ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.coalgebraEquivOver_inverse
@[simps]
def coalgebraEquivOver : Coalgebra (prodComonad X) ≌ Over X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryProducts C]
Goal: (CategoryTheory.coalgebraEquivOver X).inverse = CategoryTheory.overToCoalgebra X
-- CategoryTheory.coprodMonad_η_app
@[simps!]
def coprodMonad : Monad C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (Y : C)
Goal: (CategoryTheory.coprodMonad X).η.app Y = CategoryTheory.Limits.coprod.inr
-- CategoryTheory.coprodMonad_μ_app
@[simps!]
def coprodMonad : Monad C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (Y : C)
Goal: (CategoryTheory.coprodMonad X).μ.app Y =
  CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.id (X ⨿ Y))
-- CategoryTheory.coprodMonad_obj
@[simps!]
def coprodMonad : Monad C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (Y : C)
Goal: (CategoryTheory.coprodMonad X).obj Y = (X ⨿ Y)
-- CategoryTheory.coprodMonad_map
@[simps!]
def coprodMonad : Monad C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] {Y Z : C} (g : Y ⟶ Z)
Goal: (CategoryTheory.coprodMonad X).map g = CategoryTheory.Limits.coprod.map (CategoryTheory.CategoryStruct.id X) g
-- CategoryTheory.algebraToUnder_obj
@[simps]
def algebraToUnder : Monad.Algebra (coprodMonad X) ⥤ Under X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (A : (CategoryTheory.coprodMonad X).Algebra)
Goal: (CategoryTheory.algebraToUnder X).obj A =
  CategoryTheory.Under.mk (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl A.a)
-- CategoryTheory.algebraToUnder_map
@[simps]
def algebraToUnder : Monad.Algebra (coprodMonad X) ⥤ Under X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C]
Goal: ∀ {X_1 Y : (CategoryTheory.coprodMonad X).Algebra} (f : X_1 ⟶ Y),
  (CategoryTheory.algebraToUnder X).map f = CategoryTheory.Under.homMk f.f ⋯
-- CategoryTheory.underToAlgebra_map_f
@[simps]
def underToAlgebra : Under X ⥤ Monad.Algebra (coprodMonad X) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C]
Goal: ∀ {X_1 Y : CategoryTheory.Under X} (g : X_1 ⟶ Y), ((CategoryTheory.underToAlgebra X).map g).f = g.right
-- CategoryTheory.underToAlgebra_obj_a
@[simps]
def underToAlgebra : Under X ⥤ Monad.Algebra (coprodMonad X) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (f : CategoryTheory.Under X)
Goal: ((CategoryTheory.underToAlgebra X).obj f).a =
  CategoryTheory.Limits.coprod.desc f.hom (CategoryTheory.CategoryStruct.id f.right)
-- CategoryTheory.underToAlgebra_obj_A
@[simps]
def underToAlgebra : Under X ⥤ Monad.Algebra (coprodMonad X) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (f : CategoryTheory.Under X)
Goal: ((CategoryTheory.underToAlgebra X).obj f).A = f.right
-- CategoryTheory.algebraEquivUnder_functor
@[simps]
def algebraEquivUnder : Monad.Algebra (coprodMonad X) ≌ Under X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C]
Goal: (CategoryTheory.algebraEquivUnder X).functor = CategoryTheory.algebraToUnder X
-- CategoryTheory.algebraEquivUnder_unitIso
@[simps]
def algebraEquivUnder : Monad.Algebra (coprodMonad X) ≌ Under X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C]
Goal: (CategoryTheory.algebraEquivUnder X).unitIso =
  CategoryTheory.NatIso.ofComponents
    (fun A =>
      CategoryTheory.Monad.Algebra.isoMk
        (CategoryTheory.Iso.refl ((CategoryTheory.Functor.id (CategoryTheory.coprodMonad X).Algebra).obj A).A) ⋯)
    ⋯
-- CategoryTheory.algebraEquivUnder_counitIso
@[simps]
def algebraEquivUnder : Monad.Algebra (coprodMonad X) ≌ Under X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C]
Goal: (CategoryTheory.algebraEquivUnder X).counitIso =
  CategoryTheory.NatIso.ofComponents
    (fun f =>
      CategoryTheory.Under.isoMk
        (CategoryTheory.Iso.refl
          (((CategoryTheory.underToAlgebra X).comp (CategoryTheory.algebraToUnder X)).obj f).right)
        ⋯)
    ⋯
-- CategoryTheory.algebraEquivUnder_inverse
@[simps]
def algebraEquivUnder : Monad.Algebra (coprodMonad X) ≌ Under X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C]
Goal: (CategoryTheory.algebraEquivUnder X).inverse = CategoryTheory.underToAlgebra X
-- CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_comp_d
@[simp, reassoc]
theorem p_comp_d : P.p.f n ≫ f.f.f n = f.f.f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {P Q : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)} (f : P ⟶ Q) (n : ι)
Goal: CategoryTheory.CategoryStruct.comp (P.p.f n) (f.f.f n) = f.f.f n
-- CategoryTheory.Idempotents.Karoubi.HomologicalComplex.comp_p_d
@[simp, reassoc]
theorem comp_p_d : f.f.f n ≫ Q.p.f n = f.f.f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {P Q : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)} (f : P ⟶ Q) (n : ι)
Goal: CategoryTheory.CategoryStruct.comp (f.f.f n) (Q.p.f n) = f.f.f n
-- CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_comm_f
@[reassoc]
theorem p_comm_f : P.p.f n ≫ f.f.f n = f.f.f n ≫ Q.p.f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {P Q : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)} (f : P ⟶ Q) (n : ι)
Goal: CategoryTheory.CategoryStruct.comp (P.p.f n) (f.f.f n) = CategoryTheory.CategoryStruct.comp (f.f.f n) (Q.p.f n)
-- CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_idem
@[simp, reassoc]
theorem p_idem : P.p.f n ≫ P.p.f n = P.p.f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)) (n : ι)
Goal: CategoryTheory.CategoryStruct.comp (P.p.f n) (P.p.f n) = P.p.f n
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj_d_f
@[simps]
def obj (P : Karoubi (HomologicalComplex C c)) : HomologicalComplex (Karoubi C) c 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)) (i j : ι)
Goal: ((CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj P).d i j).f =
  CategoryTheory.CategoryStruct.comp (P.p.f i) (P.X.d i j)
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj_X_p
@[simps]
def obj (P : Karoubi (HomologicalComplex C c)) : HomologicalComplex (Karoubi C) c 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)) (n : ι)
Goal: ((CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj P).X n).p = P.p.f n
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj_X_X
@[simps]
def obj (P : Karoubi (HomologicalComplex C c)) : HomologicalComplex (Karoubi C) c 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)) (n : ι)
Goal: ((CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj P).X n).X = P.X.X n
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.map_f_f
@[simps]
def map {P Q : Karoubi (HomologicalComplex C c)} (f : P ⟶ Q) : obj P ⟶ obj Q 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {P Q : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)} (f : P ⟶ Q) (n : ι)
Goal: ((CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.map f).f n).f = f.f.f n
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor_map
@[simps]
def functor : Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι}
Goal: ∀ {X Y : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)} (f : X ⟶ Y),
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor.map f =
    CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.map f
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor_obj
@[simps]
def functor : Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c))
Goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor.obj P =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj P
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj_p_f
@[simps]
def obj (K : HomologicalComplex (Karoubi C) c) : Karoubi (HomologicalComplex C c) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c) (n : ι)
Goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj K).p.f n = (K.X n).p
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj_X_X
@[simps]
def obj (K : HomologicalComplex (Karoubi C) c) : Karoubi (HomologicalComplex C c) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c) (n : ι)
Goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj K).X.X n = (K.X n).X
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj_X_d
@[simps]
def obj (K : HomologicalComplex (Karoubi C) c) : Karoubi (HomologicalComplex C c) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c) (i j : ι)
Goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj K).X.d i j = (K.d i j).f
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.map_f_f
@[simps]
def map {K L : HomologicalComplex (Karoubi C) c} (f : K ⟶ L) : obj K ⟶ obj L 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {K L : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c} (f : K ⟶ L) (n : ι)
Goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.map f).f.f n = (f.f n).f
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse_map
@[simps]
def inverse : HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι}
Goal: ∀ {X Y : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c} (f : X ⟶ Y),
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse.map f =
    CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.map f
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse_obj
@[simps]
def inverse : HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c)
Goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse.obj K =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj K
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso_hom
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (HomologicalComplex (Karoubi C) c) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι}
Goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso.hom = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso_inv
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (HomologicalComplex (Karoubi C) c) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι}
Goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso.inv = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso_hom_app_f_f
@[simps]
def unitIso : 𝟭 (Karoubi (HomologicalComplex C c)) ≅ functor ⋙ inverse 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)) (n : ι)
Goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso.hom.app P).f.f n = P.p.f n
-- CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso_inv_app_f_f
@[simps]
def unitIso : 𝟭 (Karoubi (HomologicalComplex C c)) ≅ functor ⋙ inverse 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)) (n : ι)
Goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso.inv.app P).f.f n = P.p.f n
-- CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_counitIso
@[simps]
def karoubiHomologicalComplexEquivalence :
    Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} (c : ComplexShape ι)
Goal: (CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence C c).counitIso =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso
-- CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_unitIso
@[simps]
def karoubiHomologicalComplexEquivalence :
    Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} (c : ComplexShape ι)
Goal: (CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence C c).unitIso =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso
-- CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_functor
@[simps]
def karoubiHomologicalComplexEquivalence :
    Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} (c : ComplexShape ι)
Goal: (CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence C c).functor =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor
-- CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_inverse
@[simps]
def karoubiHomologicalComplexEquivalence :
    Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} (c : ComplexShape ι)
Goal: (CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence C c).inverse =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_obj_X_X
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α] (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C (ComplexShape.down α))) (n : α)
Goal: (((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).functor.obj P).X n).X = P.X.X n
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_map_f_f
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α]
Goal: ∀ {X Y : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) (ComplexShape.down α)} (f : X ⟶ Y) (n : α),
  ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).inverse.map f).f.f n = (f.f n).f
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_map_f_f
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α]
Goal: ∀ {X Y : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C (ComplexShape.down α))} (f : X ⟶ Y) (n : α),
  (((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).functor.map f).f n).f = f.f.f n
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_obj_p_f
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α] (K : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) (ComplexShape.down α)) (n : α)
Goal: ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).inverse.obj K).p.f n = (K.X n).p
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_obj_X_d
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α] (K : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) (ComplexShape.down α)) (i j : α)
Goal: ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).inverse.obj K).X.d i j = (K.d i j).f
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_unitIso_hom_app_f_f
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α] (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C (ComplexShape.down α))) (n : α)
Goal: ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).unitIso.hom.app P).f.f n = P.p.f n
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_obj_X_p
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α] (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C (ComplexShape.down α))) (n : α)
Goal: (((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).functor.obj P).X n).p = P.p.f n
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_unitIso_inv_app_f_f
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α] (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C (ComplexShape.down α))) (n : α)
Goal: ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).unitIso.inv.app P).f.f n = P.p.f n
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_obj_d_f
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α] (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C (ComplexShape.down α))) (i j : α)
Goal: (((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).functor.obj P).d i j).f =
  CategoryTheory.CategoryStruct.comp (P.X.d i j) (P.p.f j)
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_counitIso_hom
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α]
Goal: (CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).counitIso.hom = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_obj_X_X
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α] (K : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) (ComplexShape.down α)) (n : α)
Goal: ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).inverse.obj K).X.X n = (K.X n).X
-- CategoryTheory.Idempotents.karoubiChainComplexEquivalence_counitIso_inv
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α]
Goal: (CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).counitIso.inv = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_map_f_f
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α]
Goal: ∀ {X Y : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) (ComplexShape.up α)} (f : X ⟶ Y) (n : α),
  ((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).inverse.map f).f.f n = (f.f n).f
-- CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_counitIso_inv
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α]
Goal: (CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).counitIso.inv = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_obj_p_f
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] (α : Type u_3) [AddRightCancelSemigroup α] [One α] (K : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) (ComplexShape.up α)) (n : α)
Goal: ((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).inverse.obj K).p.f n = (K.X n).p