ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HurwitzZeta.differentiable_completedHurwitzZetaEven₀
lemma differentiable_completedHurwitzZetaEven₀ (a : UnitAddCircle) :
    Differentiable ℂ (completedHurwitzZetaEven₀ a) 
Arguments: (a : UnitAddCircle)
Goal: Differentiable ℂ (HurwitzZeta.completedHurwitzZetaEven₀ a)
-- HurwitzZeta.differentiableAt_one_completedHurwitzZetaEven_sub_completedHurwitzZetaEven
lemma differentiableAt_one_completedHurwitzZetaEven_sub_completedHurwitzZetaEven
    (a b : UnitAddCircle) :
    DifferentiableAt ℂ (fun s ↦ completedHurwitzZetaEven a s - completedHurwitzZetaEven b s) 1 
Arguments: (a b : UnitAddCircle)
Goal: DifferentiableAt ℂ (fun s => HurwitzZeta.completedHurwitzZetaEven a s - HurwitzZeta.completedHurwitzZetaEven b s) 1
-- HurwitzZeta.differentiableAt_completedCosZeta
lemma differentiableAt_completedCosZeta
    (a : UnitAddCircle) {s : ℂ} (hs : s ≠ 0) (hs' : s ≠ 1 ∨ a ≠ 0) :
    DifferentiableAt ℂ (completedCosZeta a) s 
Arguments: (a : UnitAddCircle) {s : ℂ} (hs : s ≠ 0) (hs' : s ≠ 1 ∨ a ≠ 0)
Goal: DifferentiableAt ℂ (HurwitzZeta.completedCosZeta a) s
-- HurwitzZeta.differentiable_completedCosZeta₀
lemma differentiable_completedCosZeta₀ (a : UnitAddCircle) :
    Differentiable ℂ (completedCosZeta₀ a) 
Arguments: (a : UnitAddCircle)
Goal: Differentiable ℂ (HurwitzZeta.completedCosZeta₀ a)
-- _private.Mathlib.NumberTheory.LSeries.HurwitzZetaEven.0.HurwitzZeta.tendsto_div_two_punctured_nhds
private lemma tendsto_div_two_punctured_nhds (a : ℂ) :
    Tendsto (fun s : ℂ ↦ s / 2) (𝓝[≠] a) (𝓝[≠] (a / 2)) 
Arguments: (a : ℂ)
Goal: Filter.Tendsto (fun s => s / 2) (𝓝[≠] a) (𝓝[≠] (a / 2))
-- HurwitzZeta.completedHurwitzZetaEven_residue_one
lemma completedHurwitzZetaEven_residue_one (a : UnitAddCircle) :
    Tendsto (fun s ↦ (s - 1) * completedHurwitzZetaEven a s) (𝓝[≠] 1) (𝓝 1) 
Arguments: (a : UnitAddCircle)
Goal: Filter.Tendsto (fun s => (s - 1) * HurwitzZeta.completedHurwitzZetaEven a s) (𝓝[≠] 1) (𝓝 1)
-- HurwitzZeta.completedHurwitzZetaEven_residue_zero
lemma completedHurwitzZetaEven_residue_zero (a : UnitAddCircle) :
    Tendsto (fun s ↦ s * completedHurwitzZetaEven a s) (𝓝[≠] 0) (𝓝 (if a = 0 then -1 else 0)) 
Arguments: (a : UnitAddCircle)
Goal: Filter.Tendsto (fun s => s * HurwitzZeta.completedHurwitzZetaEven a s) (𝓝[≠] 0) (𝓝 (if a = 0 then -1 else 0))
-- HurwitzZeta.completedCosZeta_residue_zero
lemma completedCosZeta_residue_zero (a : UnitAddCircle) :
    Tendsto (fun s ↦ s * completedCosZeta a s) (𝓝[≠] 0) (𝓝 (-1)) 
Arguments: (a : UnitAddCircle)
Goal: Filter.Tendsto (fun s => s * HurwitzZeta.completedCosZeta a s) (𝓝[≠] 0) (𝓝 (-1))
-- HurwitzZeta.hasSum_int_completedCosZeta
lemma hasSum_int_completedCosZeta (a : ℝ) {s : ℂ} (hs : 1 < re s) :
    HasSum (fun n : ℤ ↦ Gammaℝ s * cexp (2 * π * I * a * n) / (↑|n| : ℂ) ^ s / 2)
    (completedCosZeta a s) 
Arguments: (a : ℝ) {s : ℂ} (hs : 1 < s.re)
Goal: HasSum (fun n => s.Gammaℝ * cexp (2 * ↑π * Complex.I * ↑a * ↑n) / ↑|n| ^ s / 2) (HurwitzZeta.completedCosZeta (↑a) s)
-- HurwitzZeta.hasSum_nat_completedCosZeta
lemma hasSum_nat_completedCosZeta (a : ℝ) {s : ℂ} (hs : 1 < re s) :
    HasSum (fun n : ℕ ↦ if n = 0 then 0 else Gammaℝ s * Real.cos (2 * π * a * n) / (n : ℂ) ^ s)
    (completedCosZeta a s) 
Arguments: (a : ℝ) {s : ℂ} (hs : 1 < s.re)
Goal: HasSum (fun n => if n = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s)
  (HurwitzZeta.completedCosZeta (↑a) s)
-- HurwitzZeta.hasSum_int_completedHurwitzZetaEven
lemma hasSum_int_completedHurwitzZetaEven (a : ℝ) {s : ℂ} (hs : 1 < re s) :
    HasSum (fun n : ℤ ↦ Gammaℝ s / (↑|n + a| : ℂ) ^ s / 2) (completedHurwitzZetaEven a s) 
Arguments: (a : ℝ) {s : ℂ} (hs : 1 < s.re)
Goal: HasSum (fun n => s.Gammaℝ / ↑|↑n + a| ^ s / 2) (HurwitzZeta.completedHurwitzZetaEven (↑a) s)
-- HurwitzZeta.differentiableAt_update_of_residue
lemma differentiableAt_update_of_residue
    {Λ : ℂ → ℂ} (hf : ∀ (s : ℂ) (_ : s ≠ 0) (_ : s ≠ 1), DifferentiableAt ℂ Λ s)
    {L : ℂ} (h_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)) (s : ℂ) (hs' : s ≠ 1) :
    DifferentiableAt ℂ (Function.update (fun s ↦ Λ s / Gammaℝ s) 0 (L / 2)) s 
Arguments: {Λ : ℂ → ℂ} (hf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s) {L : ℂ} (h_lim : Filter.Tendsto (fun s => s * Λ s) (𝓝[≠] 0) (𝓝 L)) (s : ℂ) (hs' : s ≠ 1)
Goal: DifferentiableAt ℂ (Function.update (fun s => Λ s / s.Gammaℝ) 0 (L / 2)) s
-- HurwitzZeta.hurwitzZetaEven_def_of_ne_or_ne
lemma hurwitzZetaEven_def_of_ne_or_ne {a : UnitAddCircle} {s : ℂ} (h : a ≠ 0 ∨ s ≠ 0) :
    hurwitzZetaEven a s = completedHurwitzZetaEven a s / Gammaℝ s 
Arguments: {a : UnitAddCircle} {s : ℂ} (h : a ≠ 0 ∨ s ≠ 0)
Goal: HurwitzZeta.hurwitzZetaEven a s = HurwitzZeta.completedHurwitzZetaEven a s / s.Gammaℝ
-- HurwitzZeta.hurwitzZetaEven_apply_zero
lemma hurwitzZetaEven_apply_zero (a : UnitAddCircle) :
    hurwitzZetaEven a 0 = if a = 0 then -1 / 2 else 0 
Arguments: (a : UnitAddCircle)
Goal: HurwitzZeta.hurwitzZetaEven a 0 = if a = 0 then -1 / 2 else 0
-- HurwitzZeta.hurwitzZetaEven_neg
lemma hurwitzZetaEven_neg (a : UnitAddCircle) (s : ℂ) :
    hurwitzZetaEven (-a) s = hurwitzZetaEven a s 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.hurwitzZetaEven (-a) s = HurwitzZeta.hurwitzZetaEven a s
-- HurwitzZeta.hurwitzZetaEven_neg_two_mul_nat_add_one
theorem hurwitzZetaEven_neg_two_mul_nat_add_one (a : UnitAddCircle) (n : ℕ) :
    hurwitzZetaEven a (-2 * (n + 1)) = 0 
Arguments: (a : UnitAddCircle) (n : ℕ)
Goal: HurwitzZeta.hurwitzZetaEven a (-2 * (↑n + 1)) = 0
-- HurwitzZeta.differentiableAt_hurwitzZetaEven
lemma differentiableAt_hurwitzZetaEven (a : UnitAddCircle) {s : ℂ} (hs' : s ≠ 1) :
    DifferentiableAt ℂ (hurwitzZetaEven a) s 
Arguments: (a : UnitAddCircle) {s : ℂ} (hs' : s ≠ 1)
Goal: DifferentiableAt ℂ (HurwitzZeta.hurwitzZetaEven a) s
-- HurwitzZeta.hurwitzZetaEven_residue_one
lemma hurwitzZetaEven_residue_one (a : UnitAddCircle) :
    Tendsto (fun s ↦ (s - 1) * hurwitzZetaEven a s) (𝓝[≠] 1) (𝓝 1) 
Arguments: (a : UnitAddCircle)
Goal: Filter.Tendsto (fun s => (s - 1) * HurwitzZeta.hurwitzZetaEven a s) (𝓝[≠] 1) (𝓝 1)
-- HurwitzZeta.differentiableAt_hurwitzZetaEven_sub_one_div
lemma differentiableAt_hurwitzZetaEven_sub_one_div (a : UnitAddCircle) :
    DifferentiableAt ℂ (fun s ↦ hurwitzZetaEven a s - 1 / (s - 1) / Gammaℝ s) 1 
Arguments: (a : UnitAddCircle)
Goal: DifferentiableAt ℂ (fun s => HurwitzZeta.hurwitzZetaEven a s - 1 / (s - 1) / s.Gammaℝ) 1
-- HurwitzZeta.tendsto_hurwitzZetaEven_sub_one_div_nhds_one
lemma tendsto_hurwitzZetaEven_sub_one_div_nhds_one (a : UnitAddCircle) :
    Tendsto (fun s ↦ hurwitzZetaEven a s - 1 / (s - 1) / Gammaℝ s) (𝓝 1)
    (𝓝 (hurwitzZetaEven a 1)) 
Arguments: (a : UnitAddCircle)
Goal: Filter.Tendsto (fun s => HurwitzZeta.hurwitzZetaEven a s - 1 / (s - 1) / s.Gammaℝ) (𝓝 1)
  (𝓝 (HurwitzZeta.hurwitzZetaEven a 1))
-- HurwitzZeta.differentiable_hurwitzZetaEven_sub_hurwitzZetaEven
lemma differentiable_hurwitzZetaEven_sub_hurwitzZetaEven (a b : UnitAddCircle) :
    Differentiable ℂ (fun s ↦ hurwitzZetaEven a s - hurwitzZetaEven b s) 
Arguments: (a b : UnitAddCircle)
Goal: Differentiable ℂ fun s => HurwitzZeta.hurwitzZetaEven a s - HurwitzZeta.hurwitzZetaEven b s
-- HurwitzZeta.hasSum_int_hurwitzZetaEven
lemma hasSum_int_hurwitzZetaEven (a : ℝ) {s : ℂ} (hs : 1 < re s) :
    HasSum (fun n : ℤ ↦ 1 / (↑|n + a| : ℂ) ^ s / 2) (hurwitzZetaEven a s) 
Arguments: (a : ℝ) {s : ℂ} (hs : 1 < s.re)
Goal: HasSum (fun n => 1 / ↑|↑n + a| ^ s / 2) (HurwitzZeta.hurwitzZetaEven (↑a) s)
-- HurwitzZeta.hasSum_nat_hurwitzZetaEven
lemma hasSum_nat_hurwitzZetaEven (a : ℝ) {s : ℂ} (hs : 1 < re s) :
    HasSum (fun n : ℕ ↦ (1 / (↑|n + a| : ℂ) ^ s + 1 / (↑|n + 1 - a| : ℂ) ^ s) / 2)
    (hurwitzZetaEven a s) 
Arguments: (a : ℝ) {s : ℂ} (hs : 1 < s.re)
Goal: HasSum (fun n => (1 / ↑|↑n + a| ^ s + 1 / ↑|↑n + 1 - a| ^ s) / 2) (HurwitzZeta.hurwitzZetaEven (↑a) s)
-- HurwitzZeta.hasSum_nat_hurwitzZetaEven_of_mem_Icc
lemma hasSum_nat_hurwitzZetaEven_of_mem_Icc {a : ℝ} (ha : a ∈ Icc 0 1) {s : ℂ} (hs : 1 < re s) :
    HasSum (fun n : ℕ ↦ (1 / (n + a : ℂ) ^ s + 1 / (n + 1 - a : ℂ) ^ s) / 2)
    (hurwitzZetaEven 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 + 1 / (↑n + 1 - ↑a) ^ s) / 2) (HurwitzZeta.hurwitzZetaEven (↑a) s)
-- HurwitzZeta.cosZeta_apply_zero
lemma cosZeta_apply_zero (a : UnitAddCircle) : cosZeta a 0 = -1 / 2 
Arguments: (a : UnitAddCircle)
Goal: HurwitzZeta.cosZeta a 0 = -1 / 2
-- HurwitzZeta.cosZeta_neg
lemma cosZeta_neg (a : UnitAddCircle) (s : ℂ) :
    cosZeta (-a) s = cosZeta a s 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.cosZeta (-a) s = HurwitzZeta.cosZeta a s
-- HurwitzZeta.cosZeta_neg_two_mul_nat_add_one
theorem cosZeta_neg_two_mul_nat_add_one (a : UnitAddCircle) (n : ℕ) :
    cosZeta a (-2 * (n + 1)) = 0 
Arguments: (a : UnitAddCircle) (n : ℕ)
Goal: HurwitzZeta.cosZeta a (-2 * (↑n + 1)) = 0
-- HurwitzZeta.differentiableAt_cosZeta
lemma differentiableAt_cosZeta (a : UnitAddCircle) {s : ℂ} (hs' : s ≠ 1 ∨ a ≠ 0) :
    DifferentiableAt ℂ (cosZeta a) s 
Arguments: (a : UnitAddCircle) {s : ℂ} (hs' : s ≠ 1 ∨ a ≠ 0)
Goal: DifferentiableAt ℂ (HurwitzZeta.cosZeta a) s
-- HurwitzZeta.differentiable_cosZeta_of_ne_zero
lemma differentiable_cosZeta_of_ne_zero {a : UnitAddCircle} (ha : a ≠ 0) :
    Differentiable ℂ (cosZeta a) 
Arguments: {a : UnitAddCircle} (ha : a ≠ 0)
Goal: Differentiable ℂ (HurwitzZeta.cosZeta a)
-- HurwitzZeta.hasSum_int_cosZeta
lemma hasSum_int_cosZeta (a : ℝ) {s : ℂ} (hs : 1 < re s) :
    HasSum (fun n : ℤ ↦ cexp (2 * π * I * a * n) / ↑|n| ^ s / 2) (cosZeta a s) 
Arguments: (a : ℝ) {s : ℂ} (hs : 1 < s.re)
Goal: HasSum (fun n => cexp (2 * ↑π * Complex.I * ↑a * ↑n) / ↑|n| ^ s / 2) (HurwitzZeta.cosZeta (↑a) s)
-- HurwitzZeta.hasSum_nat_cosZeta
lemma hasSum_nat_cosZeta (a : ℝ) {s : ℂ} (hs : 1 < re s) :
    HasSum (fun n : ℕ ↦ Real.cos (2 * π * a * n) / (n : ℂ) ^ s) (cosZeta a s) 
Arguments: (a : ℝ) {s : ℂ} (hs : 1 < s.re)
Goal: HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (HurwitzZeta.cosZeta (↑a) s)
-- HurwitzZeta.LSeriesHasSum_cos
lemma LSeriesHasSum_cos (a : ℝ) {s : ℂ} (hs : 1 < re s) :
    LSeriesHasSum (Real.cos <| 2 * π * a * ·) s (cosZeta a s) 
Arguments: (a : ℝ) {s : ℂ} (hs : 1 < s.re)
Goal: LSeriesHasSum (fun x => ↑(Real.cos (2 * π * a * ↑x))) s (HurwitzZeta.cosZeta (↑a) s)
-- HurwitzZeta.hurwitzZetaEven_one_sub
lemma hurwitzZetaEven_one_sub (a : UnitAddCircle) {s : ℂ}
    (hs : ∀ (n : ℕ), s ≠ -n) (hs' : a ≠ 0 ∨ s ≠ 1) :
    hurwitzZetaEven a (1 - s) = 2 * (2 * π) ^ (-s) * Gamma s * cos (π * s / 2) * cosZeta a s 
Arguments: (a : UnitAddCircle) {s : ℂ} (hs : ∀ (n : ℕ), s ≠ -↑n) (hs' : a ≠ 0 ∨ s ≠ 1)
Goal: HurwitzZeta.hurwitzZetaEven a (1 - s) =
  2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * Complex.cos (↑π * s / 2) * HurwitzZeta.cosZeta a s
-- HurwitzZeta.cosZeta_one_sub
lemma cosZeta_one_sub (a : UnitAddCircle) {s : ℂ} (hs : ∀ (n : ℕ), s ≠ 1 - n) :
    cosZeta a (1 - s) = 2 * (2 * π) ^ (-s) * Gamma s * cos (π * s / 2) * hurwitzZetaEven a s 
Arguments: (a : UnitAddCircle) {s : ℂ} (hs : ∀ (n : ℕ), s ≠ 1 - ↑n)
Goal: HurwitzZeta.cosZeta a (1 - s) =
  2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * Complex.cos (↑π * s / 2) * HurwitzZeta.hurwitzZetaEven a s
-- FunctionField.classNumber_eq_one_iff
theorem classNumber_eq_one_iff :
    classNumber Fq F = 1 ↔ IsPrincipalIdealRing (ringOfIntegers Fq F) 
Arguments: (Fq F : Type) [Field Fq] [Fintype Fq] [Field F] [Algebra (Polynomial Fq) F] [Algebra (RatFunc Fq) F] [IsScalarTower (Polynomial Fq) (RatFunc Fq) F] [FunctionField Fq F] [Algebra.IsSeparable (RatFunc Fq) F]
Goal: FunctionField.classNumber Fq F = 1 ↔ IsPrincipalIdealRing ↥(FunctionField.ringOfIntegers Fq F)
-- AbsoluteValue.IsAdmissible.exists_partition'
structure IsAdmissible extends IsEuclidean abv where
  protected card : ℝ → ℕ
    exists_partition' :
    ∀ (n : ℕ) {ε : ℝ} (_ : 0 < ε) {b : R} (_ : b ≠ 0) (A : Fin n → R),
      ∃ t : Fin n → Fin (card ε), ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ % b - A i₀ % b) : ℝ) < abv b • ε
Arguments: {R : Type u_1} [EuclideanDomain R] {abv : AbsoluteValue R ℤ} (self : abv.IsAdmissible) (n : ℕ) {ε : ℝ}
Goal: 0 < ε →
  ∀ {b : R}, b ≠ 0 → ∀ (A : Fin n → R), ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ → ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε
-- AbsoluteValue.IsAdmissible.exists_partition
theorem exists_partition {ι : Type*} [Finite ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0)
    (A : ι → R) (h : abv.IsAdmissible) : ∃ t : ι → Fin (h.card ε),
      ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ % b - A i₀ % b) : ℝ) < abv b • ε 
Arguments: {R : Type u_1} [EuclideanDomain R] {abv : AbsoluteValue R ℤ} {ι : Type u_2} [Finite ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0) (A : ι → R) (h : abv.IsAdmissible)
Goal: ∃ t, ∀ (i₀ i₁ : ι), t i₀ = t i₁ → ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε
-- AbsoluteValue.IsAdmissible.exists_approx_aux
theorem exists_approx_aux (n : ℕ) (h : abv.IsAdmissible) :
    ∀ {ε : ℝ} (_hε : 0 < ε) {b : R} (_hb : b ≠ 0) (A : Fin (h.card ε ^ n).succ → Fin n → R),
      ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε 
Arguments: {R : Type u_1} [EuclideanDomain R] {abv : AbsoluteValue R ℤ} (n : ℕ) (h : abv.IsAdmissible) {ε : ℝ} (_hε : 0 < ε) {b : R} (_hb : b ≠ 0) (A : Fin (h.card ε ^ n).succ → Fin n → R)
Goal: ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n), ↑(abv (A i₁ k % b - A i₀ k % b)) < abv b • ε
-- AbsoluteValue.IsAdmissible.exists_approx
theorem exists_approx {ι : Type*} [Fintype ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0)
    (h : abv.IsAdmissible) (A : Fin (h.card ε ^ Fintype.card ι).succ → ι → R) :
    ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε 
Arguments: {R : Type u_1} [EuclideanDomain R] {abv : AbsoluteValue R ℤ} {ι : Type u_2} [Fintype ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0) (h : abv.IsAdmissible) (A : Fin (h.card ε ^ Fintype.card ι).succ → ι → R)
Goal: ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : ι), ↑(abv (A i₁ k % b - A i₀ k % b)) < abv b • ε
-- AbsoluteValue.exists_partition_int
theorem exists_partition_int (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : ℤ} (hb : b ≠ 0) (A : Fin n → ℤ) :
    ∃ t : Fin n → Fin ⌈1 / ε⌉₊,
    ∀ i₀ i₁, t i₀ = t i₁ → ↑(abs (A i₁ % b - A i₀ % b)) < abs b • ε 
Arguments: (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : ℤ} (hb : b ≠ 0) (A : Fin n → ℤ)
Goal: ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ → ↑|A i₁ % b - A i₀ % b| < |b| • ε
-- padic_polynomial_dist
theorem padic_polynomial_dist {p : ℕ} [Fact p.Prime] (F : Polynomial ℤ_[p]) (x y : ℤ_[p]) :
    ‖F.eval x - F.eval y‖ ≤ ‖x - y‖ 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] (F : Polynomial ℤ_[p]) (x y : ℤ_[p])
Goal: ‖Polynomial.eval x F - Polynomial.eval y F‖ ≤ ‖x - y‖
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.comp_tendsto_lim
private theorem comp_tendsto_lim {p : ℕ} [Fact p.Prime] {F : Polynomial ℤ_[p]}
    (ncs : CauSeq ℤ_[p] norm) : Tendsto (fun i => F.eval (ncs i)) atTop (𝓝 (F.eval ncs.lim)) 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} (ncs : CauSeq ℤ_[p] norm)
Goal: Filter.Tendsto (fun i => Polynomial.eval (↑ncs i) F) Filter.atTop (𝓝 (Polynomial.eval ncs.lim F))
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.ncs_tendsto_const
private theorem ncs_tendsto_const :
    Tendsto (fun i => ‖F.derivative.eval (ncs i)‖) atTop (𝓝 ‖F.derivative.eval a‖) 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {ncs : CauSeq ℤ_[p] norm} {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (ncs_der_val :
  ∀ (n : ℕ), ‖Polynomial.eval (↑ncs n) (Polynomial.derivative F)‖ = ‖Polynomial.eval a (Polynomial.derivative F)‖)
Goal: Filter.Tendsto (fun i => ‖Polynomial.eval (↑ncs i) (Polynomial.derivative F)‖) Filter.atTop
  (𝓝 ‖Polynomial.eval a (Polynomial.derivative F)‖)
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.ncs_tendsto_lim
private theorem ncs_tendsto_lim :
    Tendsto (fun i => ‖F.derivative.eval (ncs i)‖) atTop (𝓝 ‖F.derivative.eval ncs.lim‖) 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {ncs : CauSeq ℤ_[p] norm} {F : Polynomial ℤ_[p]}
Goal: Filter.Tendsto (fun i => ‖Polynomial.eval (↑ncs i) (Polynomial.derivative F)‖) Filter.atTop
  (𝓝 ‖Polynomial.eval ncs.lim (Polynomial.derivative F)‖)
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.norm_deriv_eq
private theorem norm_deriv_eq : ‖F.derivative.eval ncs.lim‖ = ‖F.derivative.eval a‖ 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {ncs : CauSeq ℤ_[p] norm} {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (ncs_der_val :
  ∀ (n : ℕ), ‖Polynomial.eval (↑ncs n) (Polynomial.derivative F)‖ = ‖Polynomial.eval a (Polynomial.derivative F)‖)
Goal: ‖Polynomial.eval ncs.lim (Polynomial.derivative F)‖ = ‖Polynomial.eval a (Polynomial.derivative F)‖
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.tendsto_zero_of_norm_tendsto_zero
private theorem tendsto_zero_of_norm_tendsto_zero : Tendsto (fun i => F.eval (ncs i)) atTop (𝓝 0) 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {ncs : CauSeq ℤ_[p] norm} {F : Polynomial ℤ_[p]} (hnorm : Filter.Tendsto (fun i => ‖Polynomial.eval (↑ncs i) F‖) Filter.atTop (𝓝 0))
Goal: Filter.Tendsto (fun i => Polynomial.eval (↑ncs i) F) Filter.atTop (𝓝 0)
-- limit_zero_of_norm_tendsto_zero
theorem limit_zero_of_norm_tendsto_zero : F.eval ncs.lim = 0 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {ncs : CauSeq ℤ_[p] norm} {F : Polynomial ℤ_[p]} (hnorm : Filter.Tendsto (fun i => ‖Polynomial.eval (↑ncs i) F‖) Filter.atTop (𝓝 0))
Goal: Polynomial.eval ncs.lim F = 0
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.deriv_sq_norm_pos
private theorem deriv_sq_norm_pos : 0 < ‖F.derivative.eval a‖ ^ 2 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2)
Goal: 0 < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.deriv_sq_norm_ne_zero
private theorem deriv_sq_norm_ne_zero : ‖F.derivative.eval a‖ ^ 2 ≠ 0 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2)
Goal: ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 ≠ 0
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.deriv_norm_ne_zero
private theorem deriv_norm_ne_zero : ‖F.derivative.eval a‖ ≠ 0 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2)
Goal: ‖Polynomial.eval a (Polynomial.derivative F)‖ ≠ 0
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.deriv_norm_pos
private theorem deriv_norm_pos : 0 < ‖F.derivative.eval a‖ 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2)
Goal: 0 < ‖Polynomial.eval a (Polynomial.derivative F)‖
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.deriv_ne_zero
private theorem deriv_ne_zero : F.derivative.eval a ≠ 0 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2)
Goal: Polynomial.eval a (Polynomial.derivative F) ≠ 0
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.T_def
private theorem T_def : T = ‖F.eval a‖ / ‖F.derivative.eval a‖ ^ 2 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]}
Goal: T_gen p F a = ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.T_lt_one
private theorem T_lt_one : T < 1 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2)
Goal: T_gen p F a < 1
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.T_nonneg
private theorem T_nonneg : 0 ≤ T 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]}
Goal: 0 ≤ T_gen p F a
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.T_pow_nonneg
private theorem T_pow_nonneg (n : ℕ) : 0 ≤ T ^ n 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (n : ℕ)
Goal: 0 ≤ T_gen p F a ^ n
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.T_pow
private theorem T_pow {n : ℕ} (hn : n ≠ 0) : T ^ n < 1 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2) {n : ℕ} (hn : n ≠ 0)
Goal: T_gen p F a ^ n < 1
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.T_pow'
private theorem T_pow' (n : ℕ) : T ^ 2 ^ n < 1 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2) (n : ℕ)
Goal: T_gen p F a ^ 2 ^ n < 1
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.ih_0
private theorem ih_0 : ih 0 a 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2)
Goal: ih_gen 0 a
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.calc_norm_le_one
private theorem calc_norm_le_one {n : ℕ} {z : ℤ_[p]} (hz : ih n z) :
    ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2) {n : ℕ} {z : ℤ_[p]} (hz : ih_gen n z)
Goal: ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤ 1
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.calc_deriv_dist
private theorem calc_deriv_dist {z z' z1 : ℤ_[p]} (hz' : z' = z - z1)
    (hz1 : ‖z1‖ = ‖F.eval z‖ / ‖F.derivative.eval a‖) {n} (hz : ih n z) :
    ‖F.derivative.eval z' - F.derivative.eval z‖ < ‖F.derivative.eval a‖ 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2) {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) (hz1 : ‖z1‖ = ‖Polynomial.eval z F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖) {n : ℕ} (hz : ih_gen n z)
Goal: ‖Polynomial.eval z' (Polynomial.derivative F) - Polynomial.eval z (Polynomial.derivative F)‖ <
  ‖Polynomial.eval a (Polynomial.derivative F)‖
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.newton_seq_deriv_norm
private theorem newton_seq_deriv_norm (n : ℕ) :
    ‖F.derivative.eval (newton_seq n)‖ = ‖F.derivative.eval a‖ 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2) (n : ℕ)
Goal: ‖Polynomial.eval (newton_seq_gen hnorm n) (Polynomial.derivative F)‖ = ‖Polynomial.eval a (Polynomial.derivative F)‖
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.newton_seq_norm_le
private theorem newton_seq_norm_le (n : ℕ) :
    ‖F.eval (newton_seq n)‖ ≤ ‖F.derivative.eval a‖ ^ 2 * T ^ 2 ^ n 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2) (n : ℕ)
Goal: ‖Polynomial.eval (newton_seq_gen hnorm n) F‖ ≤ ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 * T_gen p F a ^ 2 ^ n
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.newton_seq_norm_eq
private theorem newton_seq_norm_eq (n : ℕ) :
    ‖newton_seq (n + 1) - newton_seq n‖ =
    ‖F.eval (newton_seq n)‖ / ‖F.derivative.eval (newton_seq n)‖ 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2) (n : ℕ)
Goal: ‖newton_seq_gen hnorm (n + 1) - newton_seq_gen hnorm n‖ =
  ‖Polynomial.eval (newton_seq_gen hnorm n) F‖ / ‖Polynomial.eval (newton_seq_gen hnorm n) (Polynomial.derivative F)‖
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.newton_seq_succ_dist
private theorem newton_seq_succ_dist (n : ℕ) :
    ‖newton_seq (n + 1) - newton_seq n‖ ≤ ‖F.derivative.eval a‖ * T ^ 2 ^ n 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2) (n : ℕ)
Goal: ‖newton_seq_gen hnorm (n + 1) - newton_seq_gen hnorm n‖ ≤
  ‖Polynomial.eval a (Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n
-- _private.Mathlib.NumberTheory.Padics.Hensel.0.T_pos
private theorem T_pos : T > 0 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {F : Polynomial ℤ_[p]} {a : ℤ_[p]} (hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2) (hnsol : Polynomial.eval a F ≠ 0)
Goal: T_gen p F a > 0