ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LSeriesSummable.sub
lemma LSeriesSummable.sub {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s)
    (hg : LSeriesSummable g s) :
    LSeriesSummable (f - g) s 
Arguments: {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) (hg : LSeriesSummable g s)
Goal: LSeriesSummable (f - g) s
-- LSeries_sub
@[simp]
lemma LSeries_sub {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) (hg : LSeriesSummable g s) :
    LSeries (f - g) s = LSeries f s - LSeries g s 
Arguments: {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) (hg : LSeriesSummable g s)
Goal: LSeries (f - g) s = LSeries f s - LSeries g s
-- LSeries.term_smul
lemma LSeries.term_smul (f : ℕ → ℂ) (c s : ℂ) : term (c • f) s = c • term f s 
Arguments: (f : ℕ → ℂ) (c s : ℂ)
Goal: LSeries.term (c • f) s = c • LSeries.term f s
-- LSeries.term_smul_apply
lemma LSeries.term_smul_apply (f : ℕ → ℂ) (c s : ℂ) (n : ℕ) :
    term (c • f) s n = c * term f s n 
Arguments: (f : ℕ → ℂ) (c s : ℂ) (n : ℕ)
Goal: LSeries.term (c • f) s n = c * LSeries.term f s n
-- LSeriesHasSum.smul
lemma LSeriesHasSum.smul {f : ℕ → ℂ} (c : ℂ) {s a : ℂ} (hf : LSeriesHasSum f s a) :
    LSeriesHasSum (c • f) s (c * a) 
Arguments: {f : ℕ → ℂ} (c : ℂ) {s a : ℂ} (hf : LSeriesHasSum f s a)
Goal: LSeriesHasSum (c • f) s (c * a)
-- LSeriesSummable.smul
lemma LSeriesSummable.smul {f : ℕ → ℂ} (c : ℂ) {s : ℂ} (hf : LSeriesSummable f s) :
    LSeriesSummable (c • f) s 
Arguments: {f : ℕ → ℂ} (c : ℂ) {s : ℂ} (hf : LSeriesSummable f s)
Goal: LSeriesSummable (c • f) s
-- LSeriesSummable.of_smul
lemma LSeriesSummable.of_smul {f : ℕ → ℂ} {c s : ℂ} (hc : c ≠ 0) (hf : LSeriesSummable (c • f) s) :
    LSeriesSummable f s 
Arguments: {f : ℕ → ℂ} {c s : ℂ} (hc : c ≠ 0) (hf : LSeriesSummable (c • f) s)
Goal: LSeriesSummable f s
-- LSeriesSummable.smul_iff
lemma LSeriesSummable.smul_iff {f : ℕ → ℂ} {c s : ℂ} (hc : c ≠ 0) :
    LSeriesSummable (c • f) s ↔ LSeriesSummable f s 
Arguments: {f : ℕ → ℂ} {c s : ℂ} (hc : c ≠ 0)
Goal: LSeriesSummable (c • f) s ↔ LSeriesSummable f s
-- LSeries_smul
@[simp]
lemma LSeries_smul (f : ℕ → ℂ) (c s : ℂ) : LSeries (c • f) s = c * LSeries f s 
Arguments: (f : ℕ → ℂ) (c s : ℂ)
Goal: LSeries (c • f) s = c * LSeries f s
-- Summable.norm_lt_one
lemma Summable.norm_lt_one {F : Type*} [NormedField F] [CompleteSpace F] {f : ℕ →* F}
    (hsum : Summable f) {p : ℕ} (hp : 1 < p) :
    ‖f p‖ < 1 
Arguments: {F : Type u_1} [NormedField F] [CompleteSpace F] {f : ℕ →* F} (hsum : Summable ⇑f) {p : ℕ} (hp : 1 < p)
Goal: ‖f p‖ < 1
-- _private.Mathlib.NumberTheory.EulerProduct.Basic.0.instT0Space
@[local instance] private lemma instT0Space : T0Space R 
Arguments: {R : Type u_1} [NormedCommRing R]
Goal: T0Space R
-- EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum
lemma summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum
    (hsum : ∀ {p : ℕ}, p.Prime → Summable (fun n : ℕ ↦ ‖f (p ^ n)‖)) (s : Finset ℕ) :
    Summable (fun m : factoredNumbers s ↦ ‖f m‖) ∧
      HasSum (fun m : factoredNumbers s ↦ f m)
        (∏ p ∈ s.filter Nat.Prime, ∑' n : ℕ, f (p ^ n)) 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} (hf₁ : f 1 = 1) (hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n) (hsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n => ‖f (p ^ n)‖) (s : Finset ℕ)
Goal: (Summable fun m => ‖f ↑m‖) ∧ HasSum (fun m => f ↑m) (∏ p ∈ Finset.filter Nat.Prime s, ∑' (n : ℕ), f (p ^ n))
-- EulerProduct.prod_filter_prime_tsum_eq_tsum_factoredNumbers
lemma prod_filter_prime_tsum_eq_tsum_factoredNumbers (hsum : Summable (‖f ·‖)) (s : Finset ℕ) :
    ∏ p ∈ s.filter Nat.Prime, ∑' n : ℕ, f (p ^ n) = ∑' m : factoredNumbers s, f m 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} (hf₁ : f 1 = 1) (hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n) (hsum : Summable fun x => ‖f x‖) (s : Finset ℕ)
Goal: ∏ p ∈ Finset.filter Nat.Prime s, ∑' (n : ℕ), f (p ^ n) = ∑' (m : ↑(Nat.factoredNumbers s)), f ↑m
-- EulerProduct.norm_tsum_factoredNumbers_sub_tsum_lt
lemma norm_tsum_factoredNumbers_sub_tsum_lt (hsum : Summable f) (hf₀ : f 0 = 0) {ε : ℝ}
    (εpos : 0 < ε) :
    ∃ N : ℕ, ∀ s : Finset ℕ, primesBelow N ≤ s →
      ‖(∑' m : ℕ, f m) - ∑' m : factoredNumbers s, f m‖ < ε 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} (hsum : Summable f) (hf₀ : f 0 = 0) {ε : ℝ} (εpos : 0 < ε)
Goal: ∃ N, ∀ (s : Finset ℕ), N.primesBelow ≤ s → ‖∑' (m : ℕ), f m - ∑' (m : ↑(Nat.factoredNumbers s)), f ↑m‖ < ε
-- EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum
lemma summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum
    (hsum : ∀ {p : ℕ}, p.Prime → Summable (fun n : ℕ ↦ ‖f (p ^ n)‖)) (N : ℕ) :
    Summable (fun m : N.smoothNumbers ↦ ‖f m‖) ∧
      HasSum (fun m : N.smoothNumbers ↦ f m) (∏ p ∈ N.primesBelow, ∑' n : ℕ, f (p ^ n)) 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} (hf₁ : f 1 = 1) (hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n) (hsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n => ‖f (p ^ n)‖) (N : ℕ)
Goal: (Summable fun m => ‖f ↑m‖) ∧ HasSum (fun m => f ↑m) (∏ p ∈ N.primesBelow, ∑' (n : ℕ), f (p ^ n))
-- EulerProduct.prod_primesBelow_tsum_eq_tsum_smoothNumbers
lemma prod_primesBelow_tsum_eq_tsum_smoothNumbers (hsum : Summable (‖f ·‖)) (N : ℕ) :
    ∏ p ∈ N.primesBelow, ∑' n : ℕ, f (p ^ n) = ∑' m : N.smoothNumbers, f m 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} (hf₁ : f 1 = 1) (hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n) (hsum : Summable fun x => ‖f x‖) (N : ℕ)
Goal: ∏ p ∈ N.primesBelow, ∑' (n : ℕ), f (p ^ n) = ∑' (m : ↑N.smoothNumbers), f ↑m
-- EulerProduct.norm_tsum_smoothNumbers_sub_tsum_lt
lemma norm_tsum_smoothNumbers_sub_tsum_lt (hsum : Summable f) (hf₀ : f 0 = 0)
    {ε : ℝ} (εpos : 0 < ε) :
    ∃ N₀ : ℕ, ∀ N ≥ N₀, ‖(∑' m : ℕ, f m) - ∑' m : N.smoothNumbers, f m‖ < ε 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} (hsum : Summable f) (hf₀ : f 0 = 0) {ε : ℝ} (εpos : 0 < ε)
Goal: ∃ N₀, ∀ N ≥ N₀, ‖∑' (m : ℕ), f m - ∑' (m : ↑N.smoothNumbers), f ↑m‖ < ε
-- EulerProduct.eulerProduct_hasProd
theorem eulerProduct_hasProd (hsum : Summable (‖f ·‖)) (hf₀ : f 0 = 0) :
    HasProd (fun p : Primes ↦ ∑' e, f (p ^ e)) (∑' n, f n) 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} (hf₁ : f 1 = 1) (hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n) (hsum : Summable fun x => ‖f x‖) (hf₀ : f 0 = 0)
Goal: HasProd (fun p => ∑' (e : ℕ), f (↑p ^ e)) (∑' (n : ℕ), f n)
-- EulerProduct.eulerProduct_hasProd_mulIndicator
theorem eulerProduct_hasProd_mulIndicator (hsum : Summable (‖f ·‖)) (hf₀ : f 0 = 0) :
    HasProd (Set.mulIndicator {p | Nat.Prime p} fun p ↦  ∑' e, f (p ^ e)) (∑' n, f n) 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} (hf₁ : f 1 = 1) (hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n) (hsum : Summable fun x => ‖f x‖) (hf₀ : f 0 = 0)
Goal: HasProd ({p | Nat.Prime p}.mulIndicator fun p => ∑' (e : ℕ), f (p ^ e)) (∑' (n : ℕ), f n)
-- EulerProduct.eulerProduct
theorem eulerProduct (hsum : Summable (‖f ·‖)) (hf₀ : f 0 = 0) :
    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, ∑' e, f (p ^ e)) atTop (𝓝 (∑' n, f n)) 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} (hf₁ : f 1 = 1) (hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n) (hsum : Summable fun x => ‖f x‖) (hf₀ : f 0 = 0)
Goal: Filter.Tendsto (fun n => ∏ p ∈ n.primesBelow, ∑' (e : ℕ), f (p ^ e)) Filter.atTop (𝓝 (∑' (n : ℕ), f n))
-- EulerProduct.eulerProduct_tprod
theorem eulerProduct_tprod (hsum : Summable (‖f ·‖)) (hf₀ : f 0 = 0) :
    ∏' p : Primes, ∑' e, f (p ^ e) = ∑' n, f n 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} (hf₁ : f 1 = 1) (hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n) (hsum : Summable fun x => ‖f x‖) (hf₀ : f 0 = 0)
Goal: ∏' (p : Nat.Primes), ∑' (e : ℕ), f (↑p ^ e) = ∑' (n : ℕ), f n
-- ArithmeticFunction.IsMultiplicative.eulerProduct_hasProd
nonrec theorem IsMultiplicative.eulerProduct_hasProd {f : ArithmeticFunction R}
    (hf : f.IsMultiplicative) (hsum : Summable (‖f ·‖)) :
    HasProd (fun p : Primes ↦ ∑' e, f (p ^ e)) (∑' n, f n) 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ArithmeticFunction R} (hf : f.IsMultiplicative) (hsum : Summable fun x => ‖f x‖)
Goal: HasProd (fun p => ∑' (e : ℕ), f (↑p ^ e)) (∑' (n : ℕ), f n)
-- ArithmeticFunction.IsMultiplicative.eulerProduct
nonrec theorem IsMultiplicative.eulerProduct {f : ArithmeticFunction R} (hf : f.IsMultiplicative)
    (hsum : Summable (‖f ·‖)) :
    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, ∑' e, f (p ^ e)) atTop (𝓝 (∑' n, f n)) 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ArithmeticFunction R} (hf : f.IsMultiplicative) (hsum : Summable fun x => ‖f x‖)
Goal: Filter.Tendsto (fun n => ∏ p ∈ n.primesBelow, ∑' (e : ℕ), f (p ^ e)) Filter.atTop (𝓝 (∑' (n : ℕ), f n))
-- ArithmeticFunction.IsMultiplicative.eulerProduct_tprod
nonrec theorem IsMultiplicative.eulerProduct_tprod {f : ArithmeticFunction R}
    (hf : f.IsMultiplicative) (hsum : Summable (‖f ·‖)) :
    ∏' p : Primes, ∑' e, f (p ^ e) = ∑' n, f n 
Arguments: {R : Type u_1} [NormedCommRing R] [CompleteSpace R] {f : ArithmeticFunction R} (hf : f.IsMultiplicative) (hsum : Summable fun x => ‖f x‖)
Goal: ∏' (p : Nat.Primes), ∑' (e : ℕ), f (↑p ^ e) = ∑' (n : ℕ), f n
-- EulerProduct.one_sub_inv_eq_geometric_of_summable_norm
lemma one_sub_inv_eq_geometric_of_summable_norm {f : ℕ →*₀ F} {p : ℕ} (hp : p.Prime)
    (hsum : Summable fun x ↦ ‖f x‖) :
    (1 - f p)⁻¹ = ∑' (e : ℕ), f (p ^ e) 
Arguments: {F : Type u_1} [NormedField F] [CompleteSpace F] {f : ℕ →*₀ F} {p : ℕ} (hp : Nat.Prime p) (hsum : Summable fun x => ‖f x‖)
Goal: (1 - f p)⁻¹ = ∑' (e : ℕ), f (p ^ e)
-- EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_geometric
lemma summable_and_hasSum_factoredNumbers_prod_filter_prime_geometric {f : ℕ →* F}
    (h : ∀ {p : ℕ}, p.Prime → ‖f p‖ < 1) (s : Finset ℕ) :
    Summable (fun m : factoredNumbers s ↦ ‖f m‖) ∧
      HasSum (fun m : factoredNumbers s ↦ f m) (∏ p ∈ s.filter Nat.Prime, (1 - f p)⁻¹) 
Arguments: {F : Type u_1} [NormedField F] [CompleteSpace F] {f : ℕ →* F} (h : ∀ {p : ℕ}, Nat.Prime p → ‖f p‖ < 1) (s : Finset ℕ)
Goal: (Summable fun m => ‖f ↑m‖) ∧ HasSum (fun m => f ↑m) (∏ p ∈ Finset.filter Nat.Prime s, (1 - f p)⁻¹)
-- EulerProduct.prod_filter_prime_geometric_eq_tsum_factoredNumbers
lemma prod_filter_prime_geometric_eq_tsum_factoredNumbers {f : ℕ →* F} (hsum : Summable f)
    (s : Finset ℕ) :
    ∏ p ∈ s.filter Nat.Prime, (1 - f p)⁻¹ = ∑' m : factoredNumbers s, f m 
Arguments: {F : Type u_1} [NormedField F] [CompleteSpace F] {f : ℕ →* F} (hsum : Summable ⇑f) (s : Finset ℕ)
Goal: ∏ p ∈ Finset.filter Nat.Prime s, (1 - f p)⁻¹ = ∑' (m : ↑(Nat.factoredNumbers s)), f ↑m
-- EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_geometric
lemma summable_and_hasSum_smoothNumbers_prod_primesBelow_geometric {f : ℕ →* F}
    (h : ∀ {p : ℕ}, p.Prime → ‖f p‖ < 1) (N : ℕ) :
    Summable (fun m : N.smoothNumbers ↦ ‖f m‖) ∧
      HasSum (fun m : N.smoothNumbers ↦ f m) (∏ p ∈ N.primesBelow, (1 - f p)⁻¹) 
Arguments: {F : Type u_1} [NormedField F] [CompleteSpace F] {f : ℕ →* F} (h : ∀ {p : ℕ}, Nat.Prime p → ‖f p‖ < 1) (N : ℕ)
Goal: (Summable fun m => ‖f ↑m‖) ∧ HasSum (fun m => f ↑m) (∏ p ∈ N.primesBelow, (1 - f p)⁻¹)
-- EulerProduct.prod_primesBelow_geometric_eq_tsum_smoothNumbers
lemma prod_primesBelow_geometric_eq_tsum_smoothNumbers {f : ℕ →* F} (hsum : Summable f) (N : ℕ) :
    ∏ p ∈ N.primesBelow, (1 - f p)⁻¹ = ∑' m : N.smoothNumbers, f m 
Arguments: {F : Type u_1} [NormedField F] [CompleteSpace F] {f : ℕ →* F} (hsum : Summable ⇑f) (N : ℕ)
Goal: ∏ p ∈ N.primesBelow, (1 - f p)⁻¹ = ∑' (m : ↑N.smoothNumbers), f ↑m
-- EulerProduct.eulerProduct_completely_multiplicative_hasProd
theorem eulerProduct_completely_multiplicative_hasProd {f : ℕ →*₀ F} (hsum : Summable (‖f ·‖)) :
    HasProd (fun p : Primes ↦ (1 - f p)⁻¹) (∑' n, f n) 
Arguments: {F : Type u_1} [NormedField F] [CompleteSpace F] {f : ℕ →*₀ F} (hsum : Summable fun x => ‖f x‖)
Goal: HasProd (fun p => (1 - f ↑p)⁻¹) (∑' (n : ℕ), f n)
-- EulerProduct.eulerProduct_completely_multiplicative_tprod
theorem eulerProduct_completely_multiplicative_tprod {f : ℕ →*₀ F} (hsum : Summable (‖f ·‖)) :
    ∏' p : Primes, (1 - f p)⁻¹ = ∑' n, f n 
Arguments: {F : Type u_1} [NormedField F] [CompleteSpace F] {f : ℕ →*₀ F} (hsum : Summable fun x => ‖f x‖)
Goal: ∏' (p : Nat.Primes), (1 - f ↑p)⁻¹ = ∑' (n : ℕ), f n
-- EulerProduct.eulerProduct_completely_multiplicative
theorem eulerProduct_completely_multiplicative {f : ℕ →*₀ F} (hsum : Summable (‖f ·‖)) :
    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - f p)⁻¹) atTop (𝓝 (∑' n, f n)) 
Arguments: {F : Type u_1} [NormedField F] [CompleteSpace F] {f : ℕ →*₀ F} (hsum : Summable fun x => ‖f x‖)
Goal: Filter.Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - f p)⁻¹) Filter.atTop (𝓝 (∑' (n : ℕ), f n))
-- Polynomial.bernoulli_def
theorem bernoulli_def (n : ℕ) : bernoulli n =
    ∑ i ∈ range (n + 1), Polynomial.monomial i (_root_.bernoulli (n - i) * choose n i) 
Arguments: (n : ℕ)
Goal: Polynomial.bernoulli n = ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (bernoulli (n - i) * ↑(n.choose i))
-- Polynomial.bernoulli_zero
@[simp]
theorem bernoulli_zero : bernoulli 0 = 1 
Goal: Polynomial.bernoulli 0 = 1
-- Polynomial.bernoulli_eval_zero
@[simp]
theorem bernoulli_eval_zero (n : ℕ) : (bernoulli n).eval 0 = _root_.bernoulli n 
Arguments: (n : ℕ)
Goal: Polynomial.eval 0 (Polynomial.bernoulli n) = bernoulli n
-- Polynomial.bernoulli_eval_one
@[simp]
theorem bernoulli_eval_one (n : ℕ) : (bernoulli n).eval 1 = bernoulli' n 
Arguments: (n : ℕ)
Goal: Polynomial.eval 1 (Polynomial.bernoulli n) = bernoulli' n
-- Polynomial.derivative_bernoulli_add_one
theorem derivative_bernoulli_add_one (k : ℕ) :
    Polynomial.derivative (bernoulli (k + 1)) = (k + 1) * bernoulli k 
Arguments: (k : ℕ)
Goal: Polynomial.derivative (Polynomial.bernoulli (k + 1)) = (↑k + 1) * Polynomial.bernoulli k
-- Polynomial.derivative_bernoulli
theorem derivative_bernoulli (k : ℕ) :
    Polynomial.derivative (bernoulli k) = k * bernoulli (k - 1) 
Arguments: (k : ℕ)
Goal: Polynomial.derivative (Polynomial.bernoulli k) = ↑k * Polynomial.bernoulli (k - 1)
-- Polynomial.sum_bernoulli
@[simp]
nonrec theorem sum_bernoulli (n : ℕ) :
    (∑ k ∈ range (n + 1), ((n + 1).choose k : ℚ) • bernoulli k) = monomial n (n + 1 : ℚ) 
Arguments: (n : ℕ)
Goal: ∑ k ∈ Finset.range (n + 1), ↑((n + 1).choose k) • Polynomial.bernoulli k = (Polynomial.monomial n) (↑n + 1)
-- Polynomial.bernoulli_eq_sub_sum
theorem bernoulli_eq_sub_sum (n : ℕ) :
    (n.succ : ℚ) • bernoulli n =
      monomial n (n.succ : ℚ) - ∑ k ∈ Finset.range n, ((n + 1).choose k : ℚ) • bernoulli k 
Arguments: (n : ℕ)
Goal: ↑n.succ • Polynomial.bernoulli n =
  (Polynomial.monomial n) ↑n.succ - ∑ k ∈ Finset.range n, ↑((n + 1).choose k) • Polynomial.bernoulli k
-- Polynomial.sum_range_pow_eq_bernoulli_sub
theorem sum_range_pow_eq_bernoulli_sub (n p : ℕ) :
    ((p + 1 : ℚ) * ∑ k ∈ range n, (k : ℚ) ^ p) = (bernoulli p.succ).eval (n : ℚ) -
    _root_.bernoulli p.succ 
Arguments: (n p : ℕ)
Goal: (↑p + 1) * ∑ k ∈ Finset.range n, ↑k ^ p = Polynomial.eval (↑n) (Polynomial.bernoulli p.succ) - bernoulli p.succ
-- Polynomial.bernoulli_succ_eval
theorem bernoulli_succ_eval (n p : ℕ) : (bernoulli p.succ).eval (n : ℚ) =
    _root_.bernoulli p.succ + (p + 1 : ℚ) * ∑ k ∈ range n, (k : ℚ) ^ p 
Arguments: (n p : ℕ)
Goal: Polynomial.eval (↑n) (Polynomial.bernoulli p.succ) = bernoulli p.succ + (↑p + 1) * ∑ k ∈ Finset.range n, ↑k ^ p
-- Polynomial.bernoulli_eval_one_add
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
    (bernoulli n).eval (1 + x) = (bernoulli n).eval x + n * x ^ (n - 1) 
Arguments: (n : ℕ) (x : ℚ)
Goal: Polynomial.eval (1 + x) (Polynomial.bernoulli n) = Polynomial.eval x (Polynomial.bernoulli n) + ↑n * x ^ (n - 1)
-- Polynomial.bernoulli_generating_function
theorem bernoulli_generating_function (t : A) :
    (mk fun n => aeval t ((1 / n ! : ℚ) • bernoulli n)) * (exp A - 1) =
      PowerSeries.X * rescale t (exp A) 
Arguments: {A : Type u_1} [CommRing A] [Algebra ℚ A] (t : A)
Goal: (PowerSeries.mk fun n => (Polynomial.aeval t) ((1 / ↑n !) • Polynomial.bernoulli n)) * (PowerSeries.exp A - 1) =
  PowerSeries.X * (PowerSeries.rescale t) (PowerSeries.exp A)
-- LSeries.term_def
lemma term_def (f : ℕ → ℂ) (s : ℂ) (n : ℕ) :
    term f s n = if n = 0 then 0 else f n / n ^ s 
Arguments: (f : ℕ → ℂ) (s : ℂ) (n : ℕ)
Goal: LSeries.term f s n = if n = 0 then 0 else f n / ↑n ^ s
-- LSeries.term_zero
@[simp]
lemma term_zero (f : ℕ → ℂ) (s : ℂ) : term f s 0 = 0 
Arguments: (f : ℕ → ℂ) (s : ℂ)
Goal: LSeries.term f s 0 = 0
-- LSeries.term_of_ne_zero
@[simp]
lemma term_of_ne_zero {n : ℕ} (hn : n ≠ 0) (f : ℕ → ℂ) (s : ℂ) :
    term f s n = f n / n ^ s 
Arguments: {n : ℕ} (hn : n ≠ 0) (f : ℕ → ℂ) (s : ℂ)
Goal: LSeries.term f s n = f n / ↑n ^ s
-- LSeries.term_congr
lemma term_congr {f g : ℕ → ℂ} (h : ∀ {n}, n ≠ 0 → f n = g n) (s : ℂ) (n : ℕ) :
    term f s n = term g s n 
Arguments: {f g : ℕ → ℂ} (h : ∀ {n : ℕ}, n ≠ 0 → f n = g n) (s : ℂ) (n : ℕ)
Goal: LSeries.term f s n = LSeries.term g s n
-- LSeries.norm_term_eq
lemma norm_term_eq (f : ℕ → ℂ) (s : ℂ) (n : ℕ) :
    ‖term f s n‖ = if n = 0 then 0 else ‖f n‖ / n ^ s.re 
Arguments: (f : ℕ → ℂ) (s : ℂ) (n : ℕ)
Goal: ‖LSeries.term f s n‖ = if n = 0 then 0 else ‖f n‖ / ↑n ^ s.re
-- LSeries.norm_term_le
lemma norm_term_le {f g : ℕ → ℂ} (s : ℂ) {n : ℕ} (h : ‖f n‖ ≤ ‖g n‖) :
    ‖term f s n‖ ≤ ‖term g s n‖ 
Arguments: {f g : ℕ → ℂ} (s : ℂ) {n : ℕ} (h : ‖f n‖ ≤ ‖g n‖)
Goal: ‖LSeries.term f s n‖ ≤ ‖LSeries.term g s n‖
-- LSeries.norm_term_le_of_re_le_re
lemma norm_term_le_of_re_le_re (f : ℕ → ℂ) {s s' : ℂ} (h : s.re ≤ s'.re) (n : ℕ) :
    ‖term f s' n‖ ≤ ‖term f s n‖ 
Arguments: (f : ℕ → ℂ) {s s' : ℂ} (h : s.re ≤ s'.re) (n : ℕ)
Goal: ‖LSeries.term f s' n‖ ≤ ‖LSeries.term f s n‖
-- LSeries_congr
lemma LSeries_congr {f g : ℕ → ℂ} (s : ℂ) (h : ∀ {n}, n ≠ 0 → f n = g n) :
    LSeries f s = LSeries g s 
Arguments: {f g : ℕ → ℂ} (s : ℂ) (h : ∀ {n : ℕ}, n ≠ 0 → f n = g n)
Goal: LSeries f s = LSeries g s
-- LSeriesSummable_congr
lemma LSeriesSummable_congr {f g : ℕ → ℂ} (s : ℂ) (h : ∀ {n}, n ≠ 0 → f n = g n) :
    LSeriesSummable f s ↔ LSeriesSummable g s 
Arguments: {f g : ℕ → ℂ} (s : ℂ) (h : ∀ {n : ℕ}, n ≠ 0 → f n = g n)
Goal: LSeriesSummable f s ↔ LSeriesSummable g s
-- LSeriesSummable.congr'
lemma LSeriesSummable.congr' {f g : ℕ → ℂ} (s : ℂ) (h : f =ᶠ[atTop] g) (hf : LSeriesSummable f s) :
    LSeriesSummable g s 
Arguments: {f g : ℕ → ℂ} (s : ℂ) (h : f =ᶠ[Filter.atTop] g) (hf : LSeriesSummable f s)
Goal: LSeriesSummable g s
-- LSeriesSummable_congr'
lemma LSeriesSummable_congr' {f g : ℕ → ℂ} (s : ℂ) (h : f =ᶠ[atTop] g) :
    LSeriesSummable f s ↔ LSeriesSummable g s 
Arguments: {f g : ℕ → ℂ} (s : ℂ) (h : f =ᶠ[Filter.atTop] g)
Goal: LSeriesSummable f s ↔ LSeriesSummable g s
-- LSeries.eq_zero_of_not_LSeriesSummable
theorem LSeries.eq_zero_of_not_LSeriesSummable (f : ℕ → ℂ) (s : ℂ) :
    ¬ LSeriesSummable f s → LSeries f s = 0 
Arguments: (f : ℕ → ℂ) (s : ℂ)
Goal: ¬LSeriesSummable f s → LSeries f s = 0
-- LSeriesSummable_zero
@[simp]
theorem LSeriesSummable_zero {s : ℂ} : LSeriesSummable 0 s 
Arguments: {s : ℂ}
Goal: LSeriesSummable 0 s
-- LSeriesHasSum.LSeriesSummable
lemma LSeriesHasSum.LSeriesSummable {f : ℕ → ℂ} {s a : ℂ}
    (h : LSeriesHasSum f s a) : LSeriesSummable f s 
Arguments: {f : ℕ → ℂ} {s a : ℂ} (h : LSeriesHasSum f s a)
Goal: LSeriesSummable f s
-- LSeriesHasSum.LSeries_eq
lemma LSeriesHasSum.LSeries_eq {f : ℕ → ℂ} {s a : ℂ}
    (h : LSeriesHasSum f s a) : LSeries f s = a 
Arguments: {f : ℕ → ℂ} {s a : ℂ} (h : LSeriesHasSum f s a)
Goal: LSeries f s = a
-- LSeriesSummable.LSeriesHasSum
lemma LSeriesSummable.LSeriesHasSum {f : ℕ → ℂ} {s : ℂ} (h : LSeriesSummable f s) :
    LSeriesHasSum f s (LSeries f s) 
Arguments: {f : ℕ → ℂ} {s : ℂ} (h : LSeriesSummable f s)
Goal: LSeriesHasSum f s (LSeries f s)
-- LSeriesHasSum_iff
lemma LSeriesHasSum_iff {f : ℕ → ℂ} {s a : ℂ} :
    LSeriesHasSum f s a ↔ LSeriesSummable f s ∧ LSeries f s = a 
Arguments: {f : ℕ → ℂ} {s a : ℂ}
Goal: LSeriesHasSum f s a ↔ LSeriesSummable f s ∧ LSeries f s = a
-- LSeriesHasSum_congr
lemma LSeriesHasSum_congr {f g : ℕ → ℂ} (s a : ℂ) (h : ∀ {n}, n ≠ 0 → f n = g n) :
    LSeriesHasSum f s a ↔ LSeriesHasSum g s a 
Arguments: {f g : ℕ → ℂ} (s a : ℂ) (h : ∀ {n : ℕ}, n ≠ 0 → f n = g n)
Goal: LSeriesHasSum f s a ↔ LSeriesHasSum g s a
-- LSeriesSummable.of_re_le_re
lemma LSeriesSummable.of_re_le_re {f : ℕ → ℂ} {s s' : ℂ} (h : s.re ≤ s'.re)
    (hf : LSeriesSummable f s) : LSeriesSummable f s' 
Arguments: {f : ℕ → ℂ} {s s' : ℂ} (h : s.re ≤ s'.re) (hf : LSeriesSummable f s)
Goal: LSeriesSummable f s'
-- LSeriesSummable_iff_of_re_eq_re
theorem LSeriesSummable_iff_of_re_eq_re {f : ℕ → ℂ} {s s' : ℂ} (h : s.re = s'.re) :
    LSeriesSummable f s ↔ LSeriesSummable f s' 
Arguments: {f : ℕ → ℂ} {s s' : ℂ} (h : s.re = s'.re)
Goal: LSeriesSummable f s ↔ LSeriesSummable f s'
-- LSeries_zero
@[simp]
lemma LSeries_zero : LSeries 0 = 0 
Goal: LSeries 0 = 0
-- LSeries.term_delta
lemma term_delta (s : ℂ) (n : ℕ) : term δ s n = if n = 1 then 1 else 0 
Arguments: (s : ℂ) (n : ℕ)
Goal: LSeries.term LSeries.delta s n = if n = 1 then 1 else 0
-- LSeries.mul_delta_eq_smul_delta
lemma mul_delta_eq_smul_delta {f : ℕ → ℂ} : f * δ = f 1 • δ 
Arguments: {f : ℕ → ℂ}
Goal: f * LSeries.delta = f 1 • LSeries.delta
-- LSeries.mul_delta
lemma mul_delta {f : ℕ → ℂ} (h : f 1 = 1) : f * δ = δ 
Arguments: {f : ℕ → ℂ} (h : f 1 = 1)
Goal: f * LSeries.delta = LSeries.delta
-- LSeries.delta_mul_eq_smul_delta
lemma delta_mul_eq_smul_delta {f : ℕ → ℂ} : δ * f = f 1 • δ 
Arguments: {f : ℕ → ℂ}
Goal: LSeries.delta * f = f 1 • LSeries.delta
-- LSeries.delta_mul
lemma delta_mul {f : ℕ → ℂ} (h : f 1 = 1) : δ * f = δ 
Arguments: {f : ℕ → ℂ} (h : f 1 = 1)
Goal: LSeries.delta * f = LSeries.delta
-- LSeries_delta
lemma LSeries_delta : LSeries δ = 1 
Goal: LSeries LSeries.delta = 1
-- LSeriesSummable.le_const_mul_rpow
lemma LSeriesSummable.le_const_mul_rpow {f : ℕ → ℂ} {s : ℂ} (h : LSeriesSummable f s) :
    ∃ C, ∀ n ≠ 0, ‖f n‖ ≤ C * n ^ s.re 
Arguments: {f : ℕ → ℂ} {s : ℂ} (h : LSeriesSummable f s)
Goal: ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ s.re
-- LSeriesSummable.isBigO_rpow
lemma LSeriesSummable.isBigO_rpow {f : ℕ → ℂ} {s : ℂ} (h : LSeriesSummable f s) :
    f =O[atTop] fun n ↦ (n : ℝ) ^ s.re 
Arguments: {f : ℕ → ℂ} {s : ℂ} (h : LSeriesSummable f s)
Goal: f =O[Filter.atTop] fun n => ↑n ^ s.re
-- LSeriesSummable_of_le_const_mul_rpow
lemma LSeriesSummable_of_le_const_mul_rpow {f : ℕ → ℂ} {x : ℝ} {s : ℂ} (hs : x < s.re)
    (h : ∃ C, ∀ n ≠ 0, ‖f n‖ ≤ C * n ^ (x - 1)) :
    LSeriesSummable f s 
Arguments: {f : ℕ → ℂ} {x : ℝ} {s : ℂ} (hs : x < s.re) (h : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1))
Goal: LSeriesSummable f s
-- LSeriesSummable_of_isBigO_rpow
lemma LSeriesSummable_of_isBigO_rpow {f : ℕ → ℂ} {x : ℝ} {s : ℂ} (hs : x < s.re)
    (h : f =O[atTop] fun n ↦ (n : ℝ) ^ (x - 1)) :
    LSeriesSummable f s 
Arguments: {f : ℕ → ℂ} {x : ℝ} {s : ℂ} (hs : x < s.re) (h : f =O[Filter.atTop] fun n => ↑n ^ (x - 1))
Goal: LSeriesSummable f s
-- LSeriesSummable_of_bounded_of_one_lt_re
theorem LSeriesSummable_of_bounded_of_one_lt_re {f : ℕ → ℂ} {m : ℝ}
    (h : ∀ n ≠ 0, Complex.abs (f n) ≤ m) {s : ℂ} (hs : 1 < s.re) :
    LSeriesSummable f s 
Arguments: {f : ℕ → ℂ} {m : ℝ} (h : ∀ (n : ℕ), n ≠ 0 → Complex.abs (f n) ≤ m) {s : ℂ} (hs : 1 < s.re)
Goal: LSeriesSummable f s
-- LSeriesSummable_of_bounded_of_one_lt_real
theorem LSeriesSummable_of_bounded_of_one_lt_real {f : ℕ → ℂ} {m : ℝ}
    (h : ∀ n ≠ 0, Complex.abs (f n) ≤ m) {s : ℝ} (hs : 1 < s) :
    LSeriesSummable f s 
Arguments: {f : ℕ → ℂ} {m : ℝ} (h : ∀ (n : ℕ), n ≠ 0 → Complex.abs (f n) ≤ m) {s : ℝ} (hs : 1 < s)
Goal: LSeriesSummable f ↑s
-- LSeries.abscissaOfAbsConv_congr
lemma LSeries.abscissaOfAbsConv_congr {f g : ℕ → ℂ} (h : ∀ {n}, n ≠ 0 → f n = g n) :
    abscissaOfAbsConv f = abscissaOfAbsConv g 
Arguments: {f g : ℕ → ℂ} (h : ∀ {n : ℕ}, n ≠ 0 → f n = g n)
Goal: LSeries.abscissaOfAbsConv f = LSeries.abscissaOfAbsConv g
-- LSeries.abscissaOfAbsConv_congr'
lemma LSeries.abscissaOfAbsConv_congr' {f g : ℕ → ℂ} (h : f =ᶠ[atTop] g) :
    abscissaOfAbsConv f = abscissaOfAbsConv g 
Arguments: {f g : ℕ → ℂ} (h : f =ᶠ[Filter.atTop] g)
Goal: LSeries.abscissaOfAbsConv f = LSeries.abscissaOfAbsConv g