ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NNReal.inner_le_Lp_mul_Lq_tsum
theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjExponent q)
    (hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
    (Summable fun i => f i * g i) ∧
      ∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) 
Arguments: {ι : Type u} {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjExponent q) (hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q)
Goal: (Summable fun i => f i * g i) ∧
  ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)
-- NNReal.summable_mul_of_Lp_Lq
theorem summable_mul_of_Lp_Lq {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjExponent q)
    (hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
    Summable fun i => f i * g i 
Arguments: {ι : Type u} {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjExponent q) (hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q)
Goal: Summable fun i => f i * g i
-- NNReal.inner_le_Lp_mul_Lq_tsum'
theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjExponent q)
    (hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
    ∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) 
Arguments: {ι : Type u} {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjExponent q) (hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q)
Goal: ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)
-- NNReal.inner_le_Lp_mul_Lq_hasSum
theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
    (hpq : p.IsConjExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
    (hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C 
Arguments: {ι : Type u} {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ} (hpq : p.IsConjExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ q) (B ^ q))
Goal: ∃ C ≤ A * B, HasSum (fun i => f i * g i) C
-- NNReal.rpow_sum_le_const_mul_sum_rpow
theorem rpow_sum_le_const_mul_sum_rpow (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
    (∑ i ∈ s, f i) ^ p ≤ (card s : ℝ≥0) ^ (p - 1) * ∑ i ∈ s, f i ^ p 
Arguments: {ι : Type u} (s : Finset ι) (f : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p)
Goal: (∑ i ∈ s, f i) ^ p ≤ ↑s.card ^ (p - 1) * ∑ i ∈ s, f i ^ p
-- NNReal.isGreatest_Lp
theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q) :
    IsGreatest ((fun g : ι → ℝ≥0 => ∑ i ∈ s, f i * g i) '' { g | ∑ i ∈ s, g i ^ q ≤ 1 })
      ((∑ i ∈ s, f i ^ p) ^ (1 / p)) 
Arguments: {ι : Type u} (s : Finset ι) (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q)
Goal: IsGreatest ((fun g => ∑ i ∈ s, f i * g i) '' {g | ∑ i ∈ s, g i ^ q ≤ 1}) ((∑ i ∈ s, f i ^ p) ^ (1 / p))
-- NNReal.Lp_add_le
theorem Lp_add_le (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
    (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤
      (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p) 
Arguments: {ι : Type u} (s : Finset ι) (f g : ι → ℝ≥0) {p : ℝ} (hp : 1 ≤ p)
Goal: (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)
-- NNReal.Lp_add_le_tsum
theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
    (hg : Summable fun i => g i ^ p) :
    (Summable fun i => (f i + g i) ^ p) ∧
      (∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
        (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) 
Arguments: {ι : Type u} {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ p)
Goal: (Summable fun i => (f i + g i) ^ p) ∧
  (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)
-- NNReal.summable_Lp_add
theorem summable_Lp_add {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
    (hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p 
Arguments: {ι : Type u} {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ p)
Goal: Summable fun i => (f i + g i) ^ p
-- NNReal.Lp_add_le_tsum'
theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
    (hg : Summable fun i => g i ^ p) :
    (∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) 
Arguments: {ι : Type u} {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ p)
Goal: (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)
-- NNReal.Lp_add_le_hasSum
theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
    (hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
    ∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) 
Arguments: {ι : Type u} {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p))
Goal: ∃ C ≤ A + B, HasSum (fun i => (f i + g i) ^ p) (C ^ p)
-- Real.inner_le_Lp_mul_Lq
theorem inner_le_Lp_mul_Lq (hpq : IsConjExponent p q) :
    ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, |f i| ^ p) ^ (1 / p) * (∑ i ∈ s, |g i| ^ q) ^ (1 / q) 
Arguments: {ι : Type u} (s : Finset ι) (f g : ι → ℝ) {p q : ℝ} (hpq : p.IsConjExponent q)
Goal: ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, |f i| ^ p) ^ (1 / p) * (∑ i ∈ s, |g i| ^ q) ^ (1 / q)
-- Real.rpow_sum_le_const_mul_sum_rpow
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
    (∑ i ∈ s, |f i|) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i ∈ s, |f i| ^ p 
Arguments: {ι : Type u} (s : Finset ι) (f : ι → ℝ) {p : ℝ} (hp : 1 ≤ p)
Goal: (∑ i ∈ s, |f i|) ^ p ≤ ↑s.card ^ (p - 1) * ∑ i ∈ s, |f i| ^ p
-- Real.Lp_add_le
theorem Lp_add_le (hp : 1 ≤ p) :
    (∑ i ∈ s, |f i + g i| ^ p) ^ (1 / p) ≤
      (∑ i ∈ s, |f i| ^ p) ^ (1 / p) + (∑ i ∈ s, |g i| ^ p) ^ (1 / p) 
Arguments: {ι : Type u} (s : Finset ι) (f g : ι → ℝ) {p : ℝ} (hp : 1 ≤ p)
Goal: (∑ i ∈ s, |f i + g i| ^ p) ^ (1 / p) ≤ (∑ i ∈ s, |f i| ^ p) ^ (1 / p) + (∑ i ∈ s, |g i| ^ p) ^ (1 / p)
-- Real.inner_le_Lp_mul_Lq_of_nonneg
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : IsConjExponent p q) (hf : ∀ i ∈ s, 0 ≤ f i)
    (hg : ∀ i ∈ s, 0 ≤ g i) :
    ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) 
Arguments: {ι : Type u} (s : Finset ι) {f g : ι → ℝ} {p q : ℝ} (hpq : p.IsConjExponent q) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i)
Goal: ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)
-- Real.inner_le_weight_mul_Lp_of_nonneg
lemma inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) (w f : ι → ℝ)
    (hw : ∀ i, 0 ≤ w i) (hf : ∀ i, 0 ≤ f i) :
    ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ 
Arguments: {ι : Type u} (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) (w f : ι → ℝ) (hw : ∀ (i : ι), 0 ≤ w i) (hf : ∀ (i : ι), 0 ≤ f i)
Goal: ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
-- Real.inner_le_Lp_mul_Lq_tsum_of_nonneg
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.IsConjExponent q) (hf : ∀ i, 0 ≤ f i)
    (hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
    (Summable fun i => f i * g i) ∧
      ∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) 
Arguments: {ι : Type u} {f g : ι → ℝ} {p q : ℝ} (hpq : p.IsConjExponent q) (hf : ∀ (i : ι), 0 ≤ f i) (hg : ∀ (i : ι), 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q)
Goal: (Summable fun i => f i * g i) ∧
  ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)
-- Real.summable_mul_of_Lp_Lq_of_nonneg
theorem summable_mul_of_Lp_Lq_of_nonneg (hpq : p.IsConjExponent q) (hf : ∀ i, 0 ≤ f i)
    (hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
    Summable fun i => f i * g i 
Arguments: {ι : Type u} {f g : ι → ℝ} {p q : ℝ} (hpq : p.IsConjExponent q) (hf : ∀ (i : ι), 0 ≤ f i) (hg : ∀ (i : ι), 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q)
Goal: Summable fun i => f i * g i
-- Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjExponent q) (hf : ∀ i, 0 ≤ f i)
    (hg : ∀ i, 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q) :
    ∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) 
Arguments: {ι : Type u} {f g : ι → ℝ} {p q : ℝ} (hpq : p.IsConjExponent q) (hf : ∀ (i : ι), 0 ≤ f i) (hg : ∀ (i : ι), 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ q)
Goal: ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)
-- Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg
theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjExponent q) {A B : ℝ} (hA : 0 ≤ A)
    (hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
    (hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
    ∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C 
Arguments: {ι : Type u} {f g : ι → ℝ} {p q : ℝ} (hpq : p.IsConjExponent q) {A B : ℝ} (hA : 0 ≤ A) (hB : 0 ≤ B) (hf : ∀ (i : ι), 0 ≤ f i) (hg : ∀ (i : ι), 0 ≤ g i) (hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q))
Goal: ∃ C, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C
-- Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
theorem rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
    (∑ i ∈ s, f i) ^ p ≤ (card s : ℝ) ^ (p - 1) * ∑ i ∈ s, f i ^ p 
Arguments: {ι : Type u} (s : Finset ι) {f : ι → ℝ} {p : ℝ} (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i)
Goal: (∑ i ∈ s, f i) ^ p ≤ ↑s.card ^ (p - 1) * ∑ i ∈ s, f i ^ p
-- Real.Lp_add_le_of_nonneg
theorem Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
    (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤
      (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p) 
Arguments: {ι : Type u} (s : Finset ι) {f g : ι → ℝ} {p : ℝ} (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i)
Goal: (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)
-- Real.Lp_add_le_tsum_of_nonneg
theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
    (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
    (Summable fun i => (f i + g i) ^ p) ∧
      (∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
        (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) 
Arguments: {ι : Type u} {f g : ι → ℝ} {p : ℝ} (hp : 1 ≤ p) (hf : ∀ (i : ι), 0 ≤ f i) (hg : ∀ (i : ι), 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p)
Goal: (Summable fun i => (f i + g i) ^ p) ∧
  (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)
-- Real.summable_Lp_add_of_nonneg
theorem summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
    (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
    Summable fun i => (f i + g i) ^ p 
Arguments: {ι : Type u} {f g : ι → ℝ} {p : ℝ} (hp : 1 ≤ p) (hf : ∀ (i : ι), 0 ≤ f i) (hg : ∀ (i : ι), 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p)
Goal: Summable fun i => (f i + g i) ^ p
-- Real.Lp_add_le_tsum_of_nonneg'
theorem Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
    (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
    (∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) 
Arguments: {ι : Type u} {f g : ι → ℝ} {p : ℝ} (hp : 1 ≤ p) (hf : ∀ (i : ι), 0 ≤ f i) (hg : ∀ (i : ι), 0 ≤ g i) (hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p)
Goal: (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)
-- Real.Lp_add_le_hasSum_of_nonneg
theorem Lp_add_le_hasSum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
    (hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p))
    (hgB : HasSum (fun i => g i ^ p) (B ^ p)) :
    ∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) 
Arguments: {ι : Type u} {f g : ι → ℝ} {p : ℝ} (hp : 1 ≤ p) (hf : ∀ (i : ι), 0 ≤ f i) (hg : ∀ (i : ι), 0 ≤ g i) {A B : ℝ} (hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : HasSum (fun i => f i ^ p) (A ^ p)) (hgB : HasSum (fun i => g i ^ p) (B ^ p))
Goal: ∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p)
-- ENNReal.inner_le_Lp_mul_Lq
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjExponent q) :
    ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) 
Arguments: {ι : Type u} (s : Finset ι) (f g : ι → ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjExponent q)
Goal: ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)
-- ENNReal.inner_le_weight_mul_Lp_of_nonneg
lemma inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) (w f : ι → ℝ≥0∞) :
    ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ 
Arguments: {ι : Type u} (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) (w f : ι → ℝ≥0∞)
Goal: ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
-- ENNReal.rpow_sum_le_const_mul_sum_rpow
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
    (∑ i ∈ s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i ∈ s, f i ^ p 
Arguments: {ι : Type u} (s : Finset ι) (f : ι → ℝ≥0∞) {p : ℝ} (hp : 1 ≤ p)
Goal: (∑ i ∈ s, f i) ^ p ≤ ↑s.card ^ (p - 1) * ∑ i ∈ s, f i ^ p
-- ENNReal.Lp_add_le
theorem Lp_add_le (hp : 1 ≤ p) :
    (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤
      (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p) 
Arguments: {ι : Type u} (s : Finset ι) (f g : ι → ℝ≥0∞) {p : ℝ} (hp : 1 ≤ p)
Goal: (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)
-- zero_cpow_eq_nhds
theorem zero_cpow_eq_nhds {b : ℂ} (hb : b ≠ 0) : (fun x : ℂ => (0 : ℂ) ^ x) =ᶠ[𝓝 b] 0 
Arguments: {b : ℂ} (hb : b ≠ 0)
Goal: (fun x => 0 ^ x) =ᶠ[𝓝 b] 0
-- cpow_eq_nhds
theorem cpow_eq_nhds {a b : ℂ} (ha : a ≠ 0) :
    (fun x => x ^ b) =ᶠ[𝓝 a] fun x => exp (log x * b) 
Arguments: {a b : ℂ} (ha : a ≠ 0)
Goal: (fun x => x ^ b) =ᶠ[𝓝 a] fun x => Complex.exp (Complex.log x * b)
-- cpow_eq_nhds'
theorem cpow_eq_nhds' {p : ℂ × ℂ} (hp_fst : p.fst ≠ 0) :
    (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) 
Arguments: {p : ℂ × ℂ} (hp_fst : p.1 ≠ 0)
Goal: (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => Complex.exp (Complex.log x.1 * x.2)
-- continuousAt_const_cpow
theorem continuousAt_const_cpow {a b : ℂ} (ha : a ≠ 0) : ContinuousAt (fun x : ℂ => a ^ x) b 
Arguments: {a b : ℂ} (ha : a ≠ 0)
Goal: ContinuousAt (fun x => a ^ x) b
-- continuousAt_const_cpow'
theorem continuousAt_const_cpow' {a b : ℂ} (h : b ≠ 0) : ContinuousAt (fun x : ℂ => a ^ x) b 
Arguments: {a b : ℂ} (h : b ≠ 0)
Goal: ContinuousAt (fun x => a ^ x) b
-- continuousAt_cpow
theorem continuousAt_cpow {p : ℂ × ℂ} (hp_fst : p.fst ∈ slitPlane) :
    ContinuousAt (fun x : ℂ × ℂ => x.1 ^ x.2) p 
Arguments: {p : ℂ × ℂ} (hp_fst : p.1 ∈ Complex.slitPlane)
Goal: ContinuousAt (fun x => x.1 ^ x.2) p
-- continuousAt_cpow_const
theorem continuousAt_cpow_const {a b : ℂ} (ha : a ∈ slitPlane) :
    ContinuousAt (· ^ b) a 
Arguments: {a b : ℂ} (ha : a ∈ Complex.slitPlane)
Goal: ContinuousAt (fun x => x ^ b) a
-- Filter.Tendsto.cpow
theorem Filter.Tendsto.cpow {l : Filter α} {f g : α → ℂ} {a b : ℂ} (hf : Tendsto f l (𝓝 a))
    (hg : Tendsto g l (𝓝 b)) (ha : a ∈ slitPlane) :
    Tendsto (fun x => f x ^ g x) l (𝓝 (a ^ b)) 
Arguments: {α : Type u_1} {l : Filter α} {f g : α → ℂ} {a b : ℂ} (hf : Filter.Tendsto f l (𝓝 a)) (hg : Filter.Tendsto g l (𝓝 b)) (ha : a ∈ Complex.slitPlane)
Goal: Filter.Tendsto (fun x => f x ^ g x) l (𝓝 (a ^ b))
-- Filter.Tendsto.const_cpow
theorem Filter.Tendsto.const_cpow {l : Filter α} {f : α → ℂ} {a b : ℂ} (hf : Tendsto f l (𝓝 b))
    (h : a ≠ 0 ∨ b ≠ 0) : Tendsto (fun x => a ^ f x) l (𝓝 (a ^ b)) 
Arguments: {α : Type u_1} {l : Filter α} {f : α → ℂ} {a b : ℂ} (hf : Filter.Tendsto f l (𝓝 b)) (h : a ≠ 0 ∨ b ≠ 0)
Goal: Filter.Tendsto (fun x => a ^ f x) l (𝓝 (a ^ b))
-- ContinuousWithinAt.cpow
nonrec theorem ContinuousWithinAt.cpow (hf : ContinuousWithinAt f s a)
    (hg : ContinuousWithinAt g s a) (h0 : f a ∈ slitPlane) :
    ContinuousWithinAt (fun x => f x ^ g x) s a 
Arguments: {α : Type u_1} [TopologicalSpace α] {f g : α → ℂ} {s : Set α} {a : α} (hf : ContinuousWithinAt f s a) (hg : ContinuousWithinAt g s a) (h0 : f a ∈ Complex.slitPlane)
Goal: ContinuousWithinAt (fun x => f x ^ g x) s a
-- ContinuousWithinAt.const_cpow
nonrec theorem ContinuousWithinAt.const_cpow {b : ℂ} (hf : ContinuousWithinAt f s a)
    (h : b ≠ 0 ∨ f a ≠ 0) : ContinuousWithinAt (fun x => b ^ f x) s a 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} {s : Set α} {a : α} {b : ℂ} (hf : ContinuousWithinAt f s a) (h : b ≠ 0 ∨ f a ≠ 0)
Goal: ContinuousWithinAt (fun x => b ^ f x) s a
-- ContinuousAt.cpow
nonrec theorem ContinuousAt.cpow (hf : ContinuousAt f a) (hg : ContinuousAt g a)
    (h0 : f a ∈ slitPlane) : ContinuousAt (fun x => f x ^ g x) a 
Arguments: {α : Type u_1} [TopologicalSpace α] {f g : α → ℂ} {a : α} (hf : ContinuousAt f a) (hg : ContinuousAt g a) (h0 : f a ∈ Complex.slitPlane)
Goal: ContinuousAt (fun x => f x ^ g x) a
-- ContinuousAt.const_cpow
nonrec theorem ContinuousAt.const_cpow {b : ℂ} (hf : ContinuousAt f a) (h : b ≠ 0 ∨ f a ≠ 0) :
    ContinuousAt (fun x => b ^ f x) a 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} {a : α} {b : ℂ} (hf : ContinuousAt f a) (h : b ≠ 0 ∨ f a ≠ 0)
Goal: ContinuousAt (fun x => b ^ f x) a
-- ContinuousOn.cpow
theorem ContinuousOn.cpow (hf : ContinuousOn f s) (hg : ContinuousOn g s)
    (h0 : ∀ a ∈ s, f a ∈ slitPlane) : ContinuousOn (fun x => f x ^ g x) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {f g : α → ℂ} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (h0 : ∀ a ∈ s, f a ∈ Complex.slitPlane)
Goal: ContinuousOn (fun x => f x ^ g x) s
-- ContinuousOn.const_cpow
theorem ContinuousOn.const_cpow {b : ℂ} (hf : ContinuousOn f s) (h : b ≠ 0 ∨ ∀ a ∈ s, f a ≠ 0) :
    ContinuousOn (fun x => b ^ f x) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} {s : Set α} {b : ℂ} (hf : ContinuousOn f s) (h : b ≠ 0 ∨ ∀ a ∈ s, f a ≠ 0)
Goal: ContinuousOn (fun x => b ^ f x) s
-- Continuous.cpow
theorem Continuous.cpow (hf : Continuous f) (hg : Continuous g)
    (h0 : ∀ a, f a ∈ slitPlane) : Continuous fun x => f x ^ g x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f g : α → ℂ} (hf : Continuous f) (hg : Continuous g) (h0 : ∀ (a : α), f a ∈ Complex.slitPlane)
Goal: Continuous fun x => f x ^ g x
-- Continuous.const_cpow
theorem Continuous.const_cpow {b : ℂ} (hf : Continuous f) (h : b ≠ 0 ∨ ∀ a, f a ≠ 0) :
    Continuous fun x => b ^ f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} {b : ℂ} (hf : Continuous f) (h : b ≠ 0 ∨ ∀ (a : α), f a ≠ 0)
Goal: Continuous fun x => b ^ f x
-- ContinuousOn.cpow_const
theorem ContinuousOn.cpow_const {b : ℂ} (hf : ContinuousOn f s)
    (h : ∀ a : α, a ∈ s → f a ∈ slitPlane) : ContinuousOn (fun x => f x ^ b) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℂ} {s : Set α} {b : ℂ} (hf : ContinuousOn f s) (h : ∀ a ∈ s, f a ∈ Complex.slitPlane)
Goal: ContinuousOn (fun x => f x ^ b) s
-- Real.continuousAt_const_rpow
theorem continuousAt_const_rpow {a b : ℝ} (h : a ≠ 0) : ContinuousAt (a ^ ·) b 
Arguments: {a b : ℝ} (h : a ≠ 0)
Goal: ContinuousAt (fun x => a ^ x) b
-- Real.continuousAt_const_rpow'
theorem continuousAt_const_rpow' {a b : ℝ} (h : b ≠ 0) : ContinuousAt (a ^ ·) b 
Arguments: {a b : ℝ} (h : b ≠ 0)
Goal: ContinuousAt (fun x => a ^ x) b
-- Real.rpow_eq_nhds_of_neg
theorem rpow_eq_nhds_of_neg {p : ℝ × ℝ} (hp_fst : p.fst < 0) :
    (fun x : ℝ × ℝ => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) * cos (x.2 * π) 
Arguments: {p : ℝ × ℝ} (hp_fst : p.1 < 0)
Goal: (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => rexp (Real.log x.1 * x.2) * Real.cos (x.2 * π)
-- Real.rpow_eq_nhds_of_pos
theorem rpow_eq_nhds_of_pos {p : ℝ × ℝ} (hp_fst : 0 < p.fst) :
    (fun x : ℝ × ℝ => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) 
Arguments: {p : ℝ × ℝ} (hp_fst : 0 < p.1)
Goal: (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => rexp (Real.log x.1 * x.2)
-- Real.continuousAt_rpow_of_ne
theorem continuousAt_rpow_of_ne (p : ℝ × ℝ) (hp : p.1 ≠ 0) :
    ContinuousAt (fun p : ℝ × ℝ => p.1 ^ p.2) p 
Arguments: (p : ℝ × ℝ) (hp : p.1 ≠ 0)
Goal: ContinuousAt (fun p => p.1 ^ p.2) p
-- Real.continuousAt_rpow_of_pos
theorem continuousAt_rpow_of_pos (p : ℝ × ℝ) (hp : 0 < p.2) :
    ContinuousAt (fun p : ℝ × ℝ => p.1 ^ p.2) p 
Arguments: (p : ℝ × ℝ) (hp : 0 < p.2)
Goal: ContinuousAt (fun p => p.1 ^ p.2) p
-- Real.continuousAt_rpow
theorem continuousAt_rpow (p : ℝ × ℝ) (h : p.1 ≠ 0 ∨ 0 < p.2) :
    ContinuousAt (fun p : ℝ × ℝ => p.1 ^ p.2) p 
Arguments: (p : ℝ × ℝ) (h : p.1 ≠ 0 ∨ 0 < p.2)
Goal: ContinuousAt (fun p => p.1 ^ p.2) p
-- Real.continuousAt_rpow_const
@[fun_prop]
theorem continuousAt_rpow_const (x : ℝ) (q : ℝ) (h : x ≠ 0 ∨ 0 ≤ q) :
    ContinuousAt (fun x : ℝ => x ^ q) x 
Arguments: (x q : ℝ) (h : x ≠ 0 ∨ 0 ≤ q)
Goal: ContinuousAt (fun x => x ^ q) x
-- Real.continuous_rpow_const
@[fun_prop]
theorem continuous_rpow_const {q : ℝ} (h : 0 ≤ q) : Continuous (fun x : ℝ => x ^ q) 
Arguments: {q : ℝ} (h : 0 ≤ q)
Goal: Continuous fun x => x ^ q
-- Filter.Tendsto.rpow
theorem Filter.Tendsto.rpow {l : Filter α} {f g : α → ℝ} {x y : ℝ} (hf : Tendsto f l (𝓝 x))
    (hg : Tendsto g l (𝓝 y)) (h : x ≠ 0 ∨ 0 < y) : Tendsto (fun t => f t ^ g t) l (𝓝 (x ^ y)) 
Arguments: {α : Type u_1} {l : Filter α} {f g : α → ℝ} {x y : ℝ} (hf : Filter.Tendsto f l (𝓝 x)) (hg : Filter.Tendsto g l (𝓝 y)) (h : x ≠ 0 ∨ 0 < y)
Goal: Filter.Tendsto (fun t => f t ^ g t) l (𝓝 (x ^ y))
-- Filter.Tendsto.rpow_const
theorem Filter.Tendsto.rpow_const {l : Filter α} {f : α → ℝ} {x p : ℝ} (hf : Tendsto f l (𝓝 x))
    (h : x ≠ 0 ∨ 0 ≤ p) : Tendsto (fun a => f a ^ p) l (𝓝 (x ^ p)) 
Arguments: {α : Type u_1} {l : Filter α} {f : α → ℝ} {x p : ℝ} (hf : Filter.Tendsto f l (𝓝 x)) (h : x ≠ 0 ∨ 0 ≤ p)
Goal: Filter.Tendsto (fun a => f a ^ p) l (𝓝 (x ^ p))
-- ContinuousAt.rpow
nonrec theorem ContinuousAt.rpow (hf : ContinuousAt f x) (hg : ContinuousAt g x)
    (h : f x ≠ 0 ∨ 0 < g x) : ContinuousAt (fun t => f t ^ g t) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f g : α → ℝ} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x) (h : f x ≠ 0 ∨ 0 < g x)
Goal: ContinuousAt (fun t => f t ^ g t) x
-- ContinuousWithinAt.rpow
nonrec theorem ContinuousWithinAt.rpow (hf : ContinuousWithinAt f s x)
    (hg : ContinuousWithinAt g s x) (h : f x ≠ 0 ∨ 0 < g x) :
    ContinuousWithinAt (fun t => f t ^ g t) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f g : α → ℝ} {s : Set α} {x : α} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) (h : f x ≠ 0 ∨ 0 < g x)
Goal: ContinuousWithinAt (fun t => f t ^ g t) s x
-- ContinuousOn.rpow
theorem ContinuousOn.rpow (hf : ContinuousOn f s) (hg : ContinuousOn g s)
    (h : ∀ x ∈ s, f x ≠ 0 ∨ 0 < g x) : ContinuousOn (fun t => f t ^ g t) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {f g : α → ℝ} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (h : ∀ x ∈ s, f x ≠ 0 ∨ 0 < g x)
Goal: ContinuousOn (fun t => f t ^ g t) s
-- Continuous.rpow
theorem Continuous.rpow (hf : Continuous f) (hg : Continuous g) (h : ∀ x, f x ≠ 0 ∨ 0 < g x) :
    Continuous fun x => f x ^ g x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f g : α → ℝ} (hf : Continuous f) (hg : Continuous g) (h : ∀ (x : α), f x ≠ 0 ∨ 0 < g x)
Goal: Continuous fun x => f x ^ g x
-- ContinuousWithinAt.rpow_const
nonrec theorem ContinuousWithinAt.rpow_const (hf : ContinuousWithinAt f s x) (h : f x ≠ 0 ∨ 0 ≤ p) :
    ContinuousWithinAt (fun x => f x ^ p) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {s : Set α} {x : α} {p : ℝ} (hf : ContinuousWithinAt f s x) (h : f x ≠ 0 ∨ 0 ≤ p)
Goal: ContinuousWithinAt (fun x => f x ^ p) s x
-- ContinuousAt.rpow_const
nonrec theorem ContinuousAt.rpow_const (hf : ContinuousAt f x) (h : f x ≠ 0 ∨ 0 ≤ p) :
    ContinuousAt (fun x => f x ^ p) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {x : α} {p : ℝ} (hf : ContinuousAt f x) (h : f x ≠ 0 ∨ 0 ≤ p)
Goal: ContinuousAt (fun x => f x ^ p) x