ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.ShortComplex.RightHomologyData.op_K
@[simps]
def op (h : S.RightHomologyData) : S.op.LeftHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData)
Goal: h.op.K = Opposite.op h.Q
-- CategoryTheory.ShortComplex.RightHomologyData.op_f'
@[simp] lemma op_f' (h : S.RightHomologyData) :
    h.op.f' = h.g'.op 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData)
Goal: h.op.f' = h.g'.op
-- CategoryTheory.ShortComplex.RightHomologyData.unop_π
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.RightHomologyData) : S.unop.LeftHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.RightHomologyData)
Goal: h.unop.π = h.ι.unop
-- CategoryTheory.ShortComplex.RightHomologyData.unop_i
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.RightHomologyData) : S.unop.LeftHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.RightHomologyData)
Goal: h.unop.i = h.p.unop
-- CategoryTheory.ShortComplex.RightHomologyData.unop_K
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.RightHomologyData) : S.unop.LeftHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.RightHomologyData)
Goal: h.unop.K = Opposite.unop h.Q
-- CategoryTheory.ShortComplex.RightHomologyData.unop_H
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.RightHomologyData) : S.unop.LeftHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.RightHomologyData)
Goal: h.unop.H = Opposite.unop h.H
-- CategoryTheory.ShortComplex.RightHomologyData.unop_f'
@[simp] lemma unop_f' {S : ShortComplex Cᵒᵖ} (h : S.RightHomologyData) :
    h.unop.f' = h.g'.unop 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.RightHomologyData)
Goal: h.unop.f' = h.g'.unop
-- CategoryTheory.ShortComplex.LeftHomologyData.op_Q
@[simps]
def op (h : S.LeftHomologyData) : S.op.RightHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData)
Goal: h.op.Q = Opposite.op h.K
-- CategoryTheory.ShortComplex.LeftHomologyData.op_ι
@[simps]
def op (h : S.LeftHomologyData) : S.op.RightHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData)
Goal: h.op.ι = h.π.op
-- CategoryTheory.ShortComplex.LeftHomologyData.op_p
@[simps]
def op (h : S.LeftHomologyData) : S.op.RightHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData)
Goal: h.op.p = h.i.op
-- CategoryTheory.ShortComplex.LeftHomologyData.op_H
@[simps]
def op (h : S.LeftHomologyData) : S.op.RightHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData)
Goal: h.op.H = Opposite.op h.H
-- CategoryTheory.ShortComplex.LeftHomologyData.op_g'
@[simp] lemma op_g' (h : S.LeftHomologyData) :
    h.op.g' = h.f'.op 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData)
Goal: h.op.g' = h.f'.op
-- CategoryTheory.ShortComplex.LeftHomologyData.unop_p
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.LeftHomologyData) : S.unop.RightHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.LeftHomologyData)
Goal: h.unop.p = h.i.unop
-- CategoryTheory.ShortComplex.LeftHomologyData.unop_Q
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.LeftHomologyData) : S.unop.RightHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.LeftHomologyData)
Goal: h.unop.Q = Opposite.unop h.K
-- CategoryTheory.ShortComplex.LeftHomologyData.unop_ι
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.LeftHomologyData) : S.unop.RightHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.LeftHomologyData)
Goal: h.unop.ι = h.π.unop
-- CategoryTheory.ShortComplex.LeftHomologyData.unop_H
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.LeftHomologyData) : S.unop.RightHomologyData 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.LeftHomologyData)
Goal: h.unop.H = Opposite.unop h.H
-- CategoryTheory.ShortComplex.LeftHomologyData.unop_g'
@[simp] lemma unop_g' {S : ShortComplex Cᵒᵖ} (h : S.LeftHomologyData) :
    h.unop.g' = h.f'.unop 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.LeftHomologyData)
Goal: h.unop.g' = h.f'.unop
-- CategoryTheory.ShortComplex.hasLeftHomology_iff_op
lemma hasLeftHomology_iff_op (S : ShortComplex C) :
    S.HasLeftHomology ↔ S.op.HasRightHomology 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: S.HasLeftHomology ↔ S.op.HasRightHomology
-- CategoryTheory.ShortComplex.hasRightHomology_iff_op
lemma hasRightHomology_iff_op (S : ShortComplex C) :
    S.HasRightHomology ↔ S.op.HasLeftHomology 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C)
Goal: S.HasRightHomology ↔ S.op.HasLeftHomology
-- CategoryTheory.ShortComplex.hasLeftHomology_iff_unop
lemma hasLeftHomology_iff_unop (S : ShortComplex Cᵒᵖ) :
    S.HasLeftHomology ↔ S.unop.HasRightHomology 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex Cᵒᵖ)
Goal: S.HasLeftHomology ↔ S.unop.HasRightHomology
-- CategoryTheory.ShortComplex.hasRightHomology_iff_unop
lemma hasRightHomology_iff_unop (S : ShortComplex Cᵒᵖ) :
    S.HasRightHomology ↔ S.unop.HasLeftHomology 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex Cᵒᵖ)
Goal: S.HasRightHomology ↔ S.unop.HasLeftHomology
-- CategoryTheory.ShortComplex.RightHomologyMapData.commι
structure RightHomologyMapData where
    φQ : h₁.Q ⟶ h₂.Q
    φH : h₁.H ⟶ h₂.H
    commp : h₁.p ≫ φQ = φ.τ₂ ≫ h₂.p := by aesop_cat
    commg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃ := by aesop_cat
    commι : φH ≫ h₂.ι = h₁.ι ≫ φQ := by aesop_cat
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (self : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂)
Goal: CategoryTheory.CategoryStruct.comp self.φH h₂.ι = CategoryTheory.CategoryStruct.comp h₁.ι self.φQ
-- CategoryTheory.ShortComplex.RightHomologyMapData.commp
structure RightHomologyMapData where
    φQ : h₁.Q ⟶ h₂.Q
    φH : h₁.H ⟶ h₂.H
    commp : h₁.p ≫ φQ = φ.τ₂ ≫ h₂.p := by aesop_cat
    commg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃ := by aesop_cat
    commι : φH ≫ h₂.ι = h₁.ι ≫ φQ := by aesop_cat
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (self : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂)
Goal: CategoryTheory.CategoryStruct.comp h₁.p self.φQ = CategoryTheory.CategoryStruct.comp φ.τ₂ h₂.p
-- CategoryTheory.ShortComplex.RightHomologyMapData.commg'
structure RightHomologyMapData where
    φQ : h₁.Q ⟶ h₂.Q
    φH : h₁.H ⟶ h₂.H
    commp : h₁.p ≫ φQ = φ.τ₂ ≫ h₂.p := by aesop_cat
    commg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃ := by aesop_cat
    commι : φH ≫ h₂.ι = h₁.ι ≫ φQ := by aesop_cat
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (self : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂)
Goal: CategoryTheory.CategoryStruct.comp self.φQ h₂.g' = CategoryTheory.CategoryStruct.comp h₁.g' φ.τ₃
-- CategoryTheory.ShortComplex.RightHomologyMapData.zero_φQ
@[simps]
def zero (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) :
    RightHomologyMapData 0 h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: (CategoryTheory.ShortComplex.RightHomologyMapData.zero h₁ h₂).φQ = 0
-- CategoryTheory.ShortComplex.RightHomologyMapData.zero_φH
@[simps]
def zero (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) :
    RightHomologyMapData 0 h₁ h₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData)
Goal: (CategoryTheory.ShortComplex.RightHomologyMapData.zero h₁ h₂).φH = 0
-- CategoryTheory.ShortComplex.RightHomologyMapData.id_φH
@[simps]
def id (h : S.RightHomologyData) : RightHomologyMapData (𝟙 S) h h 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData)
Goal: (CategoryTheory.ShortComplex.RightHomologyMapData.id h).φH = CategoryTheory.CategoryStruct.id h.H
-- CategoryTheory.ShortComplex.RightHomologyMapData.id_φQ
@[simps]
def id (h : S.RightHomologyData) : RightHomologyMapData (𝟙 S) h h 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData)
Goal: (CategoryTheory.ShortComplex.RightHomologyMapData.id h).φQ = CategoryTheory.CategoryStruct.id h.Q
-- CategoryTheory.ShortComplex.RightHomologyMapData.comp_φH
@[simps]
def comp {φ : S₁ ⟶ S₂} {φ' : S₂ ⟶ S₃} {h₁ : S₁.RightHomologyData}
    {h₂ : S₂.RightHomologyData} {h₃ : S₃.RightHomologyData}
    (ψ : RightHomologyMapData φ h₁ h₂) (ψ' : RightHomologyMapData φ' h₂ h₃) :
    RightHomologyMapData (φ ≫ φ') h₁ h₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {φ' : S₂ ⟶ S₃} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} {h₃ : S₃.RightHomologyData} (ψ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (ψ' : CategoryTheory.ShortComplex.RightHomologyMapData φ' h₂ h₃)
Goal: (ψ.comp ψ').φH = CategoryTheory.CategoryStruct.comp ψ.φH ψ'.φH
-- CategoryTheory.ShortComplex.RightHomologyMapData.comp_φQ
@[simps]
def comp {φ : S₁ ⟶ S₂} {φ' : S₂ ⟶ S₃} {h₁ : S₁.RightHomologyData}
    {h₂ : S₂.RightHomologyData} {h₃ : S₃.RightHomologyData}
    (ψ : RightHomologyMapData φ h₁ h₂) (ψ' : RightHomologyMapData φ' h₂ h₃) :
    RightHomologyMapData (φ ≫ φ') h₁ h₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {φ' : S₂ ⟶ S₃} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} {h₃ : S₃.RightHomologyData} (ψ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (ψ' : CategoryTheory.ShortComplex.RightHomologyMapData φ' h₂ h₃)
Goal: (ψ.comp ψ').φQ = CategoryTheory.CategoryStruct.comp ψ.φQ ψ'.φQ
-- CategoryTheory.ShortComplex.RightHomologyMapData.congr_φH
lemma congr_φH {γ₁ γ₂ : RightHomologyMapData φ h₁ h₂} (eq : γ₁ = γ₂) : γ₁.φH = γ₂.φH 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} {γ₁ γ₂ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂} (eq : γ₁ = γ₂)
Goal: γ₁.φH = γ₂.φH
-- CategoryTheory.ShortComplex.RightHomologyMapData.congr_φQ
lemma congr_φQ {γ₁ γ₂ : RightHomologyMapData φ h₁ h₂} (eq : γ₁ = γ₂) : γ₁.φQ = γ₂.φQ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} {γ₁ γ₂ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂} (eq : γ₁ = γ₂)
Goal: γ₁.φQ = γ₂.φQ
-- CategoryTheory.ShortComplex.RightHomologyMapData.ofZeros_φQ
@[simps]
def ofZeros (φ : S₁ ⟶ S₂) (hf₁ : S₁.f = 0) (hg₁ : S₁.g = 0) (hf₂ : S₂.f = 0) (hg₂ : S₂.g = 0) :
    RightHomologyMapData φ (RightHomologyData.ofZeros S₁ hf₁ hg₁)
    (RightHomologyData.ofZeros S₂ hf₂ hg₂) 
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) (hg₁ : S₁.g = 0) (hf₂ : S₂.f = 0) (hg₂ : S₂.g = 0)
Goal: (CategoryTheory.ShortComplex.RightHomologyMapData.ofZeros φ hf₁ hg₁ hf₂ hg₂).φQ = φ.τ₂
-- CategoryTheory.ShortComplex.RightHomologyMapData.ofZeros_φH
@[simps]
def ofZeros (φ : S₁ ⟶ S₂) (hf₁ : S₁.f = 0) (hg₁ : S₁.g = 0) (hf₂ : S₂.f = 0) (hg₂ : S₂.g = 0) :
    RightHomologyMapData φ (RightHomologyData.ofZeros S₁ hf₁ hg₁)
    (RightHomologyData.ofZeros S₂ hf₂ hg₂) 
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) (hg₁ : S₁.g = 0) (hf₂ : S₂.f = 0) (hg₂ : S₂.g = 0)
Goal: (CategoryTheory.ShortComplex.RightHomologyMapData.ofZeros φ hf₁ hg₁ hf₂ hg₂).φH = φ.τ₂
-- CategoryTheory.ShortComplex.RightHomologyMapData.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) :
    RightHomologyMapData φ (RightHomologyData.ofIsLimitKernelFork S₁ hf₁ c₁ hc₁)
      (RightHomologyData.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.RightHomologyMapData.ofIsLimitKernelFork φ hf₁ c₁ hc₁ hf₂ c₂ hc₂ f comm).φH = f
-- CategoryTheory.ShortComplex.RightHomologyMapData.ofIsLimitKernelFork_φQ
@[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) :
    RightHomologyMapData φ (RightHomologyData.ofIsLimitKernelFork S₁ hf₁ c₁ hc₁)
      (RightHomologyData.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.RightHomologyMapData.ofIsLimitKernelFork φ hf₁ c₁ hc₁ hf₂ c₂ hc₂ f comm).φQ = φ.τ₂
-- CategoryTheory.ShortComplex.RightHomologyMapData.ofIsColimitCokernelCofork_φQ
@[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) :
    RightHomologyMapData φ (RightHomologyData.ofIsColimitCokernelCofork S₁ hg₁ c₁ hc₁)
      (RightHomologyData.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.RightHomologyMapData.ofIsColimitCokernelCofork φ hg₁ c₁ hc₁ hg₂ c₂ hc₂ f comm).φQ = f
-- CategoryTheory.ShortComplex.RightHomologyMapData.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) :
    RightHomologyMapData φ (RightHomologyData.ofIsColimitCokernelCofork S₁ hg₁ c₁ hc₁)
      (RightHomologyData.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.RightHomologyMapData.ofIsColimitCokernelCofork φ hg₁ c₁ hc₁ hg₂ c₂ hc₂ f comm).φH = f
-- CategoryTheory.ShortComplex.RightHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_φQ
@[simps]
def compatibilityOfZerosOfIsLimitKernelFork (hf : S.f = 0) (hg : S.g = 0)
    (c : KernelFork S.g) (hc : IsLimit c) :
    RightHomologyMapData (𝟙 S)
      (RightHomologyData.ofIsLimitKernelFork S hf c hc)
      (RightHomologyData.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.RightHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork S hf hg c hc).φQ =
  CategoryTheory.CategoryStruct.id (CategoryTheory.ShortComplex.RightHomologyData.ofIsLimitKernelFork S hf c hc).Q
-- CategoryTheory.ShortComplex.RightHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_φH
@[simps]
def compatibilityOfZerosOfIsLimitKernelFork (hf : S.f = 0) (hg : S.g = 0)
    (c : KernelFork S.g) (hc : IsLimit c) :
    RightHomologyMapData (𝟙 S)
      (RightHomologyData.ofIsLimitKernelFork S hf c hc)
      (RightHomologyData.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.RightHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork S hf hg c hc).φH =
  CategoryTheory.Limits.Fork.ι c
-- CategoryTheory.ShortComplex.RightHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork_φH
@[simps]
def compatibilityOfZerosOfIsColimitCokernelCofork (hf : S.f = 0) (hg : S.g = 0)
    (c : CokernelCofork S.f) (hc : IsColimit c) :
    RightHomologyMapData (𝟙 S)
      (RightHomologyData.ofZeros S hf hg)
      (RightHomologyData.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.RightHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork S hf hg c hc).φH =
  CategoryTheory.Limits.Cofork.π c
-- CategoryTheory.ShortComplex.RightHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork_φQ
@[simps]
def compatibilityOfZerosOfIsColimitCokernelCofork (hf : S.f = 0) (hg : S.g = 0)
    (c : CokernelCofork S.f) (hc : IsColimit c) :
    RightHomologyMapData (𝟙 S)
      (RightHomologyData.ofZeros S hf hg)
      (RightHomologyData.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.RightHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork S hf hg c hc).φQ =
  CategoryTheory.Limits.Cofork.π c
-- CategoryTheory.ShortComplex.f_pOpcycles
@[reassoc (attr := simp)]
lemma f_pOpcycles : S.f ≫ S.pOpcycles = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology]
Goal: CategoryTheory.CategoryStruct.comp S.f S.pOpcycles = 0
-- CategoryTheory.ShortComplex.p_fromOpcycles
@[reassoc (attr := simp)]
lemma p_fromOpcycles : S.pOpcycles ≫ S.fromOpcycles = S.g 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology]
Goal: CategoryTheory.CategoryStruct.comp S.pOpcycles S.fromOpcycles = S.g
-- CategoryTheory.ShortComplex.rightHomology_ext_iff
lemma rightHomology_ext_iff {A : C} (f₁ f₂ : A ⟶ S.rightHomology) :
    f₁ = f₂ ↔ f₁ ≫ S.rightHomologyι = f₂ ≫ S.rightHomologyι 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] {A : C} (f₁ f₂ : A ⟶ S.rightHomology)
Goal: f₁ = f₂ ↔
  CategoryTheory.CategoryStruct.comp f₁ S.rightHomologyι = CategoryTheory.CategoryStruct.comp f₂ S.rightHomologyι
-- CategoryTheory.ShortComplex.rightHomology_ext
@[ext]
lemma rightHomology_ext {A : C} (f₁ f₂ : A ⟶ S.rightHomology)
    (h : f₁ ≫ S.rightHomologyι = f₂ ≫ S.rightHomologyι) : f₁ = f₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasRightHomology] {A : C} (f₁ f₂ : A ⟶ S.rightHomology) (h : CategoryTheory.CategoryStruct.comp f₁ S.rightHomologyι = CategoryTheory.CategoryStruct.comp f₂ S.rightHomologyι)
Goal: f₁ = f₂