ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgebraicTopology.normalizedMooreComplex_objD
theorem normalizedMooreComplex_objD (X : SimplicialObject C) (n : ℕ) :
    ((normalizedMooreComplex C).obj X).d (n + 1) n = NormalizedMooreComplex.objD X n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Abelian C] (X : CategoryTheory.SimplicialObject C) (n : ℕ)
Goal: ((AlgebraicTopology.normalizedMooreComplex C).obj X).d (n + 1) n = AlgebraicTopology.NormalizedMooreComplex.objD X n
-- ContinuousMap.Homotopy.hcast_def
@[simp]
theorem hcast_def {X : TopCat} {x₀ x₁ : X} (hx₀ : x₀ = x₁) :
    hcast hx₀ = eqToHom (FundamentalGroupoid.ext _ _ hx₀) 
Arguments: {X : TopCat} {x₀ x₁ : ↑X} (hx₀ : x₀ = x₁)
Goal: ContinuousMap.Homotopy.hcast hx₀ = CategoryTheory.eqToHom ⋯
-- ContinuousMap.Homotopy.heq_path_of_eq_image
theorem heq_path_of_eq_image : HEq ((πₘ f).map ⟦p⟧) ((πₘ g).map ⟦q⟧) 
Arguments: {X₁ X₂ Y : TopCat} {f : C(↑X₁, ↑Y)} {g : C(↑X₂, ↑Y)} {x₀ x₁ : ↑X₁} {x₂ x₃ : ↑X₂} {p : Path x₀ x₁} {q : Path x₂ x₃} (hfg : ∀ (t : ↑I), f (p t) = g (q t))
Goal: HEq ((π.map f).map ⟦p⟧) ((π.map g).map ⟦q⟧)
-- _private.Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps.0.ContinuousMap.Homotopy.start_path
private theorem start_path : f x₀ = g x₂ 
Arguments: {X₁ X₂ Y : TopCat} {f : C(↑X₁, ↑Y)} {g : C(↑X₂, ↑Y)} {x₀ x₁ : ↑X₁} {x₂ x₃ : ↑X₂} {p : Path x₀ x₁} {q : Path x₂ x₃} (hfg : ∀ (t : ↑I), f (p t) = g (q t))
Goal: f x₀ = g x₂
-- _private.Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps.0.ContinuousMap.Homotopy.end_path
private theorem end_path : f x₁ = g x₃ 
Arguments: {X₁ X₂ Y : TopCat} {f : C(↑X₁, ↑Y)} {g : C(↑X₂, ↑Y)} {x₀ x₁ : ↑X₁} {x₂ x₃ : ↑X₂} {p : Path x₀ x₁} {q : Path x₂ x₃} (hfg : ∀ (t : ↑I), f (p t) = g (q t))
Goal: f x₁ = g x₃
-- ContinuousMap.Homotopy.eq_path_of_eq_image
theorem eq_path_of_eq_image :
    (πₘ f).map ⟦p⟧ = hcast (start_path hfg) ≫ (πₘ g).map ⟦q⟧ ≫ hcast (end_path hfg).symm 
Arguments: {X₁ X₂ Y : TopCat} {f : C(↑X₁, ↑Y)} {g : C(↑X₂, ↑Y)} {x₀ x₁ : ↑X₁} {x₂ x₃ : ↑X₂} {p : Path x₀ x₁} {q : Path x₂ x₃} (hfg : ∀ (t : ↑I), f (p t) = g (q t))
Goal: (π.map f).map ⟦p⟧ = ContinuousMap.Homotopy.hcast ⋯ ≫ (π.map g).map ⟦q⟧ ≫ ContinuousMap.Homotopy.hcast ⋯
-- ContinuousMap.Homotopy.ulift_apply
@[simp, nolint simpNF]
theorem ulift_apply (i : ULift.{u} I) (x : X) : H.uliftMap (i, x) = H (i.down, x) 
Arguments: {X Y : TopCat} {f g : C(↑X, ↑Y)} (H : f.Homotopy g) (i : ULift.{u, 0} ↑I) (x : ↑X)
Goal: H.uliftMap (i, x) = H (i.down, x)
-- ContinuousMap.Homotopy.apply_zero_path
theorem apply_zero_path : (πₘ f).map p = hcast (H.apply_zero x₀).symm ≫
    (πₘ H.uliftMap).map (prodToProdTopI (𝟙 (@fromTop (TopCat.of _) (ULift.up 0))) p) ≫
    hcast (H.apply_zero x₁) 
Arguments: {X Y : TopCat} {f g : C(↑X, ↑Y)} (H : f.Homotopy g) {x₀ x₁ : ↑X} (p : FundamentalGroupoid.fromTop x₀ ⟶ FundamentalGroupoid.fromTop x₁)
Goal: (π.map f).map p =
  ContinuousMap.Homotopy.hcast ⋯ ≫
    (π.map H.uliftMap).map (ContinuousMap.Homotopy.prodToProdTopI (𝟙 (FundamentalGroupoid.fromTop { down := 0 })) p) ≫
      ContinuousMap.Homotopy.hcast ⋯
-- ContinuousMap.Homotopy.apply_one_path
theorem apply_one_path : (πₘ g).map p = hcast (H.apply_one x₀).symm ≫
    (πₘ H.uliftMap).map (prodToProdTopI (𝟙 (@fromTop (TopCat.of _) (ULift.up 1))) p) ≫
    hcast (H.apply_one x₁) 
Arguments: {X Y : TopCat} {f g : C(↑X, ↑Y)} (H : f.Homotopy g) {x₀ x₁ : ↑X} (p : FundamentalGroupoid.fromTop x₀ ⟶ FundamentalGroupoid.fromTop x₁)
Goal: (π.map g).map p =
  ContinuousMap.Homotopy.hcast ⋯ ≫
    (π.map H.uliftMap).map (ContinuousMap.Homotopy.prodToProdTopI (𝟙 (FundamentalGroupoid.fromTop { down := 1 })) p) ≫
      ContinuousMap.Homotopy.hcast ⋯
-- ContinuousMap.Homotopy.evalAt_eq
theorem evalAt_eq (x : X) : ⟦H.evalAt x⟧ = hcast (H.apply_zero x).symm ≫
    (πₘ H.uliftMap).map (prodToProdTopI uhpath01 (𝟙 (fromTop x))) ≫
      hcast (H.apply_one x).symm.symm 
Arguments: {X Y : TopCat} {f g : C(↑X, ↑Y)} (H : f.Homotopy g) (x : ↑X)
Goal: ⟦H.evalAt x⟧ =
  ContinuousMap.Homotopy.hcast ⋯ ≫
    (π.map H.uliftMap).map
        (ContinuousMap.Homotopy.prodToProdTopI unitInterval.uhpath01 (𝟙 (FundamentalGroupoid.fromTop x))) ≫
      ContinuousMap.Homotopy.hcast ⋯
-- ContinuousMap.Homotopy.eq_diag_path
theorem eq_diag_path : (πₘ f).map p ≫ ⟦H.evalAt x₁⟧ = H.diagonalPath' p ∧
    (⟦H.evalAt x₀⟧ ≫ (πₘ g).map p : fromTop (f x₀) ⟶ fromTop (g x₁)) = H.diagonalPath' p 
Arguments: {X Y : TopCat} {f g : C(↑X, ↑Y)} (H : f.Homotopy g) {x₀ x₁ : ↑X} (p : FundamentalGroupoid.fromTop x₀ ⟶ FundamentalGroupoid.fromTop x₁)
Goal: (π.map f).map p ≫ ⟦H.evalAt x₁⟧ = H.diagonalPath' p ∧ ⟦H.evalAt x₀⟧ ≫ (π.map g).map p = H.diagonalPath' p
-- AlgebraicTopology.DoldKan.homotopyPToId_eventually_constant
theorem homotopyPToId_eventually_constant {q n : ℕ} (hqn : n < q) :
    ((homotopyPToId X (q + 1)).hom n (n + 1) : X _[n] ⟶ X _[n + 1]) =
      (homotopyPToId X q).hom n (n + 1) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) {q n : ℕ} (hqn : n < q)
Goal: (AlgebraicTopology.DoldKan.homotopyPToId X (q + 1)).hom n (n + 1) =
  (AlgebraicTopology.DoldKan.homotopyPToId X q).hom n (n + 1)
-- AlgebraicTopology.DoldKan.homotopyPInftyToId_hom
@[simps]
def homotopyPInftyToId : Homotopy (PInfty : K[X] ⟶ _) (𝟙 _) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (i j : ℕ)
Goal: (AlgebraicTopology.DoldKan.homotopyPInftyToId X).hom i j = (AlgebraicTopology.DoldKan.homotopyPToId X (j + 1)).hom i j
-- AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_homotopyHomInvId
@[simps]
def homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex {A : Type*} [Category A]
    [Abelian A] {Y : SimplicialObject A} :
    HomotopyEquiv ((normalizedMooreComplex A).obj Y) ((alternatingFaceMapComplex A).obj Y) 
Arguments: {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] {Y : CategoryTheory.SimplicialObject A}
Goal: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.homotopyHomInvId =
  Homotopy.ofEq ⋯
-- AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_inv
@[simps]
def homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex {A : Type*} [Category A]
    [Abelian A] {Y : SimplicialObject A} :
    HomotopyEquiv ((normalizedMooreComplex A).obj Y) ((alternatingFaceMapComplex A).obj Y) 
Arguments: {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] {Y : CategoryTheory.SimplicialObject A}
Goal: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.inv =
  AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex Y
-- AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_hom
@[simps]
def homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex {A : Type*} [Category A]
    [Abelian A] {Y : SimplicialObject A} :
    HomotopyEquiv ((normalizedMooreComplex A).obj Y) ((alternatingFaceMapComplex A).obj Y) 
Arguments: {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] {Y : CategoryTheory.SimplicialObject A}
Goal: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.hom =
  AlgebraicTopology.inclusionOfMooreComplexMap Y
-- AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_homotopyInvHomId
@[simps]
def homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex {A : Type*} [Category A]
    [Abelian A] {Y : SimplicialObject A} :
    HomotopyEquiv ((normalizedMooreComplex A).obj Y) ((alternatingFaceMapComplex A).obj Y) 
Arguments: {A : Type u_2} [CategoryTheory.Category.{u_3, u_2} A] [CategoryTheory.Abelian A] {Y : CategoryTheory.SimplicialObject A}
Goal: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.homotopyInvHomId =
  (Homotopy.ofEq ⋯).trans (AlgebraicTopology.DoldKan.homotopyPInftyToId Y)
-- AlgebraicTopology.DoldKan.HigherFacesVanish.inclusionOfMooreComplexMap
theorem HigherFacesVanish.inclusionOfMooreComplexMap (n : ℕ) :
    HigherFacesVanish (n + 1) ((inclusionOfMooreComplexMap X).f (n + 1)) 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] {X : CategoryTheory.SimplicialObject A} (n : ℕ)
Goal: AlgebraicTopology.DoldKan.HigherFacesVanish (n + 1) ((AlgebraicTopology.inclusionOfMooreComplexMap X).f (n + 1))
-- AlgebraicTopology.DoldKan.factors_normalizedMooreComplex_PInfty
theorem factors_normalizedMooreComplex_PInfty (n : ℕ) :
    Subobject.Factors (NormalizedMooreComplex.objX X n) (PInfty.f n) 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] {X : CategoryTheory.SimplicialObject A} (n : ℕ)
Goal: (AlgebraicTopology.NormalizedMooreComplex.objX X n).Factors (AlgebraicTopology.DoldKan.PInfty.f n)
-- AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_f
@[simps!]
def PInftyToNormalizedMooreComplex (X : SimplicialObject A) : K[X] ⟶ N[X] 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] (X : CategoryTheory.SimplicialObject A) (n : ℕ)
Goal: (AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex X).f n =
  (AlgebraicTopology.NormalizedMooreComplex.objX X n).factorThru (AlgebraicTopology.DoldKan.PInfty.f n) ⋯
-- AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap
@[reassoc (attr := simp)]
theorem PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap (X : SimplicialObject A) :
    PInftyToNormalizedMooreComplex X ≫ inclusionOfMooreComplexMap X = PInfty 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] (X : CategoryTheory.SimplicialObject A)
Goal: AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex X ≫ AlgebraicTopology.inclusionOfMooreComplexMap X =
  AlgebraicTopology.DoldKan.PInfty
-- AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_naturality
@[reassoc (attr := simp)]
theorem PInftyToNormalizedMooreComplex_naturality {X Y : SimplicialObject A} (f : X ⟶ Y) :
    AlternatingFaceMapComplex.map f ≫ PInftyToNormalizedMooreComplex Y =
      PInftyToNormalizedMooreComplex X ≫ NormalizedMooreComplex.map f 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] {X Y : CategoryTheory.SimplicialObject A} (f : X ⟶ Y)
Goal: AlgebraicTopology.AlternatingFaceMapComplex.map f ≫ AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex Y =
  AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex X ≫ AlgebraicTopology.NormalizedMooreComplex.map f
-- AlgebraicTopology.DoldKan.PInfty_comp_PInftyToNormalizedMooreComplex
@[reassoc (attr := simp)]
theorem PInfty_comp_PInftyToNormalizedMooreComplex (X : SimplicialObject A) :
    PInfty ≫ PInftyToNormalizedMooreComplex X = PInftyToNormalizedMooreComplex X 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] (X : CategoryTheory.SimplicialObject A)
Goal: AlgebraicTopology.DoldKan.PInfty ≫ AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex X =
  AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex X
-- AlgebraicTopology.DoldKan.inclusionOfMooreComplexMap_comp_PInfty
@[reassoc (attr := simp)]
theorem inclusionOfMooreComplexMap_comp_PInfty (X : SimplicialObject A) :
    inclusionOfMooreComplexMap X ≫ PInfty = inclusionOfMooreComplexMap X 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.Abelian A] (X : CategoryTheory.SimplicialObject A)
Goal: AlgebraicTopology.inclusionOfMooreComplexMap X ≫ AlgebraicTopology.DoldKan.PInfty =
  AlgebraicTopology.inclusionOfMooreComplexMap X
-- Complex.stolzSet_empty
theorem stolzSet_empty {M : ℝ} (hM : M ≤ 1) : stolzSet M = ∅ 
Arguments: {M : ℝ} (hM : M ≤ 1)
Goal: Complex.stolzSet M = ∅
-- Complex.nhdsWithin_lt_le_nhdsWithin_stolzSet
theorem nhdsWithin_lt_le_nhdsWithin_stolzSet {M : ℝ} (hM : 1 < M) :
    (𝓝[<] 1).map ofReal' ≤ 𝓝[stolzSet M] 1 
Arguments: {M : ℝ} (hM : 1 < M)
Goal: Filter.map Complex.ofReal' (𝓝[<] 1) ≤ 𝓝[Complex.stolzSet M] 1
-- _private.Mathlib.Analysis.Complex.AbelLimit.0.Complex.stolzCone_subset_stolzSet_aux'
private lemma stolzCone_subset_stolzSet_aux' (s : ℝ) :
    ∃ M ε, 0 < M ∧ 0 < ε ∧ ∀ x y, 0 < x → x < ε → |y| < s * x →
      sqrt (x ^ 2 + y ^ 2) < M * (1 - sqrt ((1 - x) ^ 2 + y ^ 2)) 
Arguments: (s : ℝ)
Goal: ∃ M ε, 0 < M ∧ 0 < ε ∧ ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
-- Complex.stolzCone_subset_stolzSet_aux
lemma stolzCone_subset_stolzSet_aux {s : ℝ} (hs : 0 < s) :
    ∃ M ε, 0 < M ∧ 0 < ε ∧ {z : ℂ | 1 - ε < z.re} ∩ stolzCone s ⊆ stolzSet M 
Arguments: {s : ℝ} (hs : 0 < s)
Goal: ∃ M ε, 0 < M ∧ 0 < ε ∧ {z | 1 - ε < z.re} ∩ Complex.stolzCone s ⊆ Complex.stolzSet M
-- Complex.nhdsWithin_stolzCone_le_nhdsWithin_stolzSet
lemma nhdsWithin_stolzCone_le_nhdsWithin_stolzSet {s : ℝ} (hs : 0 < s) :
    ∃ M, 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1 
Arguments: {s : ℝ} (hs : 0 < s)
Goal: ∃ M, 𝓝[Complex.stolzCone s] 1 ≤ 𝓝[Complex.stolzSet M] 1
-- Complex.abel_aux
lemma abel_aux (h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {z : ℂ} (hz : ‖z‖ < 1) :
    Tendsto (fun n ↦ (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i)
      atTop (𝓝 (l - ∑' n, f n * z ^ n)) 
Arguments: {f : ℕ → ℂ} {l : ℂ} (h : Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 l)) {z : ℂ} (hz : ‖z‖ < 1)
Goal: Filter.Tendsto (fun n => (1 - z) * ∑ i ∈ Finset.range n, (l - ∑ j ∈ Finset.range (i + 1), f j) * z ^ i) Filter.atTop
  (𝓝 (l - ∑' (n : ℕ), f n * z ^ n))
-- Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet
theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
    (h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
    Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l) 
Arguments: {f : ℕ → ℂ} {l : ℂ} (h : Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 l)) {M : ℝ}
Goal: Filter.Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[Complex.stolzSet M] 1) (𝓝 l)
-- Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzCone
theorem tendsto_tsum_powerSeries_nhdsWithin_stolzCone
    (h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {s : ℝ} (hs : 0 < s) :
    Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzCone s] 1) (𝓝 l) 
Arguments: {f : ℕ → ℂ} {l : ℂ} (h : Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 l)) {s : ℝ} (hs : 0 < s)
Goal: Filter.Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[Complex.stolzCone s] 1) (𝓝 l)
-- Complex.tendsto_tsum_powerSeries_nhdsWithin_lt
theorem tendsto_tsum_powerSeries_nhdsWithin_lt
    (h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) :
    Tendsto (fun z ↦ ∑' n, f n * z ^ n) ((𝓝[<] 1).map ofReal') (𝓝 l) 
Arguments: {f : ℕ → ℂ} {l : ℂ} (h : Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 l))
Goal: Filter.Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (Filter.map Complex.ofReal' (𝓝[<] 1)) (𝓝 l)
-- Real.tendsto_tsum_powerSeries_nhdsWithin_lt
theorem tendsto_tsum_powerSeries_nhdsWithin_lt
    (h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) :
    Tendsto (fun x ↦ ∑' n, f n * x ^ n) (𝓝[<] 1) (𝓝 l) 
Arguments: {f : ℕ → ℝ} {l : ℝ} (h : Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 l))
Goal: Filter.Tendsto (fun x => ∑' (n : ℕ), f n * x ^ n) (𝓝[<] 1) (𝓝 l)
-- AlgebraicTopology.DoldKan.decomposition_Q
theorem decomposition_Q (n q : ℕ) :
    ((Q q).f (n + 1) : X _[n + 1] ⟶ X _[n + 1]) =
      ∑ i ∈ Finset.filter (fun i : Fin (n + 1) => (i : ℕ) < q) Finset.univ,
        (P i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ (Fin.rev i) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (n q : ℕ)
Goal: (AlgebraicTopology.DoldKan.Q q).f (n + 1) =
  ∑ i ∈ Finset.filter (fun i => ↑i < q) Finset.univ,
    (AlgebraicTopology.DoldKan.P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev
-- AlgebraicTopology.DoldKan.MorphComponents.id_a
@[simps]
def id : MorphComponents X n (X _[n + 1]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (n : ℕ)
Goal: (AlgebraicTopology.DoldKan.MorphComponents.id X n).a = AlgebraicTopology.DoldKan.PInfty.f (n + 1)
-- AlgebraicTopology.DoldKan.MorphComponents.id_b
@[simps]
def id : MorphComponents X n (X _[n + 1]) 
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: (AlgebraicTopology.DoldKan.MorphComponents.id X n).b i = X.σ i
-- AlgebraicTopology.DoldKan.MorphComponents.id_φ
@[simp]
theorem id_φ : (id X n).φ = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) (n : ℕ)
Goal: (AlgebraicTopology.DoldKan.MorphComponents.id X n).φ = 𝟙 (X _[n + 1])
-- AlgebraicTopology.DoldKan.MorphComponents.postComp_a
@[simps]
def postComp : MorphComponents X n Z' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject C} {n : ℕ} {Z Z' : C} (f : AlgebraicTopology.DoldKan.MorphComponents X n Z) (h : Z ⟶ Z')
Goal: (f.postComp h).a = f.a ≫ h
-- AlgebraicTopology.DoldKan.MorphComponents.postComp_b
@[simps]
def postComp : MorphComponents X n Z' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject C} {n : ℕ} {Z Z' : C} (f : AlgebraicTopology.DoldKan.MorphComponents X n Z) (h : Z ⟶ Z') (i : Fin (n + 1))
Goal: (f.postComp h).b i = f.b i ≫ h
-- AlgebraicTopology.DoldKan.MorphComponents.postComp_φ
@[simp]
theorem postComp_φ : (f.postComp h).φ = f.φ ≫ h 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} {n : ℕ} {Z Z' : C} (f : AlgebraicTopology.DoldKan.MorphComponents X n Z) (h : Z ⟶ Z')
Goal: (f.postComp h).φ = f.φ ≫ h
-- AlgebraicTopology.DoldKan.MorphComponents.preComp_a
@[simps]
def preComp : MorphComponents X' n Z 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X X' : CategoryTheory.SimplicialObject C} {n : ℕ} {Z : C} (f : AlgebraicTopology.DoldKan.MorphComponents X n Z) (g : X' ⟶ X)
Goal: (f.preComp g).a = g.app (Opposite.op [n + 1]) ≫ f.a
-- AlgebraicTopology.DoldKan.MorphComponents.preComp_b
@[simps]
def preComp : MorphComponents X' n Z 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X X' : CategoryTheory.SimplicialObject C} {n : ℕ} {Z : C} (f : AlgebraicTopology.DoldKan.MorphComponents X n Z) (g : X' ⟶ X) (i : Fin (n + 1))
Goal: (f.preComp g).b i = g.app (Opposite.op [n]) ≫ f.b i
-- AlgebraicTopology.DoldKan.MorphComponents.preComp_φ
@[simp]
theorem preComp_φ : (f.preComp g).φ = g.app (op [n + 1]) ≫ f.φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X X' : CategoryTheory.SimplicialObject C} {n : ℕ} {Z : C} (f : AlgebraicTopology.DoldKan.MorphComponents X n Z) (g : X' ⟶ X)
Goal: (f.preComp g).φ = g.app (Opposite.op [n + 1]) ≫ f.φ
-- CategoryTheory.Preadditive.DoldKan.equivalence_functor
@[simps]
def equivalence : Karoubi (SimplicialObject C) ≌ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: CategoryTheory.Preadditive.DoldKan.equivalence.functor = CategoryTheory.Preadditive.DoldKan.N
-- CategoryTheory.Preadditive.DoldKan.equivalence_inverse
@[simps]
def equivalence : Karoubi (SimplicialObject C) ≌ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: CategoryTheory.Preadditive.DoldKan.equivalence.inverse = CategoryTheory.Preadditive.DoldKan.Γ
-- CategoryTheory.Preadditive.DoldKan.equivalence_unitIso
@[simps]
def equivalence : Karoubi (SimplicialObject C) ≌ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: CategoryTheory.Preadditive.DoldKan.equivalence.unitIso = AlgebraicTopology.DoldKan.Γ₂N₂
-- CategoryTheory.Preadditive.DoldKan.equivalence_counitIso
@[simps]
def equivalence : Karoubi (SimplicialObject C) ≌ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: CategoryTheory.Preadditive.DoldKan.equivalence.counitIso = AlgebraicTopology.DoldKan.N₂Γ₂
-- AlgebraicTopology.DoldKan.N₁_map_f
@[simps]
def N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C]
Goal: ∀ {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y),
  (AlgebraicTopology.DoldKan.N₁.map f).f =
    AlgebraicTopology.DoldKan.PInfty ≫ AlgebraicTopology.AlternatingFaceMapComplex.map f
-- AlgebraicTopology.DoldKan.N₁_obj_X
@[simps]
def N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C)
Goal: (AlgebraicTopology.DoldKan.N₁.obj X).X = AlgebraicTopology.AlternatingFaceMapComplex.obj X
-- AlgebraicTopology.DoldKan.N₁_obj_p
@[simps]
def N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C)
Goal: (AlgebraicTopology.DoldKan.N₁.obj X).p = AlgebraicTopology.DoldKan.PInfty
-- AlgebraicTopology.DoldKan.N₂_map_f_f
@[simps!]
def N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C]
Goal: ∀ {X Y : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)} (f : X ⟶ Y) (i : ℕ),
  (AlgebraicTopology.DoldKan.N₂.map f).f.f i =
    AlgebraicTopology.DoldKan.PInfty.f i ≫ f.f.app (Opposite.op (SimplexCategory.mk i))
-- AlgebraicTopology.DoldKan.N₂_obj_p_f
@[simps!]
def N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) (i : ℕ)
Goal: (AlgebraicTopology.DoldKan.N₂.obj P).p.f i =
  AlgebraicTopology.DoldKan.PInfty.f i ≫ P.p.app (Opposite.op (SimplexCategory.mk i))
-- AlgebraicTopology.DoldKan.N₂_obj_X_d
@[simps!]
def N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) (i j : ℕ)
Goal: (AlgebraicTopology.DoldKan.N₂.obj P).X.d i j =
  if h : i = j + 1 then CategoryTheory.eqToHom ⋯ ≫ ∑ i : Fin (j + 2), (-1) ^ ↑i • P.X.δ i else 0
-- AlgebraicTopology.DoldKan.N₂_obj_X_X
@[simps!]
def N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)) (n : ℕ)
Goal: (AlgebraicTopology.DoldKan.N₂.obj P).X.X n = P.X.obj (Opposite.op (SimplexCategory.mk n))
-- AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁_hom_app
@[simp]
lemma toKaroubiCompN₂IsoN₁_hom_app (X : SimplicialObject C) :
    (toKaroubiCompN₂IsoN₁.hom.app X).f = PInfty 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C)
Goal: (AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁.hom.app X).f = AlgebraicTopology.DoldKan.PInfty
-- AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁_inv_app
@[simp]
lemma toKaroubiCompN₂IsoN₁_inv_app (X : SimplicialObject C) :
    (toKaroubiCompN₂IsoN₁.inv.app X).f = PInfty 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C)
Goal: (AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁.inv.app X).f = AlgebraicTopology.DoldKan.PInfty