ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsPrimitiveRoot.card_quotient_toInteger_sub_one
lemma card_quotient_toInteger_sub_one [NumberField K] {k : ℕ+} (hk : 1 < k)
    (hζ : IsPrimitiveRoot ζ k) :
    Nat.card (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1}) =
      (Algebra.norm ℤ (hζ.toInteger - 1)).natAbs 
Arguments: {K : Type u} [Field K] {ζ : K} [NumberField K] {k : ℕ+} (hk : 1 < k) (hζ : IsPrimitiveRoot ζ ↑k)
Goal: Nat.card (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1}) = ((Algebra.norm ℤ) (hζ.toInteger - 1)).natAbs
-- IsPrimitiveRoot.toInteger_isPrimitiveRoot
lemma toInteger_isPrimitiveRoot {k : ℕ+} (hζ : IsPrimitiveRoot ζ k) :
    IsPrimitiveRoot hζ.toInteger k 
Arguments: {K : Type u} [Field K] {ζ : K} {k : ℕ+} (hζ : IsPrimitiveRoot ζ ↑k)
Goal: IsPrimitiveRoot hζ.toInteger ↑k
-- IsPrimitiveRoot.integralPowerBasis_gen
@[simp]
theorem integralPowerBasis_gen [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ k)) :
    hζ.integralPowerBasis.gen = hζ.toInteger 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p ^ k} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ k))
Goal: hζ.integralPowerBasis.gen = hζ.toInteger
-- IsPrimitiveRoot.integralPowerBasis_dim
@[simp]
theorem integralPowerBasis_dim [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ k)) : hζ.integralPowerBasis.dim = φ (p ^ k) 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p ^ k} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ k))
Goal: hζ.integralPowerBasis.dim = φ (↑p ^ k)
-- IsPrimitiveRoot.adjoinEquivRingOfIntegers'_symm_apply
@[simps!]
noncomputable def _root_.IsPrimitiveRoot.adjoinEquivRingOfIntegers'
    [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ p) :
    adjoin ℤ ({ζ} : Set K) ≃ₐ[ℤ] 𝓞 K 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p) (a : 𝓞 K)
Goal: hζ.adjoinEquivRingOfIntegers'.symm a = (IsIntegralClosure.lift ℤ (↥(Algebra.adjoin ℤ {ζ})) K) a
-- IsPrimitiveRoot.adjoinEquivRingOfIntegers'_apply
@[simps!]
noncomputable def _root_.IsPrimitiveRoot.adjoinEquivRingOfIntegers'
    [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ p) :
    adjoin ℤ ({ζ} : Set K) ≃ₐ[ℤ] 𝓞 K 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p) (a : ↥(Algebra.adjoin ℤ {ζ}))
Goal: hζ.adjoinEquivRingOfIntegers' a = (IsIntegralClosure.lift ℤ (𝓞 K) K) a
-- IsPrimitiveRoot.integralPowerBasis'_gen
@[simp]
theorem integralPowerBasis'_gen [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ p) :
    hζ.integralPowerBasis'.gen = hζ.toInteger 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p)
Goal: hζ.integralPowerBasis'.gen = hζ.toInteger
-- IsPrimitiveRoot.power_basis_int'_dim
@[simp]
theorem power_basis_int'_dim [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ p) :
    hζ.integralPowerBasis'.dim = φ p 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p)
Goal: hζ.integralPowerBasis'.dim = φ ↑p
-- IsPrimitiveRoot.subOneIntegralPowerBasis_gen
@[simp]
theorem subOneIntegralPowerBasis_gen [IsCyclotomicExtension {p ^ k} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ k)) :
    hζ.subOneIntegralPowerBasis.gen =
      ⟨ζ - 1, Subalgebra.sub_mem _ (hζ.isIntegral (p ^ k).pos) (Subalgebra.one_mem _)⟩ 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ k} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ k))
Goal: hζ.subOneIntegralPowerBasis.gen = ⟨ζ - 1, ⋯⟩
-- IsPrimitiveRoot.subOneIntegralPowerBasis'_gen
@[simp]
theorem subOneIntegralPowerBasis'_gen [hcycl : IsCyclotomicExtension {p} ℚ K]
    (hζ : IsPrimitiveRoot ζ p) :
    hζ.subOneIntegralPowerBasis'.gen = hζ.toInteger - 1 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p)
Goal: hζ.subOneIntegralPowerBasis'.gen = hζ.toInteger - 1
-- IsPrimitiveRoot.zeta_sub_one_prime_of_ne_two
theorem zeta_sub_one_prime_of_ne_two [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2) :
    Prime (hζ.toInteger - 1) 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2)
Goal: Prime (hζ.toInteger - 1)
-- IsPrimitiveRoot.zeta_sub_one_prime_of_two_pow
theorem zeta_sub_one_prime_of_two_pow [IsCyclotomicExtension {(2 : ℕ+) ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑((2 : ℕ+) ^ (k + 1))) :
    Prime (hζ.toInteger - 1) 
Arguments: {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [IsCyclotomicExtension {2 ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1)))
Goal: Prime (hζ.toInteger - 1)
-- IsPrimitiveRoot.zeta_sub_one_prime
theorem zeta_sub_one_prime [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) : Prime (hζ.toInteger - 1) 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1)))
Goal: Prime (hζ.toInteger - 1)
-- IsPrimitiveRoot.zeta_sub_one_prime'
theorem zeta_sub_one_prime' [h : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ p) :
    Prime ((hζ.toInteger - 1)) 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [h : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p)
Goal: Prime (hζ.toInteger - 1)
-- IsPrimitiveRoot.subOneIntegralPowerBasis_gen_prime
theorem subOneIntegralPowerBasis_gen_prime [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) :
    Prime hζ.subOneIntegralPowerBasis.gen 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1)))
Goal: Prime hζ.subOneIntegralPowerBasis.gen
-- IsPrimitiveRoot.subOneIntegralPowerBasis'_gen_prime
theorem subOneIntegralPowerBasis'_gen_prime [IsCyclotomicExtension {p} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑p) :
    Prime hζ.subOneIntegralPowerBasis'.gen 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p)
Goal: Prime hζ.subOneIntegralPowerBasis'.gen
-- IsPrimitiveRoot.norm_toInteger_pow_sub_one_of_prime_pow_ne_two
lemma norm_toInteger_pow_sub_one_of_prime_pow_ne_two [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) {s : ℕ} (hs : s ≤ k) (htwo : p ^ (k - s + 1) ≠ 2) :
    Algebra.norm ℤ (hζ.toInteger ^ (p : ℕ) ^ s - 1) = p ^ (p : ℕ) ^ s 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) {s : ℕ} (hs : s ≤ k) (htwo : p ^ (k - s + 1) ≠ 2)
Goal: (Algebra.norm ℤ) (hζ.toInteger ^ ↑p ^ s - 1) = ↑↑p ^ ↑p ^ s
-- IsPrimitiveRoot.norm_toInteger_pow_sub_one_of_two
lemma norm_toInteger_pow_sub_one_of_two [IsCyclotomicExtension {2 ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑((2 : ℕ+) ^ (k + 1))) :
    Algebra.norm ℤ (hζ.toInteger ^ 2 ^ k - 1) = (-2) ^ (2 : ℕ) ^ k 
Arguments: {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [IsCyclotomicExtension {2 ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1)))
Goal: (Algebra.norm ℤ) (hζ.toInteger ^ 2 ^ k - 1) = (-2) ^ 2 ^ k
-- IsPrimitiveRoot.norm_toInteger_pow_sub_one_of_prime_ne_two
lemma norm_toInteger_pow_sub_one_of_prime_ne_two [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) {s : ℕ} (hs : s ≤ k) (hodd : p ≠ 2) :
    Algebra.norm ℤ (hζ.toInteger ^ (p : ℕ) ^ s - 1) = p ^ (p : ℕ) ^ s 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) {s : ℕ} (hs : s ≤ k) (hodd : p ≠ 2)
Goal: (Algebra.norm ℤ) (hζ.toInteger ^ ↑p ^ s - 1) = ↑↑p ^ ↑p ^ s
-- IsPrimitiveRoot.norm_toInteger_sub_one_of_prime_ne_two
lemma norm_toInteger_sub_one_of_prime_ne_two [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2) :
    Algebra.norm ℤ (hζ.toInteger - 1) = p 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2)
Goal: (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑↑p
-- IsPrimitiveRoot.norm_toInteger_sub_one_of_prime_ne_two'
lemma norm_toInteger_sub_one_of_prime_ne_two' [hcycl : IsCyclotomicExtension {p} ℚ K]
    (hζ : IsPrimitiveRoot ζ p) (h : p ≠ 2) : Algebra.norm ℤ (hζ.toInteger - 1) = p 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p) (h : p ≠ 2)
Goal: (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑↑p
-- IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_pow_ne_two
lemma prime_norm_toInteger_sub_one_of_prime_pow_ne_two [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (htwo : p ^ (k + 1) ≠ 2) :
    Prime (Algebra.norm ℤ (hζ.toInteger - 1)) 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (htwo : p ^ (k + 1) ≠ 2)
Goal: Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))
-- IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_ne_two
lemma prime_norm_toInteger_sub_one_of_prime_ne_two [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2) :
    Prime (Algebra.norm ℤ (hζ.toInteger - 1)) 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2)
Goal: Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))
-- IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_ne_two'
lemma prime_norm_toInteger_sub_one_of_prime_ne_two' [hcycl : IsCyclotomicExtension {p} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑p) (hodd : p ≠ 2) :
    Prime (Algebra.norm ℤ (hζ.toInteger - 1)) 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p) (hodd : p ≠ 2)
Goal: Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))
-- IsPrimitiveRoot.not_exists_int_prime_dvd_sub_of_prime_pow_ne_two
theorem not_exists_int_prime_dvd_sub_of_prime_pow_ne_two
    [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (htwo : p ^ (k + 1) ≠ 2) :
    ¬(∃ n : ℤ, (p : 𝓞 K) ∣ (hζ.toInteger - n : 𝓞 K)) 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (htwo : p ^ (k + 1) ≠ 2)
Goal: ¬∃ n, ↑↑p ∣ hζ.toInteger - ↑n
-- IsPrimitiveRoot.not_exists_int_prime_dvd_sub_of_prime_ne_two
theorem not_exists_int_prime_dvd_sub_of_prime_ne_two
    [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2) :
    ¬(∃ n : ℤ, (p : 𝓞 K) ∣ (hζ.toInteger - n : 𝓞 K)) 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2)
Goal: ¬∃ n, ↑↑p ∣ hζ.toInteger - ↑n
-- IsPrimitiveRoot.not_exists_int_prime_dvd_sub_of_prime_ne_two'
theorem not_exists_int_prime_dvd_sub_of_prime_ne_two'
    [hcycl : IsCyclotomicExtension {p} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑p) (hodd : p ≠ 2) :
    ¬(∃ n : ℤ, (p : 𝓞 K) ∣ (hζ.toInteger - n : 𝓞 K)) 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p) (hodd : p ≠ 2)
Goal: ¬∃ n, ↑↑p ∣ hζ.toInteger - ↑n
-- IsPrimitiveRoot.finite_quotient_span_sub_one
theorem finite_quotient_span_sub_one [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) :
    Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1}) 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1)))
Goal: Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1})
-- IsPrimitiveRoot.finite_quotient_span_sub_one'
theorem finite_quotient_span_sub_one' [hcycl : IsCyclotomicExtension {p} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑p) :
    Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1}) 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p)
Goal: Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1})
-- IsPrimitiveRoot.toInteger_sub_one_dvd_prime
lemma toInteger_sub_one_dvd_prime [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) : ((hζ.toInteger - 1)) ∣ p 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1)))
Goal: hζ.toInteger - 1 ∣ ↑↑p
-- IsPrimitiveRoot.toInteger_sub_one_dvd_prime'
lemma toInteger_sub_one_dvd_prime' [hcycl : IsCyclotomicExtension {p} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑p) : ((hζ.toInteger - 1)) ∣ p 
Arguments: {p : ℕ+} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ ↑p)
Goal: hζ.toInteger - 1 ∣ ↑↑p
-- IsPrimitiveRoot.toInteger_sub_one_not_dvd_two
lemma toInteger_sub_one_not_dvd_two [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2) : ¬ hζ.toInteger - 1 ∣ 2 
Arguments: {p : ℕ+} {k : ℕ} {K : Type u} [Field K] [CharZero K] {ζ : K} [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2)
Goal: ¬hζ.toInteger - 1 ∣ 2
-- IsCyclotomicExtension.Rat.absdiscr_prime_pow
theorem absdiscr_prime_pow [IsCyclotomicExtension {p ^ k} ℚ K] :
    haveI : NumberField K := IsCyclotomicExtension.numberField {p ^ k} ℚ K
    NumberField.discr K =
    (-1) ^ ((p ^ k : ℕ).totient / 2) * p ^ ((p : ℕ) ^ (k - 1) * ((p - 1) * k - 1)) 
Arguments: (p : ℕ+) (k : ℕ) (K : Type u) [Field K] [CharZero K] [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ k} ℚ K]
Goal: NumberField.discr K = (-1) ^ (φ (↑p ^ k) / 2) * ↑↑p ^ (↑p ^ (k - 1) * ((↑p - 1) * k - 1))
-- IsCyclotomicExtension.Rat.absdiscr_prime_pow_succ
theorem absdiscr_prime_pow_succ [IsCyclotomicExtension {p ^ (k + 1)} ℚ K] :
    haveI : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
    NumberField.discr K =
    (-1) ^ ((p : ℕ) ^ k * (p - 1) / 2) * p ^ ((p : ℕ) ^ k * ((p - 1) * (k + 1) - 1)) 
Arguments: (p : ℕ+) (k : ℕ) (K : Type u) [Field K] [CharZero K] [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
Goal: NumberField.discr K = (-1) ^ (↑p ^ k * (↑p - 1) / 2) * ↑↑p ^ (↑p ^ k * ((↑p - 1) * (k + 1) - 1))
-- IsCyclotomicExtension.Rat.absdiscr_prime
theorem absdiscr_prime [IsCyclotomicExtension {p} ℚ K] :
    haveI : NumberField K := IsCyclotomicExtension.numberField {p} ℚ K
    NumberField.discr K = (-1) ^ (((p : ℕ) - 1) / 2) * p ^ ((p : ℕ) - 2) 
Arguments: (p : ℕ+) (K : Type u) [Field K] [CharZero K] [hp : Fact (Nat.Prime ↑p)] [IsCyclotomicExtension {p} ℚ K]
Goal: NumberField.discr K = (-1) ^ ((↑p - 1) / 2) * ↑↑p ^ (↑p - 2)
-- ClassGroup.normBound_pos
theorem normBound_pos : 0 < normBound abv bS 
Arguments: {R : Type u_1} {S : Type u_2} [EuclideanDomain R] [CommRing S] [IsDomain S] [Algebra R S] (abv : AbsoluteValue R ℤ) {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S)
Goal: 0 < ClassGroup.normBound abv bS
-- ClassGroup.norm_le
theorem norm_le (a : S) {y : ℤ} (hy : ∀ k, abv (bS.repr a k) ≤ y) :
    abv (Algebra.norm R a) ≤ normBound abv bS * y ^ Fintype.card ι 
Arguments: {R : Type u_1} {S : Type u_2} [EuclideanDomain R] [CommRing S] [IsDomain S] [Algebra R S] (abv : AbsoluteValue R ℤ) {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S) (a : S) {y : ℤ} (hy : ∀ (k : ι), abv ((bS.repr a) k) ≤ y)
Goal: abv ((Algebra.norm R) a) ≤ ClassGroup.normBound abv bS * y ^ Fintype.card ι
-- ClassGroup.norm_lt
theorem norm_lt {T : Type*} [LinearOrderedRing T] (a : S) {y : T}
    (hy : ∀ k, (abv (bS.repr a k) : T) < y) :
    (abv (Algebra.norm R a) : T) < normBound abv bS * y ^ Fintype.card ι 
Arguments: {R : Type u_1} {S : Type u_2} [EuclideanDomain R] [CommRing S] [IsDomain S] [Algebra R S] (abv : AbsoluteValue R ℤ) {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S) {T : Type u_6} [LinearOrderedRing T] (a : S) {y : T} (hy : ∀ (k : ι), ↑(abv ((bS.repr a) k)) < y)
Goal: ↑(abv ((Algebra.norm R) a)) < ↑(ClassGroup.normBound abv bS) * y ^ Fintype.card ι
-- ClassGroup.exists_min
theorem exists_min (I : (Ideal S)⁰) :
    ∃ b ∈ (I : Ideal S),
      b ≠ 0 ∧ ∀ c ∈ (I : Ideal S), abv (Algebra.norm R c) < abv (Algebra.norm R b) → c =
      (0 : S) 
Arguments: {R : Type u_1} {S : Type u_2} [EuclideanDomain R] [CommRing S] [IsDomain S] [Algebra R S] (abv : AbsoluteValue R ℤ) (I : ↥(Ideal S)⁰)
Goal: ∃ b ∈ ↑I, b ≠ 0 ∧ ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0
-- ClassGroup.finsetApprox.zero_not_mem
theorem finsetApprox.zero_not_mem : (0 : R) ∉ finsetApprox bS adm 
Arguments: {R : Type u_1} {S : Type u_2} [EuclideanDomain R] [CommRing S] [IsDomain S] [Algebra R S] {abv : AbsoluteValue R ℤ} {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S) (adm : abv.IsAdmissible) [Infinite R] [DecidableEq R]
Goal: 0 ∉ ClassGroup.finsetApprox bS adm
-- ClassGroup.mem_finsetApprox
@[simp]
theorem mem_finsetApprox {x : R} :
    x ∈ finsetApprox bS adm ↔ ∃ i j, i ≠ j ∧ distinctElems bS adm i - distinctElems bS adm j =
    x 
Arguments: {R : Type u_1} {S : Type u_2} [EuclideanDomain R] [CommRing S] [IsDomain S] [Algebra R S] {abv : AbsoluteValue R ℤ} {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S) (adm : abv.IsAdmissible) [Infinite R] [DecidableEq R] {x : R}
Goal: x ∈ ClassGroup.finsetApprox bS adm ↔
  ∃ i j, i ≠ j ∧ (ClassGroup.distinctElems bS adm) i - (ClassGroup.distinctElems bS adm) j = x
-- ClassGroup.exists_mem_finsetApprox
theorem exists_mem_finsetApprox (a : S) {b} (hb : b ≠ (0 : R)) :
    ∃ q : S,
      ∃ r ∈ finsetApprox bS adm, abv (Algebra.norm R (r • a - b • q)) <
      abv (Algebra.norm R (algebraMap R S b)) 
Arguments: {R : Type u_1} {S : Type u_2} [EuclideanDomain R] [CommRing S] [IsDomain S] [Algebra R S] {abv : AbsoluteValue R ℤ} {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S) (adm : abv.IsAdmissible) [Infinite R] [DecidableEq R] (a : S) {b : R} (hb : b ≠ 0)
Goal: ∃ q,
  ∃ r ∈ ClassGroup.finsetApprox bS adm,
    abv ((Algebra.norm R) (r • a - b • q)) < abv ((Algebra.norm R) ((algebraMap R S) b))
-- ClassGroup.exists_mem_finset_approx'
theorem exists_mem_finset_approx' [Algebra.IsAlgebraic R L] (a : S) {b : S} (hb : b ≠ 0) :
    ∃ q : S,
      ∃ r ∈ finsetApprox bS adm, abv (Algebra.norm R (r • a - q * b)) < abv (Algebra.norm R b) 
Arguments: {R : Type u_1} {S : Type u_2} (L : Type u_4) [EuclideanDomain R] [CommRing S] [IsDomain S] [Field L] [algRL : Algebra R L] [Algebra R S] [Algebra S L] [ist : IsScalarTower R S L] [iic : IsIntegralClosure S R L] {abv : AbsoluteValue R ℤ} {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S) (adm : abv.IsAdmissible) [Infinite R] [DecidableEq R] [Algebra.IsAlgebraic R L] (a : S) {b : S} (hb : b ≠ 0)
Goal: ∃ q, ∃ r ∈ ClassGroup.finsetApprox bS adm, abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) b)
-- ClassGroup.prod_finsetApprox_ne_zero
theorem prod_finsetApprox_ne_zero : algebraMap R S (∏ m ∈ finsetApprox bS adm, m) ≠ 0 
Arguments: {R : Type u_1} {S : Type u_2} [EuclideanDomain R] [CommRing S] [IsDomain S] [Algebra R S] {abv : AbsoluteValue R ℤ} {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S) (adm : abv.IsAdmissible) [Infinite R] [DecidableEq R]
Goal: (algebraMap R S) (∏ m ∈ ClassGroup.finsetApprox bS adm, m) ≠ 0
-- ClassGroup.ne_bot_of_prod_finsetApprox_mem
theorem ne_bot_of_prod_finsetApprox_mem (J : Ideal S)
    (h : algebraMap _ _ (∏ m ∈ finsetApprox bS adm, m) ∈ J) : J ≠ ⊥ 
Arguments: {R : Type u_1} {S : Type u_2} [EuclideanDomain R] [CommRing S] [IsDomain S] [Algebra R S] {abv : AbsoluteValue R ℤ} {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S) (adm : abv.IsAdmissible) [Infinite R] [DecidableEq R] (J : Ideal S) (h : (algebraMap R S) (∏ m ∈ ClassGroup.finsetApprox bS adm, m) ∈ J)
Goal: J ≠ ⊥
-- ClassGroup.exists_mk0_eq_mk0
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] [Algebra.IsAlgebraic R L] (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m ∈ finsetApprox bS adm, m) ∈ (J : Ideal S) 
Arguments: {R : Type u_1} {S : Type u_2} (L : Type u_4) [EuclideanDomain R] [CommRing S] [IsDomain S] [Field L] [algRL : Algebra R L] [Algebra R S] [Algebra S L] [ist : IsScalarTower R S L] [iic : IsIntegralClosure S R L] {abv : AbsoluteValue R ℤ} {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S) (adm : abv.IsAdmissible) [Infinite R] [DecidableEq R] [IsDedekindDomain S] [Algebra.IsAlgebraic R L] (I : ↥(Ideal S)⁰)
Goal: ∃ J, ClassGroup.mk0 I = ClassGroup.mk0 J ∧ (algebraMap R S) (∏ m ∈ ClassGroup.finsetApprox bS adm, m) ∈ ↑J
-- ClassGroup.mkMMem_surjective
theorem mkMMem_surjective [IsDedekindDomain S] [Algebra.IsAlgebraic R L] :
    Function.Surjective (ClassGroup.mkMMem bS adm) 
Arguments: {R : Type u_1} {S : Type u_2} (L : Type u_4) [EuclideanDomain R] [CommRing S] [IsDomain S] [Field L] [algRL : Algebra R L] [Algebra R S] [Algebra S L] [ist : IsScalarTower R S L] [iic : IsIntegralClosure S R L] {abv : AbsoluteValue R ℤ} {ι : Type u_5} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S) (adm : abv.IsAdmissible) [Infinite R] [DecidableEq R] [IsDedekindDomain S] [Algebra.IsAlgebraic R L]
Goal: Function.Surjective (ClassGroup.mkMMem bS adm)
-- rieszContentAux_image_nonempty
theorem rieszContentAux_image_nonempty (K : Compacts X) :
    (Λ '' { f : X →ᵇ ℝ≥0 | ∀ x ∈ K, (1 : ℝ≥0) ≤ f x }).Nonempty 
Arguments: {X : Type u_1} [TopologicalSpace X] (Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0) (K : TopologicalSpace.Compacts X)
Goal: (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x}).Nonempty
-- rieszContentAux_mono
theorem rieszContentAux_mono {K₁ K₂ : Compacts X} (h : K₁ ≤ K₂) :
    rieszContentAux Λ K₁ ≤ rieszContentAux Λ K₂ 
Arguments: {X : Type u_1} [TopologicalSpace X] (Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0) {K₁ K₂ : TopologicalSpace.Compacts X} (h : K₁ ≤ K₂)
Goal: rieszContentAux Λ K₁ ≤ rieszContentAux Λ K₂
-- rieszContentAux_le
theorem rieszContentAux_le {K : Compacts X} {f : X →ᵇ ℝ≥0} (h : ∀ x ∈ K, (1 : ℝ≥0) ≤ f x) :
    rieszContentAux Λ K ≤ Λ f 
Arguments: {X : Type u_1} [TopologicalSpace X] (Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0) {K : TopologicalSpace.Compacts X} {f : X →ᵇ ℝ≥0} (h : ∀ x ∈ K, 1 ≤ f x)
Goal: rieszContentAux Λ K ≤ Λ f
-- exists_lt_rieszContentAux_add_pos
theorem exists_lt_rieszContentAux_add_pos (K : Compacts X) {ε : ℝ≥0} (εpos : 0 < ε) :
    ∃ f : X →ᵇ ℝ≥0, (∀ x ∈ K, (1 : ℝ≥0) ≤ f x) ∧ Λ f < rieszContentAux Λ K + ε 
Arguments: {X : Type u_1} [TopologicalSpace X] (Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0) (K : TopologicalSpace.Compacts X) {ε : ℝ≥0} (εpos : 0 < ε)
Goal: ∃ f, (∀ x ∈ K, 1 ≤ f x) ∧ Λ f < rieszContentAux Λ K + ε
-- rieszContentAux_sup_le
theorem rieszContentAux_sup_le (K1 K2 : Compacts X) :
    rieszContentAux Λ (K1 ⊔ K2) ≤ rieszContentAux Λ K1 + rieszContentAux Λ K2 
Arguments: {X : Type u_1} [TopologicalSpace X] (Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0) (K1 K2 : TopologicalSpace.Compacts X)
Goal: rieszContentAux Λ (K1 ⊔ K2) ≤ rieszContentAux Λ K1 + rieszContentAux Λ K2