ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.ShortComplex.HomologyMapData.neg_right
@[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.right = γ.right.neg
-- CategoryTheory.ShortComplex.HomologyMapData.add_left
@[simps]
def add : HomologyMapData (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.HomologyData} {h₂ : S₂.HomologyData} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (γ' : CategoryTheory.ShortComplex.HomologyMapData φ' h₁ h₂)
Goal: (γ.add γ').left = γ.left.add γ'.left
-- CategoryTheory.ShortComplex.HomologyMapData.add_right
@[simps]
def add : HomologyMapData (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.HomologyData} {h₂ : S₂.HomologyData} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (γ' : CategoryTheory.ShortComplex.HomologyMapData φ' h₁ h₂)
Goal: (γ.add γ').right = γ.right.add γ'.right
-- CategoryTheory.ShortComplex.homologyMap'_neg
@[simp]
lemma homologyMap'_neg :
    homologyMap' (-φ) h₁ h₂ = -homologyMap' 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)
Goal: CategoryTheory.ShortComplex.homologyMap' (-φ) h₁ h₂ = -CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂
-- CategoryTheory.ShortComplex.homologyMap'_add
@[simp]
lemma homologyMap'_add :
    homologyMap' (φ + φ') h₁ h₂ = homologyMap' φ h₁ h₂ + homologyMap' 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.HomologyData) (h₂ : S₂.HomologyData)
Goal: CategoryTheory.ShortComplex.homologyMap' (φ + φ') h₁ h₂ =
  CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂ + CategoryTheory.ShortComplex.homologyMap' φ' h₁ h₂
-- CategoryTheory.ShortComplex.homologyMap'_sub
@[simp]
lemma homologyMap'_sub :
    homologyMap' (φ - φ') h₁ h₂ = homologyMap' φ h₁ h₂ - homologyMap' 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.HomologyData) (h₂ : S₂.HomologyData)
Goal: CategoryTheory.ShortComplex.homologyMap' (φ - φ') h₁ h₂ =
  CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂ - CategoryTheory.ShortComplex.homologyMap' φ' h₁ h₂
-- CategoryTheory.ShortComplex.homologyMap_neg
@[simp]
lemma homologyMap_neg : homologyMap (-φ) = -homologyMap 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.HasHomology] [S₂.HasHomology]
Goal: CategoryTheory.ShortComplex.homologyMap (-φ) = -CategoryTheory.ShortComplex.homologyMap φ
-- CategoryTheory.ShortComplex.homologyMap_add
@[simp]
lemma homologyMap_add : homologyMap (φ + φ') = homologyMap φ + homologyMap 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.HasHomology] [S₂.HasHomology]
Goal: CategoryTheory.ShortComplex.homologyMap (φ + φ') =
  CategoryTheory.ShortComplex.homologyMap φ + CategoryTheory.ShortComplex.homologyMap φ'
-- CategoryTheory.ShortComplex.homologyMap_sub
@[simp]
lemma homologyMap_sub : homologyMap (φ - φ') = homologyMap φ - homologyMap 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.HasHomology] [S₂.HasHomology]
Goal: CategoryTheory.ShortComplex.homologyMap (φ - φ') =
  CategoryTheory.ShortComplex.homologyMap φ - CategoryTheory.ShortComplex.homologyMap φ'
-- CategoryTheory.ShortComplex.Homotopy.g_h₃
@[ext]
structure Homotopy where
    h₀ : S₁.X₁ ⟶ S₂.X₁
  h₀_f : h₀ ≫ S₂.f = 0 := by aesop_cat
    h₁ : S₁.X₂ ⟶ S₂.X₁
    h₂ : S₁.X₃ ⟶ S₂.X₂
    h₃ : S₁.X₃ ⟶ S₂.X₃
  g_h₃ : S₁.g ≫ h₃ = 0 := by aesop_cat
  comm₁ : φ₁.τ₁ = S₁.f ≫ h₁ + h₀ + φ₂.τ₁ := by aesop_cat
  comm₂ : φ₁.τ₂ = S₁.g ≫ h₂ + h₁ ≫ S₂.f + φ₂.τ₂ := by aesop_cat
  comm₃ : φ₁.τ₃ = h₃ + h₂ ≫ S₂.g + φ₂.τ₃ := by aesop_cat
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (self : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: CategoryTheory.CategoryStruct.comp S₁.g self.h₃ = 0
-- CategoryTheory.ShortComplex.Homotopy.comm₁
@[ext]
structure Homotopy where
    h₀ : S₁.X₁ ⟶ S₂.X₁
  h₀_f : h₀ ≫ S₂.f = 0 := by aesop_cat
    h₁ : S₁.X₂ ⟶ S₂.X₁
    h₂ : S₁.X₃ ⟶ S₂.X₂
    h₃ : S₁.X₃ ⟶ S₂.X₃
  g_h₃ : S₁.g ≫ h₃ = 0 := by aesop_cat
  comm₁ : φ₁.τ₁ = S₁.f ≫ h₁ + h₀ + φ₂.τ₁ := by aesop_cat
  comm₂ : φ₁.τ₂ = S₁.g ≫ h₂ + h₁ ≫ S₂.f + φ₂.τ₂ := by aesop_cat
  comm₃ : φ₁.τ₃ = h₃ + h₂ ≫ S₂.g + φ₂.τ₃ := by aesop_cat
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (self : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: φ₁.τ₁ = CategoryTheory.CategoryStruct.comp S₁.f self.h₁ + self.h₀ + φ₂.τ₁
-- CategoryTheory.ShortComplex.Homotopy.comm₃
@[ext]
structure Homotopy where
    h₀ : S₁.X₁ ⟶ S₂.X₁
  h₀_f : h₀ ≫ S₂.f = 0 := by aesop_cat
    h₁ : S₁.X₂ ⟶ S₂.X₁
    h₂ : S₁.X₃ ⟶ S₂.X₂
    h₃ : S₁.X₃ ⟶ S₂.X₃
  g_h₃ : S₁.g ≫ h₃ = 0 := by aesop_cat
  comm₁ : φ₁.τ₁ = S₁.f ≫ h₁ + h₀ + φ₂.τ₁ := by aesop_cat
  comm₂ : φ₁.τ₂ = S₁.g ≫ h₂ + h₁ ≫ S₂.f + φ₂.τ₂ := by aesop_cat
  comm₃ : φ₁.τ₃ = h₃ + h₂ ≫ S₂.g + φ₂.τ₃ := by aesop_cat
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (self : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: φ₁.τ₃ = self.h₃ + CategoryTheory.CategoryStruct.comp self.h₂ S₂.g + φ₂.τ₃
-- CategoryTheory.ShortComplex.Homotopy.comm₂
@[ext]
structure Homotopy where
    h₀ : S₁.X₁ ⟶ S₂.X₁
  h₀_f : h₀ ≫ S₂.f = 0 := by aesop_cat
    h₁ : S₁.X₂ ⟶ S₂.X₁
    h₂ : S₁.X₃ ⟶ S₂.X₂
    h₃ : S₁.X₃ ⟶ S₂.X₃
  g_h₃ : S₁.g ≫ h₃ = 0 := by aesop_cat
  comm₁ : φ₁.τ₁ = S₁.f ≫ h₁ + h₀ + φ₂.τ₁ := by aesop_cat
  comm₂ : φ₁.τ₂ = S₁.g ≫ h₂ + h₁ ≫ S₂.f + φ₂.τ₂ := by aesop_cat
  comm₃ : φ₁.τ₃ = h₃ + h₂ ≫ S₂.g + φ₂.τ₃ := by aesop_cat
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (self : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: φ₁.τ₂ = CategoryTheory.CategoryStruct.comp S₁.g self.h₂ + CategoryTheory.CategoryStruct.comp self.h₁ S₂.f + φ₂.τ₂
-- CategoryTheory.ShortComplex.Homotopy.h₀_f
@[ext]
structure Homotopy where
    h₀ : S₁.X₁ ⟶ S₂.X₁
  h₀_f : h₀ ≫ S₂.f = 0 := by aesop_cat
    h₁ : S₁.X₂ ⟶ S₂.X₁
    h₂ : S₁.X₃ ⟶ S₂.X₂
    h₃ : S₁.X₃ ⟶ S₂.X₃
  g_h₃ : S₁.g ≫ h₃ = 0 := by aesop_cat
  comm₁ : φ₁.τ₁ = S₁.f ≫ h₁ + h₀ + φ₂.τ₁ := by aesop_cat
  comm₂ : φ₁.τ₂ = S₁.g ≫ h₂ + h₁ ≫ S₂.f + φ₂.τ₂ := by aesop_cat
  comm₃ : φ₁.τ₃ = h₃ + h₂ ≫ S₂.g + φ₂.τ₃ := by aesop_cat
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (self : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: CategoryTheory.CategoryStruct.comp self.h₀ S₂.f = 0
-- CategoryTheory.ShortComplex.nullHomotopic_τ₁
@[simps]
def nullHomotopic (h₀ : S₁.X₁ ⟶ S₂.X₁) (h₀_f : h₀ ≫ S₂.f = 0)
    (h₁ : S₁.X₂ ⟶ S₂.X₁) (h₂ : S₁.X₃ ⟶ S₂.X₂) (h₃ : S₁.X₃ ⟶ S₂.X₃) (g_h₃ : S₁.g ≫ h₃ = 0) :
    S₁ ⟶ S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S₁ S₂ : CategoryTheory.ShortComplex C) (h₀ : S₁.X₁ ⟶ S₂.X₁) (h₀_f : CategoryTheory.CategoryStruct.comp h₀ S₂.f = 0) (h₁ : S₁.X₂ ⟶ S₂.X₁) (h₂ : S₁.X₃ ⟶ S₂.X₂) (h₃ : S₁.X₃ ⟶ S₂.X₃) (g_h₃ : CategoryTheory.CategoryStruct.comp S₁.g h₃ = 0)
Goal: (S₁.nullHomotopic S₂ h₀ h₀_f h₁ h₂ h₃ g_h₃).τ₁ = h₀ + CategoryTheory.CategoryStruct.comp S₁.f h₁
-- CategoryTheory.ShortComplex.nullHomotopic_τ₃
@[simps]
def nullHomotopic (h₀ : S₁.X₁ ⟶ S₂.X₁) (h₀_f : h₀ ≫ S₂.f = 0)
    (h₁ : S₁.X₂ ⟶ S₂.X₁) (h₂ : S₁.X₃ ⟶ S₂.X₂) (h₃ : S₁.X₃ ⟶ S₂.X₃) (g_h₃ : S₁.g ≫ h₃ = 0) :
    S₁ ⟶ S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S₁ S₂ : CategoryTheory.ShortComplex C) (h₀ : S₁.X₁ ⟶ S₂.X₁) (h₀_f : CategoryTheory.CategoryStruct.comp h₀ S₂.f = 0) (h₁ : S₁.X₂ ⟶ S₂.X₁) (h₂ : S₁.X₃ ⟶ S₂.X₂) (h₃ : S₁.X₃ ⟶ S₂.X₃) (g_h₃ : CategoryTheory.CategoryStruct.comp S₁.g h₃ = 0)
Goal: (S₁.nullHomotopic S₂ h₀ h₀_f h₁ h₂ h₃ g_h₃).τ₃ = CategoryTheory.CategoryStruct.comp h₂ S₂.g + h₃
-- CategoryTheory.ShortComplex.nullHomotopic_τ₂
@[simps]
def nullHomotopic (h₀ : S₁.X₁ ⟶ S₂.X₁) (h₀_f : h₀ ≫ S₂.f = 0)
    (h₁ : S₁.X₂ ⟶ S₂.X₁) (h₂ : S₁.X₃ ⟶ S₂.X₂) (h₃ : S₁.X₃ ⟶ S₂.X₃) (g_h₃ : S₁.g ≫ h₃ = 0) :
    S₁ ⟶ S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (S₁ S₂ : CategoryTheory.ShortComplex C) (h₀ : S₁.X₁ ⟶ S₂.X₁) (h₀_f : CategoryTheory.CategoryStruct.comp h₀ S₂.f = 0) (h₁ : S₁.X₂ ⟶ S₂.X₁) (h₂ : S₁.X₃ ⟶ S₂.X₂) (h₃ : S₁.X₃ ⟶ S₂.X₃) (g_h₃ : CategoryTheory.CategoryStruct.comp S₁.g h₃ = 0)
Goal: (S₁.nullHomotopic S₂ h₀ h₀_f h₁ h₂ h₃ g_h₃).τ₂ =
  CategoryTheory.CategoryStruct.comp h₁ S₂.f + CategoryTheory.CategoryStruct.comp S₁.g h₂
-- CategoryTheory.ShortComplex.Homotopy.ofEq_h₁
@[simps]
def ofEq (h : φ₁ = φ₂) : Homotopy φ₁ φ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : φ₁ = φ₂)
Goal: (CategoryTheory.ShortComplex.Homotopy.ofEq h).h₁ = 0
-- CategoryTheory.ShortComplex.Homotopy.ofEq_h₂
@[simps]
def ofEq (h : φ₁ = φ₂) : Homotopy φ₁ φ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : φ₁ = φ₂)
Goal: (CategoryTheory.ShortComplex.Homotopy.ofEq h).h₂ = 0
-- CategoryTheory.ShortComplex.Homotopy.ofEq_h₀
@[simps]
def ofEq (h : φ₁ = φ₂) : Homotopy φ₁ φ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : φ₁ = φ₂)
Goal: (CategoryTheory.ShortComplex.Homotopy.ofEq h).h₀ = 0
-- CategoryTheory.ShortComplex.Homotopy.ofEq_h₃
@[simps]
def ofEq (h : φ₁ = φ₂) : Homotopy φ₁ φ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : φ₁ = φ₂)
Goal: (CategoryTheory.ShortComplex.Homotopy.ofEq h).h₃ = 0
-- CategoryTheory.ShortComplex.Homotopy.refl_h₃
@[simps!]
def refl (φ : S₁ ⟶ S₂) : Homotopy φ φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂)
Goal: (CategoryTheory.ShortComplex.Homotopy.refl φ).h₃ = 0
-- CategoryTheory.ShortComplex.Homotopy.refl_h₁
@[simps!]
def refl (φ : S₁ ⟶ S₂) : Homotopy φ φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂)
Goal: (CategoryTheory.ShortComplex.Homotopy.refl φ).h₁ = 0
-- CategoryTheory.ShortComplex.Homotopy.refl_h₂
@[simps!]
def refl (φ : S₁ ⟶ S₂) : Homotopy φ φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂)
Goal: (CategoryTheory.ShortComplex.Homotopy.refl φ).h₂ = 0
-- CategoryTheory.ShortComplex.Homotopy.refl_h₀
@[simps!]
def refl (φ : S₁ ⟶ S₂) : Homotopy φ φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂)
Goal: (CategoryTheory.ShortComplex.Homotopy.refl φ).h₀ = 0
-- CategoryTheory.ShortComplex.Homotopy.symm_h₀
@[simps]
def symm (h : Homotopy φ₁ φ₂) : Homotopy φ₂ φ₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: h.symm.h₀ = -h.h₀
-- CategoryTheory.ShortComplex.Homotopy.symm_h₃
@[simps]
def symm (h : Homotopy φ₁ φ₂) : Homotopy φ₂ φ₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: h.symm.h₃ = -h.h₃
-- CategoryTheory.ShortComplex.Homotopy.symm_h₂
@[simps]
def symm (h : Homotopy φ₁ φ₂) : Homotopy φ₂ φ₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: h.symm.h₂ = -h.h₂
-- CategoryTheory.ShortComplex.Homotopy.symm_h₁
@[simps]
def symm (h : Homotopy φ₁ φ₂) : Homotopy φ₂ φ₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: h.symm.h₁ = -h.h₁
-- CategoryTheory.ShortComplex.Homotopy.neg_h₃
@[simps]
def neg (h : Homotopy φ₁ φ₂) : Homotopy (-φ₁) (-φ₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: h.neg.h₃ = -h.h₃
-- CategoryTheory.ShortComplex.Homotopy.neg_h₀
@[simps]
def neg (h : Homotopy φ₁ φ₂) : Homotopy (-φ₁) (-φ₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: h.neg.h₀ = -h.h₀
-- CategoryTheory.ShortComplex.Homotopy.neg_h₁
@[simps]
def neg (h : Homotopy φ₁ φ₂) : Homotopy (-φ₁) (-φ₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: h.neg.h₁ = -h.h₁
-- CategoryTheory.ShortComplex.Homotopy.neg_h₂
@[simps]
def neg (h : Homotopy φ₁ φ₂) : Homotopy (-φ₁) (-φ₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂)
Goal: h.neg.h₂ = -h.h₂
-- CategoryTheory.ShortComplex.Homotopy.trans_h₁
@[simps]
def trans (h₁₂ : Homotopy φ₁ φ₂) (h₂₃ : Homotopy φ₂ φ₃) : Homotopy φ₁ φ₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ : S₁ ⟶ S₂} (h₁₂ : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h₂₃ : CategoryTheory.ShortComplex.Homotopy φ₂ φ₃)
Goal: (h₁₂.trans h₂₃).h₁ = h₁₂.h₁ + h₂₃.h₁
-- CategoryTheory.ShortComplex.Homotopy.trans_h₃
@[simps]
def trans (h₁₂ : Homotopy φ₁ φ₂) (h₂₃ : Homotopy φ₂ φ₃) : Homotopy φ₁ φ₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ : S₁ ⟶ S₂} (h₁₂ : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h₂₃ : CategoryTheory.ShortComplex.Homotopy φ₂ φ₃)
Goal: (h₁₂.trans h₂₃).h₃ = h₁₂.h₃ + h₂₃.h₃
-- CategoryTheory.ShortComplex.Homotopy.trans_h₀
@[simps]
def trans (h₁₂ : Homotopy φ₁ φ₂) (h₂₃ : Homotopy φ₂ φ₃) : Homotopy φ₁ φ₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ : S₁ ⟶ S₂} (h₁₂ : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h₂₃ : CategoryTheory.ShortComplex.Homotopy φ₂ φ₃)
Goal: (h₁₂.trans h₂₃).h₀ = h₁₂.h₀ + h₂₃.h₀
-- CategoryTheory.ShortComplex.Homotopy.trans_h₂
@[simps]
def trans (h₁₂ : Homotopy φ₁ φ₂) (h₂₃ : Homotopy φ₂ φ₃) : Homotopy φ₁ φ₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ : S₁ ⟶ S₂} (h₁₂ : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h₂₃ : CategoryTheory.ShortComplex.Homotopy φ₂ φ₃)
Goal: (h₁₂.trans h₂₃).h₂ = h₁₂.h₂ + h₂₃.h₂
-- CategoryTheory.ShortComplex.Homotopy.add_h₀
@[simps]
def add (h : Homotopy φ₁ φ₂) (h' : Homotopy φ₃ φ₄) : Homotopy (φ₁ + φ₃) (φ₂ + φ₄) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h' : CategoryTheory.ShortComplex.Homotopy φ₃ φ₄)
Goal: (h.add h').h₀ = h.h₀ + h'.h₀
-- CategoryTheory.ShortComplex.Homotopy.add_h₃
@[simps]
def add (h : Homotopy φ₁ φ₂) (h' : Homotopy φ₃ φ₄) : Homotopy (φ₁ + φ₃) (φ₂ + φ₄) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h' : CategoryTheory.ShortComplex.Homotopy φ₃ φ₄)
Goal: (h.add h').h₃ = h.h₃ + h'.h₃
-- CategoryTheory.ShortComplex.Homotopy.add_h₁
@[simps]
def add (h : Homotopy φ₁ φ₂) (h' : Homotopy φ₃ φ₄) : Homotopy (φ₁ + φ₃) (φ₂ + φ₄) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h' : CategoryTheory.ShortComplex.Homotopy φ₃ φ₄)
Goal: (h.add h').h₁ = h.h₁ + h'.h₁
-- CategoryTheory.ShortComplex.Homotopy.add_h₂
@[simps]
def add (h : Homotopy φ₁ φ₂) (h' : Homotopy φ₃ φ₄) : Homotopy (φ₁ + φ₃) (φ₂ + φ₄) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h' : CategoryTheory.ShortComplex.Homotopy φ₃ φ₄)
Goal: (h.add h').h₂ = h.h₂ + h'.h₂
-- CategoryTheory.ShortComplex.Homotopy.sub_h₀
@[simps]
def sub (h : Homotopy φ₁ φ₂) (h' : Homotopy φ₃ φ₄) : Homotopy (φ₁ - φ₃) (φ₂ - φ₄) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h' : CategoryTheory.ShortComplex.Homotopy φ₃ φ₄)
Goal: (h.sub h').h₀ = h.h₀ - h'.h₀
-- CategoryTheory.ShortComplex.Homotopy.sub_h₃
@[simps]
def sub (h : Homotopy φ₁ φ₂) (h' : Homotopy φ₃ φ₄) : Homotopy (φ₁ - φ₃) (φ₂ - φ₄) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h' : CategoryTheory.ShortComplex.Homotopy φ₃ φ₄)
Goal: (h.sub h').h₃ = h.h₃ - h'.h₃
-- CategoryTheory.ShortComplex.Homotopy.sub_h₁
@[simps]
def sub (h : Homotopy φ₁ φ₂) (h' : Homotopy φ₃ φ₄) : Homotopy (φ₁ - φ₃) (φ₂ - φ₄) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h' : CategoryTheory.ShortComplex.Homotopy φ₃ φ₄)
Goal: (h.sub h').h₁ = h.h₁ - h'.h₁
-- CategoryTheory.ShortComplex.Homotopy.sub_h₂
@[simps]
def sub (h : Homotopy φ₁ φ₂) (h' : Homotopy φ₃ φ₄) : Homotopy (φ₁ - φ₃) (φ₂ - φ₄) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (h' : CategoryTheory.ShortComplex.Homotopy φ₃ φ₄)
Goal: (h.sub h').h₂ = h.h₂ - h'.h₂
-- CategoryTheory.ShortComplex.Homotopy.compLeft_h₁
@[simps]
def compLeft (h : Homotopy φ₁ φ₂) (ψ : S₃ ⟶ S₁) : Homotopy (ψ ≫ φ₁) (ψ ≫ φ₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (ψ : S₃ ⟶ S₁)
Goal: (h.compLeft ψ).h₁ = CategoryTheory.CategoryStruct.comp ψ.τ₂ h.h₁
-- CategoryTheory.ShortComplex.Homotopy.compLeft_h₀
@[simps]
def compLeft (h : Homotopy φ₁ φ₂) (ψ : S₃ ⟶ S₁) : Homotopy (ψ ≫ φ₁) (ψ ≫ φ₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (ψ : S₃ ⟶ S₁)
Goal: (h.compLeft ψ).h₀ = CategoryTheory.CategoryStruct.comp ψ.τ₁ h.h₀
-- CategoryTheory.ShortComplex.Homotopy.compLeft_h₂
@[simps]
def compLeft (h : Homotopy φ₁ φ₂) (ψ : S₃ ⟶ S₁) : Homotopy (ψ ≫ φ₁) (ψ ≫ φ₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (ψ : S₃ ⟶ S₁)
Goal: (h.compLeft ψ).h₂ = CategoryTheory.CategoryStruct.comp ψ.τ₃ h.h₂
-- CategoryTheory.ShortComplex.Homotopy.compLeft_h₃
@[simps]
def compLeft (h : Homotopy φ₁ φ₂) (ψ : S₃ ⟶ S₁) : Homotopy (ψ ≫ φ₁) (ψ ≫ φ₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (ψ : S₃ ⟶ S₁)
Goal: (h.compLeft ψ).h₃ = CategoryTheory.CategoryStruct.comp ψ.τ₃ h.h₃
-- CategoryTheory.ShortComplex.Homotopy.compRight_h₂
@[simps]
def compRight (h : Homotopy φ₁ φ₂) (ψ : S₂ ⟶ S₃) : Homotopy (φ₁ ≫ ψ) (φ₂ ≫ ψ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (ψ : S₂ ⟶ S₃)
Goal: (h.compRight ψ).h₂ = CategoryTheory.CategoryStruct.comp h.h₂ ψ.τ₂
-- CategoryTheory.ShortComplex.Homotopy.compRight_h₃
@[simps]
def compRight (h : Homotopy φ₁ φ₂) (ψ : S₂ ⟶ S₃) : Homotopy (φ₁ ≫ ψ) (φ₂ ≫ ψ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (ψ : S₂ ⟶ S₃)
Goal: (h.compRight ψ).h₃ = CategoryTheory.CategoryStruct.comp h.h₃ ψ.τ₃
-- CategoryTheory.ShortComplex.Homotopy.compRight_h₀
@[simps]
def compRight (h : Homotopy φ₁ φ₂) (ψ : S₂ ⟶ S₃) : Homotopy (φ₁ ≫ ψ) (φ₂ ≫ ψ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (ψ : S₂ ⟶ S₃)
Goal: (h.compRight ψ).h₀ = CategoryTheory.CategoryStruct.comp h.h₀ ψ.τ₁
-- CategoryTheory.ShortComplex.Homotopy.compRight_h₁
@[simps]
def compRight (h : Homotopy φ₁ φ₂) (ψ : S₂ ⟶ S₃) : Homotopy (φ₁ ≫ ψ) (φ₂ ≫ ψ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (ψ : S₂ ⟶ S₃)
Goal: (h.compRight ψ).h₁ = CategoryTheory.CategoryStruct.comp h.h₁ ψ.τ₁
-- CategoryTheory.ShortComplex.Homotopy.comp_h₀
@[simps!]
def comp (h : Homotopy φ₁ φ₂) {ψ₁ ψ₂ : S₂ ⟶ S₃} (h' : Homotopy ψ₁ ψ₂) :
    Homotopy (φ₁ ≫ ψ₁) (φ₂ ≫ ψ₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) {ψ₁ ψ₂ : S₂ ⟶ S₃} (h' : CategoryTheory.ShortComplex.Homotopy ψ₁ ψ₂)
Goal: (h.comp h').h₀ = CategoryTheory.CategoryStruct.comp h.h₀ ψ₁.τ₁ + CategoryTheory.CategoryStruct.comp φ₂.τ₁ h'.h₀