ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Polynomial.cyclotomic.roots_eq_primitiveRoots_val
theorem cyclotomic.roots_eq_primitiveRoots_val [NeZero (n : R)] :
    (cyclotomic n R).roots = (primitiveRoots n R).val 
Arguments: {R : Type u_1} [CommRing R] {n : ℕ} [IsDomain R] [NeZero ↑n]
Goal: (Polynomial.cyclotomic n R).roots = (primitiveRoots n R).val
-- Polynomial.isRoot_cyclotomic_iff_charZero
theorem isRoot_cyclotomic_iff_charZero {n : ℕ} {R : Type*} [CommRing R] [IsDomain R] [CharZero R]
    {μ : R} (hn : 0 < n) : (Polynomial.cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n 
Arguments: {n : ℕ} {R : Type u_2} [CommRing R] [IsDomain R] [CharZero R] {μ : R} (hn : 0 < n)
Goal: (Polynomial.cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n
-- Polynomial.cyclotomic_injective
theorem cyclotomic_injective [CharZero R] : Function.Injective fun n => cyclotomic n R 
Arguments: {R : Type u_1} [CommRing R] [CharZero R]
Goal: Function.Injective fun n => Polynomial.cyclotomic n R
-- IsPrimitiveRoot.minpoly_dvd_cyclotomic
theorem _root_.IsPrimitiveRoot.minpoly_dvd_cyclotomic {n : ℕ} {K : Type*} [Field K] {μ : K}
    (h : IsPrimitiveRoot μ n) (hpos : 0 < n) [CharZero K] : minpoly ℤ μ ∣ cyclotomic n ℤ 
Arguments: {n : ℕ} {K : Type u_2} [Field K] {μ : K} (h : IsPrimitiveRoot μ n) (hpos : 0 < n) [CharZero K]
Goal: minpoly ℤ μ ∣ Polynomial.cyclotomic n ℤ
-- IsPrimitiveRoot.minpoly_eq_cyclotomic_of_irreducible
theorem _root_.IsPrimitiveRoot.minpoly_eq_cyclotomic_of_irreducible {K : Type*} [Field K]
    {R : Type*} [CommRing R] [IsDomain R] {μ : R} {n : ℕ} [Algebra K R] (hμ : IsPrimitiveRoot μ n)
    (h : Irreducible <| cyclotomic n K) [NeZero (n : K)] : cyclotomic n K = minpoly K μ 
Arguments: {K : Type u_2} [Field K] {R : Type u_3} [CommRing R] [IsDomain R] {μ : R} {n : ℕ} [Algebra K R] (hμ : IsPrimitiveRoot μ n) (h : Irreducible (Polynomial.cyclotomic n K)) [NeZero ↑n]
Goal: Polynomial.cyclotomic n K = minpoly K μ
-- Polynomial.cyclotomic_eq_minpoly
theorem cyclotomic_eq_minpoly {n : ℕ} {K : Type*} [Field K] {μ : K} (h : IsPrimitiveRoot μ n)
    (hpos : 0 < n) [CharZero K] : cyclotomic n ℤ = minpoly ℤ μ 
Arguments: {n : ℕ} {K : Type u_2} [Field K] {μ : K} (h : IsPrimitiveRoot μ n) (hpos : 0 < n) [CharZero K]
Goal: Polynomial.cyclotomic n ℤ = minpoly ℤ μ
-- Polynomial.cyclotomic_eq_minpoly_rat
theorem cyclotomic_eq_minpoly_rat {n : ℕ} {K : Type*} [Field K] {μ : K} (h : IsPrimitiveRoot μ n)
    (hpos : 0 < n) [CharZero K] : cyclotomic n ℚ = minpoly ℚ μ 
Arguments: {n : ℕ} {K : Type u_2} [Field K] {μ : K} (h : IsPrimitiveRoot μ n) (hpos : 0 < n) [CharZero K]
Goal: Polynomial.cyclotomic n ℚ = minpoly ℚ μ
-- Polynomial.cyclotomic.irreducible
theorem cyclotomic.irreducible {n : ℕ} (hpos : 0 < n) : Irreducible (cyclotomic n ℤ) 
Arguments: {n : ℕ} (hpos : 0 < n)
Goal: Irreducible (Polynomial.cyclotomic n ℤ)
-- Polynomial.cyclotomic.irreducible_rat
theorem cyclotomic.irreducible_rat {n : ℕ} (hpos : 0 < n) : Irreducible (cyclotomic n ℚ) 
Arguments: {n : ℕ} (hpos : 0 < n)
Goal: Irreducible (Polynomial.cyclotomic n ℚ)
-- Polynomial.cyclotomic.isCoprime_rat
theorem cyclotomic.isCoprime_rat {n m : ℕ} (h : n ≠ m) :
    IsCoprime (cyclotomic n ℚ) (cyclotomic m ℚ) 
Arguments: {n m : ℕ} (h : n ≠ m)
Goal: IsCoprime (Polynomial.cyclotomic n ℚ) (Polynomial.cyclotomic m ℚ)
-- Polynomial.isWeaklyEisensteinAt_iff
@[mk_iff]
structure IsWeaklyEisensteinAt [CommSemiring R] (f : R[X]) (𝓟 : Ideal R) : Prop where
  mem : ∀ {n}, n < f.natDegree → f.coeff n ∈ 𝓟
Arguments: {R : Type u} [CommSemiring R] (f : R[X]) (𝓟 : Ideal R)
Goal: f.IsWeaklyEisensteinAt 𝓟 ↔ ∀ {n : ℕ}, n < f.natDegree → f.coeff n ∈ 𝓟
-- Polynomial.IsWeaklyEisensteinAt.mem
@[mk_iff]
structure IsWeaklyEisensteinAt [CommSemiring R] (f : R[X]) (𝓟 : Ideal R) : Prop where
  mem : ∀ {n}, n < f.natDegree → f.coeff n ∈ 𝓟
Arguments: {R : Type u} [CommSemiring R] {f : R[X]} {𝓟 : Ideal R} (self : f.IsWeaklyEisensteinAt 𝓟) {n : ℕ}
Goal: n < f.natDegree → f.coeff n ∈ 𝓟
-- Polynomial.IsEisensteinAt.mem
@[mk_iff]
structure IsEisensteinAt [CommSemiring R] (f : R[X]) (𝓟 : Ideal R) : Prop where
  leading : f.leadingCoeff ∉ 𝓟
  mem : ∀ {n}, n < f.natDegree → f.coeff n ∈ 𝓟
  not_mem : f.coeff 0 ∉ 𝓟 ^ 2
Arguments: {R : Type u} [CommSemiring R] {f : R[X]} {𝓟 : Ideal R} (self : f.IsEisensteinAt 𝓟) {n : ℕ}
Goal: n < f.natDegree → f.coeff n ∈ 𝓟
-- Polynomial.isEisensteinAt_iff
@[mk_iff]
structure IsEisensteinAt [CommSemiring R] (f : R[X]) (𝓟 : Ideal R) : Prop where
  leading : f.leadingCoeff ∉ 𝓟
  mem : ∀ {n}, n < f.natDegree → f.coeff n ∈ 𝓟
  not_mem : f.coeff 0 ∉ 𝓟 ^ 2
Arguments: {R : Type u} [CommSemiring R] (f : R[X]) (𝓟 : Ideal R)
Goal: f.IsEisensteinAt 𝓟 ↔ f.leadingCoeff ∉ 𝓟 ∧ (∀ {n : ℕ}, n < f.natDegree → f.coeff n ∈ 𝓟) ∧ f.coeff 0 ∉ 𝓟 ^ 2
-- Polynomial.IsEisensteinAt.leading
@[mk_iff]
structure IsEisensteinAt [CommSemiring R] (f : R[X]) (𝓟 : Ideal R) : Prop where
  leading : f.leadingCoeff ∉ 𝓟
  mem : ∀ {n}, n < f.natDegree → f.coeff n ∈ 𝓟
  not_mem : f.coeff 0 ∉ 𝓟 ^ 2
Arguments: {R : Type u} [CommSemiring R] {f : R[X]} {𝓟 : Ideal R} (self : f.IsEisensteinAt 𝓟)
Goal: f.leadingCoeff ∉ 𝓟