ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SSet.horn.primitiveEdge_coe_down
@[simps!]
def primitiveEdge {n : ℕ} {i : Fin (n+1)}
    (h₀ : 0 < i) (hₙ : i < Fin.last n) (j : Fin n) :
    Λ[n, i] _[1] 
Arguments: {n : ℕ} {i : Fin (n + 1)} (h₀ : 0 < i) (hₙ : i < Fin.last n) (j : Fin n)
Goal: (↑(SSet.horn.primitiveEdge h₀ hₙ j)).down = SimplexCategory.Hom.mk { toFun := ![j.castSucc, j.succ], monotone' := ⋯ }
-- SSet.horn.primitiveTriangle_coe
@[simps]
def primitiveTriangle {n : ℕ} (i : Fin (n+4))
    (h₀ : 0 < i) (hₙ : i < Fin.last (n+3))
    (k : ℕ) (h : k < n+2) : Λ[n+3, i] _[2] 
Arguments: {n : ℕ} (i : Fin (n + 4)) (h₀ : 0 < i) (hₙ : i < Fin.last (n + 3)) (k : ℕ) (h : k < n + 2)
Goal: ↑(SSet.horn.primitiveTriangle i h₀ hₙ k h) = SSet.standardSimplex.triangle ⟨k, ⋯⟩ ⟨k + 1, ⋯⟩ ⟨k + 2, ⋯⟩ ⋯ ⋯
-- SSet.horn.face_coe
@[simps]
def face {n : ℕ} (i j : Fin (n+2)) (h : j ≠ i) : Λ[n+1, i] _[n] 
Arguments: {n : ℕ} (i j : Fin (n + 2)) (h : j ≠ i)
Goal: ↑(SSet.horn.face i j h) = (SSet.standardSimplex.objEquiv [n + 1] (Opposite.op [n])).symm (SimplexCategory.δ j)
-- SSet.horn.hom_ext
protected
lemma hom_ext {n : ℕ} {i : Fin (n+2)} {S : SSet} (σ₁ σ₂ : Λ[n+1, i] ⟶ S)
    (h : ∀ (j) (h : j ≠ i), σ₁.app _ (face i j h) = σ₂.app _ (face i j h)) :
    σ₁ = σ₂ 
Arguments: {n : ℕ} {i : Fin (n + 2)} {S : SSet} (σ₁ σ₂ : Λ[n + 1, i] ⟶ S) (h :
  ∀ (j : Fin (n + 2)) (h : j ≠ i),
    σ₁.app (Opposite.op [n]) (SSet.horn.face i j h) = σ₂.app (Opposite.op [n]) (SSet.horn.face i j h))
Goal: σ₁ = σ₂
-- SSet.Truncated.hom_ext
@[ext]
lemma Truncated.hom_ext {n : ℕ} {X Y : Truncated n} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) :
    f = g 
Arguments: {n : ℕ} {X Y : SSet.Truncated n} {f g : X ⟶ Y} (w : ∀ (n_1 : (SimplexCategory.Truncated n)ᵒᵖ), f.app n_1 = g.app n_1)
Goal: f = g
-- SSet.Augmented.standardSimplex_obj_hom_app
@[simps]
noncomputable def standardSimplex : SimplexCategory ⥤ SSet.Augmented.{u} 
Arguments: (Δ : SimplexCategory) (Δ' : SimplexCategoryᵒᵖ)
Goal: ∀ (a : ((𝟭 (CategoryTheory.SimplicialObject (Type u))).obj (SSet.standardSimplex.obj Δ)).obj Δ'),
  (SSet.Augmented.standardSimplex.obj Δ).hom.app Δ' a =
    CategoryTheory.Limits.terminal.from
      (((𝟭 (CategoryTheory.SimplicialObject (Type u))).obj (SSet.standardSimplex.obj Δ)).obj Δ') a
-- SSet.Augmented.standardSimplex_obj_left
@[simps]
noncomputable def standardSimplex : SimplexCategory ⥤ SSet.Augmented.{u} 
Arguments: (Δ : SimplexCategory)
Goal: (SSet.Augmented.standardSimplex.obj Δ).left = SSet.standardSimplex.obj Δ
-- SSet.Augmented.standardSimplex_map_right
@[simps]
noncomputable def standardSimplex : SimplexCategory ⥤ SSet.Augmented.{u} 
Goal: ∀ {X Y : SimplexCategory} (θ : X ⟶ Y)
  (a :
    ((fun Δ =>
          { left := SSet.standardSimplex.obj Δ, right := ⊤_ Type u,
            hom :=
              {
                app := fun Δ' =>
                  CategoryTheory.Limits.terminal.from
                    (((𝟭 (CategoryTheory.SimplicialObject (Type u))).obj (SSet.standardSimplex.obj Δ)).obj Δ'),
                naturality := ⋯ } })
        X).right),
  (SSet.Augmented.standardSimplex.map θ).right a =
    CategoryTheory.Limits.terminal.from
      ((fun Δ =>
            { left := SSet.standardSimplex.obj Δ, right := ⊤_ Type u,
              hom :=
                {
                  app := fun Δ' =>
                    CategoryTheory.Limits.terminal.from
                      (((𝟭 (CategoryTheory.SimplicialObject (Type u))).obj (SSet.standardSimplex.obj Δ)).obj Δ'),
                  naturality := ⋯ } })
          X).right
      a
-- SSet.Augmented.standardSimplex_obj_right
@[simps]
noncomputable def standardSimplex : SimplexCategory ⥤ SSet.Augmented.{u} 
Arguments: (Δ : SimplexCategory)
Goal: (SSet.Augmented.standardSimplex.obj Δ).right = ⊤_ Type u
-- SSet.Augmented.standardSimplex_map_left
@[simps]
noncomputable def standardSimplex : SimplexCategory ⥤ SSet.Augmented.{u} 
Goal: ∀ {X Y : SimplexCategory} (θ : X ⟶ Y), (SSet.Augmented.standardSimplex.map θ).left = SSet.standardSimplex.map θ
-- ZFSet.empty_isTransitive
@[simp]
theorem empty_isTransitive : IsTransitive ∅ 
Goal: ∅.IsTransitive
-- ZFSet.IsTransitive.subset_of_mem
theorem IsTransitive.subset_of_mem (h : x.IsTransitive) : y ∈ x → y ⊆ x 
Arguments: {x y : ZFSet} (h : x.IsTransitive)
Goal: y ∈ x → y ⊆ x
-- ZFSet.isTransitive_iff_mem_trans
theorem isTransitive_iff_mem_trans : z.IsTransitive ↔ ∀ {x y : ZFSet}, x ∈ y → y ∈ z → x ∈ z 
Arguments: {z : ZFSet}
Goal: z.IsTransitive ↔ ∀ {x y : ZFSet}, x ∈ y → y ∈ z → x ∈ z
-- ZFSet.IsTransitive.inter
protected theorem IsTransitive.inter (hx : x.IsTransitive) (hy : y.IsTransitive) :
    (x ∩ y).IsTransitive 
Arguments: {x y : ZFSet} (hx : x.IsTransitive) (hy : y.IsTransitive)
Goal: (x ∩ y).IsTransitive
-- ZFSet.IsTransitive.sUnion
protected theorem IsTransitive.sUnion (h : x.IsTransitive) :
    (⋃₀ x : ZFSet).IsTransitive 
Arguments: {x : ZFSet} (h : x.IsTransitive)
Goal: (⋃₀ x).IsTransitive
-- ZFSet.IsTransitive.sUnion'
theorem IsTransitive.sUnion' (H : ∀ y ∈ x, IsTransitive y) :
    (⋃₀ x : ZFSet).IsTransitive 
Arguments: {x : ZFSet} (H : ∀ y ∈ x, y.IsTransitive)
Goal: (⋃₀ x).IsTransitive
-- ZFSet.IsTransitive.union
protected theorem IsTransitive.union (hx : x.IsTransitive) (hy : y.IsTransitive) :
    (x ∪ y).IsTransitive 
Arguments: {x y : ZFSet} (hx : x.IsTransitive) (hy : y.IsTransitive)
Goal: (x ∪ y).IsTransitive
-- ZFSet.IsTransitive.powerset
protected theorem IsTransitive.powerset (h : x.IsTransitive) : (powerset x).IsTransitive 
Arguments: {x : ZFSet} (h : x.IsTransitive)
Goal: x.powerset.IsTransitive
-- ZFSet.isTransitive_iff_sUnion_subset
theorem isTransitive_iff_sUnion_subset : x.IsTransitive ↔ (⋃₀ x : ZFSet) ⊆ x 
Arguments: {x : ZFSet}
Goal: x.IsTransitive ↔ ⋃₀ x ⊆ x
-- ZFSet.isTransitive_iff_subset_powerset
theorem isTransitive_iff_subset_powerset : x.IsTransitive ↔ x ⊆ powerset x 
Arguments: {x : ZFSet}
Goal: x.IsTransitive ↔ x ⊆ x.powerset
-- AlgebraicTopology.DoldKan.P_zero
lemma P_zero : (P 0 : K[X] ⟶ K[X]) = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C}
Goal: AlgebraicTopology.DoldKan.P 0 = 𝟙 K[X]
-- AlgebraicTopology.DoldKan.P_succ
lemma P_succ (q : ℕ) : (P (q+1) : K[X] ⟶ K[X]) = P q ≫ (𝟙 _ + Hσ q) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q : ℕ)
Goal: AlgebraicTopology.DoldKan.P (q + 1) = AlgebraicTopology.DoldKan.P q ≫ (𝟙 K[X] + AlgebraicTopology.DoldKan.Hσ q)
-- AlgebraicTopology.DoldKan.P_f_0_eq
@[simp]
theorem P_f_0_eq (q : ℕ) : ((P q).f 0 : X _[0] ⟶ X _[0]) = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q : ℕ)
Goal: (AlgebraicTopology.DoldKan.P q).f 0 = 𝟙 (K[X].X 0)
-- AlgebraicTopology.DoldKan.P_add_Q
theorem P_add_Q (q : ℕ) : P q + Q q = 𝟙 K[X] 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q : ℕ)
Goal: AlgebraicTopology.DoldKan.P q + AlgebraicTopology.DoldKan.Q q = 𝟙 K[X]
-- AlgebraicTopology.DoldKan.P_add_Q_f
theorem P_add_Q_f (q n : ℕ) : (P q).f n + (Q q).f n = 𝟙 (X _[n]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q n : ℕ)
Goal: (AlgebraicTopology.DoldKan.P q).f n + (AlgebraicTopology.DoldKan.Q q).f n = 𝟙 (X _[n])
-- AlgebraicTopology.DoldKan.Q_zero
@[simp]
theorem Q_zero : (Q 0 : K[X] ⟶ _) = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C}
Goal: AlgebraicTopology.DoldKan.Q 0 = 0
-- AlgebraicTopology.DoldKan.Q_succ
theorem Q_succ (q : ℕ) : (Q (q + 1) : K[X] ⟶ _) = Q q - P q ≫ Hσ q 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q : ℕ)
Goal: AlgebraicTopology.DoldKan.Q (q + 1) =
  AlgebraicTopology.DoldKan.Q q - AlgebraicTopology.DoldKan.P q ≫ AlgebraicTopology.DoldKan.Hσ q
-- AlgebraicTopology.DoldKan.Q_f_0_eq
@[simp]
theorem Q_f_0_eq (q : ℕ) : ((Q q).f 0 : X _[0] ⟶ X _[0]) = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q : ℕ)
Goal: (AlgebraicTopology.DoldKan.Q q).f 0 = 0
-- AlgebraicTopology.DoldKan.HigherFacesVanish.of_P
theorem of_P : ∀ q n : ℕ, HigherFacesVanish q ((P q).f (n + 1) : X _[n + 1] ⟶ X _[n + 1])
  
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q n : ℕ)
Goal: AlgebraicTopology.DoldKan.HigherFacesVanish q ((AlgebraicTopology.DoldKan.P q).f (n + 1))
-- AlgebraicTopology.DoldKan.HigherFacesVanish.comp_P_eq_self
@[reassoc]
theorem comp_P_eq_self {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ) :
    φ ≫ (P q).f (n + 1) = φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : AlgebraicTopology.DoldKan.HigherFacesVanish q φ)
Goal: φ ≫ (AlgebraicTopology.DoldKan.P q).f (n + 1) = φ
-- AlgebraicTopology.DoldKan.comp_P_eq_self_iff
theorem comp_P_eq_self_iff {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]} :
    φ ≫ (P q).f (n + 1) = φ ↔ HigherFacesVanish q φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]}
Goal: φ ≫ (AlgebraicTopology.DoldKan.P q).f (n + 1) = φ ↔ AlgebraicTopology.DoldKan.HigherFacesVanish q φ
-- AlgebraicTopology.DoldKan.P_f_idem
@[reassoc (attr := simp)]
theorem P_f_idem (q n : ℕ) : ((P q).f n : X _[n] ⟶ _) ≫ (P q).f n = (P q).f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q n : ℕ)
Goal: (AlgebraicTopology.DoldKan.P q).f n ≫ (AlgebraicTopology.DoldKan.P q).f n = (AlgebraicTopology.DoldKan.P q).f n
-- AlgebraicTopology.DoldKan.Q_f_idem
@[reassoc (attr := simp)]
theorem Q_f_idem (q n : ℕ) : ((Q q).f n : X _[n] ⟶ _) ≫ (Q q).f n = (Q q).f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q n : ℕ)
Goal: (AlgebraicTopology.DoldKan.Q q).f n ≫ (AlgebraicTopology.DoldKan.Q q).f n = (AlgebraicTopology.DoldKan.Q q).f n
-- AlgebraicTopology.DoldKan.P_idem
@[reassoc (attr := simp)]
theorem P_idem (q : ℕ) : (P q : K[X] ⟶ K[X]) ≫ P q = P q 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q : ℕ)
Goal: AlgebraicTopology.DoldKan.P q ≫ AlgebraicTopology.DoldKan.P q = AlgebraicTopology.DoldKan.P q
-- AlgebraicTopology.DoldKan.Q_idem
@[reassoc (attr := simp)]
theorem Q_idem (q : ℕ) : (Q q : K[X] ⟶ K[X]) ≫ Q q = Q q 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q : ℕ)
Goal: AlgebraicTopology.DoldKan.Q q ≫ AlgebraicTopology.DoldKan.Q q = AlgebraicTopology.DoldKan.Q q
-- AlgebraicTopology.DoldKan.natTransP_app
@[simps]
def natTransP (q : ℕ) : alternatingFaceMapComplex C ⟶ alternatingFaceMapComplex C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (q : ℕ) (X : CategoryTheory.SimplicialObject C)
Goal: (AlgebraicTopology.DoldKan.natTransP q).app X = AlgebraicTopology.DoldKan.P q
-- AlgebraicTopology.DoldKan.P_f_naturality
@[reassoc (attr := simp)]
theorem P_f_naturality (q n : ℕ) {X Y : SimplicialObject C} (f : X ⟶ Y) :
    f.app (op [n]) ≫ (P q).f n = (P q).f n ≫ f.app (op [n]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (q n : ℕ) {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y)
Goal: f.app (Opposite.op [n]) ≫ (AlgebraicTopology.DoldKan.P q).f n =
  (AlgebraicTopology.DoldKan.P q).f n ≫ f.app (Opposite.op [n])
-- AlgebraicTopology.DoldKan.Q_f_naturality
@[reassoc (attr := simp)]
theorem Q_f_naturality (q n : ℕ) {X Y : SimplicialObject C} (f : X ⟶ Y) :
    f.app (op [n]) ≫ (Q q).f n = (Q q).f n ≫ f.app (op [n]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (q n : ℕ) {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y)
Goal: f.app (Opposite.op [n]) ≫ (AlgebraicTopology.DoldKan.Q q).f n =
  (AlgebraicTopology.DoldKan.Q q).f n ≫ f.app (Opposite.op [n])
-- AlgebraicTopology.DoldKan.natTransQ_app
@[simps]
def natTransQ (q : ℕ) : alternatingFaceMapComplex C ⟶ alternatingFaceMapComplex C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (q : ℕ) (X : CategoryTheory.SimplicialObject C)
Goal: (AlgebraicTopology.DoldKan.natTransQ q).app X = AlgebraicTopology.DoldKan.Q q
-- AlgebraicTopology.DoldKan.map_P
theorem map_P {D : Type*} [Category D] [Preadditive D] (G : C ⥤ D) [G.Additive]
    (X : SimplicialObject C) (q n : ℕ) :
    G.map ((P q : K[X] ⟶ _).f n) = (P q : K[((whiskering C D).obj G).obj X] ⟶ _).f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] {D : Type u_2} [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Preadditive D] (G : C ⥤ D) [G.Additive] (X : CategoryTheory.SimplicialObject C) (q n : ℕ)
Goal: G.map ((AlgebraicTopology.DoldKan.P q).f n) = (AlgebraicTopology.DoldKan.P q).f n
-- AlgebraicTopology.DoldKan.map_Q
theorem map_Q {D : Type*} [Category D] [Preadditive D] (G : C ⥤ D) [G.Additive]
    (X : SimplicialObject C) (q n : ℕ) :
    G.map ((Q q : K[X] ⟶ _).f n) = (Q q : K[((whiskering C D).obj G).obj X] ⟶ _).f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] {D : Type u_2} [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Preadditive D] (G : C ⥤ D) [G.Additive] (X : CategoryTheory.SimplicialObject C) (q n : ℕ)
Goal: G.map ((AlgebraicTopology.DoldKan.Q q).f n) = (AlgebraicTopology.DoldKan.Q q).f n
-- AlgebraicTopology.DoldKan.c_mk
theorem c_mk (i j : ℕ) (h : j + 1 = i) : c.Rel i j 
Arguments: (i j : ℕ) (h : j + 1 = i)
Goal: AlgebraicTopology.DoldKan.c.Rel i j
-- AlgebraicTopology.DoldKan.cs_down_0_not_rel_left
theorem cs_down_0_not_rel_left (j : ℕ) : ¬c.Rel 0 j 
Arguments: (j : ℕ)
Goal: ¬AlgebraicTopology.DoldKan.c.Rel 0 j
-- AlgebraicTopology.DoldKan.hσ'_eq_zero
theorem hσ'_eq_zero {q n m : ℕ} (hnq : n < q) (hnm : c.Rel m n) :
    (hσ' q n m hnm : X _[n] ⟶ X _[m]) = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} {q n m : ℕ} (hnq : n < q) (hnm : AlgebraicTopology.DoldKan.c.Rel m n)
Goal: AlgebraicTopology.DoldKan.hσ' q n m hnm = 0
-- AlgebraicTopology.DoldKan.hσ'_eq
theorem hσ'_eq {q n a m : ℕ} (ha : n = a + q) (hnm : c.Rel m n) :
    (hσ' q n m hnm : X _[n] ⟶ X _[m]) =
      ((-1 : ℤ) ^ a • X.σ ⟨a, Nat.lt_succ_iff.mpr (Nat.le.intro (Eq.symm ha))⟩) ≫
        eqToHom (by congr) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} {q n a m : ℕ} (ha : n = a + q) (hnm : AlgebraicTopology.DoldKan.c.Rel m n)
Goal: AlgebraicTopology.DoldKan.hσ' q n m hnm = ((-1) ^ a • X.σ ⟨a, ⋯⟩) ≫ CategoryTheory.eqToHom ⋯
-- AlgebraicTopology.DoldKan.hσ'_eq'
theorem hσ'_eq' {q n a : ℕ} (ha : n = a + q) :
    (hσ' q n (n + 1) rfl : X _[n] ⟶ X _[n + 1]) =
      (-1 : ℤ) ^ a • X.σ ⟨a, Nat.lt_succ_iff.mpr (Nat.le.intro (Eq.symm ha))⟩ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} {q n a : ℕ} (ha : n = a + q)
Goal: AlgebraicTopology.DoldKan.hσ' q n (n + 1) ⋯ = (-1) ^ a • X.σ ⟨a, ⋯⟩
-- AlgebraicTopology.DoldKan.Hσ_eq_zero
theorem Hσ_eq_zero (q : ℕ) : (Hσ q : K[X] ⟶ K[X]).f 0 = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (q : ℕ)
Goal: (AlgebraicTopology.DoldKan.Hσ q).f 0 = 0
-- AlgebraicTopology.DoldKan.hσ'_naturality
theorem hσ'_naturality (q : ℕ) (n m : ℕ) (hnm : c.Rel m n) {X Y : SimplicialObject C} (f : X ⟶ Y) :
    f.app (op [n]) ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app (op [m]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (q n m : ℕ) (hnm : AlgebraicTopology.DoldKan.c.Rel m n) {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y)
Goal: f.app (Opposite.op [n]) ≫ AlgebraicTopology.DoldKan.hσ' q n m hnm =
  AlgebraicTopology.DoldKan.hσ' q n m hnm ≫ f.app (Opposite.op [m])
-- AlgebraicTopology.DoldKan.map_hσ'
theorem map_hσ' {D : Type*} [Category D] [Preadditive D] (G : C ⥤ D) [G.Additive]
    (X : SimplicialObject C) (q n m : ℕ) (hnm : c.Rel m n) :
    (hσ' q n m hnm : K[((whiskering _ _).obj G).obj X].X n ⟶ _) =
      G.map (hσ' q n m hnm : K[X].X n ⟶ _) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] {D : Type u_2} [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Preadditive D] (G : C ⥤ D) [G.Additive] (X : CategoryTheory.SimplicialObject C) (q n m : ℕ) (hnm : AlgebraicTopology.DoldKan.c.Rel m n)
Goal: AlgebraicTopology.DoldKan.hσ' q n m hnm = G.map (AlgebraicTopology.DoldKan.hσ' q n m hnm)
-- AlgebraicTopology.DoldKan.map_Hσ
theorem map_Hσ {D : Type*} [Category D] [Preadditive D] (G : C ⥤ D) [G.Additive]
    (X : SimplicialObject C) (q n : ℕ) :
    (Hσ q : K[((whiskering C D).obj G).obj X] ⟶ _).f n = G.map ((Hσ q : K[X] ⟶ _).f n) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Preadditive C] {D : Type u_2} [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Preadditive D] (G : C ⥤ D) [G.Additive] (X : CategoryTheory.SimplicialObject C) (q n : ℕ)
Goal: (AlgebraicTopology.DoldKan.Hσ q).f n = G.map ((AlgebraicTopology.DoldKan.Hσ q).f n)
-- CategoryTheory.Idempotents.DoldKan.N_map
@[simps!, nolint unusedArguments]
def N [IsIdempotentComplete C] [HasFiniteCoproducts C] : SimplicialObject C ⥤ ChainComplex C ℕ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: ∀ {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y),
  CategoryTheory.Idempotents.DoldKan.N.map f =
    (CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex C ℕ)).inverse.map
      (AlgebraicTopology.DoldKan.N₁.map f)
-- CategoryTheory.Idempotents.DoldKan.N_obj
@[simps!, nolint unusedArguments]
def N [IsIdempotentComplete C] [HasFiniteCoproducts C] : SimplicialObject C ⥤ ChainComplex C ℕ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : CategoryTheory.SimplicialObject C)
Goal: CategoryTheory.Idempotents.DoldKan.N.obj X =
  (CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex C ℕ)).inverse.obj (AlgebraicTopology.DoldKan.N₁.obj X)
-- CategoryTheory.Idempotents.DoldKan.Γ_obj_map
@[simps!, nolint unusedArguments]
def Γ [IsIdempotentComplete C] [HasFiniteCoproducts C] : ChainComplex C ℕ ⥤ SimplicialObject C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : ChainComplex C ℕ)
Goal: ∀ {X_1 Y : SimplexCategoryᵒᵖ} (θ : X_1 ⟶ Y),
  (CategoryTheory.Idempotents.DoldKan.Γ.obj X).map θ = AlgebraicTopology.DoldKan.Γ₀.Obj.map X θ
-- CategoryTheory.Idempotents.DoldKan.Γ_map_app
@[simps!, nolint unusedArguments]
def Γ [IsIdempotentComplete C] [HasFiniteCoproducts C] : ChainComplex C ℕ ⥤ SimplicialObject C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: ∀ {X Y : ChainComplex C ℕ} (f : X ⟶ Y) (Δ : SimplexCategoryᵒᵖ),
  (CategoryTheory.Idempotents.DoldKan.Γ.map f).app Δ =
    (AlgebraicTopology.DoldKan.Γ₀.splitting X).desc Δ fun A =>
      f.f (Opposite.unop A.fst).len ≫ ((AlgebraicTopology.DoldKan.Γ₀.splitting Y).cofan Δ).inj A
-- CategoryTheory.Idempotents.DoldKan.Γ_obj_obj
@[simps!, nolint unusedArguments]
def Γ [IsIdempotentComplete C] [HasFiniteCoproducts C] : ChainComplex C ℕ ⥤ SimplicialObject C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : ChainComplex C ℕ) (Δ : SimplexCategoryᵒᵖ)
Goal: (CategoryTheory.Idempotents.DoldKan.Γ.obj X).obj Δ = AlgebraicTopology.DoldKan.Γ₀.Obj.obj₂ X Δ
-- CategoryTheory.Idempotents.DoldKan.isoN₁_hom_app_f
@[simp]
lemma isoN₁_hom_app_f (X : SimplicialObject C) :
    (isoN₁.hom.app X).f = PInfty 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : CategoryTheory.SimplicialObject C)
Goal: (CategoryTheory.Idempotents.DoldKan.isoN₁.hom.app X).f = AlgebraicTopology.DoldKan.PInfty
-- CategoryTheory.Idempotents.DoldKan.N₂_map_isoΓ₀_hom_app_f
@[simp]
lemma N₂_map_isoΓ₀_hom_app_f (X : ChainComplex C ℕ) :
    (N₂.map (isoΓ₀.hom.app X)).f = PInfty 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : ChainComplex C ℕ)
Goal: (AlgebraicTopology.DoldKan.N₂.map (CategoryTheory.Idempotents.DoldKan.isoΓ₀.hom.app X)).f =
  AlgebraicTopology.DoldKan.PInfty
-- CategoryTheory.Idempotents.DoldKan.equivalence_functor
theorem equivalence_functor : (equivalence : SimplicialObject C ≌ _).functor = N 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: CategoryTheory.Idempotents.DoldKan.equivalence.functor = CategoryTheory.Idempotents.DoldKan.N
-- CategoryTheory.Idempotents.DoldKan.equivalence_inverse
theorem equivalence_inverse : (equivalence : SimplicialObject C ≌ _).inverse = Γ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: CategoryTheory.Idempotents.DoldKan.equivalence.inverse = CategoryTheory.Idempotents.DoldKan.Γ
-- CategoryTheory.Idempotents.DoldKan.hη
theorem hη :
    Compatibility.τ₀ =
      Compatibility.τ₁ isoN₁ isoΓ₀
        (N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubiEquivalence (ChainComplex C ℕ)).functor) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C]
Goal: AlgebraicTopology.DoldKan.Compatibility.τ₀ =
  AlgebraicTopology.DoldKan.Compatibility.τ₁ CategoryTheory.Idempotents.DoldKan.isoN₁
    CategoryTheory.Idempotents.DoldKan.isoΓ₀ AlgebraicTopology.DoldKan.N₁Γ₀
-- CategoryTheory.Idempotents.DoldKan.η_inv_app_f
@[simps!]
def η : Γ ⋙ N ≅ 𝟭 (ChainComplex C ℕ) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.IsIdempotentComplete C] [CategoryTheory.Limits.HasFiniteCoproducts C] (X : ChainComplex C ℕ) (i : ℕ)
Goal: (CategoryTheory.Idempotents.DoldKan.η.inv.app X).f i =
  ((CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex C ℕ)).unitIso.hom.app X).f i ≫
    ((CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex C ℕ)).inverse.map
          (AlgebraicTopology.DoldKan.N₁Γ₀.inv.app X)).f
      i