ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ChainComplex.toSingle₀Equiv_apply_coe
@[simps apply_coe]
noncomputable def toSingle₀Equiv (C : ChainComplex V ℕ) (X : V) :
    (C ⟶ (single₀ V).obj X) ≃ { f : C.X 0 ⟶ X // C.d 1 0 ≫ f = 0 } 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] (C : ChainComplex V ℕ) (X : V) (φ : C ⟶ (ChainComplex.single₀ V).obj X)
Goal: ↑((C.toSingle₀Equiv X) φ) = φ.f 0
-- ChainComplex.toSingle₀Equiv_symm_apply_f_zero
@[simp]
lemma toSingle₀Equiv_symm_apply_f_zero {C : ChainComplex V ℕ} {X : V}
    (f : C.X 0 ⟶ X) (hf : C.d 1 0 ≫ f = 0) :
    ((toSingle₀Equiv C X).symm ⟨f, hf⟩).f 0 = f 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {C : ChainComplex V ℕ} {X : V} (f : C.X 0 ⟶ X) (hf : C.d 1 0 ≫ f = 0)
Goal: ((C.toSingle₀Equiv X).symm ⟨f, hf⟩).f 0 = f
-- ChainComplex.fromSingle₀Equiv_apply
@[simps apply]
noncomputable def fromSingle₀Equiv (C : ChainComplex V ℕ) (X : V) :
    ((single₀ V).obj X ⟶ C) ≃ (X ⟶ C.X 0) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] (C : ChainComplex V ℕ) (X : V) (f : (ChainComplex.single₀ V).obj X ⟶ C)
Goal: (C.fromSingle₀Equiv X) f = f.f 0
-- ChainComplex.fromSingle₀Equiv_symm_apply_f_zero
@[simp]
lemma fromSingle₀Equiv_symm_apply_f_zero
    {C : ChainComplex V ℕ} {X : V} (f : X ⟶ C.X 0) :
    ((fromSingle₀Equiv C X).symm f).f 0 = f 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {C : ChainComplex V ℕ} {X : V} (f : X ⟶ C.X 0)
Goal: ((C.fromSingle₀Equiv X).symm f).f 0 = f
-- ChainComplex.fromSingle₀Equiv_symm_apply_f_succ
@[simp]
lemma fromSingle₀Equiv_symm_apply_f_succ
    {C : ChainComplex V ℕ} {X : V} (f : X ⟶ C.X 0) (n : ℕ) :
    ((fromSingle₀Equiv C X).symm f).f (n + 1) = 0 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {C : ChainComplex V ℕ} {X : V} (f : X ⟶ C.X 0) (n : ℕ)
Goal: ((C.fromSingle₀Equiv X).symm f).f (n + 1) = 0
-- CochainComplex.single₀_obj_zero
@[simp]
lemma single₀_obj_zero (A : V) :
    ((single₀ V).obj A).X 0 = A 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] (A : V)
Goal: ((CochainComplex.single₀ V).obj A).X 0 = A
-- CochainComplex.single₀_map_f_zero
@[simp]
lemma single₀_map_f_zero {A B : V} (f : A ⟶ B) :
    ((single₀ V).map f).f 0 = f 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {A B : V} (f : A ⟶ B)
Goal: ((CochainComplex.single₀ V).map f).f 0 = f
-- CochainComplex.single₀ObjXSelf
@[simp]
lemma single₀ObjXSelf (X : V) :
    HomologicalComplex.singleObjXSelf (ComplexShape.up ℕ) 0 X = Iso.refl _ 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] (X : V)
Goal: HomologicalComplex.singleObjXSelf (ComplexShape.up ℕ) 0 X =
  CategoryTheory.Iso.refl (((HomologicalComplex.single V (ComplexShape.up ℕ) 0).obj X).X 0)
-- CochainComplex.fromSingle₀Equiv_apply_coe
@[simps apply_coe]
noncomputable def fromSingle₀Equiv (C : CochainComplex V ℕ) (X : V) :
    ((single₀ V).obj X ⟶ C) ≃ { f : X ⟶ C.X 0 // f ≫ C.d 0 1 = 0 } 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] (C : CochainComplex V ℕ) (X : V) (φ : (CochainComplex.single₀ V).obj X ⟶ C)
Goal: ↑((C.fromSingle₀Equiv X) φ) = φ.f 0
-- CochainComplex.fromSingle₀Equiv_symm_apply_f_zero
@[simp]
lemma fromSingle₀Equiv_symm_apply_f_zero {C : CochainComplex V ℕ} {X : V}
    (f : X ⟶ C.X 0) (hf : f ≫ C.d 0 1 = 0) :
    ((fromSingle₀Equiv C X).symm ⟨f, hf⟩).f 0 = f 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {C : CochainComplex V ℕ} {X : V} (f : X ⟶ C.X 0) (hf : f ≫ C.d 0 1 = 0)
Goal: ((C.fromSingle₀Equiv X).symm ⟨f, hf⟩).f 0 = f
-- CochainComplex.toSingle₀Equiv_apply
@[simps apply]
noncomputable def toSingle₀Equiv (C : CochainComplex V ℕ) (X : V) :
    (C ⟶ (single₀ V).obj X) ≃ (C.X 0 ⟶ X) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] (C : CochainComplex V ℕ) (X : V) (f : C ⟶ (CochainComplex.single₀ V).obj X)
Goal: (C.toSingle₀Equiv X) f = f.f 0
-- CochainComplex.toSingle₀Equiv_symm_apply_f_zero
@[simp]
lemma toSingle₀Equiv_symm_apply_f_zero
    {C : CochainComplex V ℕ} {X : V} (f : C.X 0 ⟶ X) :
    ((toSingle₀Equiv C X).symm f).f 0 = f 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {C : CochainComplex V ℕ} {X : V} (f : C.X 0 ⟶ X)
Goal: ((C.toSingle₀Equiv X).symm f).f 0 = f
-- CochainComplex.toSingle₀Equiv_symm_apply_f_succ
@[simp]
lemma toSingle₀Equiv_symm_apply_f_succ
    {C : CochainComplex V ℕ} {X : V} (f : C.X 0 ⟶ X) (n : ℕ) :
    ((toSingle₀Equiv C X).symm f).f (n + 1) = 0 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {C : CochainComplex V ℕ} {X : V} (f : C.X 0 ⟶ X) (n : ℕ)
Goal: ((C.toSingle₀Equiv X).symm f).f (n + 1) = 0
-- CategoryTheory.ShortComplex.zero
structure ShortComplex [HasZeroMorphisms C] where
    {X₁ : C}
    {X₂ : C}
    {X₃ : C}
    f : X₁ ⟶ X₂
    g : X₂ ⟶ X₃
    zero : f ≫ g = 0
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (self : CategoryTheory.ShortComplex C)
Goal: CategoryTheory.CategoryStruct.comp self.f self.g = 0
-- CategoryTheory.ShortComplex.Hom.comm₁₂
@[ext]
structure Hom (S₁ S₂ : ShortComplex C) where
    τ₁ : S₁.X₁ ⟶ S₂.X₁
    τ₂ : S₁.X₂ ⟶ S₂.X₂
    τ₃ : S₁.X₃ ⟶ S₂.X₃
    comm₁₂ : τ₁ ≫ S₂.f = S₁.f ≫ τ₂ := by aesop_cat
    comm₂₃ : τ₂ ≫ S₂.g = S₁.g ≫ τ₃ := by aesop_cat
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (self : S₁.Hom S₂)
Goal: CategoryTheory.CategoryStruct.comp self.τ₁ S₂.f = CategoryTheory.CategoryStruct.comp S₁.f self.τ₂
-- CategoryTheory.ShortComplex.Hom.comm₂₃
@[ext]
structure Hom (S₁ S₂ : ShortComplex C) where
    τ₁ : S₁.X₁ ⟶ S₂.X₁
    τ₂ : S₁.X₂ ⟶ S₂.X₂
    τ₃ : S₁.X₃ ⟶ S₂.X₃
    comm₁₂ : τ₁ ≫ S₂.f = S₁.f ≫ τ₂ := by aesop_cat
    comm₂₃ : τ₂ ≫ S₂.g = S₁.g ≫ τ₃ := by aesop_cat
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (self : S₁.Hom S₂)
Goal: CategoryTheory.CategoryStruct.comp self.τ₂ S₂.g = CategoryTheory.CategoryStruct.comp S₁.g self.τ₃
-- CategoryTheory.ShortComplex.Hom.id_τ₃
@[simps]
def Hom.id : Hom S S 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.ShortComplex.Hom.id S).τ₃ = CategoryTheory.CategoryStruct.id S.X₃
-- CategoryTheory.ShortComplex.Hom.id_τ₂
@[simps]
def Hom.id : Hom S S 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.ShortComplex.Hom.id S).τ₂ = CategoryTheory.CategoryStruct.id S.X₂
-- CategoryTheory.ShortComplex.Hom.id_τ₁
@[simps]
def Hom.id : Hom S S 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.ShortComplex.Hom.id S).τ₁ = CategoryTheory.CategoryStruct.id S.X₁
-- CategoryTheory.ShortComplex.Hom.comp_τ₃
@[simps]
def Hom.comp (φ₁₂ : Hom S₁ S₂) (φ₂₃ : Hom S₂ S₃) : Hom S₁ S₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} (φ₁₂ : S₁.Hom S₂) (φ₂₃ : S₂.Hom S₃)
Goal: (φ₁₂.comp φ₂₃).τ₃ = CategoryTheory.CategoryStruct.comp φ₁₂.τ₃ φ₂₃.τ₃
-- CategoryTheory.ShortComplex.Hom.comp_τ₁
@[simps]
def Hom.comp (φ₁₂ : Hom S₁ S₂) (φ₂₃ : Hom S₂ S₃) : Hom S₁ S₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} (φ₁₂ : S₁.Hom S₂) (φ₂₃ : S₂.Hom S₃)
Goal: (φ₁₂.comp φ₂₃).τ₁ = CategoryTheory.CategoryStruct.comp φ₁₂.τ₁ φ₂₃.τ₁
-- CategoryTheory.ShortComplex.Hom.comp_τ₂
@[simps]
def Hom.comp (φ₁₂ : Hom S₁ S₂) (φ₂₃ : Hom S₂ S₃) : Hom S₁ S₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} (φ₁₂ : S₁.Hom S₂) (φ₂₃ : S₂.Hom S₃)
Goal: (φ₁₂.comp φ₂₃).τ₂ = CategoryTheory.CategoryStruct.comp φ₁₂.τ₂ φ₂₃.τ₂
-- CategoryTheory.ShortComplex.hom_ext
@[ext]
lemma hom_ext (f g : S₁ ⟶ S₂) (h₁ : f.τ₁ = g.τ₁) (h₂ : f.τ₂ = g.τ₂) (h₃ : f.τ₃ = g.τ₃) : f = g 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (f g : S₁ ⟶ S₂) (h₁ : f.τ₁ = g.τ₁) (h₂ : f.τ₂ = g.τ₂) (h₃ : f.τ₃ = g.τ₃)
Goal: f = g
-- CategoryTheory.ShortComplex.homMk_τ₂
@[simps]
def homMk {S₁ S₂ : ShortComplex C} (τ₁ : S₁.X₁ ⟶ S₂.X₁) (τ₂ : S₁.X₂ ⟶ S₂.X₂)
    (τ₃ : S₁.X₃ ⟶ S₂.X₃) (comm₁₂ : τ₁ ≫ S₂.f = S₁.f ≫ τ₂)
    (comm₂₃ : τ₂ ≫ S₂.g = S₁.g ≫ τ₃) : S₁ ⟶ S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (τ₁ : S₁.X₁ ⟶ S₂.X₁) (τ₂ : S₁.X₂ ⟶ S₂.X₂) (τ₃ : S₁.X₃ ⟶ S₂.X₃) (comm₁₂ : CategoryTheory.CategoryStruct.comp τ₁ S₂.f = CategoryTheory.CategoryStruct.comp S₁.f τ₂) (comm₂₃ : CategoryTheory.CategoryStruct.comp τ₂ S₂.g = CategoryTheory.CategoryStruct.comp S₁.g τ₃)
Goal: (CategoryTheory.ShortComplex.homMk τ₁ τ₂ τ₃ comm₁₂ comm₂₃).τ₂ = τ₂
-- CategoryTheory.ShortComplex.homMk_τ₃
@[simps]
def homMk {S₁ S₂ : ShortComplex C} (τ₁ : S₁.X₁ ⟶ S₂.X₁) (τ₂ : S₁.X₂ ⟶ S₂.X₂)
    (τ₃ : S₁.X₃ ⟶ S₂.X₃) (comm₁₂ : τ₁ ≫ S₂.f = S₁.f ≫ τ₂)
    (comm₂₃ : τ₂ ≫ S₂.g = S₁.g ≫ τ₃) : S₁ ⟶ S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (τ₁ : S₁.X₁ ⟶ S₂.X₁) (τ₂ : S₁.X₂ ⟶ S₂.X₂) (τ₃ : S₁.X₃ ⟶ S₂.X₃) (comm₁₂ : CategoryTheory.CategoryStruct.comp τ₁ S₂.f = CategoryTheory.CategoryStruct.comp S₁.f τ₂) (comm₂₃ : CategoryTheory.CategoryStruct.comp τ₂ S₂.g = CategoryTheory.CategoryStruct.comp S₁.g τ₃)
Goal: (CategoryTheory.ShortComplex.homMk τ₁ τ₂ τ₃ comm₁₂ comm₂₃).τ₃ = τ₃
-- CategoryTheory.ShortComplex.homMk_τ₁
@[simps]
def homMk {S₁ S₂ : ShortComplex C} (τ₁ : S₁.X₁ ⟶ S₂.X₁) (τ₂ : S₁.X₂ ⟶ S₂.X₂)
    (τ₃ : S₁.X₃ ⟶ S₂.X₃) (comm₁₂ : τ₁ ≫ S₂.f = S₁.f ≫ τ₂)
    (comm₂₃ : τ₂ ≫ S₂.g = S₁.g ≫ τ₃) : S₁ ⟶ S₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (τ₁ : S₁.X₁ ⟶ S₂.X₁) (τ₂ : S₁.X₂ ⟶ S₂.X₂) (τ₃ : S₁.X₃ ⟶ S₂.X₃) (comm₁₂ : CategoryTheory.CategoryStruct.comp τ₁ S₂.f = CategoryTheory.CategoryStruct.comp S₁.f τ₂) (comm₂₃ : CategoryTheory.CategoryStruct.comp τ₂ S₂.g = CategoryTheory.CategoryStruct.comp S₁.g τ₃)
Goal: (CategoryTheory.ShortComplex.homMk τ₁ τ₂ τ₃ comm₁₂ comm₂₃).τ₁ = τ₁
-- CategoryTheory.ShortComplex.id_τ₁
@[simp] lemma id_τ₁ : Hom.τ₁ (𝟙 S) = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.CategoryStruct.id S).τ₁ = CategoryTheory.CategoryStruct.id S.X₁
-- CategoryTheory.ShortComplex.id_τ₂
@[simp] lemma id_τ₂ : Hom.τ₂ (𝟙 S) = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.CategoryStruct.id S).τ₂ = CategoryTheory.CategoryStruct.id S.X₂
-- CategoryTheory.ShortComplex.id_τ₃
@[simp] lemma id_τ₃ : Hom.τ₃ (𝟙 S) = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: (CategoryTheory.CategoryStruct.id S).τ₃ = CategoryTheory.CategoryStruct.id S.X₃
-- CategoryTheory.ShortComplex.comp_τ₁
@[reassoc] lemma comp_τ₁ (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃) :
    (φ₁₂ ≫ φ₂₃).τ₁ = φ₁₂.τ₁ ≫ φ₂₃.τ₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃)
Goal: (CategoryTheory.CategoryStruct.comp φ₁₂ φ₂₃).τ₁ = CategoryTheory.CategoryStruct.comp φ₁₂.τ₁ φ₂₃.τ₁
-- CategoryTheory.ShortComplex.comp_τ₂
@[reassoc] lemma comp_τ₂ (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃) :
    (φ₁₂ ≫ φ₂₃).τ₂ = φ₁₂.τ₂ ≫ φ₂₃.τ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃)
Goal: (CategoryTheory.CategoryStruct.comp φ₁₂ φ₂₃).τ₂ = CategoryTheory.CategoryStruct.comp φ₁₂.τ₂ φ₂₃.τ₂
-- CategoryTheory.ShortComplex.comp_τ₃
@[reassoc] lemma comp_τ₃ (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃) :
    (φ₁₂ ≫ φ₂₃).τ₃ = φ₁₂.τ₃ ≫ φ₂₃.τ₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} (φ₁₂ : S₁ ⟶ S₂) (φ₂₃ : S₂ ⟶ S₃)
Goal: (CategoryTheory.CategoryStruct.comp φ₁₂ φ₂₃).τ₃ = CategoryTheory.CategoryStruct.comp φ₁₂.τ₃ φ₂₃.τ₃
-- CategoryTheory.ShortComplex.zero_τ₁
@[simp] lemma zero_τ₁ : Hom.τ₁ (0 : S₁ ⟶ S₂) = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S₁ S₂ : CategoryTheory.ShortComplex C)
Goal: CategoryTheory.ShortComplex.Hom.τ₁ 0 = 0
-- CategoryTheory.ShortComplex.zero_τ₂
@[simp] lemma zero_τ₂ : Hom.τ₂ (0 : S₁ ⟶ S₂) = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S₁ S₂ : CategoryTheory.ShortComplex C)
Goal: CategoryTheory.ShortComplex.Hom.τ₂ 0 = 0
-- CategoryTheory.ShortComplex.zero_τ₃
@[simp] lemma zero_τ₃ : Hom.τ₃ (0 : S₁ ⟶ S₂) = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S₁ S₂ : CategoryTheory.ShortComplex C)
Goal: CategoryTheory.ShortComplex.Hom.τ₃ 0 = 0
-- CategoryTheory.ShortComplex.π₁_obj
@[simps]
def π₁ : ShortComplex C ⥤ C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: CategoryTheory.ShortComplex.π₁.obj S = S.X₁
-- CategoryTheory.ShortComplex.π₁_map
@[simps]
def π₁ : ShortComplex C ⥤ C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C]
Goal: ∀ {X Y : CategoryTheory.ShortComplex C} (f : X ⟶ Y), CategoryTheory.ShortComplex.π₁.map f = f.τ₁
-- CategoryTheory.ShortComplex.π₂_map
@[simps]
def π₂ : ShortComplex C ⥤ C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C]
Goal: ∀ {X Y : CategoryTheory.ShortComplex C} (f : X ⟶ Y), CategoryTheory.ShortComplex.π₂.map f = f.τ₂
-- CategoryTheory.ShortComplex.π₂_obj
@[simps]
def π₂ : ShortComplex C ⥤ C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: CategoryTheory.ShortComplex.π₂.obj S = S.X₂
-- CategoryTheory.ShortComplex.π₃_obj
@[simps]
def π₃ : ShortComplex C ⥤ C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: CategoryTheory.ShortComplex.π₃.obj S = S.X₃
-- CategoryTheory.ShortComplex.π₃_map
@[simps]
def π₃ : ShortComplex C ⥤ C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C]
Goal: ∀ {X Y : CategoryTheory.ShortComplex C} (f : X ⟶ Y), CategoryTheory.ShortComplex.π₃.map f = f.τ₃
-- CategoryTheory.ShortComplex.π₁Toπ₂_app
@[simps] def π₁Toπ₂ : (π₁ : _ ⥤ C) ⟶ π₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: CategoryTheory.ShortComplex.π₁Toπ₂.app S = S.f
-- CategoryTheory.ShortComplex.π₂Toπ₃_app
@[simps] def π₂Toπ₃ : (π₂ : _ ⥤ C) ⟶ π₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: CategoryTheory.ShortComplex.π₂Toπ₃.app S = S.g
-- CategoryTheory.ShortComplex.π₁Toπ₂_comp_π₂Toπ₃
@[reassoc (attr := simp)]
lemma π₁Toπ₂_comp_π₂Toπ₃ : (π₁Toπ₂ : (_ : _ ⥤ C) ⟶ _) ≫ π₂Toπ₃ = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C]
Goal: CategoryTheory.CategoryStruct.comp CategoryTheory.ShortComplex.π₁Toπ₂ CategoryTheory.ShortComplex.π₂Toπ₃ = 0
-- CategoryTheory.ShortComplex.map_g
@[simps]
def map (F : C ⥤ D) [F.PreservesZeroMorphisms] : ShortComplex D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms]
Goal: (S.map F).g = F.map S.g
-- CategoryTheory.ShortComplex.map_X₂
@[simps]
def map (F : C ⥤ D) [F.PreservesZeroMorphisms] : ShortComplex D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms]
Goal: (S.map F).X₂ = F.obj S.X₂
-- CategoryTheory.ShortComplex.map_X₃
@[simps]
def map (F : C ⥤ D) [F.PreservesZeroMorphisms] : ShortComplex D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms]
Goal: (S.map F).X₃ = F.obj S.X₃
-- CategoryTheory.ShortComplex.map_f
@[simps]
def map (F : C ⥤ D) [F.PreservesZeroMorphisms] : ShortComplex D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms]
Goal: (S.map F).f = F.map S.f
-- CategoryTheory.ShortComplex.map_X₁
@[simps]
def map (F : C ⥤ D) [F.PreservesZeroMorphisms] : ShortComplex D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms]
Goal: (S.map F).X₁ = F.obj S.X₁
-- CategoryTheory.ShortComplex.mapNatTrans_τ₂
@[simps]
def mapNatTrans {F G : C ⥤ D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ⟶ G) :
    S.map F ⟶ S.map G 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [CategoryTheory.Limits.HasZeroMorphisms D] {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ⟶ G)
Goal: (S.mapNatTrans τ).τ₂ = τ.app S.X₂
-- CategoryTheory.ShortComplex.mapNatTrans_τ₁
@[simps]
def mapNatTrans {F G : C ⥤ D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ⟶ G) :
    S.map F ⟶ S.map G 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [CategoryTheory.Limits.HasZeroMorphisms D] {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ⟶ G)
Goal: (S.mapNatTrans τ).τ₁ = τ.app S.X₁
-- CategoryTheory.ShortComplex.mapNatTrans_τ₃
@[simps]
def mapNatTrans {F G : C ⥤ D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ⟶ G) :
    S.map F ⟶ S.map G 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [CategoryTheory.Limits.HasZeroMorphisms D] {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ⟶ G)
Goal: (S.mapNatTrans τ).τ₃ = τ.app S.X₃
-- CategoryTheory.ShortComplex.mapNatIso_inv
@[simps]
def mapNatIso {F G : C ⥤ D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ≅ G) :
    S.map F ≅ S.map G 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [CategoryTheory.Limits.HasZeroMorphisms D] {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ≅ G)
Goal: (S.mapNatIso τ).inv = S.mapNatTrans τ.inv
-- CategoryTheory.ShortComplex.mapNatIso_hom
@[simps]
def mapNatIso {F G : C ⥤ D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ≅ G) :
    S.map F ≅ S.map G 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [CategoryTheory.Limits.HasZeroMorphisms D] {F G : CategoryTheory.Functor C D} [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms] (τ : F ≅ G)
Goal: (S.mapNatIso τ).hom = S.mapNatTrans τ.hom
-- CategoryTheory.Functor.mapShortComplex_obj
@[simps]
def _root_.CategoryTheory.Functor.mapShortComplex (F : C ⥤ D) [F.PreservesZeroMorphisms] :
    ShortComplex C ⥤ ShortComplex D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] (S : CategoryTheory.ShortComplex C)
Goal: F.mapShortComplex.obj S = S.map F
-- CategoryTheory.Functor.mapShortComplex_map_τ₂
@[simps]
def _root_.CategoryTheory.Functor.mapShortComplex (F : C ⥤ D) [F.PreservesZeroMorphisms] :
    ShortComplex C ⥤ ShortComplex D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms]
Goal: ∀ {X Y : CategoryTheory.ShortComplex C} (φ : X ⟶ Y), (F.mapShortComplex.map φ).τ₂ = F.map φ.τ₂
-- CategoryTheory.Functor.mapShortComplex_map_τ₃
@[simps]
def _root_.CategoryTheory.Functor.mapShortComplex (F : C ⥤ D) [F.PreservesZeroMorphisms] :
    ShortComplex C ⥤ ShortComplex D 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms]
Goal: ∀ {X Y : CategoryTheory.ShortComplex C} (φ : X ⟶ Y), (F.mapShortComplex.map φ).τ₃ = F.map φ.τ₃