ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul
theorem lintegral_rpow_eq_lintegral_meas_le_mul :
    ∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
      ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (μ : MeasureTheory.Measure α) (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) {p : ℝ} (p_pos : 0 < p)
Goal: ∫⁻ (ω : α), ENNReal.ofReal (f ω ^ p) ∂μ =
  ENNReal.ofReal p * ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1))
-- MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul
theorem lintegral_rpow_eq_lintegral_meas_lt_mul :
    ∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
      ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1)) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (μ : MeasureTheory.Measure α) (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) {p : ℝ} (p_pos : 0 < p)
Goal: ∫⁻ (ω : α), ENNReal.ofReal (f ω ^ p) ∂μ =
  ENNReal.ofReal p * ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))
-- bernstein_apply
@[simp]
theorem bernstein_apply (n ν : ℕ) (x : I) :
    bernstein n ν x = (n.choose ν : ℝ) * (x : ℝ) ^ ν * (1 - (x : ℝ)) ^ (n - ν) 
Arguments: (n ν : ℕ) (x : ↑I)
Goal: (bernstein n ν) x = ↑(n.choose ν) * ↑x ^ ν * (1 - ↑x) ^ (n - ν)
-- bernstein_nonneg
theorem bernstein_nonneg {n ν : ℕ} {x : I} : 0 ≤ bernstein n ν x 
Arguments: {n ν : ℕ} {x : ↑I}
Goal: 0 ≤ (bernstein n ν) x
-- bernstein.probability
theorem probability (n : ℕ) (x : I) : (∑ k : Fin (n + 1), bernstein n k x) = 1 
Arguments: (n : ℕ) (x : ↑I)
Goal: ∑ k : Fin (n + 1), (bernstein n ↑k) x = 1
-- bernstein.variance
theorem variance {n : ℕ} (h : 0 < (n : ℝ)) (x : I) :
    (∑ k : Fin (n + 1), (x - k/ₙ : ℝ) ^ 2 * bernstein n k x) = (x : ℝ) * (1 - x) / n 
Arguments: {n : ℕ} (h : 0 < ↑n) (x : ↑I)
Goal: ∑ k : Fin (n + 1), (↑x - ↑k/ₙ) ^ 2 * (bernstein n ↑k) x = ↑x * (1 - ↑x) / ↑n
-- bernsteinApproximation.apply
@[simp]
theorem apply (n : ℕ) (f : C(I, ℝ)) (x : I) :
    bernsteinApproximation n f x = ∑ k : Fin (n + 1), f k/ₙ * bernstein n k x 
Arguments: (n : ℕ) (f : C(↑I, ℝ)) (x : ↑I)
Goal: (bernsteinApproximation n f) x = ∑ k : Fin (n + 1), f k/ₙ * (bernstein n ↑k) x
-- bernsteinApproximation.δ_pos
theorem δ_pos {f : C(I, ℝ)} {ε : ℝ} {h : 0 < ε} : 0 < δ f ε h 
Arguments: {f : C(↑I, ℝ)} {ε : ℝ} {h : 0 < ε}
Goal: 0 < bernsteinApproximation.δ f ε h
-- bernsteinApproximation.lt_of_mem_S
theorem lt_of_mem_S {f : C(I, ℝ)} {ε : ℝ} {h : 0 < ε} {n : ℕ} {x : I} {k : Fin (n + 1)}
    (m : k ∈ S f ε h n x) : |f k/ₙ - f x| < ε / 2 
Arguments: {f : C(↑I, ℝ)} {ε : ℝ} {h : 0 < ε} {n : ℕ} {x : ↑I} {k : Fin (n + 1)} (m : k ∈ bernsteinApproximation.S f ε h n x)
Goal: |f k/ₙ - f x| < ε / 2
-- bernsteinApproximation.le_of_mem_S_compl
theorem le_of_mem_S_compl {f : C(I, ℝ)} {ε : ℝ} {h : 0 < ε} {n : ℕ} {x : I} {k : Fin (n + 1)}
    (m : k ∈ (S f ε h n x)ᶜ) : (1 : ℝ) ≤ δ f ε h ^ (-2 : ℤ) * ((x : ℝ) - k/ₙ) ^ 2 
Arguments: {f : C(↑I, ℝ)} {ε : ℝ} {h : 0 < ε} {n : ℕ} {x : ↑I} {k : Fin (n + 1)} (m : k ∈ (bernsteinApproximation.S f ε h n x)ᶜ)
Goal: 1 ≤ bernsteinApproximation.δ f ε h ^ (-2) * (↑x - ↑k/ₙ) ^ 2
-- bernsteinApproximation_uniform
theorem bernsteinApproximation_uniform (f : C(I, ℝ)) :
    Tendsto (fun n : ℕ => bernsteinApproximation n f) atTop (𝓝 f) 
Arguments: (f : C(↑I, ℝ))
Goal: Filter.Tendsto (fun n => bernsteinApproximation n f) Filter.atTop (𝓝 f)
-- Stirling.stirlingSeq_zero
@[simp]
theorem stirlingSeq_zero : stirlingSeq 0 = 0 
Goal: Stirling.stirlingSeq 0 = 0
-- Stirling.stirlingSeq_one
@[simp]
theorem stirlingSeq_one : stirlingSeq 1 = exp 1 / √2 
Goal: Stirling.stirlingSeq 1 = rexp 1 / √2
-- Stirling.log_stirlingSeq_formula
theorem log_stirlingSeq_formula (n : ℕ) :
    log (stirlingSeq n) = Real.log n ! - 1 / 2 * Real.log (2 * n) - n * log (n / exp 1) 
Arguments: (n : ℕ)
Goal: Real.log (Stirling.stirlingSeq n) = Real.log ↑n ! - 1 / 2 * Real.log (2 * ↑n) - ↑n * Real.log (↑n / rexp 1)
-- Stirling.log_stirlingSeq_diff_hasSum
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => (1 : ℝ) / (2 * ↑(k + 1) + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) 
Arguments: (m : ℕ)
Goal: HasSum (fun k => 1 / (2 * ↑(k + 1) + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ (k + 1))
  (Real.log (Stirling.stirlingSeq (m + 1)) - Real.log (Stirling.stirlingSeq (m + 2)))
-- Stirling.log_stirlingSeq'_antitone
theorem log_stirlingSeq'_antitone : Antitone (Real.log ∘ stirlingSeq ∘ succ) 
Goal: Antitone (Real.log ∘ Stirling.stirlingSeq ∘ Nat.succ)
-- Stirling.log_stirlingSeq_diff_le_geo_sum
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1 : ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - ((1 : ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) 
Arguments: (n : ℕ)
Goal: Real.log (Stirling.stirlingSeq (n + 1)) - Real.log (Stirling.stirlingSeq (n + 2)) ≤
  (1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2)
-- Stirling.log_stirlingSeq_sub_log_stirlingSeq_succ
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1) : ℝ) ^ 2) 
Arguments: (n : ℕ)
Goal: Real.log (Stirling.stirlingSeq (n + 1)) - Real.log (Stirling.stirlingSeq (n + 2)) ≤ 1 / (4 * ↑(n + 1) ^ 2)
-- Stirling.log_stirlingSeq_bounded_aux
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c 
Goal: ∃ c, ∀ (n : ℕ), Real.log (Stirling.stirlingSeq 1) - Real.log (Stirling.stirlingSeq (n + 1)) ≤ c
-- Stirling.log_stirlingSeq_bounded_by_constant
theorem log_stirlingSeq_bounded_by_constant : ∃ c, ∀ n : ℕ, c ≤ log (stirlingSeq (n + 1)) 
Goal: ∃ c, ∀ (n : ℕ), c ≤ Real.log (Stirling.stirlingSeq (n + 1))
-- Stirling.stirlingSeq'_pos
theorem stirlingSeq'_pos (n : ℕ) : 0 < stirlingSeq (n + 1) 
Arguments: (n : ℕ)
Goal: 0 < Stirling.stirlingSeq (n + 1)
-- Stirling.stirlingSeq'_bounded_by_pos_constant
theorem stirlingSeq'_bounded_by_pos_constant : ∃ a, 0 < a ∧ ∀ n : ℕ, a ≤ stirlingSeq (n + 1) 
Goal: ∃ a, 0 < a ∧ ∀ (n : ℕ), a ≤ Stirling.stirlingSeq (n + 1)
-- Stirling.stirlingSeq'_antitone
theorem stirlingSeq'_antitone : Antitone (stirlingSeq ∘ succ) 
Goal: Antitone (Stirling.stirlingSeq ∘ Nat.succ)
-- Stirling.stirlingSeq_has_pos_limit_a
theorem stirlingSeq_has_pos_limit_a : ∃ a : ℝ, 0 < a ∧ Tendsto stirlingSeq atTop (𝓝 a) 
Goal: ∃ a, 0 < a ∧ Filter.Tendsto Stirling.stirlingSeq Filter.atTop (𝓝 a)
-- Stirling.tendsto_self_div_two_mul_self_add_one
theorem tendsto_self_div_two_mul_self_add_one :
    Tendsto (fun n : ℕ => (n : ℝ) / (2 * n + 1)) atTop (𝓝 (1 / 2)) 
Goal: Filter.Tendsto (fun n => ↑n / (2 * ↑n + 1)) Filter.atTop (𝓝 (1 / 2))
-- Stirling.stirlingSeq_pow_four_div_stirlingSeq_pow_two_eq
theorem stirlingSeq_pow_four_div_stirlingSeq_pow_two_eq (n : ℕ) (hn : n ≠ 0) :
    stirlingSeq n ^ 4 / stirlingSeq (2 * n) ^ 2 * (n / (2 * n + 1)) = Wallis.W n 
Arguments: (n : ℕ) (hn : n ≠ 0)
Goal: Stirling.stirlingSeq n ^ 4 / Stirling.stirlingSeq (2 * n) ^ 2 * (↑n / (2 * ↑n + 1)) = Real.Wallis.W n
-- Stirling.second_wallis_limit
theorem second_wallis_limit (a : ℝ) (hane : a ≠ 0) (ha : Tendsto stirlingSeq atTop (𝓝 a)) :
    Tendsto Wallis.W atTop (𝓝 (a ^ 2 / 2)) 
Arguments: (a : ℝ) (hane : a ≠ 0) (ha : Filter.Tendsto Stirling.stirlingSeq Filter.atTop (𝓝 a))
Goal: Filter.Tendsto Real.Wallis.W Filter.atTop (𝓝 (a ^ 2 / 2))
-- Stirling.tendsto_stirlingSeq_sqrt_pi
theorem tendsto_stirlingSeq_sqrt_pi : Tendsto stirlingSeq atTop (𝓝 (√π)) 
Goal: Filter.Tendsto Stirling.stirlingSeq Filter.atTop (𝓝 √π)
-- Stirling.factorial_isEquivalent_stirling
lemma factorial_isEquivalent_stirling :
    (fun n ↦ n ! : ℕ → ℝ) ~[atTop] fun n ↦ Real.sqrt (2 * n * π) * (n / exp 1) ^ n 
Goal: (fun n => ↑n !) ~[Filter.atTop] fun n => √(2 * ↑n * π) * (↑n / rexp 1) ^ n
-- tendsto_rpow_atTop
theorem tendsto_rpow_atTop {y : ℝ} (hy : 0 < y) : Tendsto (fun x : ℝ => x ^ y) atTop atTop 
Arguments: {y : ℝ} (hy : 0 < y)
Goal: Filter.Tendsto (fun x => x ^ y) Filter.atTop Filter.atTop
-- tendsto_rpow_neg_atTop
theorem tendsto_rpow_neg_atTop {y : ℝ} (hy : 0 < y) : Tendsto (fun x : ℝ => x ^ (-y)) atTop (𝓝 0) 
Arguments: {y : ℝ} (hy : 0 < y)
Goal: Filter.Tendsto (fun x => x ^ (-y)) Filter.atTop (𝓝 0)
-- tendsto_rpow_atTop_of_base_lt_one
lemma tendsto_rpow_atTop_of_base_lt_one (b : ℝ) (hb₀ : -1 < b) (hb₁ : b < 1) :
    Tendsto (b ^ · : ℝ → ℝ) atTop (𝓝 (0 : ℝ)) 
Arguments: (b : ℝ) (hb₀ : -1 < b) (hb₁ : b < 1)
Goal: Filter.Tendsto (fun x => b ^ x) Filter.atTop (𝓝 0)
-- tendsto_rpow_atTop_of_base_gt_one
lemma tendsto_rpow_atTop_of_base_gt_one (b : ℝ) (hb : 1 < b) :
    Tendsto (b ^ · : ℝ → ℝ) atBot (𝓝 (0 : ℝ)) 
Arguments: (b : ℝ) (hb : 1 < b)
Goal: Filter.Tendsto (fun x => b ^ x) Filter.atBot (𝓝 0)
-- tendsto_rpow_atBot_of_base_lt_one
lemma tendsto_rpow_atBot_of_base_lt_one (b : ℝ) (hb₀ : 0 < b) (hb₁ : b < 1) :
    Tendsto (b ^ · : ℝ → ℝ) atBot atTop 
Arguments: (b : ℝ) (hb₀ : 0 < b) (hb₁ : b < 1)
Goal: Filter.Tendsto (fun x => b ^ x) Filter.atBot Filter.atTop
-- tendsto_rpow_atBot_of_base_gt_one
lemma tendsto_rpow_atBot_of_base_gt_one (b : ℝ) (hb : 1 < b) :
    Tendsto (b ^ · : ℝ → ℝ) atBot (𝓝 0) 
Arguments: (b : ℝ) (hb : 1 < b)
Goal: Filter.Tendsto (fun x => b ^ x) Filter.atBot (𝓝 0)
-- tendsto_rpow_div_mul_add
theorem tendsto_rpow_div_mul_add (a b c : ℝ) (hb : 0 ≠ b) :
    Tendsto (fun x => x ^ (a / (b * x + c))) atTop (𝓝 1) 
Arguments: (a b c : ℝ) (hb : 0 ≠ b)
Goal: Filter.Tendsto (fun x => x ^ (a / (b * x + c))) Filter.atTop (𝓝 1)
-- tendsto_rpow_div
theorem tendsto_rpow_div : Tendsto (fun x => x ^ ((1 : ℝ) / x)) atTop (𝓝 1) 
Goal: Filter.Tendsto (fun x => x ^ (1 / x)) Filter.atTop (𝓝 1)
-- tendsto_rpow_neg_div
theorem tendsto_rpow_neg_div : Tendsto (fun x => x ^ (-(1 : ℝ) / x)) atTop (𝓝 1) 
Goal: Filter.Tendsto (fun x => x ^ (-1 / x)) Filter.atTop (𝓝 1)
-- tendsto_exp_div_rpow_atTop
theorem tendsto_exp_div_rpow_atTop (s : ℝ) : Tendsto (fun x : ℝ => exp x / x ^ s) atTop atTop 
Arguments: (s : ℝ)
Goal: Filter.Tendsto (fun x => rexp x / x ^ s) Filter.atTop Filter.atTop
-- tendsto_exp_mul_div_rpow_atTop
theorem tendsto_exp_mul_div_rpow_atTop (s : ℝ) (b : ℝ) (hb : 0 < b) :
    Tendsto (fun x : ℝ => exp (b * x) / x ^ s) atTop atTop 
Arguments: (s b : ℝ) (hb : 0 < b)
Goal: Filter.Tendsto (fun x => rexp (b * x) / x ^ s) Filter.atTop Filter.atTop
-- tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero
theorem tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero (s : ℝ) (b : ℝ) (hb : 0 < b) :
    Tendsto (fun x : ℝ => x ^ s * exp (-b * x)) atTop (𝓝 0) 
Arguments: (s b : ℝ) (hb : 0 < b)
Goal: Filter.Tendsto (fun x => x ^ s * rexp (-b * x)) Filter.atTop (𝓝 0)
-- NNReal.tendsto_rpow_atTop
nonrec theorem NNReal.tendsto_rpow_atTop {y : ℝ} (hy : 0 < y) :
    Tendsto (fun x : ℝ≥0 => x ^ y) atTop atTop 
Arguments: {y : ℝ} (hy : 0 < y)
Goal: Filter.Tendsto (fun x => x ^ y) Filter.atTop Filter.atTop
-- ENNReal.tendsto_rpow_at_top
theorem ENNReal.tendsto_rpow_at_top {y : ℝ} (hy : 0 < y) :
    Tendsto (fun x : ℝ≥0∞ => x ^ y) (𝓝 ⊤) (𝓝 ⊤) 
Arguments: {y : ℝ} (hy : 0 < y)
Goal: Filter.Tendsto (fun x => x ^ y) (𝓝 ⊤) (𝓝 ⊤)
-- Complex.isTheta_exp_arg_mul_im
theorem isTheta_exp_arg_mul_im (hl : IsBoundedUnder (· ≤ ·) l fun x => |(g x).im|) :
    (fun x => Real.exp (arg (f x) * im (g x))) =Θ[l] fun _ => (1 : ℝ) 
Arguments: {α : Type u_1} {l : Filter α} {f g : α → ℂ} (hl : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => |(g x).im|)
Goal: (fun x => rexp ((f x).arg * (g x).im)) =Θ[l] fun x => 1
-- Complex.isBigO_cpow_rpow
theorem isBigO_cpow_rpow (hl : IsBoundedUnder (· ≤ ·) l fun x => |(g x).im|) :
    (fun x => f x ^ g x) =O[l] fun x => abs (f x) ^ (g x).re 
Arguments: {α : Type u_1} {l : Filter α} {f g : α → ℂ} (hl : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => |(g x).im|)
Goal: (fun x => f x ^ g x) =O[l] fun x => Complex.abs (f x) ^ (g x).re
-- Complex.isTheta_cpow_rpow
theorem isTheta_cpow_rpow (hl_im : IsBoundedUnder (· ≤ ·) l fun x => |(g x).im|)
    (hl : ∀ᶠ x in l, f x = 0 → re (g x) = 0 → g x = 0) :
    (fun x => f x ^ g x) =Θ[l] fun x => abs (f x) ^ (g x).re 
Arguments: {α : Type u_1} {l : Filter α} {f g : α → ℂ} (hl_im : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => |(g x).im|) (hl : ∀ᶠ (x : α) in l, f x = 0 → (g x).re = 0 → g x = 0)
Goal: (fun x => f x ^ g x) =Θ[l] fun x => Complex.abs (f x) ^ (g x).re
-- Complex.isTheta_cpow_const_rpow
theorem isTheta_cpow_const_rpow {b : ℂ} (hl : b.re = 0 → b ≠ 0 → ∀ᶠ x in l, f x ≠ 0) :
    (fun x => f x ^ b) =Θ[l] fun x => abs (f x) ^ b.re 
Arguments: {α : Type u_1} {l : Filter α} {f : α → ℂ} {b : ℂ} (hl : b.re = 0 → b ≠ 0 → ∀ᶠ (x : α) in l, f x ≠ 0)
Goal: (fun x => f x ^ b) =Θ[l] fun x => Complex.abs (f x) ^ b.re
-- Asymptotics.IsBigOWith.rpow
theorem IsBigOWith.rpow (h : IsBigOWith c l f g) (hc : 0 ≤ c) (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g) :
    IsBigOWith (c ^ r) l (fun x => f x ^ r) fun x => g x ^ r 
Arguments: {α : Type u_1} {r c : ℝ} {l : Filter α} {f g : α → ℝ} (h : Asymptotics.IsBigOWith c l f g) (hc : 0 ≤ c) (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g)
Goal: Asymptotics.IsBigOWith (c ^ r) l (fun x => f x ^ r) fun x => g x ^ r
-- Asymptotics.IsBigO.rpow
theorem IsBigO.rpow (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g) (h : f =O[l] g) :
    (fun x => f x ^ r) =O[l] fun x => g x ^ r 
Arguments: {α : Type u_1} {r : ℝ} {l : Filter α} {f g : α → ℝ} (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g) (h : f =O[l] g)
Goal: (fun x => f x ^ r) =O[l] fun x => g x ^ r
-- Asymptotics.IsTheta.rpow
theorem IsTheta.rpow (hr : 0 ≤ r) (hf : 0 ≤ᶠ[l] f) (hg : 0 ≤ᶠ[l] g) (h : f =Θ[l] g) :
    (fun x => f x ^ r) =Θ[l] fun x => g x ^ r 
Arguments: {α : Type u_1} {r : ℝ} {l : Filter α} {f g : α → ℝ} (hr : 0 ≤ r) (hf : 0 ≤ᶠ[l] f) (hg : 0 ≤ᶠ[l] g) (h : f =Θ[l] g)
Goal: (fun x => f x ^ r) =Θ[l] fun x => g x ^ r
-- Asymptotics.IsLittleO.rpow
theorem IsLittleO.rpow (hr : 0 < r) (hg : 0 ≤ᶠ[l] g) (h : f =o[l] g) :
    (fun x => f x ^ r) =o[l] fun x => g x ^ r 
Arguments: {α : Type u_1} {r : ℝ} {l : Filter α} {f g : α → ℝ} (hr : 0 < r) (hg : 0 ≤ᶠ[l] g) (h : f =o[l] g)
Goal: (fun x => f x ^ r) =o[l] fun x => g x ^ r
-- Asymptotics.IsBigO.sqrt
protected lemma IsBigO.sqrt (hfg : f =O[l] g) (hg : 0 ≤ᶠ[l] g) :
    (Real.sqrt <| f ·) =O[l] (Real.sqrt <| g ·) 
Arguments: {α : Type u_1} {l : Filter α} {f g : α → ℝ} (hfg : f =O[l] g) (hg : 0 ≤ᶠ[l] g)
Goal: (fun x => √(f x)) =O[l] fun x => √(g x)
-- Asymptotics.IsLittleO.sqrt
protected lemma IsLittleO.sqrt (hfg : f =o[l] g) (hg : 0 ≤ᶠ[l] g) :
    (Real.sqrt <| f ·) =o[l] (Real.sqrt <| g ·) 
Arguments: {α : Type u_1} {l : Filter α} {f g : α → ℝ} (hfg : f =o[l] g) (hg : 0 ≤ᶠ[l] g)
Goal: (fun x => √(f x)) =o[l] fun x => √(g x)
-- Asymptotics.IsTheta.sqrt
protected lemma IsTheta.sqrt (hfg : f =Θ[l] g) (hf : 0 ≤ᶠ[l] f) (hg : 0 ≤ᶠ[l] g) :
    (Real.sqrt <| f ·) =Θ[l] (Real.sqrt <| g ·) 
Arguments: {α : Type u_1} {l : Filter α} {f g : α → ℝ} (hfg : f =Θ[l] g) (hf : 0 ≤ᶠ[l] f) (hg : 0 ≤ᶠ[l] g)
Goal: (fun x => √(f x)) =Θ[l] fun x => √(g x)
-- isLittleO_rpow_exp_pos_mul_atTop
theorem isLittleO_rpow_exp_pos_mul_atTop (s : ℝ) {b : ℝ} (hb : 0 < b) :
    (fun x : ℝ => x ^ s) =o[atTop] fun x => exp (b * x) 
Arguments: (s : ℝ) {b : ℝ} (hb : 0 < b)
Goal: (fun x => x ^ s) =o[Filter.atTop] fun x => rexp (b * x)
-- isLittleO_zpow_exp_pos_mul_atTop
theorem isLittleO_zpow_exp_pos_mul_atTop (k : ℤ) {b : ℝ} (hb : 0 < b) :
    (fun x : ℝ => x ^ k) =o[atTop] fun x => exp (b * x) 
Arguments: (k : ℤ) {b : ℝ} (hb : 0 < b)
Goal: (fun x => x ^ k) =o[Filter.atTop] fun x => rexp (b * x)
-- isLittleO_pow_exp_pos_mul_atTop
theorem isLittleO_pow_exp_pos_mul_atTop (k : ℕ) {b : ℝ} (hb : 0 < b) :
    (fun x : ℝ => x ^ k) =o[atTop] fun x => exp (b * x) 
Arguments: (k : ℕ) {b : ℝ} (hb : 0 < b)
Goal: (fun x => x ^ k) =o[Filter.atTop] fun x => rexp (b * x)
-- isLittleO_rpow_exp_atTop
theorem isLittleO_rpow_exp_atTop (s : ℝ) : (fun x : ℝ => x ^ s) =o[atTop] exp 
Arguments: (s : ℝ)
Goal: (fun x => x ^ s) =o[Filter.atTop] rexp
-- isLittleO_exp_neg_mul_rpow_atTop
theorem isLittleO_exp_neg_mul_rpow_atTop {a : ℝ} (ha : 0 < a) (b : ℝ) :
    IsLittleO atTop (fun x : ℝ => exp (-a * x)) fun x : ℝ => x ^ b 
Arguments: {a : ℝ} (ha : 0 < a) (b : ℝ)
Goal: (fun x => rexp (-a * x)) =o[Filter.atTop] fun x => x ^ b
-- isLittleO_log_rpow_atTop
theorem isLittleO_log_rpow_atTop {r : ℝ} (hr : 0 < r) : log =o[atTop] fun x => x ^ r 
Arguments: {r : ℝ} (hr : 0 < r)
Goal: Real.log =o[Filter.atTop] fun x => x ^ r
-- isLittleO_log_rpow_rpow_atTop
theorem isLittleO_log_rpow_rpow_atTop {s : ℝ} (r : ℝ) (hs : 0 < s) :
    (fun x => log x ^ r) =o[atTop] fun x => x ^ s 
Arguments: {s : ℝ} (r : ℝ) (hs : 0 < s)
Goal: (fun x => Real.log x ^ r) =o[Filter.atTop] fun x => x ^ s
-- isLittleO_abs_log_rpow_rpow_nhds_zero
theorem isLittleO_abs_log_rpow_rpow_nhds_zero {s : ℝ} (r : ℝ) (hs : s < 0) :
    (fun x => |log x| ^ r) =o[𝓝[>] 0] fun x => x ^ s 
Arguments: {s : ℝ} (r : ℝ) (hs : s < 0)
Goal: (fun x => |Real.log x| ^ r) =o[𝓝[>] 0] fun x => x ^ s
-- isLittleO_log_rpow_nhds_zero
theorem isLittleO_log_rpow_nhds_zero {r : ℝ} (hr : r < 0) : log =o[𝓝[>] 0] fun x => x ^ r 
Arguments: {r : ℝ} (hr : r < 0)
Goal: Real.log =o[𝓝[>] 0] fun x => x ^ r
-- tendsto_log_div_rpow_nhds_zero
theorem tendsto_log_div_rpow_nhds_zero {r : ℝ} (hr : r < 0) :
    Tendsto (fun x => log x / x ^ r) (𝓝[>] 0) (𝓝 0) 
Arguments: {r : ℝ} (hr : r < 0)
Goal: Filter.Tendsto (fun x => Real.log x / x ^ r) (𝓝[>] 0) (𝓝 0)
-- tendsto_log_mul_rpow_nhds_zero
theorem tendsto_log_mul_rpow_nhds_zero {r : ℝ} (hr : 0 < r) :
    Tendsto (fun x => log x * x ^ r) (𝓝[>] 0) (𝓝 0) 
Arguments: {r : ℝ} (hr : 0 < r)
Goal: Filter.Tendsto (fun x => Real.log x * x ^ r) (𝓝[>] 0) (𝓝 0)
-- tendsto_log_mul_self_nhds_zero_left
lemma tendsto_log_mul_self_nhds_zero_left : Filter.Tendsto (fun x ↦ log x * x) (𝓝[<] 0) (𝓝 0) 
Goal: Filter.Tendsto (fun x => Real.log x * x) (𝓝[<] 0) (𝓝 0)
-- Polynomial.eventually_no_roots
theorem eventually_no_roots (hP : P ≠ 0) : ∀ᶠ x in atTop, ¬P.IsRoot x 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P : 𝕜[X]) (hP : P ≠ 0)
Goal: ∀ᶠ (x : 𝕜) in Filter.atTop, ¬P.IsRoot x
-- Polynomial.isEquivalent_atTop_lead
theorem isEquivalent_atTop_lead :
    (fun x => eval x P) ~[atTop] fun x => P.leadingCoeff * x ^ P.natDegree 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P : 𝕜[X]) [OrderTopology 𝕜]
Goal: (fun x => Polynomial.eval x P) ~[Filter.atTop] fun x => P.leadingCoeff * x ^ P.natDegree
-- Polynomial.tendsto_atTop_of_leadingCoeff_nonneg
theorem tendsto_atTop_of_leadingCoeff_nonneg (hdeg : 0 < P.degree) (hnng : 0 ≤ P.leadingCoeff) :
    Tendsto (fun x => eval x P) atTop atTop 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P : 𝕜[X]) [OrderTopology 𝕜] (hdeg : 0 < P.degree) (hnng : 0 ≤ P.leadingCoeff)
Goal: Filter.Tendsto (fun x => Polynomial.eval x P) Filter.atTop Filter.atTop
-- Polynomial.tendsto_atTop_iff_leadingCoeff_nonneg
theorem tendsto_atTop_iff_leadingCoeff_nonneg :
    Tendsto (fun x => eval x P) atTop atTop ↔ 0 < P.degree ∧ 0 ≤ P.leadingCoeff 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P : 𝕜[X]) [OrderTopology 𝕜]
Goal: Filter.Tendsto (fun x => Polynomial.eval x P) Filter.atTop Filter.atTop ↔ 0 < P.degree ∧ 0 ≤ P.leadingCoeff
-- Polynomial.tendsto_atBot_iff_leadingCoeff_nonpos
theorem tendsto_atBot_iff_leadingCoeff_nonpos :
    Tendsto (fun x => eval x P) atTop atBot ↔ 0 < P.degree ∧ P.leadingCoeff ≤ 0 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P : 𝕜[X]) [OrderTopology 𝕜]
Goal: Filter.Tendsto (fun x => Polynomial.eval x P) Filter.atTop Filter.atBot ↔ 0 < P.degree ∧ P.leadingCoeff ≤ 0
-- Polynomial.tendsto_atBot_of_leadingCoeff_nonpos
theorem tendsto_atBot_of_leadingCoeff_nonpos (hdeg : 0 < P.degree) (hnps : P.leadingCoeff ≤ 0) :
    Tendsto (fun x => eval x P) atTop atBot 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P : 𝕜[X]) [OrderTopology 𝕜] (hdeg : 0 < P.degree) (hnps : P.leadingCoeff ≤ 0)
Goal: Filter.Tendsto (fun x => Polynomial.eval x P) Filter.atTop Filter.atBot
-- Polynomial.abs_tendsto_atTop
theorem abs_tendsto_atTop (hdeg : 0 < P.degree) :
    Tendsto (fun x => abs <| eval x P) atTop atTop 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P : 𝕜[X]) [OrderTopology 𝕜] (hdeg : 0 < P.degree)
Goal: Filter.Tendsto (fun x => |Polynomial.eval x P|) Filter.atTop Filter.atTop
-- Polynomial.abs_isBoundedUnder_iff
theorem abs_isBoundedUnder_iff :
    (IsBoundedUnder (· ≤ ·) atTop fun x => |eval x P|) ↔ P.degree ≤ 0 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P : 𝕜[X]) [OrderTopology 𝕜]
Goal: (Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) Filter.atTop fun x => |Polynomial.eval x P|) ↔ P.degree ≤ 0
-- Polynomial.abs_tendsto_atTop_iff
theorem abs_tendsto_atTop_iff : Tendsto (fun x => abs <| eval x P) atTop atTop ↔ 0 < P.degree 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P : 𝕜[X]) [OrderTopology 𝕜]
Goal: Filter.Tendsto (fun x => |Polynomial.eval x P|) Filter.atTop Filter.atTop ↔ 0 < P.degree
-- Polynomial.tendsto_nhds_iff
theorem tendsto_nhds_iff {c : 𝕜} :
    Tendsto (fun x => eval x P) atTop (𝓝 c) ↔ P.leadingCoeff = c ∧ P.degree ≤ 0 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P : 𝕜[X]) [OrderTopology 𝕜] {c : 𝕜}
Goal: Filter.Tendsto (fun x => Polynomial.eval x P) Filter.atTop (𝓝 c) ↔ P.leadingCoeff = c ∧ P.degree ≤ 0
-- Polynomial.isEquivalent_atTop_div
theorem isEquivalent_atTop_div :
    (fun x => eval x P / eval x Q) ~[atTop] fun x =>
      P.leadingCoeff / Q.leadingCoeff * x ^ (P.natDegree - Q.natDegree : ℤ) 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P Q : 𝕜[X]) [OrderTopology 𝕜]
Goal: (fun x => Polynomial.eval x P / Polynomial.eval x Q) ~[Filter.atTop] fun x =>
  P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)
-- Polynomial.div_tendsto_zero_of_degree_lt
theorem div_tendsto_zero_of_degree_lt (hdeg : P.degree < Q.degree) :
    Tendsto (fun x => eval x P / eval x Q) atTop (𝓝 0) 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P Q : 𝕜[X]) [OrderTopology 𝕜] (hdeg : P.degree < Q.degree)
Goal: Filter.Tendsto (fun x => Polynomial.eval x P / Polynomial.eval x Q) Filter.atTop (𝓝 0)
-- Polynomial.div_tendsto_zero_iff_degree_lt
theorem div_tendsto_zero_iff_degree_lt (hQ : Q ≠ 0) :
    Tendsto (fun x => eval x P / eval x Q) atTop (𝓝 0) ↔ P.degree < Q.degree 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P Q : 𝕜[X]) [OrderTopology 𝕜] (hQ : Q ≠ 0)
Goal: Filter.Tendsto (fun x => Polynomial.eval x P / Polynomial.eval x Q) Filter.atTop (𝓝 0) ↔ P.degree < Q.degree
-- Polynomial.div_tendsto_leadingCoeff_div_of_degree_eq
theorem div_tendsto_leadingCoeff_div_of_degree_eq (hdeg : P.degree = Q.degree) :
    Tendsto (fun x => eval x P / eval x Q) atTop (𝓝 <| P.leadingCoeff / Q.leadingCoeff) 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P Q : 𝕜[X]) [OrderTopology 𝕜] (hdeg : P.degree = Q.degree)
Goal: Filter.Tendsto (fun x => Polynomial.eval x P / Polynomial.eval x Q) Filter.atTop (𝓝 (P.leadingCoeff / Q.leadingCoeff))
-- Polynomial.div_tendsto_atTop_of_degree_gt'
theorem div_tendsto_atTop_of_degree_gt' (hdeg : Q.degree < P.degree)
    (hpos : 0 < P.leadingCoeff / Q.leadingCoeff) :
    Tendsto (fun x => eval x P / eval x Q) atTop atTop 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P Q : 𝕜[X]) [OrderTopology 𝕜] (hdeg : Q.degree < P.degree) (hpos : 0 < P.leadingCoeff / Q.leadingCoeff)
Goal: Filter.Tendsto (fun x => Polynomial.eval x P / Polynomial.eval x Q) Filter.atTop Filter.atTop
-- Polynomial.div_tendsto_atTop_of_degree_gt
theorem div_tendsto_atTop_of_degree_gt (hdeg : Q.degree < P.degree) (hQ : Q ≠ 0)
    (hnng : 0 ≤ P.leadingCoeff / Q.leadingCoeff) :
    Tendsto (fun x => eval x P / eval x Q) atTop atTop 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P Q : 𝕜[X]) [OrderTopology 𝕜] (hdeg : Q.degree < P.degree) (hQ : Q ≠ 0) (hnng : 0 ≤ P.leadingCoeff / Q.leadingCoeff)
Goal: Filter.Tendsto (fun x => Polynomial.eval x P / Polynomial.eval x Q) Filter.atTop Filter.atTop
-- Polynomial.div_tendsto_atBot_of_degree_gt'
theorem div_tendsto_atBot_of_degree_gt' (hdeg : Q.degree < P.degree)
    (hneg : P.leadingCoeff / Q.leadingCoeff < 0) :
    Tendsto (fun x => eval x P / eval x Q) atTop atBot 
Arguments: {𝕜 : Type u_1} [NormedLinearOrderedField 𝕜] (P Q : 𝕜[X]) [OrderTopology 𝕜] (hdeg : Q.degree < P.degree) (hneg : P.leadingCoeff / Q.leadingCoeff < 0)
Goal: Filter.Tendsto (fun x => Polynomial.eval x P / Polynomial.eval x Q) Filter.atTop Filter.atBot