ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsColimitCokernelCofork_φK
@[simps]
def ofIsColimitCokernelCofork (φ : S₁ ⟶ S₂)
    (hg₁ : S₁.g = 0) (c₁ : CokernelCofork S₁.f) (hc₁ : IsColimit c₁)
    (hg₂ : S₂.g = 0) (c₂ : CokernelCofork S₂.f) (hc₂ : IsColimit c₂) (f : c₁.pt ⟶ c₂.pt)
    (comm : φ.τ₂ ≫ c₂.π = c₁.π ≫ f) :
    LeftHomologyMapData φ (LeftHomologyData.ofIsColimitCokernelCofork S₁ hg₁ c₁ hc₁)
      (LeftHomologyData.ofIsColimitCokernelCofork S₂ hg₂ c₂ hc₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (hg₁ : S₁.g = 0) (c₁ : CategoryTheory.Limits.CokernelCofork S₁.f) (hc₁ : CategoryTheory.Limits.IsColimit c₁) (hg₂ : S₂.g = 0) (c₂ : CategoryTheory.Limits.CokernelCofork S₂.f) (hc₂ : CategoryTheory.Limits.IsColimit c₂) (f : c₁.pt ⟶ c₂.pt) (comm :
  CategoryTheory.CategoryStruct.comp φ.τ₂ (CategoryTheory.Limits.Cofork.π c₂) =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Cofork.π c₁) f)
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsColimitCokernelCofork φ hg₁ c₁ hc₁ hg₂ c₂ hc₂ f comm).φK = φ.τ₂
-- CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsColimitCokernelCofork_φH
@[simps]
def ofIsColimitCokernelCofork (φ : S₁ ⟶ S₂)
    (hg₁ : S₁.g = 0) (c₁ : CokernelCofork S₁.f) (hc₁ : IsColimit c₁)
    (hg₂ : S₂.g = 0) (c₂ : CokernelCofork S₂.f) (hc₂ : IsColimit c₂) (f : c₁.pt ⟶ c₂.pt)
    (comm : φ.τ₂ ≫ c₂.π = c₁.π ≫ f) :
    LeftHomologyMapData φ (LeftHomologyData.ofIsColimitCokernelCofork S₁ hg₁ c₁ hc₁)
      (LeftHomologyData.ofIsColimitCokernelCofork S₂ hg₂ c₂ hc₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (hg₁ : S₁.g = 0) (c₁ : CategoryTheory.Limits.CokernelCofork S₁.f) (hc₁ : CategoryTheory.Limits.IsColimit c₁) (hg₂ : S₂.g = 0) (c₂ : CategoryTheory.Limits.CokernelCofork S₂.f) (hc₂ : CategoryTheory.Limits.IsColimit c₂) (f : c₁.pt ⟶ c₂.pt) (comm :
  CategoryTheory.CategoryStruct.comp φ.τ₂ (CategoryTheory.Limits.Cofork.π c₂) =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Cofork.π c₁) f)
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsColimitCokernelCofork φ hg₁ c₁ hc₁ hg₂ c₂ hc₂ f comm).φH = f
-- CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsLimitKernelFork_φH
@[simps]
def ofIsLimitKernelFork (φ : S₁ ⟶ S₂)
    (hf₁ : S₁.f = 0) (c₁ : KernelFork S₁.g) (hc₁ : IsLimit c₁)
    (hf₂ : S₂.f = 0) (c₂ : KernelFork S₂.g) (hc₂ : IsLimit c₂) (f : c₁.pt ⟶ c₂.pt)
    (comm : c₁.ι ≫ φ.τ₂ = f ≫ c₂.ι) :
    LeftHomologyMapData φ (LeftHomologyData.ofIsLimitKernelFork S₁ hf₁ c₁ hc₁)
      (LeftHomologyData.ofIsLimitKernelFork S₂ hf₂ c₂ hc₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (hf₁ : S₁.f = 0) (c₁ : CategoryTheory.Limits.KernelFork S₁.g) (hc₁ : CategoryTheory.Limits.IsLimit c₁) (hf₂ : S₂.f = 0) (c₂ : CategoryTheory.Limits.KernelFork S₂.g) (hc₂ : CategoryTheory.Limits.IsLimit c₂) (f : c₁.pt ⟶ c₂.pt) (comm :
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Fork.ι c₁) φ.τ₂ =
    CategoryTheory.CategoryStruct.comp f (CategoryTheory.Limits.Fork.ι c₂))
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsLimitKernelFork φ hf₁ c₁ hc₁ hf₂ c₂ hc₂ f comm).φH = f
-- CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsLimitKernelFork_φK
@[simps]
def ofIsLimitKernelFork (φ : S₁ ⟶ S₂)
    (hf₁ : S₁.f = 0) (c₁ : KernelFork S₁.g) (hc₁ : IsLimit c₁)
    (hf₂ : S₂.f = 0) (c₂ : KernelFork S₂.g) (hc₂ : IsLimit c₂) (f : c₁.pt ⟶ c₂.pt)
    (comm : c₁.ι ≫ φ.τ₂ = f ≫ c₂.ι) :
    LeftHomologyMapData φ (LeftHomologyData.ofIsLimitKernelFork S₁ hf₁ c₁ hc₁)
      (LeftHomologyData.ofIsLimitKernelFork S₂ hf₂ c₂ hc₂) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (hf₁ : S₁.f = 0) (c₁ : CategoryTheory.Limits.KernelFork S₁.g) (hc₁ : CategoryTheory.Limits.IsLimit c₁) (hf₂ : S₂.f = 0) (c₂ : CategoryTheory.Limits.KernelFork S₂.g) (hc₂ : CategoryTheory.Limits.IsLimit c₂) (f : c₁.pt ⟶ c₂.pt) (comm :
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Fork.ι c₁) φ.τ₂ =
    CategoryTheory.CategoryStruct.comp f (CategoryTheory.Limits.Fork.ι c₂))
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsLimitKernelFork φ hf₁ c₁ hc₁ hf₂ c₂ hc₂ f comm).φK = f
-- CategoryTheory.ShortComplex.LeftHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork_φH
@[simps]
def compatibilityOfZerosOfIsColimitCokernelCofork (hf : S.f = 0) (hg : S.g = 0)
    (c : CokernelCofork S.f) (hc : IsColimit c) :
    LeftHomologyMapData (𝟙 S) (LeftHomologyData.ofZeros S hf hg)
      (LeftHomologyData.ofIsColimitCokernelCofork S hg c hc) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) (hf : S.f = 0) (hg : S.g = 0) (c : CategoryTheory.Limits.CokernelCofork S.f) (hc : CategoryTheory.Limits.IsColimit c)
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork S hf hg c hc).φH =
  CategoryTheory.Limits.Cofork.π c
-- CategoryTheory.ShortComplex.LeftHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork_φK
@[simps]
def compatibilityOfZerosOfIsColimitCokernelCofork (hf : S.f = 0) (hg : S.g = 0)
    (c : CokernelCofork S.f) (hc : IsColimit c) :
    LeftHomologyMapData (𝟙 S) (LeftHomologyData.ofZeros S hf hg)
      (LeftHomologyData.ofIsColimitCokernelCofork S hg c hc) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) (hf : S.f = 0) (hg : S.g = 0) (c : CategoryTheory.Limits.CokernelCofork S.f) (hc : CategoryTheory.Limits.IsColimit c)
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork S hf hg c hc).φK =
  CategoryTheory.CategoryStruct.id (CategoryTheory.ShortComplex.LeftHomologyData.ofZeros S hf hg).K
-- CategoryTheory.ShortComplex.LeftHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_φH
@[simps]
def compatibilityOfZerosOfIsLimitKernelFork (hf : S.f = 0) (hg : S.g = 0)
    (c : KernelFork S.g) (hc : IsLimit c) :
    LeftHomologyMapData (𝟙 S) (LeftHomologyData.ofIsLimitKernelFork S hf c hc)
      (LeftHomologyData.ofZeros S hf hg) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) (hf : S.f = 0) (hg : S.g = 0) (c : CategoryTheory.Limits.KernelFork S.g) (hc : CategoryTheory.Limits.IsLimit c)
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork S hf hg c hc).φH =
  CategoryTheory.Limits.Fork.ι c
-- CategoryTheory.ShortComplex.LeftHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_φK
@[simps]
def compatibilityOfZerosOfIsLimitKernelFork (hf : S.f = 0) (hg : S.g = 0)
    (c : KernelFork S.g) (hc : IsLimit c) :
    LeftHomologyMapData (𝟙 S) (LeftHomologyData.ofIsLimitKernelFork S hf c hc)
      (LeftHomologyData.ofZeros S hf hg) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) (hf : S.f = 0) (hg : S.g = 0) (c : CategoryTheory.Limits.KernelFork S.g) (hc : CategoryTheory.Limits.IsLimit c)
Goal: (CategoryTheory.ShortComplex.LeftHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork S hf hg c hc).φK =
  CategoryTheory.Limits.Fork.ι c
-- CategoryTheory.ShortComplex.iCycles_g
@[reassoc (attr := simp)]
lemma iCycles_g : S.iCycles ≫ S.g = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology]
Goal: CategoryTheory.CategoryStruct.comp S.iCycles S.g = 0
-- CategoryTheory.ShortComplex.toCycles_i
@[reassoc (attr := simp)]
lemma toCycles_i : S.toCycles ≫ S.iCycles = S.f 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology]
Goal: CategoryTheory.CategoryStruct.comp S.toCycles S.iCycles = S.f
-- CategoryTheory.ShortComplex.leftHomology_ext_iff
lemma leftHomology_ext_iff {A : C} (f₁ f₂ : S.leftHomology ⟶ A) :
    f₁ = f₂ ↔ S.leftHomologyπ ≫ f₁ = S.leftHomologyπ ≫ f₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] {A : C} (f₁ f₂ : S.leftHomology ⟶ A)
Goal: f₁ = f₂ ↔ CategoryTheory.CategoryStruct.comp S.leftHomologyπ f₁ = CategoryTheory.CategoryStruct.comp S.leftHomologyπ f₂
-- CategoryTheory.ShortComplex.leftHomology_ext
@[ext]
lemma leftHomology_ext {A : C} (f₁ f₂ : S.leftHomology ⟶ A)
    (h : S.leftHomologyπ ≫ f₁ = S.leftHomologyπ ≫ f₂) : f₁ = f₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] {A : C} (f₁ f₂ : S.leftHomology ⟶ A) (h : CategoryTheory.CategoryStruct.comp S.leftHomologyπ f₁ = CategoryTheory.CategoryStruct.comp S.leftHomologyπ f₂)
Goal: f₁ = f₂
-- CategoryTheory.ShortComplex.cycles_ext_iff
lemma cycles_ext_iff {A : C} (f₁ f₂ : A ⟶ S.cycles) :
    f₁ = f₂ ↔ f₁ ≫ S.iCycles = f₂ ≫ S.iCycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] {A : C} (f₁ f₂ : A ⟶ S.cycles)
Goal: f₁ = f₂ ↔ CategoryTheory.CategoryStruct.comp f₁ S.iCycles = CategoryTheory.CategoryStruct.comp f₂ S.iCycles
-- CategoryTheory.ShortComplex.cycles_ext
@[ext]
lemma cycles_ext {A : C} (f₁ f₂ : A ⟶ S.cycles) (h : f₁ ≫ S.iCycles = f₂ ≫ S.iCycles) :
    f₁ = f₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] {A : C} (f₁ f₂ : A ⟶ S.cycles) (h : CategoryTheory.CategoryStruct.comp f₁ S.iCycles = CategoryTheory.CategoryStruct.comp f₂ S.iCycles)
Goal: f₁ = f₂
-- CategoryTheory.ShortComplex.isIso_iCycles
lemma isIso_iCycles (hg : S.g = 0) : IsIso S.iCycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] (hg : S.g = 0)
Goal: CategoryTheory.IsIso S.iCycles
-- CategoryTheory.ShortComplex.cyclesIsoX₂_hom
@[simps! hom]
noncomputable def cyclesIsoX₂ (hg : S.g = 0) : S.cycles ≅ S.X₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] (hg : S.g = 0)
Goal: (S.cyclesIsoX₂ hg).hom = S.iCycles
-- CategoryTheory.ShortComplex.cyclesIsoX₂_hom_inv_id
@[reassoc (attr := simp)]
lemma cyclesIsoX₂_hom_inv_id (hg : S.g = 0) :
    S.iCycles ≫ (S.cyclesIsoX₂ hg).inv = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] (hg : S.g = 0)
Goal: CategoryTheory.CategoryStruct.comp S.iCycles (S.cyclesIsoX₂ hg).inv = CategoryTheory.CategoryStruct.id S.cycles
-- CategoryTheory.ShortComplex.cyclesIsoX₂_inv_hom_id
@[reassoc (attr := simp)]
lemma cyclesIsoX₂_inv_hom_id (hg : S.g = 0) :
    (S.cyclesIsoX₂ hg).inv ≫ S.iCycles = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] (hg : S.g = 0)
Goal: CategoryTheory.CategoryStruct.comp (S.cyclesIsoX₂ hg).inv S.iCycles = CategoryTheory.CategoryStruct.id S.X₂
-- CategoryTheory.ShortComplex.isIso_leftHomologyπ
lemma isIso_leftHomologyπ (hf : S.f = 0) : IsIso S.leftHomologyπ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] (hf : S.f = 0)
Goal: CategoryTheory.IsIso S.leftHomologyπ
-- CategoryTheory.ShortComplex.cyclesIsoLeftHomology_hom
@[simps! hom]
noncomputable def cyclesIsoLeftHomology (hf : S.f = 0) : S.cycles ≅ S.leftHomology 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] (hf : S.f = 0)
Goal: (S.cyclesIsoLeftHomology hf).hom = S.leftHomologyπ
-- CategoryTheory.ShortComplex.cyclesIsoLeftHomology_hom_inv_id
@[reassoc (attr := simp)]
lemma cyclesIsoLeftHomology_hom_inv_id (hf : S.f = 0) :
    S.leftHomologyπ ≫ (S.cyclesIsoLeftHomology hf).inv = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] (hf : S.f = 0)
Goal: CategoryTheory.CategoryStruct.comp S.leftHomologyπ (S.cyclesIsoLeftHomology hf).inv =
  CategoryTheory.CategoryStruct.id S.cycles
-- CategoryTheory.ShortComplex.cyclesIsoLeftHomology_inv_hom_id
@[reassoc (attr := simp)]
lemma cyclesIsoLeftHomology_inv_hom_id (hf : S.f = 0) :
    (S.cyclesIsoLeftHomology hf).inv ≫ S.leftHomologyπ = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] (hf : S.f = 0)
Goal: CategoryTheory.CategoryStruct.comp (S.cyclesIsoLeftHomology hf).inv S.leftHomologyπ =
  CategoryTheory.CategoryStruct.id S.leftHomology
-- CategoryTheory.ShortComplex.cyclesMap'_i
@[reassoc (attr := simp)]
lemma cyclesMap'_i : cyclesMap' φ h₁ h₂ ≫ h₂.i = h₁.i ≫ φ.τ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.cyclesMap' φ h₁ h₂) h₂.i =
  CategoryTheory.CategoryStruct.comp h₁.i φ.τ₂
-- CategoryTheory.ShortComplex.f'_cyclesMap'
@[reassoc (attr := simp)]
lemma f'_cyclesMap' : h₁.f' ≫ cyclesMap' φ h₁ h₂ = φ.τ₁ ≫ h₂.f' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.CategoryStruct.comp h₁.f' (CategoryTheory.ShortComplex.cyclesMap' φ h₁ h₂) =
  CategoryTheory.CategoryStruct.comp φ.τ₁ h₂.f'
-- CategoryTheory.ShortComplex.leftHomologyπ_naturality'
@[reassoc (attr := simp)]
lemma leftHomologyπ_naturality' :
    h₁.π ≫ leftHomologyMap' φ h₁ h₂ = cyclesMap' φ h₁ h₂ ≫ h₂.π 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.CategoryStruct.comp h₁.π (CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.cyclesMap' φ h₁ h₂) h₂.π
-- CategoryTheory.ShortComplex.cyclesMap_i
@[reassoc (attr := simp)]
lemma cyclesMap_i : cyclesMap φ ≫ S₂.iCycles = S₁.iCycles ≫ φ.τ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} [S₁.HasLeftHomology] [S₂.HasLeftHomology] (φ : S₁ ⟶ S₂)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.cyclesMap φ) S₂.iCycles =
  CategoryTheory.CategoryStruct.comp S₁.iCycles φ.τ₂
-- CategoryTheory.ShortComplex.toCycles_naturality
@[reassoc (attr := simp)]
lemma toCycles_naturality : S₁.toCycles ≫ cyclesMap φ = φ.τ₁ ≫ S₂.toCycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} [S₁.HasLeftHomology] [S₂.HasLeftHomology] (φ : S₁ ⟶ S₂)
Goal: CategoryTheory.CategoryStruct.comp S₁.toCycles (CategoryTheory.ShortComplex.cyclesMap φ) =
  CategoryTheory.CategoryStruct.comp φ.τ₁ S₂.toCycles
-- CategoryTheory.ShortComplex.leftHomologyπ_naturality
@[reassoc (attr := simp)]
lemma leftHomologyπ_naturality :
    S₁.leftHomologyπ ≫ leftHomologyMap φ = cyclesMap φ ≫ S₂.leftHomologyπ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} [S₁.HasLeftHomology] [S₂.HasLeftHomology] (φ : S₁ ⟶ S₂)
Goal: CategoryTheory.CategoryStruct.comp S₁.leftHomologyπ (CategoryTheory.ShortComplex.leftHomologyMap φ) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.cyclesMap φ) S₂.leftHomologyπ
-- CategoryTheory.ShortComplex.LeftHomologyMapData.leftHomologyMap'_eq
lemma leftHomologyMap'_eq : leftHomologyMap' 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.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂)
Goal: CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂ = γ.φH
-- CategoryTheory.ShortComplex.LeftHomologyMapData.cyclesMap'_eq
lemma cyclesMap'_eq : cyclesMap' φ h₁ h₂ = γ.φK 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂)
Goal: CategoryTheory.ShortComplex.cyclesMap' φ h₁ h₂ = γ.φK
-- CategoryTheory.ShortComplex.leftHomologyMap'_id
@[simp]
lemma leftHomologyMap'_id (h : S.LeftHomologyData) :
    leftHomologyMap' (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.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.leftHomologyMap' (CategoryTheory.CategoryStruct.id S) h h =
  CategoryTheory.CategoryStruct.id h.H
-- CategoryTheory.ShortComplex.cyclesMap'_id
@[simp]
lemma cyclesMap'_id (h : S.LeftHomologyData) :
    cyclesMap' (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.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.cyclesMap' (CategoryTheory.CategoryStruct.id S) h h = CategoryTheory.CategoryStruct.id h.K
-- CategoryTheory.ShortComplex.leftHomologyMap_id
@[simp]
lemma leftHomologyMap_id [HasLeftHomology S] :
    leftHomologyMap (𝟙 S) = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.leftHomologyMap (CategoryTheory.CategoryStruct.id S) =
  CategoryTheory.CategoryStruct.id S.leftHomology
-- CategoryTheory.ShortComplex.cyclesMap_id
@[simp]
lemma cyclesMap_id [HasLeftHomology S] :
    cyclesMap (𝟙 S) = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.cyclesMap (CategoryTheory.CategoryStruct.id S) = CategoryTheory.CategoryStruct.id S.cycles
-- CategoryTheory.ShortComplex.leftHomologyMap'_zero
@[simp]
lemma leftHomologyMap'_zero (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) :
    leftHomologyMap' 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.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.leftHomologyMap' 0 h₁ h₂ = 0
-- CategoryTheory.ShortComplex.cyclesMap'_zero
@[simp]
lemma cyclesMap'_zero (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) :
    cyclesMap' 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.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.cyclesMap' 0 h₁ h₂ = 0
-- CategoryTheory.ShortComplex.leftHomologyMap_zero
@[simp]
lemma leftHomologyMap_zero [HasLeftHomology S₁] [HasLeftHomology S₂] :
    leftHomologyMap (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.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.leftHomologyMap 0 = 0
-- CategoryTheory.ShortComplex.cyclesMap_zero
@[simp]
lemma cyclesMap_zero [HasLeftHomology S₁] [HasLeftHomology S₂] :
    cyclesMap (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.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.cyclesMap 0 = 0
-- CategoryTheory.ShortComplex.leftHomologyMap'_comp
@[reassoc]
lemma leftHomologyMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
    (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) (h₃ : S₃.LeftHomologyData) :
    leftHomologyMap' (φ₁ ≫ φ₂) h₁ h₃ = leftHomologyMap' φ₁ h₁ h₂ ≫
      leftHomologyMap' 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.LeftHomologyData) (h₂ : S₂.LeftHomologyData) (h₃ : S₃.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.leftHomologyMap' (CategoryTheory.CategoryStruct.comp φ₁ φ₂) h₁ h₃ =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftHomologyMap' φ₁ h₁ h₂)
    (CategoryTheory.ShortComplex.leftHomologyMap' φ₂ h₂ h₃)
-- CategoryTheory.ShortComplex.cyclesMap'_comp
@[reassoc]
lemma cyclesMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
    (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) (h₃ : S₃.LeftHomologyData) :
    cyclesMap' (φ₁ ≫ φ₂) h₁ h₃ = cyclesMap' φ₁ h₁ h₂ ≫ cyclesMap' 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.LeftHomologyData) (h₂ : S₂.LeftHomologyData) (h₃ : S₃.LeftHomologyData)
Goal: CategoryTheory.ShortComplex.cyclesMap' (CategoryTheory.CategoryStruct.comp φ₁ φ₂) h₁ h₃ =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.cyclesMap' φ₁ h₁ h₂)
    (CategoryTheory.ShortComplex.cyclesMap' φ₂ h₂ h₃)
-- CategoryTheory.ShortComplex.leftHomologyMap_comp
@[reassoc]
lemma leftHomologyMap_comp [HasLeftHomology S₁] [HasLeftHomology S₂] [HasLeftHomology S₃]
    (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃) :
    leftHomologyMap (φ₁ ≫ φ₂) = leftHomologyMap φ₁ ≫ leftHomologyMap φ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} [S₁.HasLeftHomology] [S₂.HasLeftHomology] [S₃.HasLeftHomology] (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
Goal: CategoryTheory.ShortComplex.leftHomologyMap (CategoryTheory.CategoryStruct.comp φ₁ φ₂) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftHomologyMap φ₁)
    (CategoryTheory.ShortComplex.leftHomologyMap φ₂)
-- CategoryTheory.ShortComplex.cyclesMap_comp
@[reassoc]
lemma cyclesMap_comp [HasLeftHomology S₁] [HasLeftHomology S₂] [HasLeftHomology S₃]
    (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃) :
    cyclesMap (φ₁ ≫ φ₂) = cyclesMap φ₁ ≫ cyclesMap φ₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} [S₁.HasLeftHomology] [S₂.HasLeftHomology] [S₃.HasLeftHomology] (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
Goal: CategoryTheory.ShortComplex.cyclesMap (CategoryTheory.CategoryStruct.comp φ₁ φ₂) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.cyclesMap φ₁)
    (CategoryTheory.ShortComplex.cyclesMap φ₂)