ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LSeriesSummable_of_abscissaOfAbsConv_lt_re
lemma LSeriesSummable_of_abscissaOfAbsConv_lt_re {f : ℕ → ℂ} {s : ℂ}
    (hs : abscissaOfAbsConv f < s.re) : LSeriesSummable f s 
Arguments: {f : ℕ → ℂ} {s : ℂ} (hs : LSeries.abscissaOfAbsConv f < ↑s.re)
Goal: LSeriesSummable f s
-- LSeriesSummable_lt_re_of_abscissaOfAbsConv_lt_re
lemma LSeriesSummable_lt_re_of_abscissaOfAbsConv_lt_re {f : ℕ → ℂ} {s : ℂ}
    (hs : abscissaOfAbsConv f < s.re) :
    ∃ x : ℝ, x < s.re ∧ LSeriesSummable f x 
Arguments: {f : ℕ → ℂ} {s : ℂ} (hs : LSeries.abscissaOfAbsConv f < ↑s.re)
Goal: ∃ x < s.re, LSeriesSummable f ↑x
-- LSeriesSummable.abscissaOfAbsConv_le
lemma LSeriesSummable.abscissaOfAbsConv_le {f : ℕ → ℂ} {s : ℂ} (h : LSeriesSummable f s) :
    abscissaOfAbsConv f ≤ s.re 
Arguments: {f : ℕ → ℂ} {s : ℂ} (h : LSeriesSummable f s)
Goal: LSeries.abscissaOfAbsConv f ≤ ↑s.re
-- LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable
lemma LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable {f : ℕ → ℂ} {x : ℝ}
    (h : ∀ y : ℝ, x < y → LSeriesSummable f y) :
    abscissaOfAbsConv f ≤ x 
Arguments: {f : ℕ → ℂ} {x : ℝ} (h : ∀ (y : ℝ), x < y → LSeriesSummable f ↑y)
Goal: LSeries.abscissaOfAbsConv f ≤ ↑x
-- LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable'
lemma LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable' {f : ℕ → ℂ} {x : EReal}
    (h : ∀ y : ℝ, x < y → LSeriesSummable f y) :
    abscissaOfAbsConv f ≤ x 
Arguments: {f : ℕ → ℂ} {x : EReal} (h : ∀ (y : ℝ), x < ↑y → LSeriesSummable f ↑y)
Goal: LSeries.abscissaOfAbsConv f ≤ x
-- LSeries.abscissaOfAbsConv_le_of_le_const_mul_rpow
lemma LSeries.abscissaOfAbsConv_le_of_le_const_mul_rpow {f : ℕ → ℂ} {x : ℝ}
    (h : ∃ C, ∀ n ≠ 0, ‖f n‖ ≤ C * n ^ x) : abscissaOfAbsConv f ≤ x + 1 
Arguments: {f : ℕ → ℂ} {x : ℝ} (h : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ x)
Goal: LSeries.abscissaOfAbsConv f ≤ ↑x + 1
-- LSeries.abscissaOfAbsConv_le_of_isBigO_rpow
lemma LSeries.abscissaOfAbsConv_le_of_isBigO_rpow {f : ℕ → ℂ} {x : ℝ}
    (h : f =O[atTop] fun n ↦ (n : ℝ) ^ x) :
    abscissaOfAbsConv f ≤ x + 1 
Arguments: {f : ℕ → ℂ} {x : ℝ} (h : f =O[Filter.atTop] fun n => ↑n ^ x)
Goal: LSeries.abscissaOfAbsConv f ≤ ↑x + 1
-- LSeries.abscissaOfAbsConv_le_of_le_const
lemma LSeries.abscissaOfAbsConv_le_of_le_const {f : ℕ → ℂ} (h : ∃ C, ∀ n ≠ 0, ‖f n‖ ≤ C) :
    abscissaOfAbsConv f ≤ 1 
Arguments: {f : ℕ → ℂ} (h : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C)
Goal: LSeries.abscissaOfAbsConv f ≤ 1
-- LSeries.abscissaOfAbsConv_le_one_of_isBigO_one
lemma LSeries.abscissaOfAbsConv_le_one_of_isBigO_one {f : ℕ → ℂ} (h : f =O[atTop] (1 : ℕ → ℝ)) :
    abscissaOfAbsConv f ≤ 1 
Arguments: {f : ℕ → ℂ} (h : f =O[Filter.atTop] 1)
Goal: LSeries.abscissaOfAbsConv f ≤ 1
-- conductor_eq_of_eq
theorem conductor_eq_of_eq {y : S} (h : (R<x> : Set S) = R<y>) : conductor R x = conductor R y 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {x y : S} (h : ↑(Algebra.adjoin R {x}) = ↑(Algebra.adjoin R {y}))
Goal: conductor R x = conductor R y
-- conductor_subset_adjoin
theorem conductor_subset_adjoin : (conductor R x : Set S) ⊆ R<x> 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {x : S}
Goal: ↑(conductor R x) ⊆ ↑(Algebra.adjoin R {x})
-- mem_conductor_iff
theorem mem_conductor_iff {y : S} : y ∈ conductor R x ↔ ∀ b : S, y * b ∈ R<x> 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {x y : S}
Goal: y ∈ conductor R x ↔ ∀ (b : S), y * b ∈ Algebra.adjoin R {x}
-- conductor_eq_top_of_adjoin_eq_top
theorem conductor_eq_top_of_adjoin_eq_top (h : R<x> = ⊤) : conductor R x = ⊤ 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {x : S} (h : Algebra.adjoin R {x} = ⊤)
Goal: conductor R x = ⊤
-- conductor_eq_top_of_powerBasis
theorem conductor_eq_top_of_powerBasis (pb : PowerBasis R S) : conductor R pb.gen = ⊤ 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (pb : PowerBasis R S)
Goal: conductor R pb.gen = ⊤
-- mem_coeSubmodule_conductor
lemma mem_coeSubmodule_conductor {L} [CommRing L] [Algebra S L] [Algebra R L]
    [IsScalarTower R S L] [NoZeroSMulDivisors S L] {x : S} {y : L} :
    y ∈ coeSubmodule L (conductor R x) ↔ ∀ z : S,
      y * (algebraMap S L) z ∈ Algebra.adjoin R {algebraMap S L x} 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {L : Type u_3} [CommRing L] [Algebra S L] [Algebra R L] [IsScalarTower R S L] [NoZeroSMulDivisors S L] {x : S} {y : L}
Goal: y ∈ IsLocalization.coeSubmodule L (conductor R x) ↔
  ∀ (z : S), y * (algebraMap S L) z ∈ Algebra.adjoin R {(algebraMap S L) x}
-- prod_mem_ideal_map_of_mem_conductor
theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
    (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
    algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {x : S} {I : Ideal R} {p : R} {z : S} (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ Ideal.map (algebraMap R S) I)
Goal: (algebraMap R S) p * z ∈
  ⇑(algebraMap (↥(Algebra.adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(Algebra.adjoin R {x})) I)
-- comap_map_eq_map_adjoin_of_coprime_conductor
theorem comap_map_eq_map_adjoin_of_coprime_conductor
    (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤)
    (h_alg : Function.Injective (algebraMap R<x> S)) :
    (I.map (algebraMap R S)).comap (algebraMap R<x> S) = I.map (algebraMap R R<x>) 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {x : S} {I : Ideal R} (hx : Ideal.comap (algebraMap R S) (conductor R x) ⊔ I = ⊤) (h_alg : Function.Injective ⇑(algebraMap (↥(Algebra.adjoin R {x})) S))
Goal: Ideal.comap (algebraMap (↥(Algebra.adjoin R {x})) S) (Ideal.map (algebraMap R S) I) =
  Ideal.map (algebraMap R ↥(Algebra.adjoin R {x})) I
-- quotAdjoinEquivQuotMap_apply_mk
@[simp]
theorem quotAdjoinEquivQuotMap_apply_mk (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤)
    (h_alg : Function.Injective (algebraMap R<x> S)) (a : R<x>) :
    quotAdjoinEquivQuotMap hx h_alg (Ideal.Quotient.mk (I.map (algebraMap R R<x>)) a) =
      Ideal.Quotient.mk (I.map (algebraMap R S)) ↑a 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {x : S} {I : Ideal R} (hx : Ideal.comap (algebraMap R S) (conductor R x) ⊔ I = ⊤) (h_alg : Function.Injective ⇑(algebraMap (↥(Algebra.adjoin R {x})) S)) (a : ↥(Algebra.adjoin R {x}))
Goal: (quotAdjoinEquivQuotMap hx h_alg) ((Ideal.Quotient.mk (Ideal.map (algebraMap R ↥(Algebra.adjoin R {x})) I)) a) =
  (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) ↑a
-- KummerDedekind.multiplicity_factors_map_eq_multiplicity
theorem multiplicity_factors_map_eq_multiplicity (hI : IsMaximal I) (hI' : I ≠ ⊥)
    (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) {J : Ideal S}
    (hJ : J ∈ normalizedFactors (I.map (algebraMap R S))) :
    multiplicity J (I.map (algebraMap R S)) =
      multiplicity (↑(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx' ⟨J, hJ⟩))
        (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {x : S} {I : Ideal R} [IsDomain R] [IsIntegrallyClosed R] [IsDedekindDomain S] [NoZeroSMulDivisors R S] (hI : I.IsMaximal) (hI' : I ≠ ⊥) (hx : Ideal.comap (algebraMap R S) (conductor R x) ⊔ I = ⊤) (hx' : IsIntegral R x) {J : Ideal S} (hJ : J ∈ UniqueFactorizationMonoid.normalizedFactors (Ideal.map (algebraMap R S) I))
Goal: multiplicity J (Ideal.map (algebraMap R S) I) =
  multiplicity (↑((KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩))
    (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
-- KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map
theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    normalizedFactors (I.map (algebraMap R S)) =
      Multiset.map
        (fun f =>
          ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {x : S} {I : Ideal R} [IsDomain R] [IsIntegrallyClosed R] [IsDedekindDomain S] [NoZeroSMulDivisors R S] (hI : I.IsMaximal) (hI' : I ≠ ⊥) (hx : Ideal.comap (algebraMap R S) (conductor R x) ⊔ I = ⊤) (hx' : IsIntegral R x)
Goal: UniqueFactorizationMonoid.normalizedFactors (Ideal.map (algebraMap R S) I) =
  Multiset.map (fun f => ↑((KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
    (UniqueFactorizationMonoid.normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach
-- KummerDedekind.Ideal.irreducible_map_of_irreducible_minpoly
theorem Ideal.irreducible_map_of_irreducible_minpoly (hI : IsMaximal I) (hI' : I ≠ ⊥)
    (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x)
    (hf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) :
    Irreducible (I.map (algebraMap R S)) 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {x : S} {I : Ideal R} [IsDomain R] [IsIntegrallyClosed R] [IsDedekindDomain S] [NoZeroSMulDivisors R S] (hI : I.IsMaximal) (hI' : I ≠ ⊥) (hx : Ideal.comap (algebraMap R S) (conductor R x) ⊔ I = ⊤) (hx' : IsIntegral R x) (hf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
Goal: Irreducible (Ideal.map (algebraMap R S) I)
-- HurwitzZeta.hurwitzZetaEven_eq
lemma hurwitzZetaEven_eq (a : UnitAddCircle) (s : ℂ) :
    hurwitzZetaEven a s = (hurwitzZeta a s + hurwitzZeta (-a) s) / 2 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.hurwitzZetaEven a s = (HurwitzZeta.hurwitzZeta a s + HurwitzZeta.hurwitzZeta (-a) s) / 2
-- HurwitzZeta.hurwitzZetaOdd_eq
lemma hurwitzZetaOdd_eq (a : UnitAddCircle) (s : ℂ) :
    hurwitzZetaOdd a s = (hurwitzZeta a s - hurwitzZeta (-a) s) / 2 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.hurwitzZetaOdd a s = (HurwitzZeta.hurwitzZeta a s - HurwitzZeta.hurwitzZeta (-a) s) / 2
-- HurwitzZeta.differentiableAt_hurwitzZeta
lemma differentiableAt_hurwitzZeta (a : UnitAddCircle) {s : ℂ} (hs : s ≠ 1) :
    DifferentiableAt ℂ (hurwitzZeta a) s 
Arguments: (a : UnitAddCircle) {s : ℂ} (hs : s ≠ 1)
Goal: DifferentiableAt ℂ (HurwitzZeta.hurwitzZeta a) s
-- HurwitzZeta.hasSum_hurwitzZeta_of_one_lt_re
lemma hasSum_hurwitzZeta_of_one_lt_re {a : ℝ} (ha : a ∈ Icc 0 1) {s : ℂ} (hs : 1 < re s) :
    HasSum (fun n : ℕ ↦ 1 / (n + a : ℂ) ^ s) (hurwitzZeta a s) 
Arguments: {a : ℝ} (ha : a ∈ Set.Icc 0 1) {s : ℂ} (hs : 1 < s.re)
Goal: HasSum (fun n => 1 / (↑n + ↑a) ^ s) (HurwitzZeta.hurwitzZeta (↑a) s)
-- HurwitzZeta.hurwitzZeta_residue_one
lemma hurwitzZeta_residue_one (a : UnitAddCircle) :
    Tendsto (fun s ↦ (s - 1) * hurwitzZeta a s) (𝓝[≠] 1) (𝓝 1) 
Arguments: (a : UnitAddCircle)
Goal: Filter.Tendsto (fun s => (s - 1) * HurwitzZeta.hurwitzZeta a s) (𝓝[≠] 1) (𝓝 1)
-- HurwitzZeta.differentiableAt_hurwitzZeta_sub_one_div
lemma differentiableAt_hurwitzZeta_sub_one_div (a : UnitAddCircle) :
    DifferentiableAt ℂ (fun s ↦ hurwitzZeta a s - 1 / (s - 1) / Gammaℝ s) 1 
Arguments: (a : UnitAddCircle)
Goal: DifferentiableAt ℂ (fun s => HurwitzZeta.hurwitzZeta a s - 1 / (s - 1) / s.Gammaℝ) 1
-- HurwitzZeta.tendsto_hurwitzZeta_sub_one_div_nhds_one
lemma tendsto_hurwitzZeta_sub_one_div_nhds_one (a : UnitAddCircle) :
    Tendsto (fun s ↦ hurwitzZeta a s - 1 / (s - 1) / Gammaℝ s) (𝓝 1) (𝓝 (hurwitzZeta a 1)) 
Arguments: (a : UnitAddCircle)
Goal: Filter.Tendsto (fun s => HurwitzZeta.hurwitzZeta a s - 1 / (s - 1) / s.Gammaℝ) (𝓝 1) (𝓝 (HurwitzZeta.hurwitzZeta a 1))
-- HurwitzZeta.differentiable_hurwitzZeta_sub_hurwitzZeta
lemma differentiable_hurwitzZeta_sub_hurwitzZeta (a b : UnitAddCircle) :
    Differentiable ℂ (fun s ↦ hurwitzZeta a s - hurwitzZeta b s) 
Arguments: (a b : UnitAddCircle)
Goal: Differentiable ℂ fun s => HurwitzZeta.hurwitzZeta a s - HurwitzZeta.hurwitzZeta b s
-- HurwitzZeta.cosZeta_eq
lemma cosZeta_eq (a : UnitAddCircle) (s : ℂ) :
    cosZeta a s = (expZeta a s + expZeta (-a) s) / 2 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.cosZeta a s = (HurwitzZeta.expZeta a s + HurwitzZeta.expZeta (-a) s) / 2
-- HurwitzZeta.sinZeta_eq
lemma sinZeta_eq (a : UnitAddCircle) (s : ℂ) :
    sinZeta a s = (expZeta a s - expZeta (-a) s) / (2 * I) 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.sinZeta a s = (HurwitzZeta.expZeta a s - HurwitzZeta.expZeta (-a) s) / (2 * Complex.I)
-- HurwitzZeta.hasSum_expZeta_of_one_lt_re
lemma hasSum_expZeta_of_one_lt_re (a : ℝ) {s : ℂ} (hs : 1 < re s) :
    HasSum (fun n : ℕ ↦ cexp (2 * π * I * a * n) / n ^ s) (expZeta a s) 
Arguments: (a : ℝ) {s : ℂ} (hs : 1 < s.re)
Goal: HasSum (fun n => cexp (2 * ↑π * Complex.I * ↑a * ↑n) / ↑n ^ s) (HurwitzZeta.expZeta (↑a) s)
-- HurwitzZeta.differentiableAt_expZeta
lemma differentiableAt_expZeta (a : UnitAddCircle) (s : ℂ) (hs : s ≠ 1 ∨ a ≠ 0) :
    DifferentiableAt ℂ (expZeta a) s 
Arguments: (a : UnitAddCircle) (s : ℂ) (hs : s ≠ 1 ∨ a ≠ 0)
Goal: DifferentiableAt ℂ (HurwitzZeta.expZeta a) s
-- HurwitzZeta.differentiable_expZeta_of_ne_zero
lemma differentiable_expZeta_of_ne_zero {a : UnitAddCircle} (ha : a ≠ 0) :
    Differentiable ℂ (expZeta a) 
Arguments: {a : UnitAddCircle} (ha : a ≠ 0)
Goal: Differentiable ℂ (HurwitzZeta.expZeta a)
-- HurwitzZeta.LSeriesHasSum_exp
lemma LSeriesHasSum_exp (a : ℝ) {s : ℂ} (hs : 1 < re s) :
    LSeriesHasSum (cexp <| 2 * π * I * a * ·) s (expZeta a s) 
Arguments: (a : ℝ) {s : ℂ} (hs : 1 < s.re)
Goal: LSeriesHasSum (fun x => cexp (2 * ↑π * Complex.I * ↑a * ↑x)) s (HurwitzZeta.expZeta (↑a) s)
-- HurwitzZeta.hurwitzZeta_one_sub
lemma hurwitzZeta_one_sub (a : UnitAddCircle) {s : ℂ}
    (hs : ∀ (n : ℕ), s ≠ -n) (hs' : a ≠ 0 ∨ s ≠ 1) :
    hurwitzZeta a (1 - s) = (2 * π) ^ (-s) * Gamma s *
    (exp (-π * I * s / 2) * expZeta a s + exp (π * I * s / 2) * expZeta (-a) s) 
Arguments: (a : UnitAddCircle) {s : ℂ} (hs : ∀ (n : ℕ), s ≠ -↑n) (hs' : a ≠ 0 ∨ s ≠ 1)
Goal: HurwitzZeta.hurwitzZeta a (1 - s) =
  (2 * ↑π) ^ (-s) * Complex.Gamma s *
    (cexp (-↑π * Complex.I * s / 2) * HurwitzZeta.expZeta a s +
      cexp (↑π * Complex.I * s / 2) * HurwitzZeta.expZeta (-a) s)
-- HurwitzZeta.expZeta_one_sub
lemma expZeta_one_sub (a : UnitAddCircle) {s : ℂ} (hs : ∀ (n : ℕ), s ≠ 1 - n) :
    expZeta a (1 - s) = (2 * π) ^ (-s) * Gamma s *
    (exp (π * I * s / 2) * hurwitzZeta a s + exp (-π * I * s / 2) * hurwitzZeta (-a) s) 
Arguments: (a : UnitAddCircle) {s : ℂ} (hs : ∀ (n : ℕ), s ≠ 1 - ↑n)
Goal: HurwitzZeta.expZeta a (1 - s) =
  (2 * ↑π) ^ (-s) * Complex.Gamma s *
    (cexp (↑π * Complex.I * s / 2) * HurwitzZeta.hurwitzZeta a s +
      cexp (-↑π * Complex.I * s / 2) * HurwitzZeta.hurwitzZeta (-a) s)
-- gaussSum_mulShift
theorem gaussSum_mulShift (χ : MulChar R R') (ψ : AddChar R R') (a : Rˣ) :
    χ a * gaussSum χ (mulShift ψ a) = gaussSum χ ψ 
Arguments: {R : Type u} [CommRing R] [Fintype R] {R' : Type v} [CommRing R'] (χ : MulChar R R') (ψ : AddChar R R') (a : Rˣ)
Goal: χ ↑a * gaussSum χ (ψ.mulShift ↑a) = gaussSum χ ψ
-- _private.Mathlib.NumberTheory.GaussSum.0.gaussSum_mul_aux
private theorem gaussSum_mul_aux {χ : MulChar R R'} (hχ : χ ≠ 1) (ψ : AddChar R R')
    (b : R) :
    ∑ a, χ (a * b⁻¹) * ψ (a - b) = ∑ c, χ c * ψ (b * (c - 1)) 
Arguments: {R : Type u} [Field R] [Fintype R] {R' : Type v} [CommRing R'] [IsDomain R'] {χ : MulChar R R'} (hχ : χ ≠ 1) (ψ : AddChar R R') (b : R)
Goal: ∑ a : R, χ (a * b⁻¹) * ψ (a - b) = ∑ c : R, χ c * ψ (b * (c - 1))
-- gaussSum_mul_gaussSum_eq_card
theorem gaussSum_mul_gaussSum_eq_card {χ : MulChar R R'} (hχ : χ ≠ 1) {ψ : AddChar R R'}
    (hψ : IsPrimitive ψ) :
    gaussSum χ ψ * gaussSum χ⁻¹ ψ⁻¹ = Fintype.card R 
Arguments: {R : Type u} [Field R] [Fintype R] {R' : Type v} [CommRing R'] [IsDomain R'] {χ : MulChar R R'} (hχ : χ ≠ 1) {ψ : AddChar R R'} (hψ : ψ.IsPrimitive)
Goal: gaussSum χ ψ * gaussSum χ⁻¹ ψ⁻¹ = ↑(Fintype.card R)
-- gaussSum_sq
theorem gaussSum_sq {χ : MulChar R R'} (hχ₁ : χ ≠ 1) (hχ₂ : IsQuadratic χ)
    {ψ : AddChar R R'} (hψ : IsPrimitive ψ) :
    gaussSum χ ψ ^ 2 = χ (-1) * Fintype.card R 
Arguments: {R : Type u} [Field R] [Fintype R] {R' : Type v} [CommRing R'] [IsDomain R'] {χ : MulChar R R'} (hχ₁ : χ ≠ 1) (hχ₂ : χ.IsQuadratic) {ψ : AddChar R R'} (hψ : ψ.IsPrimitive)
Goal: gaussSum χ ψ ^ 2 = χ (-1) * ↑(Fintype.card R)
-- gaussSum_frob
theorem gaussSum_frob (χ : MulChar R R') (ψ : AddChar R R') :
    gaussSum χ ψ ^ p = gaussSum (χ ^ p) (ψ ^ p) 
Arguments: {R : Type u} [CommRing R] [Fintype R] {R' : Type v} [CommRing R'] (p : ℕ) [fp : Fact (Nat.Prime p)] [hch : CharP R' p] (χ : MulChar R R') (ψ : AddChar R R')
Goal: gaussSum χ ψ ^ p = gaussSum (χ ^ p) (ψ ^ p)
-- MulChar.IsQuadratic.gaussSum_frob
theorem MulChar.IsQuadratic.gaussSum_frob (hp : IsUnit (p : R)) {χ : MulChar R R'}
    (hχ : IsQuadratic χ) (ψ : AddChar R R') :
    gaussSum χ ψ ^ p = χ p * gaussSum χ ψ 
Arguments: {R : Type u} [CommRing R] [Fintype R] {R' : Type v} [CommRing R'] (p : ℕ) [fp : Fact (Nat.Prime p)] [hch : CharP R' p] (hp : IsUnit ↑p) {χ : MulChar R R'} (hχ : χ.IsQuadratic) (ψ : AddChar R R')
Goal: gaussSum χ ψ ^ p = χ ↑p * gaussSum χ ψ
-- MulChar.IsQuadratic.gaussSum_frob_iter
theorem MulChar.IsQuadratic.gaussSum_frob_iter (n : ℕ) (hp : IsUnit (p : R)) {χ : MulChar R R'}
    (hχ : IsQuadratic χ) (ψ : AddChar R R') :
    gaussSum χ ψ ^ p ^ n = χ ((p : R) ^ n) * gaussSum χ ψ 
Arguments: {R : Type u} [CommRing R] [Fintype R] {R' : Type v} [CommRing R'] (p : ℕ) [fp : Fact (Nat.Prime p)] [hch : CharP R' p] (n : ℕ) (hp : IsUnit ↑p) {χ : MulChar R R'} (hχ : χ.IsQuadratic) (ψ : AddChar R R')
Goal: gaussSum χ ψ ^ p ^ n = χ (↑p ^ n) * gaussSum χ ψ
-- Char.card_pow_char_pow
theorem Char.card_pow_char_pow {χ : MulChar R R'} (hχ : IsQuadratic χ) (ψ : AddChar R R') (p n : ℕ)
    [fp : Fact p.Prime] [hch : CharP R' p] (hp : IsUnit (p : R)) (hp' : p ≠ 2)
    (hg : gaussSum χ ψ ^ 2 = χ (-1) * Fintype.card R) :
    (χ (-1) * Fintype.card R) ^ (p ^ n / 2) = χ ((p : R) ^ n) 
Arguments: {R : Type u} [CommRing R] [Fintype R] {R' : Type v} [CommRing R'] [IsDomain R'] {χ : MulChar R R'} (hχ : χ.IsQuadratic) (ψ : AddChar R R') (p n : ℕ) [fp : Fact (Nat.Prime p)] [hch : CharP R' p] (hp : IsUnit ↑p) (hp' : p ≠ 2) (hg : gaussSum χ ψ ^ 2 = χ (-1) * ↑(Fintype.card R))
Goal: (χ (-1) * ↑(Fintype.card R)) ^ (p ^ n / 2) = χ (↑p ^ n)
-- Char.card_pow_card
theorem Char.card_pow_card {F : Type*} [Field F] [Fintype F] {F' : Type*} [Field F'] [Fintype F']
    {χ : MulChar F F'} (hχ₁ : χ ≠ 1) (hχ₂ : IsQuadratic χ)
    (hch₁ : ringChar F' ≠ ringChar F) (hch₂ : ringChar F' ≠ 2) :
    (χ (-1) * Fintype.card F) ^ (Fintype.card F' / 2) = χ (Fintype.card F') 
Arguments: {F : Type u_1} [Field F] [Fintype F] {F' : Type u_2} [Field F'] [Fintype F'] {χ : MulChar F F'} (hχ₁ : χ ≠ 1) (hχ₂ : χ.IsQuadratic) (hch₁ : ringChar F' ≠ ringChar F) (hch₂ : ringChar F' ≠ 2)
Goal: (χ (-1) * ↑(Fintype.card F)) ^ (Fintype.card F' / 2) = χ ↑(Fintype.card F')
-- FiniteField.two_pow_card
theorem FiniteField.two_pow_card {F : Type*} [Fintype F] [Field F] (hF : ringChar F ≠ 2) :
    (2 : F) ^ (Fintype.card F / 2) = χ₈ (Fintype.card F) 
Arguments: {F : Type u_1} [Fintype F] [Field F] (hF : ringChar F ≠ 2)
Goal: 2 ^ (Fintype.card F / 2) = ↑(ZMod.χ₈ ↑(Fintype.card F))
-- hasSum_mellin
lemma hasSum_mellin {a : ι → ℂ} {p : ι → ℝ} {F : ℝ → ℂ} {s : ℂ}
    (hp : ∀ i, a i = 0 ∨ 0 < p i) (hs : 0 < s.re)
    (hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * rexp (-p i * t)) (F t))
    (h_sum : Summable fun i ↦ ‖a i‖ / (p i) ^ s.re) :
    HasSum (fun i ↦ Gamma s * a i / p i ^ s) (mellin F s) 
Arguments: {ι : Type u_1} [Countable ι] {a : ι → ℂ} {p : ι → ℝ} {F : ℝ → ℂ} {s : ℂ} (hp : ∀ (i : ι), a i = 0 ∨ 0 < p i) (hs : 0 < s.re) (hF : ∀ t ∈ Set.Ioi 0, HasSum (fun i => a i * ↑(rexp (-p i * t))) (F t)) (h_sum : Summable fun i => ‖a i‖ / p i ^ s.re)
Goal: HasSum (fun i => Complex.Gamma s * a i / ↑(p i) ^ s) (mellin F s)
-- hasSum_mellin_pi_mul
lemma hasSum_mellin_pi_mul {a : ι → ℂ} {q : ι → ℝ} {F : ℝ → ℂ} {s : ℂ}
    (hq : ∀ i, a i = 0 ∨ 0 < q i) (hs : 0 < s.re)
    (hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * rexp (-π * q i * t)) (F t))
    (h_sum : Summable fun i ↦ ‖a i‖ / (q i) ^ s.re) :
    HasSum (fun i ↦ π ^ (-s) * Gamma s * a i / q i ^ s) (mellin F s) 
Arguments: {ι : Type u_1} [Countable ι] {a : ι → ℂ} {q : ι → ℝ} {F : ℝ → ℂ} {s : ℂ} (hq : ∀ (i : ι), a i = 0 ∨ 0 < q i) (hs : 0 < s.re) (hF : ∀ t ∈ Set.Ioi 0, HasSum (fun i => a i * ↑(rexp (-π * q i * t))) (F t)) (h_sum : Summable fun i => ‖a i‖ / q i ^ s.re)
Goal: HasSum (fun i => ↑π ^ (-s) * Complex.Gamma s * a i / ↑(q i) ^ s) (mellin F s)
-- hasSum_mellin_pi_mul₀
lemma hasSum_mellin_pi_mul₀ {a : ι → ℂ} {p : ι → ℝ} {F : ℝ → ℂ} {s : ℂ}
    (hp : ∀ i, 0 ≤ p i) (hs : 0 < s.re)
    (hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if p i = 0 then 0 else a i * rexp (-π * p i * t)) (F t))
    (h_sum : Summable fun i ↦ ‖a i‖ / (p i) ^ s.re) :
    HasSum (fun i ↦ π ^ (-s) * Gamma s * a i / p i ^ s) (mellin F s) 
Arguments: {ι : Type u_1} [Countable ι] {a : ι → ℂ} {p : ι → ℝ} {F : ℝ → ℂ} {s : ℂ} (hp : ∀ (i : ι), 0 ≤ p i) (hs : 0 < s.re) (hF : ∀ t ∈ Set.Ioi 0, HasSum (fun i => if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) (F t)) (h_sum : Summable fun i => ‖a i‖ / p i ^ s.re)
Goal: HasSum (fun i => ↑π ^ (-s) * Complex.Gamma s * a i / ↑(p i) ^ s) (mellin F s)
-- hasSum_mellin_pi_mul_sq
lemma hasSum_mellin_pi_mul_sq {a : ι → ℂ} {r : ι → ℝ} {F : ℝ → ℂ} {s : ℂ} (hs : 0 < s.re)
    (hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i = 0 then 0 else a i * rexp (-π * r i ^ 2 * t)) (F t))
    (h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re) :
    HasSum (fun i ↦ Gammaℝ s * a i / |r i| ^ s) (mellin F (s / 2)) 
Arguments: {ι : Type u_1} [Countable ι] {a : ι → ℂ} {r : ι → ℝ} {F : ℝ → ℂ} {s : ℂ} (hs : 0 < s.re) (hF : ∀ t ∈ Set.Ioi 0, HasSum (fun i => if r i = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)) (h_sum : Summable fun i => ‖a i‖ / |r i| ^ s.re)
Goal: HasSum (fun i => s.Gammaℝ * a i / ↑|r i| ^ s) (mellin F (s / 2))
-- hasSum_mellin_pi_mul_sq'
lemma hasSum_mellin_pi_mul_sq' {a : ι → ℂ} {r : ι → ℝ} {F : ℝ → ℂ} {s : ℂ} (hs : 0 < s.re)
    (hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * r i * rexp (-π * r i ^ 2 * t)) (F t))
    (h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re) :
    HasSum (fun i ↦ Gammaℝ (s + 1) * a i * SignType.sign (r i) / |r i| ^ s)
    (mellin F ((s + 1) / 2)) 
Arguments: {ι : Type u_1} [Countable ι] {a : ι → ℂ} {r : ι → ℝ} {F : ℝ → ℂ} {s : ℂ} (hs : 0 < s.re) (hF : ∀ t ∈ Set.Ioi 0, HasSum (fun i => a i * ↑(r i) * ↑(rexp (-π * r i ^ 2 * t))) (F t)) (h_sum : Summable fun i => ‖a i‖ / |r i| ^ s.re)
Goal: HasSum (fun i => (s + 1).Gammaℝ * a i * ↑(SignType.sign (r i)) / ↑|r i| ^ s) (mellin F ((s + 1) / 2))
-- ArithmeticFunction.one_eq_delta
lemma ArithmeticFunction.one_eq_delta : ↗(1 : ArithmeticFunction ℂ) = δ 
Goal: (fun n => 1 n) = δ
-- ArithmeticFunction.not_LSeriesSummable_moebius_at_one
lemma not_LSeriesSummable_moebius_at_one : ¬ LSeriesSummable ↗μ 1 
Goal: ¬LSeriesSummable (fun n => ↑(ArithmeticFunction.moebius n)) 1
-- ArithmeticFunction.LSeriesSummable_moebius_iff
lemma LSeriesSummable_moebius_iff {s : ℂ} : LSeriesSummable ↗μ s ↔ 1 < s.re 
Arguments: {s : ℂ}
Goal: LSeriesSummable (fun n => ↑(ArithmeticFunction.moebius n)) s ↔ 1 < s.re
-- ArithmeticFunction.abscissaOfAbsConv_moebius
lemma abscissaOfAbsConv_moebius : abscissaOfAbsConv ↗μ = 1 
Goal: (LSeries.abscissaOfAbsConv fun n => ↑(ArithmeticFunction.moebius n)) = 1
-- ArithmeticFunction.const_one_eq_zeta
lemma ArithmeticFunction.const_one_eq_zeta {R : Type*} [Semiring R] {n : ℕ} (hn : n ≠ 0) :
    (1 : ℕ → R) n = (ζ ·) n 
Arguments: {R : Type u_1} [Semiring R] {n : ℕ} (hn : n ≠ 0)
Goal: 1 n = ↑((fun x => ArithmeticFunction.zeta x) n)
-- LSeries.one_convolution_eq_zeta_convolution
lemma LSeries.one_convolution_eq_zeta_convolution {R : Type*} [Semiring R] (f : ℕ → R) :
    (1 : ℕ → R) ⍟ f = ((ArithmeticFunction.zeta ·) : ℕ → R) ⍟ f 
Arguments: {R : Type u_1} [Semiring R] (f : ℕ → R)
Goal: 1 ⍟ f = (fun x => ↑(ArithmeticFunction.zeta x)) ⍟ f