ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Lists'.subset_nil
theorem subset_nil {l : Lists' α true} : l ⊆ Lists'.nil → l = Lists'.nil 
Arguments: {α : Type u_1} {l : Lists' α true}
Goal: l ⊆ Lists'.nil → l = Lists'.nil
-- Lists'.mem_of_subset'
theorem mem_of_subset' {a} : ∀ {l₁ l₂ : Lists' α true} (_ : l₁ ⊆ l₂) (_ : a ∈ l₁.toList), a ∈ l₂
  
Arguments: {α : Type u_1} {a : Lists α} {l₁ l₂ : Lists' α true}
Goal: l₁ ⊆ l₂ → a ∈ l₁.toList → a ∈ l₂
-- Lists'.subset_def
theorem subset_def {l₁ l₂ : Lists' α true} : l₁ ⊆ l₂ ↔ ∀ a ∈ l₁.toList, a ∈ l₂ 
Arguments: {α : Type u_1} {l₁ l₂ : Lists' α true}
Goal: l₁ ⊆ l₂ ↔ ∀ a ∈ l₁.toList, a ∈ l₂
-- Lists.isList_toList
theorem isList_toList (l : List (Lists α)) : IsList (ofList l) 
Arguments: {α : Type u_1} (l : List (Lists α))
Goal: (Lists.ofList l).IsList
-- Lists.to_ofList
theorem to_ofList (l : List (Lists α)) : toList (ofList l) = l 
Arguments: {α : Type u_1} (l : List (Lists α))
Goal: (Lists.ofList l).toList = l
-- Lists.of_toList
theorem of_toList : ∀ {l : Lists α}, IsList l → ofList (toList l) = l
  
Arguments: {α : Type u_1} {l : Lists α}
Goal: l.IsList → Lists.ofList l.toList = l
-- Lists.isList_of_mem
theorem isList_of_mem {a : Lists α} : ∀ {l : Lists α}, a ∈ l → IsList l
  
Arguments: {α : Type u_1} {a l : Lists α}
Goal: a ∈ l → l.IsList
-- Lists.Equiv.antisymm_iff
theorem Equiv.antisymm_iff {l₁ l₂ : Lists' α true} : of' l₁ ~ of' l₂ ↔ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁ 
Arguments: {α : Type u_1} {l₁ l₂ : Lists' α true}
Goal: Lists.of' l₁ ~ Lists.of' l₂ ↔ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁
-- Lists.equiv_atom
theorem equiv_atom {a} {l : Lists α} : atom a ~ l ↔ atom a = l 
Arguments: {α : Type u_1} {a : α} {l : Lists α}
Goal: Lists.atom a ~ l ↔ Lists.atom a = l
-- Lists.Equiv.symm
@[symm]
theorem Equiv.symm {l₁ l₂ : Lists α} (h : l₁ ~ l₂) : l₂ ~ l₁ 
Arguments: {α : Type u_1} {l₁ l₂ : Lists α} (h : l₁ ~ l₂)
Goal: l₂ ~ l₁
-- Lists.Equiv.trans
theorem Equiv.trans : ∀ {l₁ l₂ l₃ : Lists α}, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ 
Arguments: {α : Type u_1} {l₁ l₂ l₃ : Lists α}
Goal: l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃
-- Lists.sizeof_pos
theorem sizeof_pos {b} (l : Lists' α b) : 0 < SizeOf.sizeOf l 
Arguments: {α : Type u_1} {b : Bool} (l : Lists' α b)
Goal: 0 < sizeOf l
-- Lists.lt_sizeof_cons'
theorem lt_sizeof_cons' {b} (a : Lists' α b) (l) :
    SizeOf.sizeOf (⟨b, a⟩ : Lists α) < SizeOf.sizeOf (Lists'.cons' a l) 
Arguments: {α : Type u_1} {b : Bool} (a : Lists' α b) (l : Lists' α true)
Goal: sizeOf ⟨b, a⟩ < sizeOf (a.cons' l)
-- Lists'.mem_equiv_left
theorem mem_equiv_left {l : Lists' α true} : ∀ {a a'}, a ~ a' → (a ∈ l ↔ a' ∈ l) 
Arguments: {α : Type u_1} {l : Lists' α true} {a a' : Lists α}
Goal: a ~ a' → (a ∈ l ↔ a' ∈ l)
-- Lists'.mem_of_subset
theorem mem_of_subset {a} {l₁ l₂ : Lists' α true} (s : l₁ ⊆ l₂) : a ∈ l₁ → a ∈ l₂
  
Arguments: {α : Type u_1} {a : Lists α} {l₁ l₂ : Lists' α true} (s : l₁ ⊆ l₂)
Goal: a ∈ l₁ → a ∈ l₂
-- Lists'.Subset.trans
theorem Subset.trans {l₁ l₂ l₃ : Lists' α true} (h₁ : l₁ ⊆ l₂) (h₂ : l₂ ⊆ l₃) : l₁ ⊆ l₃ 
Arguments: {α : Type u_1} {l₁ l₂ l₃ : Lists' α true} (h₁ : l₁ ⊆ l₂) (h₂ : l₂ ⊆ l₃)
Goal: l₁ ⊆ l₃
-- Cardinal.two_power_aleph0
@[simp]
theorem two_power_aleph0 : 2 ^ aleph0.{u} = continuum.{u} 
Goal: 2 ^ Cardinal.aleph0 = Cardinal.continuum
-- Cardinal.lift_continuum
@[simp]
theorem lift_continuum : lift.{v} 𝔠 = 𝔠 
Goal: Cardinal.lift.{v, u_1} Cardinal.continuum = Cardinal.continuum
-- Cardinal.continuum_le_lift
@[simp]
theorem continuum_le_lift {c : Cardinal.{u}} : 𝔠 ≤ lift.{v} c ↔ 𝔠 ≤ c 
Arguments: {c : Cardinal.{u}}
Goal: Cardinal.continuum ≤ Cardinal.lift.{v, u} c ↔ Cardinal.continuum ≤ c
-- Cardinal.lift_le_continuum
@[simp]
theorem lift_le_continuum {c : Cardinal.{u}} : lift.{v} c ≤ 𝔠 ↔ c ≤ 𝔠 
Arguments: {c : Cardinal.{u}}
Goal: Cardinal.lift.{v, u} c ≤ Cardinal.continuum ↔ c ≤ Cardinal.continuum
-- Cardinal.continuum_lt_lift
@[simp]
theorem continuum_lt_lift {c : Cardinal.{u}} : 𝔠 < lift.{v} c ↔ 𝔠 < c 
Arguments: {c : Cardinal.{u}}
Goal: Cardinal.continuum < Cardinal.lift.{v, u} c ↔ Cardinal.continuum < c
-- Cardinal.lift_lt_continuum
@[simp]
theorem lift_lt_continuum {c : Cardinal.{u}} : lift.{v} c < 𝔠 ↔ c < 𝔠 
Arguments: {c : Cardinal.{u}}
Goal: Cardinal.lift.{v, u} c < Cardinal.continuum ↔ c < Cardinal.continuum
-- Cardinal.aleph0_lt_continuum
theorem aleph0_lt_continuum : ℵ₀ < 𝔠 
Goal: Cardinal.aleph0 < Cardinal.continuum
-- Cardinal.aleph0_le_continuum
theorem aleph0_le_continuum : ℵ₀ ≤ 𝔠 
Goal: Cardinal.aleph0 ≤ Cardinal.continuum
-- Cardinal.beth_one
@[simp]
theorem beth_one : beth 1 = 𝔠 
Goal: Cardinal.beth 1 = Cardinal.continuum
-- Cardinal.nat_lt_continuum
theorem nat_lt_continuum (n : ℕ) : ↑n < 𝔠 
Arguments: (n : ℕ)
Goal: ↑n < Cardinal.continuum
-- Cardinal.mk_set_nat
theorem mk_set_nat : #(Set ℕ) = 𝔠 
Goal: Cardinal.mk (Set ℕ) = Cardinal.continuum
-- Cardinal.continuum_pos
theorem continuum_pos : 0 < 𝔠 
Goal: 0 < Cardinal.continuum
-- Cardinal.continuum_ne_zero
theorem continuum_ne_zero : 𝔠 ≠ 0 
Goal: Cardinal.continuum ≠ 0
-- Cardinal.aleph_one_le_continuum
theorem aleph_one_le_continuum : aleph 1 ≤ 𝔠 
Goal: Cardinal.aleph 1 ≤ Cardinal.continuum
-- Cardinal.continuum_toNat
@[simp]
theorem continuum_toNat : toNat continuum = 0 
Goal: Cardinal.toNat Cardinal.continuum = 0
-- Cardinal.continuum_toPartENat
@[simp]
theorem continuum_toPartENat : toPartENat continuum = ⊤ 
Goal: Cardinal.toPartENat Cardinal.continuum = ⊤
-- Cardinal.aleph0_add_continuum
@[simp]
theorem aleph0_add_continuum : ℵ₀ + 𝔠 = 𝔠 
Goal: Cardinal.aleph0 + Cardinal.continuum = Cardinal.continuum
-- Cardinal.continuum_add_aleph0
@[simp]
theorem continuum_add_aleph0 : 𝔠 + ℵ₀ = 𝔠 
Goal: Cardinal.continuum + Cardinal.aleph0 = Cardinal.continuum
-- Cardinal.continuum_add_self
@[simp]
theorem continuum_add_self : 𝔠 + 𝔠 = 𝔠 
Goal: Cardinal.continuum + Cardinal.continuum = Cardinal.continuum
-- Cardinal.nat_add_continuum
@[simp]
theorem nat_add_continuum (n : ℕ) : ↑n + 𝔠 = 𝔠 
Arguments: (n : ℕ)
Goal: ↑n + Cardinal.continuum = Cardinal.continuum
-- Cardinal.continuum_add_nat
@[simp]
theorem continuum_add_nat (n : ℕ) : 𝔠 + n = 𝔠 
Arguments: (n : ℕ)
Goal: Cardinal.continuum + ↑n = Cardinal.continuum
-- Cardinal.ofNat_add_continuum
@[simp]
theorem ofNat_add_continuum {n : ℕ} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n) + 𝔠 = 𝔠 
Arguments: {n : ℕ} [n.AtLeastTwo]
Goal: OfNat.ofNat n + Cardinal.continuum = Cardinal.continuum
-- Cardinal.continuum_add_ofNat
@[simp]
theorem continuum_add_ofNat {n : ℕ} [Nat.AtLeastTwo n] : 𝔠 + no_index (OfNat.ofNat n) = 𝔠 
Arguments: {n : ℕ} [n.AtLeastTwo]
Goal: Cardinal.continuum + OfNat.ofNat n = Cardinal.continuum
-- Cardinal.continuum_mul_self
@[simp]
theorem continuum_mul_self : 𝔠 * 𝔠 = 𝔠 
Goal: Cardinal.continuum * Cardinal.continuum = Cardinal.continuum
-- Cardinal.continuum_mul_aleph0
@[simp]
theorem continuum_mul_aleph0 : 𝔠 * ℵ₀ = 𝔠 
Goal: Cardinal.continuum * Cardinal.aleph0 = Cardinal.continuum
-- Cardinal.aleph0_mul_continuum
@[simp]
theorem aleph0_mul_continuum : ℵ₀ * 𝔠 = 𝔠 
Goal: Cardinal.aleph0 * Cardinal.continuum = Cardinal.continuum
-- Cardinal.nat_mul_continuum
@[simp]
theorem nat_mul_continuum {n : ℕ} (hn : n ≠ 0) : ↑n * 𝔠 = 𝔠 
Arguments: {n : ℕ} (hn : n ≠ 0)
Goal: ↑n * Cardinal.continuum = Cardinal.continuum
-- Cardinal.continuum_mul_nat
@[simp]
theorem continuum_mul_nat {n : ℕ} (hn : n ≠ 0) : 𝔠 * n = 𝔠 
Arguments: {n : ℕ} (hn : n ≠ 0)
Goal: Cardinal.continuum * ↑n = Cardinal.continuum
-- Cardinal.ofNat_mul_continuum
@[simp]
theorem ofNat_mul_continuum {n : ℕ} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n) * 𝔠 = 𝔠 
Arguments: {n : ℕ} [n.AtLeastTwo]
Goal: OfNat.ofNat n * Cardinal.continuum = Cardinal.continuum
-- Cardinal.continuum_mul_ofNat
@[simp]
theorem continuum_mul_ofNat {n : ℕ} [Nat.AtLeastTwo n] : 𝔠 * no_index (OfNat.ofNat n) = 𝔠 
Arguments: {n : ℕ} [n.AtLeastTwo]
Goal: Cardinal.continuum * OfNat.ofNat n = Cardinal.continuum
-- Cardinal.aleph0_power_aleph0
@[simp]
theorem aleph0_power_aleph0 : aleph0.{u} ^ aleph0.{u} = 𝔠 
Goal: Cardinal.aleph0 ^ Cardinal.aleph0 = Cardinal.continuum
-- Cardinal.nat_power_aleph0
@[simp]
theorem nat_power_aleph0 {n : ℕ} (hn : 2 ≤ n) : (n ^ aleph0.{u} : Cardinal.{u}) = 𝔠 
Arguments: {n : ℕ} (hn : 2 ≤ n)
Goal: ↑n ^ Cardinal.aleph0 = Cardinal.continuum
-- Cardinal.continuum_power_aleph0
@[simp]
theorem continuum_power_aleph0 : continuum.{u} ^ aleph0.{u} = 𝔠 
Goal: Cardinal.continuum ^ Cardinal.aleph0 = Cardinal.continuum
-- univLE_iff_cardinal_le
theorem univLE_iff_cardinal_le : UnivLE.{u, v} ↔ univ.{u, v+1} ≤ univ.{v, u+1} 
Goal: UnivLE.{u, v} ↔ Cardinal.univ.{u, v + 1} ≤ Cardinal.univ.{v, u + 1}
-- univLE_total
theorem univLE_total : UnivLE.{u, v} ∨ UnivLE.{v, u} 
Goal: UnivLE.{u, v} ∨ UnivLE.{v, u}
-- SimplicialObject.Augmented.ExtraDegeneracy.s₀_comp_δ₁
@[ext]
structure ExtraDegeneracy (X : SimplicialObject.Augmented C) where
  s' : point.obj X ⟶ drop.obj X _[0]
  s : ∀ n : ℕ, drop.obj X _[n] ⟶ drop.obj X _[n + 1]
  s'_comp_ε : s' ≫ X.hom.app (op [0]) = 𝟙 _
  s₀_comp_δ₁ : s 0 ≫ X.left.δ 1 = X.hom.app (op [0]) ≫ s'
  s_comp_δ₀ : ∀ n : ℕ, s n ≫ X.left.δ 0 = 𝟙 _
  s_comp_δ :
    ∀ (n : ℕ) (i : Fin (n + 2)), s (n + 1) ≫ X.left.δ i.succ = X.left.δ i ≫ s n
  s_comp_σ :
    ∀ (n : ℕ) (i : Fin (n + 1)), s n ≫ X.left.σ i.succ = X.left.σ i ≫ s (n + 1)
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject.Augmented C} (self : SimplicialObject.Augmented.ExtraDegeneracy X)
Goal: self.s 0 ≫ X.left.δ 1 = X.hom.app (Opposite.op [0]) ≫ self.s'
-- SimplicialObject.Augmented.ExtraDegeneracy.s'_comp_ε
@[ext]
structure ExtraDegeneracy (X : SimplicialObject.Augmented C) where
  s' : point.obj X ⟶ drop.obj X _[0]
  s : ∀ n : ℕ, drop.obj X _[n] ⟶ drop.obj X _[n + 1]
  s'_comp_ε : s' ≫ X.hom.app (op [0]) = 𝟙 _
  s₀_comp_δ₁ : s 0 ≫ X.left.δ 1 = X.hom.app (op [0]) ≫ s'
  s_comp_δ₀ : ∀ n : ℕ, s n ≫ X.left.δ 0 = 𝟙 _
  s_comp_δ :
    ∀ (n : ℕ) (i : Fin (n + 2)), s (n + 1) ≫ X.left.δ i.succ = X.left.δ i ≫ s n
  s_comp_σ :
    ∀ (n : ℕ) (i : Fin (n + 1)), s n ≫ X.left.σ i.succ = X.left.σ i ≫ s (n + 1)
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject.Augmented C} (self : SimplicialObject.Augmented.ExtraDegeneracy X)
Goal: self.s' ≫ X.hom.app (Opposite.op [0]) = 𝟙 (CategoryTheory.SimplicialObject.Augmented.point.obj X)
-- SimplicialObject.Augmented.ExtraDegeneracy.s_comp_σ
@[ext]
structure ExtraDegeneracy (X : SimplicialObject.Augmented C) where
  s' : point.obj X ⟶ drop.obj X _[0]
  s : ∀ n : ℕ, drop.obj X _[n] ⟶ drop.obj X _[n + 1]
  s'_comp_ε : s' ≫ X.hom.app (op [0]) = 𝟙 _
  s₀_comp_δ₁ : s 0 ≫ X.left.δ 1 = X.hom.app (op [0]) ≫ s'
  s_comp_δ₀ : ∀ n : ℕ, s n ≫ X.left.δ 0 = 𝟙 _
  s_comp_δ :
    ∀ (n : ℕ) (i : Fin (n + 2)), s (n + 1) ≫ X.left.δ i.succ = X.left.δ i ≫ s n
  s_comp_σ :
    ∀ (n : ℕ) (i : Fin (n + 1)), s n ≫ X.left.σ i.succ = X.left.σ i ≫ s (n + 1)
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject.Augmented C} (self : SimplicialObject.Augmented.ExtraDegeneracy X) (n : ℕ) (i : Fin (n + 1))
Goal: self.s n ≫ X.left.σ i.succ = X.left.σ i ≫ self.s (n + 1)
-- SimplicialObject.Augmented.ExtraDegeneracy.s_comp_δ
@[ext]
structure ExtraDegeneracy (X : SimplicialObject.Augmented C) where
  s' : point.obj X ⟶ drop.obj X _[0]
  s : ∀ n : ℕ, drop.obj X _[n] ⟶ drop.obj X _[n + 1]
  s'_comp_ε : s' ≫ X.hom.app (op [0]) = 𝟙 _
  s₀_comp_δ₁ : s 0 ≫ X.left.δ 1 = X.hom.app (op [0]) ≫ s'
  s_comp_δ₀ : ∀ n : ℕ, s n ≫ X.left.δ 0 = 𝟙 _
  s_comp_δ :
    ∀ (n : ℕ) (i : Fin (n + 2)), s (n + 1) ≫ X.left.δ i.succ = X.left.δ i ≫ s n
  s_comp_σ :
    ∀ (n : ℕ) (i : Fin (n + 1)), s n ≫ X.left.σ i.succ = X.left.σ i ≫ s (n + 1)
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject.Augmented C} (self : SimplicialObject.Augmented.ExtraDegeneracy X) (n : ℕ) (i : Fin (n + 2))
Goal: self.s (n + 1) ≫ X.left.δ i.succ = X.left.δ i ≫ self.s n
-- SimplicialObject.Augmented.ExtraDegeneracy.s_comp_δ₀
@[ext]
structure ExtraDegeneracy (X : SimplicialObject.Augmented C) where
  s' : point.obj X ⟶ drop.obj X _[0]
  s : ∀ n : ℕ, drop.obj X _[n] ⟶ drop.obj X _[n + 1]
  s'_comp_ε : s' ≫ X.hom.app (op [0]) = 𝟙 _
  s₀_comp_δ₁ : s 0 ≫ X.left.δ 1 = X.hom.app (op [0]) ≫ s'
  s_comp_δ₀ : ∀ n : ℕ, s n ≫ X.left.δ 0 = 𝟙 _
  s_comp_δ :
    ∀ (n : ℕ) (i : Fin (n + 2)), s (n + 1) ≫ X.left.δ i.succ = X.left.δ i ≫ s n
  s_comp_σ :
    ∀ (n : ℕ) (i : Fin (n + 1)), s n ≫ X.left.σ i.succ = X.left.σ i ≫ s (n + 1)
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] {X : CategoryTheory.SimplicialObject.Augmented C} (self : SimplicialObject.Augmented.ExtraDegeneracy X) (n : ℕ)
Goal: self.s n ≫ X.left.δ 0 = 𝟙 (CategoryTheory.SimplicialObject.Augmented.drop.obj X _[n])
-- SSet.Augmented.StandardSimplex.shiftFun_0
@[simp]
theorem shiftFun_0 {n : ℕ} {X : Type*} [Zero X] (f : Fin n → X) : shiftFun f 0 = 0 
Arguments: {n : ℕ} {X : Type u_1} [Zero X] (f : Fin n → X)
Goal: SSet.Augmented.StandardSimplex.shiftFun f 0 = 0
-- SSet.Augmented.StandardSimplex.shiftFun_succ
@[simp]
theorem shiftFun_succ {n : ℕ} {X : Type*} [Zero X] (f : Fin n → X) (i : Fin n) :
    shiftFun f i.succ = f i 
Arguments: {n : ℕ} {X : Type u_1} [Zero X] (f : Fin n → X) (i : Fin n)
Goal: SSet.Augmented.StandardSimplex.shiftFun f i.succ = f i
-- CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_π_0
theorem ExtraDegeneracy.s_comp_π_0 (n : ℕ) :
    ExtraDegeneracy.s f S n ≫ WidePullback.π _ 0 =
      @WidePullback.base _ _ _ f.right (fun _ : Fin (n + 1) => f.left) (fun _ => f.hom) _ ≫
        S.section_ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (f : CategoryTheory.Arrow C) [∀ (n : ℕ), CategoryTheory.Limits.HasWidePullback f.right (fun x => f.left) fun x => f.hom] (S : CategoryTheory.SplitEpi f.hom) (n : ℕ)
Goal: CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s f S n ≫
    CategoryTheory.Limits.WidePullback.π (fun x => f.hom) 0 =
  (CategoryTheory.Limits.WidePullback.base fun x => f.hom) ≫ S.section_
-- CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_π_succ
theorem ExtraDegeneracy.s_comp_π_succ (n : ℕ) (i : Fin (n + 1)) :
    ExtraDegeneracy.s f S n ≫ WidePullback.π _ i.succ =
      @WidePullback.π _ _ _ f.right (fun _ : Fin (n + 1) => f.left) (fun _ => f.hom) _ i 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (f : CategoryTheory.Arrow C) [∀ (n : ℕ), CategoryTheory.Limits.HasWidePullback f.right (fun x => f.left) fun x => f.hom] (S : CategoryTheory.SplitEpi f.hom) (n : ℕ) (i : Fin (n + 1))
Goal: CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s f S n ≫
    CategoryTheory.Limits.WidePullback.π (fun x => f.hom) i.succ =
  CategoryTheory.Limits.WidePullback.π (fun x => f.hom) i
-- CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_base
theorem ExtraDegeneracy.s_comp_base (n : ℕ) :
    ExtraDegeneracy.s f S n ≫ WidePullback.base _ = WidePullback.base _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (f : CategoryTheory.Arrow C) [∀ (n : ℕ), CategoryTheory.Limits.HasWidePullback f.right (fun x => f.left) fun x => f.hom] (S : CategoryTheory.SplitEpi f.hom) (n : ℕ)
Goal: (CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s f S n ≫
    CategoryTheory.Limits.WidePullback.base fun x => f.hom) =
  CategoryTheory.Limits.WidePullback.base fun x => f.hom
-- _private.Mathlib.SetTheory.Cardinal.Subfield.0.Subfield.rangeOfWType_eq_top
private lemma rangeOfWType_eq_top : rangeOfWType s = ⊤ 
Arguments: {α : Type u} (s : Set α) [DivisionRing α]
Goal: Subfield.rangeOfWType s = ⊤
-- _private.Mathlib.SetTheory.Cardinal.Subfield.0.Subfield.surjective_ofWType
private lemma surjective_ofWType : Function.Surjective (WType.elim _ <| operate s) 
Arguments: {α : Type u} (s : Set α) [DivisionRing α]
Goal: Function.Surjective (WType.elim (↥(Subfield.closure s)) (Subfield.operate s))
-- Subfield.cardinal_mk_closure_le_max
lemma cardinal_mk_closure_le_max : #(closure s) ≤ max #s ℵ₀ 
Arguments: {α : Type u} (s : Set α) [DivisionRing α]
Goal: Cardinal.mk ↥(Subfield.closure s) ≤ max (Cardinal.mk ↑s) Cardinal.aleph0
-- Subfield.cardinal_mk_closure
lemma cardinal_mk_closure [Infinite s] : #(closure s) = #s 
Arguments: {α : Type u} (s : Set α) [DivisionRing α] [Infinite ↑s]
Goal: Cardinal.mk ↥(Subfield.closure s) = Cardinal.mk ↑s
-- AlgebraicTopology.DoldKan.compatibility_N₂_N₁_karoubi
theorem compatibility_N₂_N₁_karoubi :
    N₂ ⋙ (karoubiChainComplexEquivalence C ℕ).functor =
      karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙
        N₁ ⋙ (karoubiChainComplexEquivalence (Karoubi C) ℕ).functor ⋙
            Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C]
Goal: AlgebraicTopology.DoldKan.N₂ ⋙ (CategoryTheory.Idempotents.karoubiChainComplexEquivalence C ℕ).functor =
  CategoryTheory.Idempotents.karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙
    AlgebraicTopology.DoldKan.N₁ ⋙
      (CategoryTheory.Idempotents.karoubiChainComplexEquivalence (CategoryTheory.Idempotents.Karoubi C) ℕ).functor ⋙
        (CategoryTheory.Idempotents.KaroubiKaroubi.equivalence C).inverse.mapHomologicalComplex (ComplexShape.down ℕ)
-- AlgebraicTopology.DoldKan.P_is_eventually_constant
theorem P_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) :
    ((P (q + 1)).f n : X _[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} (hqn : n ≤ q)
Goal: (AlgebraicTopology.DoldKan.P (q + 1)).f n = (AlgebraicTopology.DoldKan.P q).f n
-- AlgebraicTopology.DoldKan.Q_is_eventually_constant
theorem Q_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) :
    ((Q (q + 1)).f n : X _[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} (hqn : n ≤ q)
Goal: (AlgebraicTopology.DoldKan.Q (q + 1)).f n = (AlgebraicTopology.DoldKan.Q q).f n
-- AlgebraicTopology.DoldKan.PInfty_f_0
@[simp]
theorem PInfty_f_0 : (PInfty.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}
Goal: AlgebraicTopology.DoldKan.PInfty.f 0 = 𝟙 (K[X].X 0)
-- AlgebraicTopology.DoldKan.PInfty_f
theorem PInfty_f (n : ℕ) : (PInfty.f n : X _[n] ⟶ X _[n]) = (P n).f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (n : ℕ)
Goal: AlgebraicTopology.DoldKan.PInfty.f n = (AlgebraicTopology.DoldKan.P n).f n
-- AlgebraicTopology.DoldKan.QInfty_f_0
@[simp]
theorem QInfty_f_0 : (QInfty.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}
Goal: AlgebraicTopology.DoldKan.QInfty.f 0 = 0
-- AlgebraicTopology.DoldKan.QInfty_f
theorem QInfty_f (n : ℕ) : (QInfty.f n : X _[n] ⟶ X _[n]) = (Q n).f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (n : ℕ)
Goal: AlgebraicTopology.DoldKan.QInfty.f n = (AlgebraicTopology.DoldKan.Q n).f n
-- AlgebraicTopology.DoldKan.PInfty_f_naturality
@[reassoc (attr := simp)]
theorem PInfty_f_naturality (n : ℕ) {X Y : SimplicialObject C} (f : X ⟶ Y) :
    f.app (op [n]) ≫ PInfty.f n = PInfty.f n ≫ f.app (op [n]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (n : ℕ) {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y)
Goal: f.app (Opposite.op [n]) ≫ AlgebraicTopology.DoldKan.PInfty.f n =
  AlgebraicTopology.DoldKan.PInfty.f n ≫ f.app (Opposite.op [n])
-- AlgebraicTopology.DoldKan.QInfty_f_naturality
@[reassoc (attr := simp)]
theorem QInfty_f_naturality (n : ℕ) {X Y : SimplicialObject C} (f : X ⟶ Y) :
    f.app (op [n]) ≫ QInfty.f n = QInfty.f n ≫ f.app (op [n]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (n : ℕ) {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y)
Goal: f.app (Opposite.op [n]) ≫ AlgebraicTopology.DoldKan.QInfty.f n =
  AlgebraicTopology.DoldKan.QInfty.f n ≫ f.app (Opposite.op [n])
-- AlgebraicTopology.DoldKan.PInfty_f_idem
@[reassoc (attr := simp)]
theorem PInfty_f_idem (n : ℕ) : (PInfty.f n : X _[n] ⟶ _) ≫ PInfty.f n = PInfty.f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (n : ℕ)
Goal: AlgebraicTopology.DoldKan.PInfty.f n ≫ AlgebraicTopology.DoldKan.PInfty.f n = AlgebraicTopology.DoldKan.PInfty.f n
-- AlgebraicTopology.DoldKan.PInfty_idem
@[reassoc (attr := simp)]
theorem PInfty_idem : (PInfty : K[X] ⟶ _) ≫ PInfty = PInfty 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C}
Goal: AlgebraicTopology.DoldKan.PInfty ≫ AlgebraicTopology.DoldKan.PInfty = AlgebraicTopology.DoldKan.PInfty
-- AlgebraicTopology.DoldKan.QInfty_f_idem
@[reassoc (attr := simp)]
theorem QInfty_f_idem (n : ℕ) : (QInfty.f n : X _[n] ⟶ _) ≫ QInfty.f n = QInfty.f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (n : ℕ)
Goal: AlgebraicTopology.DoldKan.QInfty.f n ≫ AlgebraicTopology.DoldKan.QInfty.f n = AlgebraicTopology.DoldKan.QInfty.f n
-- AlgebraicTopology.DoldKan.QInfty_idem
@[reassoc (attr := simp)]
theorem QInfty_idem : (QInfty : K[X] ⟶ _) ≫ QInfty = QInfty 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C}
Goal: AlgebraicTopology.DoldKan.QInfty ≫ AlgebraicTopology.DoldKan.QInfty = AlgebraicTopology.DoldKan.QInfty
-- AlgebraicTopology.DoldKan.PInfty_f_comp_QInfty_f
@[reassoc (attr := simp)]
theorem PInfty_f_comp_QInfty_f (n : ℕ) : (PInfty.f n : X _[n] ⟶ _) ≫ QInfty.f n = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (n : ℕ)
Goal: AlgebraicTopology.DoldKan.PInfty.f n ≫ AlgebraicTopology.DoldKan.QInfty.f n = 0
-- AlgebraicTopology.DoldKan.PInfty_comp_QInfty
@[reassoc (attr := simp)]
theorem PInfty_comp_QInfty : (PInfty : K[X] ⟶ _) ≫ QInfty = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C}
Goal: AlgebraicTopology.DoldKan.PInfty ≫ AlgebraicTopology.DoldKan.QInfty = 0
-- AlgebraicTopology.DoldKan.QInfty_f_comp_PInfty_f
@[reassoc (attr := simp)]
theorem QInfty_f_comp_PInfty_f (n : ℕ) : (QInfty.f n : X _[n] ⟶ _) ≫ 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)
Goal: AlgebraicTopology.DoldKan.QInfty.f n ≫ AlgebraicTopology.DoldKan.PInfty.f n = 0
-- AlgebraicTopology.DoldKan.QInfty_comp_PInfty
@[reassoc (attr := simp)]
theorem QInfty_comp_PInfty : (QInfty : K[X] ⟶ _) ≫ PInfty = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C}
Goal: AlgebraicTopology.DoldKan.QInfty ≫ AlgebraicTopology.DoldKan.PInfty = 0
-- AlgebraicTopology.DoldKan.PInfty_add_QInfty
@[simp]
theorem PInfty_add_QInfty : (PInfty : K[X] ⟶ _) + QInfty = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C}
Goal: AlgebraicTopology.DoldKan.PInfty + AlgebraicTopology.DoldKan.QInfty = 𝟙 K[X]
-- AlgebraicTopology.DoldKan.PInfty_f_add_QInfty_f
theorem PInfty_f_add_QInfty_f (n : ℕ) : (PInfty.f n : X _[n] ⟶ _) + QInfty.f 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.PInfty.f n + AlgebraicTopology.DoldKan.QInfty.f n = 𝟙 (K[X].X n)
-- AlgebraicTopology.DoldKan.natTransPInfty_app
@[simps]
noncomputable def natTransPInfty : alternatingFaceMapComplex C ⟶ alternatingFaceMapComplex C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C]
Goal: ∀ (x : CategoryTheory.SimplicialObject C),
  (AlgebraicTopology.DoldKan.natTransPInfty C).app x = AlgebraicTopology.DoldKan.PInfty