ShortComplex.rightHomologyMap' φ h₁ h₂ - CategoryTheory.ShortComplex.rightHomologyMap' φ' h₁ h₂
-- CategoryTheory.ShortComplex.opcyclesMap'_sub
@[simp]
lemma opcyclesMap'_sub :
    opcyclesMap' (φ - φ') h₁ h₂ = opcyclesMap' φ h₁ h₂ -
      opcyclesMap' φ' h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ φ' : S₁ ⟶ S₂} (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: CategoryTheory.ShortComplex.opcyclesMap' (φ - φ') h₁ h₂ =
  CategoryTheory.ShortComplex.opcyclesMap' φ h₁ h₂ - CategoryTheory.ShortComplex.opcyclesMap' φ' h₁ h₂
-- CategoryTheory.ShortComplex.rightHomologyMap_neg
@[simp]
lemma rightHomologyMap_neg : rightHomologyMap (-φ) = -rightHomologyMap φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.rightHomologyMap (-φ) = -CategoryTheory.ShortComplex.rightHomologyMap φ
-- CategoryTheory.ShortComplex.opcyclesMap_neg
@[simp]
lemma opcyclesMap_neg : opcyclesMap (-φ) = -opcyclesMap φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.opcyclesMap (-φ) = -CategoryTheory.ShortComplex.opcyclesMap φ
-- CategoryTheory.ShortComplex.rightHomologyMap_add
@[simp]
lemma rightHomologyMap_add :
    rightHomologyMap (φ + φ') = rightHomologyMap φ + rightHomologyMap φ' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.rightHomologyMap (φ + φ') =
  CategoryTheory.ShortComplex.rightHomologyMap φ + CategoryTheory.ShortComplex.rightHomologyMap φ'
-- CategoryTheory.ShortComplex.opcyclesMap_add
@[simp]
lemma opcyclesMap_add : opcyclesMap (φ + φ') = opcyclesMap φ + opcyclesMap φ' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.opcyclesMap (φ + φ') =
  CategoryTheory.ShortComplex.opcyclesMap φ + CategoryTheory.ShortComplex.opcyclesMap φ'
-- CategoryTheory.ShortComplex.rightHomologyMap_sub
@[simp]
lemma rightHomologyMap_sub :
    rightHomologyMap (φ - φ') = rightHomologyMap φ - rightHomologyMap φ' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.rightHomologyMap (φ - φ') =
  CategoryTheory.ShortComplex.rightHomologyMap φ - CategoryTheory.ShortComplex.rightHomologyMap φ'
-- CategoryTheory.ShortComplex.opcyclesMap_sub
@[simp]
lemma opcyclesMap_sub : opcyclesMap (φ - φ') = opcyclesMap φ - opcyclesMap φ' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ φ' : S₁ ⟶ S₂) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.opcyclesMap (φ - φ') =
  CategoryTheory.ShortComplex.opcyclesMap φ - CategoryTheory.ShortComplex.opcyclesMap φ'
-- CategoryTheory.ShortComplex.HomologyMapData.neg_left
@[simps]
def neg : HomologyMapData (-φ) h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂)
Goal: γ.neg.left = γ.left.neg