ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.ShortComplex.HomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_right
@[simps]
def compatibilityOfZerosOfIsLimitKernelFork (hf : S.f = 0) (hg : S.g = 0)
    (c : KernelFork S.g) (hc : IsLimit c) :
    HomologyMapData (𝟙 S)
      (HomologyData.ofIsLimitKernelFork S hf c hc)
      (HomologyData.ofZeros S hf hg) 
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) (c : CategoryTheory.Limits.KernelFork S.g) (hc : CategoryTheory.Limits.IsLimit c)
Goal: (CategoryTheory.ShortComplex.HomologyMapData.compatibilityOfZerosOfIsLimitKernelFork hf hg c hc).right =
  CategoryTheory.ShortComplex.RightHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork S hf hg c hc
-- CategoryTheory.ShortComplex.HomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_left
@[simps]
def compatibilityOfZerosOfIsLimitKernelFork (hf : S.f = 0) (hg : S.g = 0)
    (c : KernelFork S.g) (hc : IsLimit c) :
    HomologyMapData (𝟙 S)
      (HomologyData.ofIsLimitKernelFork S hf c hc)
      (HomologyData.ofZeros S hf hg) 
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) (c : CategoryTheory.Limits.KernelFork S.g) (hc : CategoryTheory.Limits.IsLimit c)
Goal: (CategoryTheory.ShortComplex.HomologyMapData.compatibilityOfZerosOfIsLimitKernelFork hf hg c hc).left =
  CategoryTheory.ShortComplex.LeftHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork S hf hg c hc
-- CategoryTheory.ShortComplex.LeftHomologyData.homologyIso_leftHomologyData
@[simp]
lemma LeftHomologyData.homologyIso_leftHomologyData [S.HasHomology] :
    S.leftHomologyData.homologyIso = S.leftHomologyIso.symm 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasHomology]
Goal: S.leftHomologyData.homologyIso = S.leftHomologyIso.symm
-- CategoryTheory.ShortComplex.RightHomologyData.homologyIso_rightHomologyData
@[simp]
lemma RightHomologyData.homologyIso_rightHomologyData [S.HasHomology] :
    S.rightHomologyData.homologyIso = S.rightHomologyIso.symm 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasHomology]
Goal: S.rightHomologyData.homologyIso = S.rightHomologyIso.symm
-- CategoryTheory.ShortComplex.HomologyMapData.homologyMap'_eq
lemma homologyMap'_eq : homologyMap' φ h₁ h₂ = γ.left.φH 
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} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂)
Goal: CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂ = γ.left.φH
-- CategoryTheory.ShortComplex.HomologyMapData.cyclesMap'_eq
lemma cyclesMap'_eq : cyclesMap' φ h₁.left h₂.left = γ.left.φK 
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} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂)
Goal: CategoryTheory.ShortComplex.cyclesMap' φ h₁.left h₂.left = γ.left.φK
-- CategoryTheory.ShortComplex.HomologyMapData.opcyclesMap'_eq
lemma opcyclesMap'_eq : opcyclesMap' φ h₁.right h₂.right = γ.right.φQ 
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} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂)
Goal: CategoryTheory.ShortComplex.opcyclesMap' φ h₁.right h₂.right = γ.right.φQ
-- CategoryTheory.ShortComplex.LeftHomologyMapData.homologyMap_eq
lemma homologyMap_eq :
    homologyMap φ = h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} 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) [S₁.HasHomology] [S₂.HasHomology]
Goal: CategoryTheory.ShortComplex.homologyMap φ =
  CategoryTheory.CategoryStruct.comp h₁.homologyIso.hom (CategoryTheory.CategoryStruct.comp γ.φH h₂.homologyIso.inv)
-- CategoryTheory.ShortComplex.LeftHomologyMapData.homologyMap_comm
lemma homologyMap_comm :
    homologyMap φ ≫ h₂.homologyIso.hom = h₁.homologyIso.hom ≫ γ.φH 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} 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) [S₁.HasHomology] [S₂.HasHomology]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap φ) h₂.homologyIso.hom =
  CategoryTheory.CategoryStruct.comp h₁.homologyIso.hom γ.φH
-- CategoryTheory.ShortComplex.RightHomologyMapData.homologyMap_eq
lemma homologyMap_eq :
    homologyMap φ = h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) [S₁.HasHomology] [S₂.HasHomology]
Goal: CategoryTheory.ShortComplex.homologyMap φ =
  CategoryTheory.CategoryStruct.comp h₁.homologyIso.hom (CategoryTheory.CategoryStruct.comp γ.φH h₂.homologyIso.inv)
-- CategoryTheory.ShortComplex.RightHomologyMapData.homologyMap_comm
lemma homologyMap_comm :
    homologyMap φ ≫ h₂.homologyIso.hom = h₁.homologyIso.hom ≫ γ.φH 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) [S₁.HasHomology] [S₂.HasHomology]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap φ) h₂.homologyIso.hom =
  CategoryTheory.CategoryStruct.comp h₁.homologyIso.hom γ.φH
-- CategoryTheory.ShortComplex.homologyMap'_id
@[simp]
lemma homologyMap'_id (h : S.HomologyData) :
    homologyMap' (𝟙 S) h h = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData)
Goal: CategoryTheory.ShortComplex.homologyMap' (CategoryTheory.CategoryStruct.id S) h h =
  CategoryTheory.CategoryStruct.id h.left.H
-- CategoryTheory.ShortComplex.homologyMap_id
@[simp]
lemma homologyMap_id [HasHomology S] :
    homologyMap (𝟙 S) = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasHomology]
Goal: CategoryTheory.ShortComplex.homologyMap (CategoryTheory.CategoryStruct.id S) =
  CategoryTheory.CategoryStruct.id S.homology
-- CategoryTheory.ShortComplex.homologyMap'_zero
@[simp]
lemma homologyMap'_zero (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData) :
    homologyMap' 0 h₁ h₂ = 0 
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.homologyMap' 0 h₁ h₂ = 0
-- CategoryTheory.ShortComplex.homologyMap_zero
@[simp]
lemma homologyMap_zero [S₁.HasHomology] [S₂.HasHomology] :
    homologyMap (0 : S₁ ⟶ S₂) = 0 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S₁ S₂ : CategoryTheory.ShortComplex C) [S₁.HasHomology] [S₂.HasHomology]
Goal: CategoryTheory.ShortComplex.homologyMap 0 = 0
-- CategoryTheory.ShortComplex.homologyMap'_comp
lemma homologyMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
    (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData) (h₃ : S₃.HomologyData) :
    homologyMap' (φ₁ ≫ φ₂) h₁ h₃ = homologyMap' φ₁ h₁ h₂ ≫
      homologyMap' φ₂ h₂ h₃ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃) (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData) (h₃ : S₃.HomologyData)
Goal: CategoryTheory.ShortComplex.homologyMap' (CategoryTheory.CategoryStruct.comp φ₁ φ₂) h₁ h₃ =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap' φ₁ h₁ h₂)
    (CategoryTheory.ShortComplex.homologyMap' φ₂ h₂ h₃)
-- CategoryTheory.ShortComplex.homologyMap_comp
@[simp]
lemma homologyMap_comp [HasHomology S₁] [HasHomology S₂] [HasHomology S₃]
    (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃) :
    homologyMap (φ₁ ≫ φ₂) = homologyMap φ₁ ≫ homologyMap φ₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} [S₁.HasHomology] [S₂.HasHomology] [S₃.HasHomology] (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
Goal: CategoryTheory.ShortComplex.homologyMap (CategoryTheory.CategoryStruct.comp φ₁ φ₂) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap φ₁)
    (CategoryTheory.ShortComplex.homologyMap φ₂)
-- CategoryTheory.ShortComplex.homologyMapIso'_hom
@[simps]
def homologyMapIso' (e : S₁ ≅ S₂) (h₁ : S₁.HomologyData)
    (h₂ : S₂.HomologyData) : h₁.left.H ≅ h₂.left.H 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData)
Goal: (CategoryTheory.ShortComplex.homologyMapIso' e h₁ h₂).hom = CategoryTheory.ShortComplex.homologyMap' e.hom h₁ h₂
-- CategoryTheory.ShortComplex.homologyMapIso'_inv
@[simps]
def homologyMapIso' (e : S₁ ≅ S₂) (h₁ : S₁.HomologyData)
    (h₂ : S₂.HomologyData) : h₁.left.H ≅ h₂.left.H 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData)
Goal: (CategoryTheory.ShortComplex.homologyMapIso' e h₁ h₂).inv = CategoryTheory.ShortComplex.homologyMap' e.inv h₂ h₁
-- CategoryTheory.ShortComplex.homologyMapIso_hom
@[simps]
noncomputable def homologyMapIso (e : S₁ ≅ S₂) [S₁.HasHomology]
    [S₂.HasHomology] : S₁.homology ≅ S₂.homology 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) [S₁.HasHomology] [S₂.HasHomology]
Goal: (CategoryTheory.ShortComplex.homologyMapIso e).hom = CategoryTheory.ShortComplex.homologyMap e.hom
-- CategoryTheory.ShortComplex.homologyMapIso_inv
@[simps]
noncomputable def homologyMapIso (e : S₁ ≅ S₂) [S₁.HasHomology]
    [S₂.HasHomology] : S₁.homology ≅ S₂.homology 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) [S₁.HasHomology] [S₂.HasHomology]
Goal: (CategoryTheory.ShortComplex.homologyMapIso e).inv = CategoryTheory.ShortComplex.homologyMap e.inv
-- CategoryTheory.ShortComplex.leftRightHomologyComparison'_eq_liftH
lemma leftRightHomologyComparison'_eq_liftH :
    leftRightHomologyComparison' h₁ h₂ =
      h₂.liftH (h₁.descH (h₁.i ≫ h₂.p) (by simp))
        (by rw [← cancel_epi h₁.π, LeftHomologyData.π_descH_assoc, assoc,
          RightHomologyData.p_g', LeftHomologyData.wi, comp_zero]) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
Goal: CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂ =
  h₂.liftH (h₁.descH (CategoryTheory.CategoryStruct.comp h₁.i h₂.p) ⋯) ⋯
-- CategoryTheory.ShortComplex.π_leftRightHomologyComparison'_ι
@[reassoc (attr := simp)]
lemma π_leftRightHomologyComparison'_ι :
    h₁.π ≫ leftRightHomologyComparison' h₁ h₂ ≫ h₂.ι = h₁.i ≫ h₂.p 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
Goal: CategoryTheory.CategoryStruct.comp h₁.π
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂) h₂.ι) =
  CategoryTheory.CategoryStruct.comp h₁.i h₂.p
-- CategoryTheory.ShortComplex.leftRightHomologyComparison'_eq_descH
lemma leftRightHomologyComparison'_eq_descH :
    leftRightHomologyComparison' h₁ h₂ =
      h₁.descH (h₂.liftH (h₁.i ≫ h₂.p) (by simp))
        (by rw [← cancel_mono h₂.ι, assoc, RightHomologyData.liftH_ι,
          LeftHomologyData.f'_i_assoc, RightHomologyData.wp, zero_comp]) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
Goal: CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂ =
  h₁.descH (h₂.liftH (CategoryTheory.CategoryStruct.comp h₁.i h₂.p) ⋯) ⋯
-- CategoryTheory.ShortComplex.π_leftRightHomologyComparison_ι
@[reassoc (attr := simp)]
lemma π_leftRightHomologyComparison_ι [S.HasLeftHomology] [S.HasRightHomology] :
    S.leftHomologyπ ≫ S.leftRightHomologyComparison ≫ S.rightHomologyι =
      S.iCycles ≫ S.pOpcycles 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasLeftHomology] [S.HasRightHomology]
Goal: CategoryTheory.CategoryStruct.comp S.leftHomologyπ
    (CategoryTheory.CategoryStruct.comp S.leftRightHomologyComparison S.rightHomologyι) =
  CategoryTheory.CategoryStruct.comp S.iCycles S.pOpcycles
-- CategoryTheory.ShortComplex.leftRightHomologyComparison'_naturality
@[reassoc]
lemma leftRightHomologyComparison'_naturality (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData)
    (h₂ : S₁.RightHomologyData) (h₁' : S₂.LeftHomologyData) (h₂' : S₂.RightHomologyData) :
    leftHomologyMap' φ h₁ h₁' ≫ leftRightHomologyComparison' h₁' h₂' =
      leftRightHomologyComparison' h₁ h₂ ≫ rightHomologyMap' φ h₂ h₂' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₁.RightHomologyData) (h₁' : S₂.LeftHomologyData) (h₂' : S₂.RightHomologyData)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₁')
    (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁' h₂') =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂)
    (CategoryTheory.ShortComplex.rightHomologyMap' φ h₂ h₂')
-- CategoryTheory.ShortComplex.leftRightHomologyComparison'_compatibility
lemma leftRightHomologyComparison'_compatibility (h₁ h₁' : S.LeftHomologyData)
    (h₂ h₂' : S.RightHomologyData) :
    leftRightHomologyComparison' h₁ h₂ = leftHomologyMap' (𝟙 S) h₁ h₁' ≫
      leftRightHomologyComparison' h₁' h₂' ≫ rightHomologyMap' (𝟙 S) _ _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h₁ h₁' : S.LeftHomologyData) (h₂ h₂' : S.RightHomologyData)
Goal: CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂ =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.ShortComplex.leftHomologyMap' (CategoryTheory.CategoryStruct.id S) h₁ h₁')
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁' h₂')
      (CategoryTheory.ShortComplex.rightHomologyMap' (CategoryTheory.CategoryStruct.id S) h₂' h₂))
-- CategoryTheory.ShortComplex.leftRightHomologyComparison_eq
lemma leftRightHomologyComparison_eq [S.HasLeftHomology] [S.HasRightHomology]
    (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData) :
    S.leftRightHomologyComparison = h₁.leftHomologyIso.hom ≫
      leftRightHomologyComparison' h₁ h₂ ≫ h₂.rightHomologyIso.inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} [S.HasLeftHomology] [S.HasRightHomology] (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
Goal: S.leftRightHomologyComparison =
  CategoryTheory.CategoryStruct.comp h₁.leftHomologyIso.hom
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂)
      h₂.rightHomologyIso.inv)
-- CategoryTheory.ShortComplex.HomologyData.leftRightHomologyComparison'_eq
@[simp]
lemma HomologyData.leftRightHomologyComparison'_eq (h : S.HomologyData) :
    leftRightHomologyComparison' h.left h.right = h.iso.hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData)
Goal: CategoryTheory.ShortComplex.leftRightHomologyComparison' h.left h.right = h.iso.hom
-- CategoryTheory.ShortComplex.HomologyData.ofIsIsoLeftRightHomologyComparison'_iso
@[simps]
noncomputable def ofIsIsoLeftRightHomologyComparison'
    (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
    [IsIso (leftRightHomologyComparison' h₁ h₂)] :
    S.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData) [CategoryTheory.IsIso (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂)]
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIsIsoLeftRightHomologyComparison' h₁ h₂).iso =
  CategoryTheory.asIso (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂)
-- CategoryTheory.ShortComplex.HomologyData.ofIsIsoLeftRightHomologyComparison'_right
@[simps]
noncomputable def ofIsIsoLeftRightHomologyComparison'
    (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
    [IsIso (leftRightHomologyComparison' h₁ h₂)] :
    S.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData) [CategoryTheory.IsIso (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂)]
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIsIsoLeftRightHomologyComparison' h₁ h₂).right = h₂
-- CategoryTheory.ShortComplex.HomologyData.ofIsIsoLeftRightHomologyComparison'_left
@[simps]
noncomputable def ofIsIsoLeftRightHomologyComparison'
    (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
    [IsIso (leftRightHomologyComparison' h₁ h₂)] :
    S.HomologyData 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData) [CategoryTheory.IsIso (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂)]
Goal: (CategoryTheory.ShortComplex.HomologyData.ofIsIsoLeftRightHomologyComparison' h₁ h₂).left = h₁
-- CategoryTheory.ShortComplex.leftRightHomologyComparison'_eq_leftHomologpMap'_comp_iso_hom_comp_rightHomologyMap'
lemma leftRightHomologyComparison'_eq_leftHomologpMap'_comp_iso_hom_comp_rightHomologyMap'
    (h : S.HomologyData) (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData) :
    leftRightHomologyComparison' h₁ h₂ =
      leftHomologyMap' (𝟙 S) h₁ h.left ≫ h.iso.hom ≫ rightHomologyMap' (𝟙 S) h.right h₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData) (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
Goal: CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂ =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.ShortComplex.leftHomologyMap' (CategoryTheory.CategoryStruct.id S) h₁ h.left)
    (CategoryTheory.CategoryStruct.comp h.iso.hom
      (CategoryTheory.ShortComplex.rightHomologyMap' (CategoryTheory.CategoryStruct.id S) h.right h₂))
-- CategoryTheory.ShortComplex.leftRightHomologyComparison'_fac
@[reassoc]
lemma leftRightHomologyComparison'_fac (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
    [S.HasHomology] :
    leftRightHomologyComparison' h₁ h₂ = h₁.homologyIso.inv ≫ h₂.homologyIso.hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData) [S.HasHomology]
Goal: CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂ =
  CategoryTheory.CategoryStruct.comp h₁.homologyIso.inv h₂.homologyIso.hom
-- CategoryTheory.ShortComplex.leftRightHomologyComparison_fac
@[reassoc]
lemma leftRightHomologyComparison_fac [S.HasHomology] :
    S.leftRightHomologyComparison = S.leftHomologyIso.hom ≫ S.rightHomologyIso.inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C) [S.HasHomology]
Goal: S.leftRightHomologyComparison = CategoryTheory.CategoryStruct.comp S.leftHomologyIso.hom S.rightHomologyIso.inv
-- CategoryTheory.ShortComplex.HomologyData.right_homologyIso_eq_left_homologyIso_trans_iso
lemma HomologyData.right_homologyIso_eq_left_homologyIso_trans_iso
    (h : S.HomologyData) [S.HasHomology] :
    h.right.homologyIso = h.left.homologyIso ≪≫ h.iso 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData) [S.HasHomology]
Goal: h.right.homologyIso = h.left.homologyIso ≪≫ h.iso
-- CategoryTheory.ShortComplex.hasHomology_of_isIso_leftRightHomologyComparison'
lemma hasHomology_of_isIso_leftRightHomologyComparison'
    (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
    [IsIso (leftRightHomologyComparison' h₁ h₂)] :
    S.HasHomology 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData) [CategoryTheory.IsIso (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂)]
Goal: S.HasHomology
-- CategoryTheory.ShortComplex.hasHomology_of_isIsoLeftRightHomologyComparison
lemma hasHomology_of_isIsoLeftRightHomologyComparison [S.HasLeftHomology]
    [S.HasRightHomology] [h : IsIso S.leftRightHomologyComparison] :
    S.HasHomology 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C} [S.HasLeftHomology] [S.HasRightHomology] [h : CategoryTheory.IsIso S.leftRightHomologyComparison]
Goal: S.HasHomology
-- CategoryTheory.ShortComplex.LeftHomologyData.leftHomologyIso_hom_naturality
@[reassoc]
lemma LeftHomologyData.leftHomologyIso_hom_naturality
    (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) :
    h₁.homologyIso.hom ≫ leftHomologyMap' φ h₁ h₂ =
      homologyMap φ ≫ h₂.homologyIso.hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} [S₁.HasHomology] [S₂.HasHomology] (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.CategoryStruct.comp h₁.homologyIso.hom (CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap φ) h₂.homologyIso.hom
-- CategoryTheory.ShortComplex.LeftHomologyData.leftHomologyIso_inv_naturality
@[reassoc]
lemma LeftHomologyData.leftHomologyIso_inv_naturality
    (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) :
    h₁.homologyIso.inv ≫ homologyMap φ =
      leftHomologyMap' φ h₁ h₂ ≫ h₂.homologyIso.inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {S₁ S₂ : CategoryTheory.ShortComplex C} [S₁.HasHomology] [S₂.HasHomology] (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData)
Goal: CategoryTheory.CategoryStruct.comp h₁.homologyIso.inv (CategoryTheory.ShortComplex.homologyMap φ) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂) h₂.homologyIso.inv