ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.ShortComplex.RightHomologyData.map_ι
@[simps]
noncomputable def map : (S.map F).RightHomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.IsPreservedBy F]
Goal: (h.map F).ι = F.map h.ι
-- CategoryTheory.ShortComplex.RightHomologyData.map_p
@[simps]
noncomputable def map : (S.map F).RightHomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.IsPreservedBy F]
Goal: (h.map F).p = F.map h.p
-- CategoryTheory.ShortComplex.RightHomologyData.map_g'
@[simp]
lemma map_g' : (h.map F).g' = F.map h.g' 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.RightHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.IsPreservedBy F]
Goal: (h.map F).g' = F.map h.g'
-- CategoryTheory.ShortComplex.RightHomologyMapData.map_φH
@[simps]
def RightHomologyMapData.map {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData}
    {h₂ : S₂.RightHomologyData} (ψ : RightHomologyMapData φ h₁ h₂) (F : C ⥤ D)
    [F.PreservesZeroMorphisms] [h₁.IsPreservedBy F] [h₂.IsPreservedBy F] :
    RightHomologyMapData (F.mapShortComplex.map φ) (h₁.map F) (h₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (ψ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h₁.IsPreservedBy F] [h₂.IsPreservedBy F]
Goal: (ψ.map F).φH = F.map ψ.φH
-- CategoryTheory.ShortComplex.RightHomologyMapData.map_φQ
@[simps]
def RightHomologyMapData.map {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData}
    {h₂ : S₂.RightHomologyData} (ψ : RightHomologyMapData φ h₁ h₂) (F : C ⥤ D)
    [F.PreservesZeroMorphisms] [h₁.IsPreservedBy F] [h₂.IsPreservedBy F] :
    RightHomologyMapData (F.mapShortComplex.map φ) (h₁.map F) (h₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (ψ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h₁.IsPreservedBy F] [h₂.IsPreservedBy F]
Goal: (ψ.map F).φQ = F.map ψ.φQ
-- CategoryTheory.ShortComplex.HomologyData.map_iso
@[simps]
noncomputable def HomologyData.map (h : S.HomologyData) (F : C ⥤ D) [F.PreservesZeroMorphisms]
    [h.left.IsPreservedBy F] [h.right.IsPreservedBy F] :
    (S.map F).HomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.left.IsPreservedBy F] [h.right.IsPreservedBy F]
Goal: (h.map F).iso = F.mapIso h.iso
-- CategoryTheory.ShortComplex.HomologyData.map_left
@[simps]
noncomputable def HomologyData.map (h : S.HomologyData) (F : C ⥤ D) [F.PreservesZeroMorphisms]
    [h.left.IsPreservedBy F] [h.right.IsPreservedBy F] :
    (S.map F).HomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.left.IsPreservedBy F] [h.right.IsPreservedBy F]
Goal: (h.map F).left = h.left.map F
-- CategoryTheory.ShortComplex.HomologyData.map_right
@[simps]
noncomputable def HomologyData.map (h : S.HomologyData) (F : C ⥤ D) [F.PreservesZeroMorphisms]
    [h.left.IsPreservedBy F] [h.right.IsPreservedBy F] :
    (S.map F).HomologyData 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.HomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h.left.IsPreservedBy F] [h.right.IsPreservedBy F]
Goal: (h.map F).right = h.right.map F
-- CategoryTheory.ShortComplex.HomologyMapData.map_right
@[simps]
def HomologyMapData.map {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData}
    (ψ : HomologyMapData φ h₁ h₂) (F : C ⥤ D) [F.PreservesZeroMorphisms]
    [h₁.left.IsPreservedBy F] [h₁.right.IsPreservedBy F]
    [h₂.left.IsPreservedBy F] [h₂.right.IsPreservedBy F] :
    HomologyMapData (F.mapShortComplex.map φ) (h₁.map F) (h₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (ψ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h₁.left.IsPreservedBy F] [h₁.right.IsPreservedBy F] [h₂.left.IsPreservedBy F] [h₂.right.IsPreservedBy F]
Goal: (ψ.map F).right = ψ.right.map F
-- CategoryTheory.ShortComplex.HomologyMapData.map_left
@[simps]
def HomologyMapData.map {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData}
    (ψ : HomologyMapData φ h₁ h₂) (F : C ⥤ D) [F.PreservesZeroMorphisms]
    [h₁.left.IsPreservedBy F] [h₁.right.IsPreservedBy F]
    [h₂.left.IsPreservedBy F] [h₂.right.IsPreservedBy F] :
    HomologyMapData (F.mapShortComplex.map φ) (h₁.map F) (h₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (ψ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h₁.left.IsPreservedBy F] [h₁.right.IsPreservedBy F] [h₂.left.IsPreservedBy F] [h₂.right.IsPreservedBy F]
Goal: (ψ.map F).left = ψ.left.map F
-- CategoryTheory.ShortComplex.LeftHomologyData.map_cyclesMap'
lemma map_cyclesMap' : F.map (ShortComplex.cyclesMap' φ hl₁ hl₂) =
    ShortComplex.cyclesMap' (F.mapShortComplex.map φ) (hl₁.map F) (hl₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (hl₁ : S₁.LeftHomologyData) (hl₂ : S₂.LeftHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [hl₁.IsPreservedBy F] [hl₂.IsPreservedBy F]
Goal: F.map (CategoryTheory.ShortComplex.cyclesMap' φ hl₁ hl₂) =
  CategoryTheory.ShortComplex.cyclesMap' (F.mapShortComplex.map φ) (hl₁.map F) (hl₂.map F)
-- CategoryTheory.ShortComplex.LeftHomologyData.map_leftHomologyMap'
lemma map_leftHomologyMap' : F.map (ShortComplex.leftHomologyMap' φ hl₁ hl₂) =
    ShortComplex.leftHomologyMap' (F.mapShortComplex.map φ) (hl₁.map F) (hl₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (hl₁ : S₁.LeftHomologyData) (hl₂ : S₂.LeftHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [hl₁.IsPreservedBy F] [hl₂.IsPreservedBy F]
Goal: F.map (CategoryTheory.ShortComplex.leftHomologyMap' φ hl₁ hl₂) =
  CategoryTheory.ShortComplex.leftHomologyMap' (F.mapShortComplex.map φ) (hl₁.map F) (hl₂.map F)
-- CategoryTheory.ShortComplex.RightHomologyData.map_opcyclesMap'
lemma map_opcyclesMap' : F.map (ShortComplex.opcyclesMap' φ hr₁ hr₂) =
    ShortComplex.opcyclesMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (hr₁ : S₁.RightHomologyData) (hr₂ : S₂.RightHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [hr₁.IsPreservedBy F] [hr₂.IsPreservedBy F]
Goal: F.map (CategoryTheory.ShortComplex.opcyclesMap' φ hr₁ hr₂) =
  CategoryTheory.ShortComplex.opcyclesMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F)
-- CategoryTheory.ShortComplex.RightHomologyData.map_rightHomologyMap'
lemma map_rightHomologyMap' : F.map (ShortComplex.rightHomologyMap' φ hr₁ hr₂) =
    ShortComplex.rightHomologyMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (hr₁ : S₁.RightHomologyData) (hr₂ : S₂.RightHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [hr₁.IsPreservedBy F] [hr₂.IsPreservedBy F]
Goal: F.map (CategoryTheory.ShortComplex.rightHomologyMap' φ hr₁ hr₂) =
  CategoryTheory.ShortComplex.rightHomologyMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F)
-- CategoryTheory.ShortComplex.HomologyData.map_homologyMap'
lemma HomologyData.map_homologyMap'
    [h₁.left.IsPreservedBy F] [h₁.right.IsPreservedBy F]
    [h₂.left.IsPreservedBy F] [h₂.right.IsPreservedBy F] :
    F.map (ShortComplex.homologyMap' φ h₁ h₂) =
      ShortComplex.homologyMap' (F.mapShortComplex.map φ) (h₁.map F) (h₂.map F) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [h₁.left.IsPreservedBy F] [h₁.right.IsPreservedBy F] [h₂.left.IsPreservedBy F] [h₂.right.IsPreservedBy F]
Goal: F.map (CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂) =
  CategoryTheory.ShortComplex.homologyMap' (F.mapShortComplex.map φ) (h₁.map F) (h₂.map F)
-- CategoryTheory.ShortComplex.mapCyclesIso_hom_iCycles
@[reassoc (attr := simp)]
lemma mapCyclesIso_hom_iCycles [S.HasLeftHomology] [F.PreservesLeftHomologyOf S] :
    (S.mapCyclesIso F).hom ≫ F.map S.iCycles = (S.map F).iCycles 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (S : CategoryTheory.ShortComplex C) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S.HasLeftHomology] [F.PreservesLeftHomologyOf S]
Goal: CategoryTheory.CategoryStruct.comp (S.mapCyclesIso F).hom (F.map S.iCycles) = (S.map F).iCycles
-- CategoryTheory.ShortComplex.LeftHomologyData.mapCyclesIso_eq
lemma LeftHomologyData.mapCyclesIso_eq [S.HasLeftHomology]
    [F.PreservesLeftHomologyOf S] :
    S.mapCyclesIso F = (hl.map F).cyclesIso ≪≫ F.mapIso hl.cyclesIso.symm 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (hl : S.LeftHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S.HasLeftHomology] [F.PreservesLeftHomologyOf S]
Goal: S.mapCyclesIso F = (hl.map F).cyclesIso ≪≫ F.mapIso hl.cyclesIso.symm
-- CategoryTheory.ShortComplex.LeftHomologyData.mapLeftHomologyIso_eq
lemma LeftHomologyData.mapLeftHomologyIso_eq [S.HasLeftHomology]
    [F.PreservesLeftHomologyOf S] :
    S.mapLeftHomologyIso F = (hl.map F).leftHomologyIso ≪≫ F.mapIso hl.leftHomologyIso.symm 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (hl : S.LeftHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S.HasLeftHomology] [F.PreservesLeftHomologyOf S]
Goal: S.mapLeftHomologyIso F = (hl.map F).leftHomologyIso ≪≫ F.mapIso hl.leftHomologyIso.symm
-- CategoryTheory.ShortComplex.RightHomologyData.mapOpcyclesIso_eq
lemma RightHomologyData.mapOpcyclesIso_eq [S.HasRightHomology]
    [F.PreservesRightHomologyOf S] :
    S.mapOpcyclesIso F = (hr.map F).opcyclesIso ≪≫ F.mapIso hr.opcyclesIso.symm 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (hr : S.RightHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S.HasRightHomology] [F.PreservesRightHomologyOf S]
Goal: S.mapOpcyclesIso F = (hr.map F).opcyclesIso ≪≫ F.mapIso hr.opcyclesIso.symm
-- CategoryTheory.ShortComplex.RightHomologyData.mapRightHomologyIso_eq
lemma RightHomologyData.mapRightHomologyIso_eq [S.HasRightHomology]
    [F.PreservesRightHomologyOf S] :
    S.mapRightHomologyIso F = (hr.map F).rightHomologyIso ≪≫
      F.mapIso hr.rightHomologyIso.symm 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (hr : S.RightHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S.HasRightHomology] [F.PreservesRightHomologyOf S]
Goal: S.mapRightHomologyIso F = (hr.map F).rightHomologyIso ≪≫ F.mapIso hr.rightHomologyIso.symm
-- CategoryTheory.ShortComplex.LeftHomologyData.mapHomologyIso_eq
lemma LeftHomologyData.mapHomologyIso_eq [S.HasHomology]
    [(S.map F).HasHomology] [F.PreservesLeftHomologyOf S] :
    S.mapHomologyIso F = (hl.map F).homologyIso ≪≫ F.mapIso hl.homologyIso.symm 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (hl : S.LeftHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S.HasHomology] [(S.map F).HasHomology] [F.PreservesLeftHomologyOf S]
Goal: S.mapHomologyIso F = (hl.map F).homologyIso ≪≫ F.mapIso hl.homologyIso.symm
-- CategoryTheory.ShortComplex.RightHomologyData.mapHomologyIso'_eq
lemma RightHomologyData.mapHomologyIso'_eq [S.HasHomology]
    [(S.map F).HasHomology] [F.PreservesRightHomologyOf S] :
    S.mapHomologyIso' F = (hr.map F).homologyIso ≪≫ F.mapIso hr.homologyIso.symm 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (hr : S.RightHomologyData) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S.HasHomology] [(S.map F).HasHomology] [F.PreservesRightHomologyOf S]
Goal: S.mapHomologyIso' F = (hr.map F).homologyIso ≪≫ F.mapIso hr.homologyIso.symm
-- CategoryTheory.ShortComplex.mapCyclesIso_hom_naturality
@[reassoc]
lemma mapCyclesIso_hom_naturality [S₁.HasLeftHomology] [S₂.HasLeftHomology]
    [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂] :
    cyclesMap (F.mapShortComplex.map φ) ≫ (S₂.mapCyclesIso F).hom =
      (S₁.mapCyclesIso F).hom ≫ F.map (cyclesMap φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasLeftHomology] [S₂.HasLeftHomology] [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.cyclesMap (F.mapShortComplex.map φ))
    (S₂.mapCyclesIso F).hom =
  CategoryTheory.CategoryStruct.comp (S₁.mapCyclesIso F).hom (F.map (CategoryTheory.ShortComplex.cyclesMap φ))
-- CategoryTheory.ShortComplex.mapCyclesIso_inv_naturality
@[reassoc]
lemma mapCyclesIso_inv_naturality [S₁.HasLeftHomology] [S₂.HasLeftHomology]
    [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂] :
    F.map (cyclesMap φ) ≫ (S₂.mapCyclesIso F).inv =
      (S₁.mapCyclesIso F).inv ≫ cyclesMap (F.mapShortComplex.map φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasLeftHomology] [S₂.HasLeftHomology] [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.ShortComplex.cyclesMap φ)) (S₂.mapCyclesIso F).inv =
  CategoryTheory.CategoryStruct.comp (S₁.mapCyclesIso F).inv
    (CategoryTheory.ShortComplex.cyclesMap (F.mapShortComplex.map φ))
-- CategoryTheory.ShortComplex.mapLeftHomologyIso_hom_naturality
@[reassoc]
lemma mapLeftHomologyIso_hom_naturality [S₁.HasLeftHomology] [S₂.HasLeftHomology]
    [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂] :
    leftHomologyMap (F.mapShortComplex.map φ) ≫ (S₂.mapLeftHomologyIso F).hom =
      (S₁.mapLeftHomologyIso F).hom ≫ F.map (leftHomologyMap φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasLeftHomology] [S₂.HasLeftHomology] [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftHomologyMap (F.mapShortComplex.map φ))
    (S₂.mapLeftHomologyIso F).hom =
  CategoryTheory.CategoryStruct.comp (S₁.mapLeftHomologyIso F).hom
    (F.map (CategoryTheory.ShortComplex.leftHomologyMap φ))
-- CategoryTheory.ShortComplex.mapLeftHomologyIso_inv_naturality
@[reassoc]
lemma mapLeftHomologyIso_inv_naturality [S₁.HasLeftHomology] [S₂.HasLeftHomology]
    [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂] :
    F.map (leftHomologyMap φ) ≫ (S₂.mapLeftHomologyIso F).inv =
      (S₁.mapLeftHomologyIso F).inv ≫ leftHomologyMap (F.mapShortComplex.map φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasLeftHomology] [S₂.HasLeftHomology] [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.ShortComplex.leftHomologyMap φ))
    (S₂.mapLeftHomologyIso F).inv =
  CategoryTheory.CategoryStruct.comp (S₁.mapLeftHomologyIso F).inv
    (CategoryTheory.ShortComplex.leftHomologyMap (F.mapShortComplex.map φ))
-- CategoryTheory.ShortComplex.mapOpcyclesIso_hom_naturality
@[reassoc]
lemma mapOpcyclesIso_hom_naturality [S₁.HasRightHomology] [S₂.HasRightHomology]
    [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂] :
    opcyclesMap (F.mapShortComplex.map φ) ≫ (S₂.mapOpcyclesIso F).hom =
      (S₁.mapOpcyclesIso F).hom ≫ F.map (opcyclesMap φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasRightHomology] [S₂.HasRightHomology] [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.opcyclesMap (F.mapShortComplex.map φ))
    (S₂.mapOpcyclesIso F).hom =
  CategoryTheory.CategoryStruct.comp (S₁.mapOpcyclesIso F).hom (F.map (CategoryTheory.ShortComplex.opcyclesMap φ))
-- CategoryTheory.ShortComplex.mapOpcyclesIso_inv_naturality
@[reassoc]
lemma mapOpcyclesIso_inv_naturality [S₁.HasRightHomology] [S₂.HasRightHomology]
    [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂] :
    F.map (opcyclesMap φ) ≫ (S₂.mapOpcyclesIso F).inv =
      (S₁.mapOpcyclesIso F).inv ≫ opcyclesMap (F.mapShortComplex.map φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasRightHomology] [S₂.HasRightHomology] [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.ShortComplex.opcyclesMap φ)) (S₂.mapOpcyclesIso F).inv =
  CategoryTheory.CategoryStruct.comp (S₁.mapOpcyclesIso F).inv
    (CategoryTheory.ShortComplex.opcyclesMap (F.mapShortComplex.map φ))
-- CategoryTheory.ShortComplex.mapRightHomologyIso_hom_naturality
@[reassoc]
lemma mapRightHomologyIso_hom_naturality [S₁.HasRightHomology] [S₂.HasRightHomology]
    [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂] :
    rightHomologyMap (F.mapShortComplex.map φ) ≫ (S₂.mapRightHomologyIso F).hom =
      (S₁.mapRightHomologyIso F).hom ≫ F.map (rightHomologyMap φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (F : CategoryTheory.Functor C D) [F.PreservesZeroMorphisms] [S₁.HasRightHomology] [S₂.HasRightHomology] [F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂]
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.rightHomologyMap (F.mapShortComplex.map φ))
    (S₂.mapRightHomologyIso F).hom =
  CategoryTheory.CategoryStruct.comp (S₁.mapRightHomologyIso F).hom
    (F.map (CategoryTheory.ShortComplex.rightHomologyMap φ))