ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- measurable_deriv_with_param
theorem measurable_deriv_with_param [LocallyCompactSpace 𝕜] [MeasurableSpace 𝕜]
    [OpensMeasurableSpace 𝕜] [MeasurableSpace F]
    [BorelSpace F] {f : α → 𝕜 → F} (hf : Continuous f.uncurry) :
    Measurable (fun (p : α × 𝕜) ↦ deriv (f p.1) p.2) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [CompleteSpace F] [LocallyCompactSpace 𝕜] [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜] [MeasurableSpace F] [BorelSpace F] {f : α → 𝕜 → F} (hf : Continuous (Function.uncurry f))
Goal: Measurable fun p => deriv (f p.1) p.2
-- stronglyMeasurable_deriv_with_param
theorem stronglyMeasurable_deriv_with_param [LocallyCompactSpace 𝕜] [MeasurableSpace 𝕜]
    [OpensMeasurableSpace 𝕜] [h : SecondCountableTopologyEither α F]
    {f : α → 𝕜 → F} (hf : Continuous f.uncurry) :
    StronglyMeasurable (fun (p : α × 𝕜) ↦ deriv (f p.1) p.2) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [CompleteSpace F] [LocallyCompactSpace 𝕜] [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜] [h : SecondCountableTopologyEither α F] {f : α → 𝕜 → F} (hf : Continuous (Function.uncurry f))
Goal: MeasureTheory.StronglyMeasurable fun p => deriv (f p.1) p.2
-- aemeasurable_deriv_with_param
theorem aemeasurable_deriv_with_param [LocallyCompactSpace 𝕜] [MeasurableSpace 𝕜]
    [OpensMeasurableSpace 𝕜] [MeasurableSpace F]
    [BorelSpace F] {f : α → 𝕜 → F} (hf : Continuous f.uncurry) (μ : Measure (α × 𝕜)) :
    AEMeasurable (fun (p : α × 𝕜) ↦ deriv (f p.1) p.2) μ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [CompleteSpace F] [LocallyCompactSpace 𝕜] [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜] [MeasurableSpace F] [BorelSpace F] {f : α → 𝕜 → F} (hf : Continuous (Function.uncurry f)) (μ : MeasureTheory.Measure (α × 𝕜))
Goal: AEMeasurable (fun p => deriv (f p.1) p.2) μ
-- aestronglyMeasurable_deriv_with_param
theorem aestronglyMeasurable_deriv_with_param [LocallyCompactSpace 𝕜] [MeasurableSpace 𝕜]
    [OpensMeasurableSpace 𝕜] [SecondCountableTopologyEither α F]
    {f : α → 𝕜 → F} (hf : Continuous f.uncurry) (μ : Measure (α × 𝕜)) :
    AEStronglyMeasurable (fun (p : α × 𝕜) ↦ deriv (f p.1) p.2) μ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {α : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [CompleteSpace F] [LocallyCompactSpace 𝕜] [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜] [SecondCountableTopologyEither α F] {f : α → 𝕜 → F} (hf : Continuous (Function.uncurry f)) (μ : MeasureTheory.Measure (α × 𝕜))
Goal: MeasureTheory.AEStronglyMeasurable (fun p => deriv (f p.1) p.2) μ
-- AlgebraicTopology.DoldKan.HigherFacesVanish.comp_σ
theorem HigherFacesVanish.comp_σ {Y : C} {X : SimplicialObject C} {n b q : ℕ} {φ : Y ⟶ X _[n + 1]}
    (v : HigherFacesVanish q φ) (hnbq : n + 1 = b + q) :
    HigherFacesVanish q
      (φ ≫
        X.σ ⟨b, by
          simp only [hnbq, Nat.lt_add_one_iff, le_add_iff_nonneg_right, zero_le]⟩) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {Y : C} {X : CategoryTheory.SimplicialObject C} {n b q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : AlgebraicTopology.DoldKan.HigherFacesVanish q φ) (hnbq : n + 1 = b + q)
Goal: AlgebraicTopology.DoldKan.HigherFacesVanish q (φ ≫ X.σ ⟨b, ⋯⟩)
-- AlgebraicTopology.DoldKan.σ_comp_P_eq_zero
theorem σ_comp_P_eq_zero (X : SimplicialObject C) {n q : ℕ} (i : Fin (n + 1)) (hi : n + 1 ≤ i + q) :
    X.σ i ≫ (P q).f (n + 1) = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) {n q : ℕ} (i : Fin (n + 1)) (hi : n + 1 ≤ ↑i + q)
Goal: X.σ i ≫ (AlgebraicTopology.DoldKan.P q).f (n + 1) = 0
-- AlgebraicTopology.DoldKan.σ_comp_PInfty
@[reassoc (attr := simp)]
theorem σ_comp_PInfty (X : SimplicialObject C) {n : ℕ} (i : Fin (n + 1)) :
    X.σ i ≫ PInfty.f (n + 1) = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} (i : Fin (n + 1))
Goal: X.σ i ≫ AlgebraicTopology.DoldKan.PInfty.f (n + 1) = 0
-- AlgebraicTopology.DoldKan.degeneracy_comp_PInfty
@[reassoc]
theorem degeneracy_comp_PInfty (X : SimplicialObject C) (n : ℕ) {Δ' : SimplexCategory}
    (θ : ([n] : SimplexCategory) ⟶ Δ') (hθ : ¬Mono θ) : X.map θ.op ≫ PInfty.f n = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (n : ℕ) {Δ' : SimplexCategory} (θ : [n] ⟶ Δ') (hθ : ¬CategoryTheory.Mono θ)
Goal: X.map θ.op ≫ AlgebraicTopology.DoldKan.PInfty.f n = 0
-- AlgebraicTopology.DoldKan.Γ₀NondegComplexIso_inv_f
@[simps!]
def Γ₀NondegComplexIso (K : ChainComplex C ℕ) : (Γ₀.splitting K).nondegComplex ≅ K 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (K : ChainComplex C ℕ) (i : ℕ)
Goal: (AlgebraicTopology.DoldKan.Γ₀NondegComplexIso K).inv.f i = 𝟙 ((AlgebraicTopology.DoldKan.Γ₀.splitting K).N i)
-- AlgebraicTopology.DoldKan.Γ₀NondegComplexIso_hom_f
@[simps!]
def Γ₀NondegComplexIso (K : ChainComplex C ℕ) : (Γ₀.splitting K).nondegComplex ≅ K 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (K : ChainComplex C ℕ) (i : ℕ)
Goal: (AlgebraicTopology.DoldKan.Γ₀NondegComplexIso K).hom.f i = 𝟙 ((AlgebraicTopology.DoldKan.Γ₀.splitting K).N i)
-- AlgebraicTopology.DoldKan.N₁Γ₀_app
theorem N₁Γ₀_app (K : ChainComplex C ℕ) :
    N₁Γ₀.app K = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.symm ≪≫
      (toKaroubi _).mapIso (Γ₀NondegComplexIso K) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (K : ChainComplex C ℕ)
Goal: AlgebraicTopology.DoldKan.N₁Γ₀.app K =
  (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.symm ≪≫
    (CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ)).mapIso (AlgebraicTopology.DoldKan.Γ₀NondegComplexIso K)
-- AlgebraicTopology.DoldKan.N₁Γ₀_hom_app
theorem N₁Γ₀_hom_app (K : ChainComplex C ℕ) :
    N₁Γ₀.hom.app K = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv ≫
        (toKaroubi _).map (Γ₀NondegComplexIso K).hom 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (K : ChainComplex C ℕ)
Goal: AlgebraicTopology.DoldKan.N₁Γ₀.hom.app K =
  (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv ≫
    (CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ)).map (AlgebraicTopology.DoldKan.Γ₀NondegComplexIso K).hom
-- AlgebraicTopology.DoldKan.N₁Γ₀_inv_app
theorem N₁Γ₀_inv_app (K : ChainComplex C ℕ) :
    N₁Γ₀.inv.app K = (toKaroubi _).map (Γ₀NondegComplexIso K).inv ≫
        (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (K : ChainComplex C ℕ)
Goal: AlgebraicTopology.DoldKan.N₁Γ₀.inv.app K =
  (CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ)).map (AlgebraicTopology.DoldKan.Γ₀NondegComplexIso K).inv ≫
    (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom
-- AlgebraicTopology.DoldKan.N₁Γ₀_hom_app_f_f
@[simp]
theorem N₁Γ₀_hom_app_f_f (K : ChainComplex C ℕ) (n : ℕ) :
    (N₁Γ₀.hom.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv.f.f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (K : ChainComplex C ℕ) (n : ℕ)
Goal: (AlgebraicTopology.DoldKan.N₁Γ₀.hom.app K).f.f n =
  (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv.f.f n
-- AlgebraicTopology.DoldKan.N₁Γ₀_inv_app_f_f
@[simp]
theorem N₁Γ₀_inv_app_f_f (K : ChainComplex C ℕ) (n : ℕ) :
    (N₁Γ₀.inv.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom.f.f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (K : ChainComplex C ℕ) (n : ℕ)
Goal: (AlgebraicTopology.DoldKan.N₁Γ₀.inv.app K).f.f n =
  (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom.f.f n
-- AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso_hom_app
@[simp]
lemma N₂Γ₂ToKaroubiIso_hom_app (X : ChainComplex C ℕ) :
    (N₂Γ₂ToKaroubiIso.hom.app X).f = PInfty 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : ChainComplex C ℕ)
Goal: (AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso.hom.app X).f = AlgebraicTopology.DoldKan.PInfty
-- AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso_inv_app
@[simp]
lemma N₂Γ₂ToKaroubiIso_inv_app (X : ChainComplex C ℕ) :
    (N₂Γ₂ToKaroubiIso.inv.app X).f = PInfty 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : ChainComplex C ℕ)
Goal: (AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso.inv.app X).f = AlgebraicTopology.DoldKan.PInfty
-- AlgebraicTopology.DoldKan.N₂Γ₂_inv_app_f_f
@[simp]
theorem N₂Γ₂_inv_app_f_f (X : Karoubi (ChainComplex C ℕ)) (n : ℕ) :
    (N₂Γ₂.inv.app X).f.f n =
      X.p.f n ≫ ((Γ₀.splitting X.X).cofan _).inj (Splitting.IndexSet.id (op [n])) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : CategoryTheory.Idempotents.Karoubi (ChainComplex C ℕ)) (n : ℕ)
Goal: (AlgebraicTopology.DoldKan.N₂Γ₂.inv.app X).f.f n =
  X.p.f n ≫
    ((AlgebraicTopology.DoldKan.Γ₀.splitting X.X).cofan (Opposite.op [n])).inj
      (SimplicialObject.Splitting.IndexSet.id (Opposite.op [n]))
-- AlgebraicTopology.DoldKan.whiskerLeft_toKaroubi_N₂Γ₂_hom
lemma whiskerLeft_toKaroubi_N₂Γ₂_hom :
    whiskerLeft (toKaroubi (ChainComplex C ℕ)) N₂Γ₂.hom = N₂Γ₂ToKaroubiIso.hom ≫ N₁Γ₀.hom 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: CategoryTheory.whiskerLeft (CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ))
    AlgebraicTopology.DoldKan.N₂Γ₂.hom =
  AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso.hom ≫ AlgebraicTopology.DoldKan.N₁Γ₀.hom
-- AlgebraicTopology.DoldKan.N₂Γ₂_compatible_with_N₁Γ₀
theorem N₂Γ₂_compatible_with_N₁Γ₀ (K : ChainComplex C ℕ) :
    N₂Γ₂.hom.app ((toKaroubi _).obj K) = N₂Γ₂ToKaroubiIso.hom.app K ≫ N₁Γ₀.hom.app K 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (K : ChainComplex C ℕ)
Goal: AlgebraicTopology.DoldKan.N₂Γ₂.hom.app ((CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ)).obj K) =
  AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso.hom.app K ≫ AlgebraicTopology.DoldKan.N₁Γ₀.hom.app K
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₀_unitIso_hom_app
@[simps! functor inverse unitIso_hom_app]
def equivalence₀ : A ≌ B' 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_4} B'] (eA : A ≌ A') (e' : A' ≌ B') (X : A)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₀ eA e').unitIso.hom.app X =
  eA.unitIso.hom.app X ≫ eA.inverse.map (e'.unitIso.hom.app (eA.functor.obj X))
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₀_functor
@[simps! functor inverse unitIso_hom_app]
def equivalence₀ : A ≌ B' 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_4} B'] (eA : A ≌ A') (e' : A' ≌ B')
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₀ eA e').functor = eA.functor ⋙ e'.functor
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₀_inverse
@[simps! functor inverse unitIso_hom_app]
def equivalence₀ : A ≌ B' 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_4} B'] (eA : A ≌ A') (e' : A' ≌ B')
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₀ eA e').inverse = e'.inverse ⋙ eA.inverse
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₁_functor
@[simps! functor]
def equivalence₁ : A ≌ B' 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_4} B'] {eA : A ≌ A'} {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₁ hF).functor = F
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₁_inverse
theorem equivalence₁_inverse : (equivalence₁ hF).inverse = e'.inverse ⋙ eA.inverse 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_7, u_2} A'] [CategoryTheory.Category.{u_6, u_4} B'] {eA : A ≌ A'} {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₁ hF).inverse = e'.inverse ⋙ eA.inverse
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso_inv_app
@[simps!]
def equivalence₁CounitIso : (e'.inverse ⋙ eA.inverse) ⋙ F ≅ 𝟭 B' 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_4} B'] {eA : A ≌ A'} {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) (X : B')
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso hF).inv.app X =
  e'.counitIso.inv.app X ≫
    e'.functor.map (eA.counitIso.inv.app (e'.inverse.obj X)) ≫ hF.hom.app (eA.inverse.obj (e'.inverse.obj X))
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso_hom_app
@[simps!]
def equivalence₁CounitIso : (e'.inverse ⋙ eA.inverse) ⋙ F ≅ 𝟭 B' 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_4} B'] {eA : A ≌ A'} {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) (X : B')
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso hF).hom.app X =
  hF.inv.app (eA.inverse.obj (e'.inverse.obj X)) ≫
    e'.functor.map (eA.counitIso.hom.app (e'.inverse.obj X)) ≫ e'.counitIso.hom.app X
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso_eq
theorem equivalence₁CounitIso_eq : (equivalence₁ hF).counitIso = equivalence₁CounitIso hF 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_6, u_1} A] [CategoryTheory.Category.{u_7, u_2} A'] [CategoryTheory.Category.{u_5, u_4} B'] {eA : A ≌ A'} {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₁ hF).counitIso =
  AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso hF
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso_inv_app
@[simps!]
def equivalence₁UnitIso : 𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_4} B'] {eA : A ≌ A'} {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) (X : A)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso hF).inv.app X =
  eA.inverse.map (e'.inverse.map (hF.inv.app X)) ≫
    eA.inverse.map (e'.unitIso.inv.app (eA.functor.obj X)) ≫ eA.unitIso.inv.app X
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso_hom_app
@[simps!]
def equivalence₁UnitIso : 𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_4} B'] {eA : A ≌ A'} {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) (X : A)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso hF).hom.app X =
  eA.unitIso.hom.app X ≫
    eA.inverse.map (e'.unitIso.hom.app (eA.functor.obj X)) ≫ eA.inverse.map (e'.inverse.map (hF.hom.app X))
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso_eq
theorem equivalence₁UnitIso_eq : (equivalence₁ hF).unitIso = equivalence₁UnitIso hF 
Arguments: {A : Type u_1} {A' : Type u_2} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_7, u_2} A'] [CategoryTheory.Category.{u_6, u_4} B'] {eA : A ≌ A'} {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₁ hF).unitIso =
  AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso hF
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₂_functor
@[simps! functor]
def equivalence₂ : A ≌ B 
Arguments: {A : Type u_1} {A' : Type u_2} {B : Type u_3} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_3} B] [CategoryTheory.Category.{u_8, u_4} B'] {eA : A ≌ A'} (eB : B ≌ B') {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₂ eB hF).functor = F ⋙ eB.inverse
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₂_inverse
theorem equivalence₂_inverse :
    (equivalence₂ eB hF).inverse = eB.functor ⋙ e'.inverse ⋙ eA.inverse 
Arguments: {A : Type u_1} {A' : Type u_2} {B : Type u_3} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_7, u_2} A'] [CategoryTheory.Category.{u_6, u_3} B] [CategoryTheory.Category.{u_8, u_4} B'] {eA : A ≌ A'} (eB : B ≌ B') {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₂ eB hF).inverse = eB.functor ⋙ e'.inverse ⋙ eA.inverse
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso_hom_app
@[simps!]
def equivalence₂CounitIso : (eB.functor ⋙ e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse ≅ 𝟭 B 
Arguments: {A : Type u_1} {A' : Type u_2} {B : Type u_3} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_3} B] [CategoryTheory.Category.{u_8, u_4} B'] {eA : A ≌ A'} (eB : B ≌ B') {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) (X : B)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso eB hF).hom.app X =
  eB.inverse.map (hF.inv.app (eA.inverse.obj (e'.inverse.obj (eB.functor.obj X)))) ≫
    eB.inverse.map (e'.functor.map (eA.counitIso.hom.app (e'.inverse.obj (eB.functor.obj X)))) ≫
      eB.inverse.map (e'.counitIso.hom.app (eB.functor.obj X)) ≫ eB.unitIso.inv.app X
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso_inv_app
@[simps!]
def equivalence₂CounitIso : (eB.functor ⋙ e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse ≅ 𝟭 B 
Arguments: {A : Type u_1} {A' : Type u_2} {B : Type u_3} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_3} B] [CategoryTheory.Category.{u_8, u_4} B'] {eA : A ≌ A'} (eB : B ≌ B') {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) (X : B)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso eB hF).inv.app X =
  eB.unitIso.hom.app X ≫
    eB.inverse.map (e'.counitIso.inv.app (eB.functor.obj X)) ≫
      eB.inverse.map (e'.functor.map (eA.counitIso.inv.app (e'.inverse.obj (eB.functor.obj X)))) ≫
        eB.inverse.map (hF.hom.app (eA.inverse.obj (e'.inverse.obj (eB.functor.obj X))))
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso_eq
theorem equivalence₂CounitIso_eq :
    (equivalence₂ eB hF).counitIso = equivalence₂CounitIso eB hF 
Arguments: {A : Type u_1} {A' : Type u_2} {B : Type u_3} {B' : Type u_4} [CategoryTheory.Category.{u_6, u_1} A] [CategoryTheory.Category.{u_7, u_2} A'] [CategoryTheory.Category.{u_5, u_3} B] [CategoryTheory.Category.{u_8, u_4} B'] {eA : A ≌ A'} (eB : B ≌ B') {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₂ eB hF).counitIso =
  AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso eB hF
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₂UnitIso_hom_app
@[simps!]
def equivalence₂UnitIso : 𝟭 A ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse 
Arguments: {A : Type u_1} {A' : Type u_2} {B : Type u_3} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_3} B] [CategoryTheory.Category.{u_8, u_4} B'] {eA : A ≌ A'} (eB : B ≌ B') {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) (X : A)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₂UnitIso eB hF).hom.app X =
  eA.unitIso.hom.app X ≫
    eA.inverse.map (e'.unitIso.hom.app (eA.functor.obj X)) ≫
      eA.inverse.map (e'.inverse.map (hF.hom.app X)) ≫ eA.inverse.map (e'.inverse.map (eB.counitIso.inv.app (F.obj X)))
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₂UnitIso_inv_app
@[simps!]
def equivalence₂UnitIso : 𝟭 A ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse 
Arguments: {A : Type u_1} {A' : Type u_2} {B : Type u_3} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_3} B] [CategoryTheory.Category.{u_8, u_4} B'] {eA : A ≌ A'} (eB : B ≌ B') {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) (X : A)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₂UnitIso eB hF).inv.app X =
  eA.inverse.map (e'.inverse.map (eB.counitIso.hom.app (F.obj X))) ≫
    eA.inverse.map (e'.inverse.map (hF.inv.app X)) ≫
      eA.inverse.map (e'.unitIso.inv.app (eA.functor.obj X)) ≫ eA.unitIso.inv.app X
-- AlgebraicTopology.DoldKan.Compatibility.equivalence₂UnitIso_eq
theorem equivalence₂UnitIso_eq : (equivalence₂ eB hF).unitIso = equivalence₂UnitIso eB hF 
Arguments: {A : Type u_1} {A' : Type u_2} {B : Type u_3} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_7, u_2} A'] [CategoryTheory.Category.{u_6, u_3} B] [CategoryTheory.Category.{u_8, u_4} B'] {eA : A ≌ A'} (eB : B ≌ B') {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence₂ eB hF).unitIso =
  AlgebraicTopology.DoldKan.Compatibility.equivalence₂UnitIso eB hF
-- AlgebraicTopology.DoldKan.Compatibility.equivalence_inverse
@[simps! inverse]
def equivalence : A ≌ B 
Arguments: {A : Type u_1} {A' : Type u_2} {B : Type u_3} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_6, u_2} A'] [CategoryTheory.Category.{u_7, u_3} B] [CategoryTheory.Category.{u_8, u_4} B'] {eA : A ≌ A'} {eB : B ≌ B'} {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) {G : B ⥤ A} (hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence hF hG).inverse = G
-- AlgebraicTopology.DoldKan.Compatibility.equivalence_functor
theorem equivalence_functor : (equivalence hF hG).functor = F ⋙ eB.inverse 
Arguments: {A : Type u_1} {A' : Type u_2} {B : Type u_3} {B' : Type u_4} [CategoryTheory.Category.{u_5, u_1} A] [CategoryTheory.Category.{u_7, u_2} A'] [CategoryTheory.Category.{u_6, u_3} B] [CategoryTheory.Category.{u_8, u_4} B'] {eA : A ≌ A'} {eB : B ≌ B'} {e' : A' ≌ B'} {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) {G : B ⥤ A} (hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor)
Goal: (AlgebraicTopology.DoldKan.Compatibility.equivalence hF hG).functor = F ⋙ eB.inverse