ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.ShortComplex.HomologyData.ofHasCokernel_iso
@[simps]
noncomputable def ofHasCokernel (hg : S.g = 0) [HasCokernel S.f] :
    S.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) (hg : S.g = 0) [CategoryTheory.Limits.HasCokernel S.f]
Goal: (CategoryTheory.ShortComplex.HomologyData.ofHasCokernel S hg).iso =
  CategoryTheory.Iso.refl (CategoryTheory.ShortComplex.LeftHomologyData.ofHasCokernel S hg).H
-- CategoryTheory.ShortComplex.HomologyData.ofZeros_iso
@[simps]
noncomputable def ofZeros (hf : S.f = 0) (hg : S.g = 0) :
    S.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) (hf : S.f = 0) (hg : S.g = 0)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofZeros S hf hg).iso =
  CategoryTheory.Iso.refl (CategoryTheory.ShortComplex.LeftHomologyData.ofZeros S hf hg).H
-- CategoryTheory.ShortComplex.HomologyData.ofZeros_right
@[simps]
noncomputable def ofZeros (hf : S.f = 0) (hg : S.g = 0) :
    S.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) (hf : S.f = 0) (hg : S.g = 0)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofZeros S hf hg).right =
  CategoryTheory.ShortComplex.RightHomologyData.ofZeros S hf hg
-- CategoryTheory.ShortComplex.HomologyData.ofZeros_left
@[simps]
noncomputable def ofZeros (hf : S.f = 0) (hg : S.g = 0) :
    S.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) (hf : S.f = 0) (hg : S.g = 0)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofZeros S hf hg).left =
  CategoryTheory.ShortComplex.LeftHomologyData.ofZeros S hf hg
-- CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono_left
@[simps]
noncomputable def ofEpiOfIsIsoOfMono (φ : S₁ ⟶ S₂) (h : HomologyData S₁)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : HomologyData S₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₁.HomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono φ h).left =
  CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono φ h.left
-- CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono_iso
@[simps]
noncomputable def ofEpiOfIsIsoOfMono (φ : S₁ ⟶ S₂) (h : HomologyData S₁)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : HomologyData S₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₁.HomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono φ h).iso = h.iso
-- CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono_right
@[simps]
noncomputable def ofEpiOfIsIsoOfMono (φ : S₁ ⟶ S₂) (h : HomologyData S₁)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : HomologyData S₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₁.HomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono φ h).right =
  CategoryTheory.ShortComplex.RightHomologyData.ofEpiOfIsIsoOfMono φ h.right
-- CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono'_right
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : HomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : HomologyData S₁ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₂.HomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono' φ h).right =
  CategoryTheory.ShortComplex.RightHomologyData.ofEpiOfIsIsoOfMono' φ h.right
-- CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono'_left
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : HomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : HomologyData S₁ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₂.HomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono' φ h).left =
  CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono' φ h.left
-- CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono'_iso
@[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : HomologyData S₂)
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : HomologyData S₁ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h : S₂.HomologyData) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: (CategoryTheory.ShortComplex.HomologyData.ofEpiOfIsIsoOfMono' φ h).iso = h.iso
-- CategoryTheory.ShortComplex.HomologyData.ofIso_left_H
@[simps!]
noncomputable def ofIso (e : S₁ ≅ S₂) (h : HomologyData S₁) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h : S₁.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIso e h).left.H = h.left.H
-- CategoryTheory.ShortComplex.HomologyData.ofIso_iso
@[simps!]
noncomputable def ofIso (e : S₁ ≅ S₂) (h : HomologyData S₁) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h : S₁.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIso e h).iso = h.iso
-- CategoryTheory.ShortComplex.HomologyData.ofIso_left_i
@[simps!]
noncomputable def ofIso (e : S₁ ≅ S₂) (h : HomologyData S₁) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h : S₁.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIso e h).left.i = CategoryTheory.CategoryStruct.comp h.left.i e.hom.τ₂
-- CategoryTheory.ShortComplex.HomologyData.ofIso_left_K
@[simps!]
noncomputable def ofIso (e : S₁ ≅ S₂) (h : HomologyData S₁) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h : S₁.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIso e h).left.K = h.left.K
-- CategoryTheory.ShortComplex.HomologyData.ofIso_right_ι
@[simps!]
noncomputable def ofIso (e : S₁ ≅ S₂) (h : HomologyData S₁) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h : S₁.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIso e h).right.ι = h.right.ι
-- CategoryTheory.ShortComplex.HomologyData.ofIso_right_H
@[simps!]
noncomputable def ofIso (e : S₁ ≅ S₂) (h : HomologyData S₁) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h : S₁.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIso e h).right.H = h.right.H
-- CategoryTheory.ShortComplex.HomologyData.ofIso_right_p
@[simps!]
noncomputable def ofIso (e : S₁ ≅ S₂) (h : HomologyData S₁) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h : S₁.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIso e h).right.p =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.inv e.hom.τ₂) h.right.p
-- CategoryTheory.ShortComplex.HomologyData.ofIso_right_Q
@[simps!]
noncomputable def ofIso (e : S₁ ≅ S₂) (h : HomologyData S₁) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h : S₁.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIso e h).right.Q = h.right.Q
-- CategoryTheory.ShortComplex.HomologyData.ofIso_left_π
@[simps!]
noncomputable def ofIso (e : S₁ ≅ S₂) (h : HomologyData S₁) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h : S₁.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIso e h).left.π = h.left.π
-- CategoryTheory.ShortComplex.HomologyData.op_left
@[simps]
def op (h : S.HomologyData) : S.op.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData)
Goal: h.op.left = h.right.op
-- CategoryTheory.ShortComplex.HomologyData.op_iso
@[simps]
def op (h : S.HomologyData) : S.op.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData)
Goal: h.op.iso = h.iso.op
-- CategoryTheory.ShortComplex.HomologyData.op_right
@[simps]
def op (h : S.HomologyData) : S.op.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData)
Goal: h.op.right = h.left.op
-- CategoryTheory.ShortComplex.HomologyData.unop_iso
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.HomologyData) : S.unop.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.HomologyData)
Goal: h.unop.iso = h.iso.unop
-- CategoryTheory.ShortComplex.HomologyData.unop_left
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.HomologyData) : S.unop.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.HomologyData)
Goal: h.unop.left = h.right.unop
-- CategoryTheory.ShortComplex.HomologyData.unop_right
@[simps]
def unop {S : ShortComplex Cᵒᵖ} (h : S.HomologyData) : S.unop.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.HomologyData)
Goal: h.unop.right = h.left.unop
-- CategoryTheory.ShortComplex.HasHomology.condition
class HasHomology : Prop where
    condition : Nonempty S.HomologyData
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} [self : S.HasHomology]
Goal: Nonempty S.HomologyData
-- CategoryTheory.ShortComplex.HasHomology.mk'
lemma HasHomology.mk' (h : S.HomologyData) : HasHomology S 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData)
Goal: S.HasHomology
-- CategoryTheory.ShortComplex.hasHomology_of_epi_of_isIso_of_mono
lemma hasHomology_of_epi_of_isIso_of_mono (φ : S₁ ⟶ S₂) [HasHomology S₁]
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : HasHomology S₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [S₁.HasHomology] [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: S₂.HasHomology
-- CategoryTheory.ShortComplex.hasHomology_of_epi_of_isIso_of_mono'
lemma hasHomology_of_epi_of_isIso_of_mono' (φ : S₁ ⟶ S₂) [HasHomology S₂]
    [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] : HasHomology S₁ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [S₂.HasHomology] [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂] [CategoryTheory.Mono φ.τ₃]
Goal: S₁.HasHomology
-- CategoryTheory.ShortComplex.hasHomology_of_iso
lemma hasHomology_of_iso (e : S₁ ≅ S₂) [HasHomology S₁] : HasHomology S₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) [S₁.HasHomology]
Goal: S₂.HasHomology
-- CategoryTheory.ShortComplex.HomologyMapData.id_left
@[simps]
def id (h : S.HomologyData) : HomologyMapData (𝟙 S) h h 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyMapData.id h).left = CategoryTheory.ShortComplex.LeftHomologyMapData.id h.left
-- CategoryTheory.ShortComplex.HomologyMapData.id_right
@[simps]
def id (h : S.HomologyData) : HomologyMapData (𝟙 S) h h 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyMapData.id h).right = CategoryTheory.ShortComplex.RightHomologyMapData.id h.right
-- CategoryTheory.ShortComplex.HomologyMapData.zero_left
@[simps]
def zero (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData) :
    HomologyMapData 0 h₁ h₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyMapData.zero h₁ h₂).left =
  CategoryTheory.ShortComplex.LeftHomologyMapData.zero h₁.left h₂.left
-- CategoryTheory.ShortComplex.HomologyMapData.zero_right
@[simps]
def zero (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData) :
    HomologyMapData 0 h₁ h₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData)
Goal: (CategoryTheory.ShortComplex.HomologyMapData.zero h₁ h₂).right =
  CategoryTheory.ShortComplex.RightHomologyMapData.zero h₁.right h₂.right
-- CategoryTheory.ShortComplex.HomologyMapData.comp_left
@[simps]
def comp {φ : S₁ ⟶ S₂} {φ' : S₂ ⟶ S₃} {h₁ : S₁.HomologyData}
    {h₂ : S₂.HomologyData} {h₃ : S₃.HomologyData}
    (ψ : HomologyMapData φ h₁ h₂) (ψ' : HomologyMapData φ' h₂ h₃) :
    HomologyMapData (φ ≫ φ') h₁ h₃ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {φ' : S₂ ⟶ S₃} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} {h₃ : S₃.HomologyData} (ψ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (ψ' : CategoryTheory.ShortComplex.HomologyMapData φ' h₂ h₃)
Goal: (ψ.comp ψ').left = ψ.left.comp ψ'.left
-- CategoryTheory.ShortComplex.HomologyMapData.comp_right
@[simps]
def comp {φ : S₁ ⟶ S₂} {φ' : S₂ ⟶ S₃} {h₁ : S₁.HomologyData}
    {h₂ : S₂.HomologyData} {h₃ : S₃.HomologyData}
    (ψ : HomologyMapData φ h₁ h₂) (ψ' : HomologyMapData φ' h₂ h₃) :
    HomologyMapData (φ ≫ φ') h₁ h₃ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {φ' : S₂ ⟶ S₃} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} {h₃ : S₃.HomologyData} (ψ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (ψ' : CategoryTheory.ShortComplex.HomologyMapData φ' h₂ h₃)
Goal: (ψ.comp ψ').right = ψ.right.comp ψ'.right
-- CategoryTheory.ShortComplex.HomologyMapData.op_right
@[simps]
def op {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData}
    (ψ : HomologyMapData φ h₁ h₂) :
    HomologyMapData (opMap φ) h₂.op h₁.op 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (ψ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂)
Goal: ψ.op.right = ψ.left.op
-- CategoryTheory.ShortComplex.HomologyMapData.op_left
@[simps]
def op {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData}
    (ψ : HomologyMapData φ h₁ h₂) :
    HomologyMapData (opMap φ) h₂.op h₁.op 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (ψ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂)
Goal: ψ.op.left = ψ.right.op
-- CategoryTheory.ShortComplex.HomologyMapData.unop_right
@[simps]
def unop {S₁ S₂ : ShortComplex Cᵒᵖ} {φ : S₁ ⟶ S₂}
    {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData}
    (ψ : HomologyMapData φ h₁ h₂) :
    HomologyMapData (unopMap φ) h₂.unop h₁.unop 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex Cᵒᵖ} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (ψ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂)
Goal: ψ.unop.right = ψ.left.unop
-- CategoryTheory.ShortComplex.HomologyMapData.unop_left
@[simps]
def unop {S₁ S₂ : ShortComplex Cᵒᵖ} {φ : S₁ ⟶ S₂}
    {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData}
    (ψ : HomologyMapData φ h₁ h₂) :
    HomologyMapData (unopMap φ) h₂.unop h₁.unop 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex Cᵒᵖ} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (ψ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂)
Goal: ψ.unop.left = ψ.right.unop
-- CategoryTheory.ShortComplex.HomologyMapData.ofZeros_right
@[simps]
def ofZeros (φ : S₁ ⟶ S₂) (hf₁ : S₁.f = 0) (hg₁ : S₁.g = 0) (hf₂ : S₂.f = 0) (hg₂ : S₂.g = 0) :
    HomologyMapData φ (HomologyData.ofZeros S₁ hf₁ hg₁) (HomologyData.ofZeros S₂ hf₂ hg₂) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} 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.HomologyMapData.ofZeros φ hf₁ hg₁ hf₂ hg₂).right =
  CategoryTheory.ShortComplex.RightHomologyMapData.ofZeros φ hf₁ hg₁ hf₂ hg₂
-- CategoryTheory.ShortComplex.HomologyMapData.ofZeros_left
@[simps]
def ofZeros (φ : S₁ ⟶ S₂) (hf₁ : S₁.f = 0) (hg₁ : S₁.g = 0) (hf₂ : S₂.f = 0) (hg₂ : S₂.g = 0) :
    HomologyMapData φ (HomologyData.ofZeros S₁ hf₁ hg₁) (HomologyData.ofZeros S₂ hf₂ hg₂) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} 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.HomologyMapData.ofZeros φ hf₁ hg₁ hf₂ hg₂).left =
  CategoryTheory.ShortComplex.LeftHomologyMapData.ofZeros φ hf₁ hg₁ hf₂ hg₂
-- CategoryTheory.ShortComplex.HomologyMapData.ofIsColimitCokernelCofork_left
@[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) :
    HomologyMapData φ (HomologyData.ofIsColimitCokernelCofork S₁ hg₁ c₁ hc₁)
      (HomologyData.ofIsColimitCokernelCofork S₂ hg₂ c₂ hc₂) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} 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.HomologyMapData.ofIsColimitCokernelCofork φ hg₁ c₁ hc₁ hg₂ c₂ hc₂ f comm).left =
  CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsColimitCokernelCofork φ hg₁ c₁ hc₁ hg₂ c₂ hc₂ f comm
-- CategoryTheory.ShortComplex.HomologyMapData.ofIsColimitCokernelCofork_right
@[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) :
    HomologyMapData φ (HomologyData.ofIsColimitCokernelCofork S₁ hg₁ c₁ hc₁)
      (HomologyData.ofIsColimitCokernelCofork S₂ hg₂ c₂ hc₂) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} 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.HomologyMapData.ofIsColimitCokernelCofork φ hg₁ c₁ hc₁ hg₂ c₂ hc₂ f comm).right =
  CategoryTheory.ShortComplex.RightHomologyMapData.ofIsColimitCokernelCofork φ hg₁ c₁ hc₁ hg₂ c₂ hc₂ f comm
-- CategoryTheory.ShortComplex.HomologyMapData.ofIsLimitKernelFork_right
@[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) :
    HomologyMapData φ (HomologyData.ofIsLimitKernelFork S₁ hf₁ c₁ hc₁)
      (HomologyData.ofIsLimitKernelFork S₂ hf₂ c₂ hc₂) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} 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.HomologyMapData.ofIsLimitKernelFork φ hf₁ c₁ hc₁ hf₂ c₂ hc₂ f comm).right =
  CategoryTheory.ShortComplex.RightHomologyMapData.ofIsLimitKernelFork φ hf₁ c₁ hc₁ hf₂ c₂ hc₂ f comm
-- CategoryTheory.ShortComplex.HomologyMapData.ofIsLimitKernelFork_left
@[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) :
    HomologyMapData φ (HomologyData.ofIsLimitKernelFork S₁ hf₁ c₁ hc₁)
      (HomologyData.ofIsLimitKernelFork S₂ hf₂ c₂ hc₂) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} 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.HomologyMapData.ofIsLimitKernelFork φ hf₁ c₁ hc₁ hf₂ c₂ hc₂ f comm).left =
  CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsLimitKernelFork φ hf₁ c₁ hc₁ hf₂ c₂ hc₂ f comm