Loogle!

Result

Found 20 declarations mentioning Real.tanh.

• Real.tanh 📋 Mathlib.Analysis.Complex.Trigonometric
  (x : ℝ) : ℝ
• Complex.ofReal_tanh 📋 Mathlib.Analysis.Complex.Trigonometric
  (x : ℝ) : ↑(Real.tanh x) = Complex.tanh ↑x
• Complex.tanh_ofReal_re 📋 Mathlib.Analysis.Complex.Trigonometric
  (x : ℝ) : (Complex.tanh ↑x).re = Real.tanh x
• Real.tanh_lt_one 📋 Mathlib.Analysis.Complex.Trigonometric
  (x : ℝ) : Real.tanh x < 1
• Real.tanh_neg 📋 Mathlib.Analysis.Complex.Trigonometric
  (x : ℝ) : Real.tanh (-x) = -Real.tanh x
• Real.neg_one_lt_tanh 📋 Mathlib.Analysis.Complex.Trigonometric
  (x : ℝ) : -1 < Real.tanh x
• Real.tanh_zero 📋 Mathlib.Analysis.Complex.Trigonometric
  : Real.tanh 0 = 0
• Real.abs_tanh_lt_one 📋 Mathlib.Analysis.Complex.Trigonometric
  (x : ℝ) : |Real.tanh x| < 1
• Real.tanh_eq_sinh_div_cosh 📋 Mathlib.Analysis.Complex.Trigonometric
  (x : ℝ) : Real.tanh x = Real.sinh x / Real.cosh x
• Real.tanh_sq_lt_one 📋 Mathlib.Analysis.Complex.Trigonometric
  (x : ℝ) : Real.tanh x ^ 2 < 1
• Real.tanh_eq 📋 Mathlib.Analysis.Complex.Trigonometric
  (x : ℝ) : Real.tanh x = (Real.exp x - Real.exp (-x)) / (Real.exp x + Real.exp (-x))
• Real.logDeriv_cosh 📋 Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
  : logDeriv Real.cosh = Real.tanh
• Real.tanh_arsinh 📋 Mathlib.Analysis.SpecialFunctions.Arsinh
  (x : ℝ) : Real.tanh (Real.arsinh x) = x / √(1 + x ^ 2)
• UpperHalfPlane.tanh_half_dist 📋 Mathlib.Analysis.Complex.UpperHalfPlane.Metric
  (z w : UpperHalfPlane) : Real.tanh (dist z w / 2) = dist ↑z ↑w / dist (↑z) ((starRingEnd ℂ) ↑w)
• Real.tanh_arcosh 📋 Mathlib.Analysis.SpecialFunctions.Arcosh
  {x : ℝ} (hx : 1 ≤ x) : Real.tanh (Real.arcosh x) = √(x ^ 2 - 1) / x
• Real.tanh_injective 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  : Function.Injective Real.tanh
• Real.artanh_tanh 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  (x : ℝ) : Real.artanh (Real.tanh x) = x
• Real.tanh_bijOn 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  : Set.BijOn Real.tanh Set.univ (Set.Ioo (-1) 1)
• Real.tanh_surjOn 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  : Set.SurjOn Real.tanh Set.univ (Set.Ioo (-1) 1)
• Real.tanh_artanh 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x : ℝ} (hx : x ∈ Set.Ioo (-1) 1) : Real.tanh (Real.artanh x) = x

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.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

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 128218b serving mathlib revision d8f2208