Loogle!

Result

Found 5 declarations mentioning HasDerivAt, Real.log, and HPow.hPow.

• Real.hasDerivAt_half_log_one_add_div_one_sub_sub_sum_range 📋 Mathlib.Analysis.SpecialFunctions.Log.Deriv
  {y : ℝ} (n : ℕ) (hy₁ : -1 < y) (hy₂ : y < 1) : HasDerivAt (fun x => 1 / 2 * Real.log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)) ((y ^ 2) ^ n / (1 - y ^ 2)) y
• HasDerivAt.const_rpow 📋 Mathlib.Analysis.SpecialFunctions.Pow.Deriv
  {f : ℝ → ℝ} {f' x a : ℝ} (ha : 0 < a) (hf : HasDerivAt f f' x) : HasDerivAt (fun x => a ^ f x) (Real.log a * f' * a ^ f x) x
• HasDerivAt.rpow 📋 Mathlib.Analysis.SpecialFunctions.Pow.Deriv
  {f g : ℝ → ℝ} {f' g' x : ℝ} (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x) (h : 0 < f x) : HasDerivAt (fun x => f x ^ g x) (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) x
• mellin_hasDerivAt_of_isBigO_rpow 📋 Mathlib.Analysis.MellinTransform
  {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} {f : ℝ → E} {s : ℂ} (hfc : MeasureTheory.LocallyIntegrableOn f (Set.Ioi 0) MeasureTheory.volume) (hf_top : f =O[Filter.atTop] fun x => x ^ (-a)) (hs_top : s.re < a) (hf_bot : f =O[nhdsWithin 0 (Set.Ioi 0)] fun x => x ^ (-b)) (hs_bot : b < s.re) : MellinConvergent (fun t => Real.log t • f t) s ∧ HasDerivAt (mellin f) (mellin (fun t => Real.log t • f t) s) s
• Complex.hasDerivAt_GammaIntegral 📋 Mathlib.Analysis.SpecialFunctions.Gamma.Deriv
  {s : ℂ} (hs : 0 < s.re) : HasDerivAt Complex.GammaIntegral (∫ (t : ℝ) in Set.Ioi 0, ↑t ^ (s - 1) * (↑(Real.log t) * ↑(Real.exp (-t)))) s

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 6ff4759 serving mathlib revision c05a1ae