Loogle!

Result

Found 16 declarations mentioning Real, Finset.prod, HPow.hPow and Finset.sum.

• Real.rpow_sum_of_pos 📋 Mathlib.Analysis.SpecialFunctions.Pow.Real
  {ι : Type u_1} {a : ℝ} (ha : 0 < a) (f : ι → ℝ) (s : Finset ι) : a ^ ∑ x ∈ s, f x = ∏ x ∈ s, a ^ f x
• Real.rpow_sum_of_nonneg 📋 Mathlib.Analysis.SpecialFunctions.Pow.Real
  {ι : Type u_1} {a : ℝ} (ha : 0 ≤ a) {s : Finset ι} {f : ι → ℝ} (h : ∀ x ∈ s, 0 ≤ f x) : a ^ ∑ x ∈ s, f x = ∏ x ∈ s, a ^ f x
• NNReal.geom_mean_le_arith_mean_weighted 📋 Mathlib.Analysis.MeanInequalities
  {ι : Type u} (s : Finset ι) (w z : ι → NNReal) (hw' : ∑ i ∈ s, w i = 1) : ∏ i ∈ s, z i ^ ↑(w i) ≤ ∑ i ∈ s, w i * z i
• Real.geom_mean_le_arith_mean_weighted 📋 Mathlib.Analysis.MeanInequalities
  {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) : ∏ i ∈ s, z i ^ w i ≤ ∑ i ∈ s, w i * z i
• Real.geom_mean_weighted_of_constant 📋 Mathlib.Analysis.MeanInequalities
  {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∏ i ∈ s, z i ^ w i = x
• Real.harm_mean_le_geom_mean_weighted 📋 Mathlib.Analysis.MeanInequalities
  {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hs : s.Nonempty) (hw : ∀ i ∈ s, 0 < w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 < z i) : (∑ i ∈ s, w i / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ w i
• Real.geom_mean_eq_arith_mean_weighted_of_constant 📋 Mathlib.Analysis.MeanInequalities
  {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i
• Real.geom_mean_eq_arith_mean_weighted_iff' 📋 Mathlib.Analysis.MeanInequalities
  {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 < w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) : ∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i ↔ ∀ j ∈ s, z j = ∑ i ∈ s, w i * z i
• Real.geom_mean_lt_arith_mean_weighted_iff_of_pos 📋 Mathlib.Analysis.MeanInequalities
  {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 < w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) : ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i ↔ ∃ j ∈ s, ∃ k ∈ s, z j ≠ z k
• Real.geom_mean_le_arith_mean 📋 Mathlib.Analysis.MeanInequalities
  {ι : Type u_1} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i ∈ s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) : (∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹ ≤ (∑ i ∈ s, w i * z i) / ∑ i ∈ s, w i
• Real.geom_mean_eq_arith_mean_weighted_iff 📋 Mathlib.Analysis.MeanInequalities
  {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) : ∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i ↔ ∀ j ∈ s, w j ≠ 0 → z j = ∑ i ∈ s, w i * z i
• Real.harm_mean_le_geom_mean 📋 Mathlib.Analysis.MeanInequalities
  {ι : Type u_1} (s : Finset ι) (hs : s.Nonempty) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 < w i) (hw' : 0 < ∑ i ∈ s, w i) (hz : ∀ i ∈ s, 0 < z i) : (∑ i ∈ s, w i) / ∑ i ∈ s, w i / z i ≤ (∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹
• ENNReal.lintegral_prod_norm_pow_le 📋 Mathlib.MeasureTheory.Integral.MeanInequalities
  {α : Type u_2} {ι : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (s : Finset ι) {f : ι → α → ENNReal} (hf : ∀ i ∈ s, AEMeasurable (f i) μ) {p : ι → ℝ} (hp : ∑ i ∈ s, p i = 1) (h2p : ∀ i ∈ s, 0 ≤ p i) : ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
• ENNReal.lintegral_mul_prod_norm_pow_le 📋 Mathlib.MeasureTheory.Integral.MeanInequalities
  {α : Type u_2} {ι : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (s : Finset ι) {g : α → ENNReal} {f : ι → α → ENNReal} (hg : AEMeasurable g μ) (hf : ∀ i ∈ s, AEMeasurable (f i) μ) (q : ℝ) {p : ι → ℝ} (hpq : q + ∑ i ∈ s, p i = 1) (hq : 0 ≤ q) (hp : ∀ i ∈ s, 0 ≤ p i) : ∫⁻ (a : α), g a ^ q * ∏ i ∈ s, f i a ^ p i ∂μ ≤ (∫⁻ (a : α), g a ∂μ) ^ q * ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
• FormalMultilinearSeries.radius_right_inv_pos_of_radius_pos_aux1 📋 Mathlib.Analysis.Analytic.Inverse
  (n : ℕ) (p : ℕ → ℝ) (hp : ∀ (k : ℕ), 0 ≤ p k) {r a : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a) : ∑ k ∈ Finset.Ico 2 (n + 1), a ^ k * ∑ c ∈ {c | 1 < c.length}.toFinset, r ^ c.length * ∏ j, p (c.blocksFun j) ≤ ∑ j ∈ Finset.Ico 2 (n + 1), r ^ j * (∑ k ∈ Finset.Ico 1 n, a ^ k * p k) ^ j
• GaussianFourier.integral_cexp_neg_sum_mul_add 📋 Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
  {ι : Type u_2} [Fintype ι] {b : ι → ℂ} (hb : ∀ (i : ι), 0 < (b i).re) (c : ι → ℂ) : ∫ (v : ι → ℝ), Complex.exp (-∑ i, b i * ↑(v i) ^ 2 + ∑ i, c i * ↑(v i)) = ∏ i, (↑Real.pi / b i) ^ (1 / 2) * Complex.exp (c i ^ 2 / (4 * b i))

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision d796b90