Loogle!

Result

Found 21 declarations mentioning Real.artanh.

• Real.artanh 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  (x : ℝ) : ℝ
• Real.artanh_tanh 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  (x : ℝ) : Real.artanh (Real.tanh x) = x
• Real.artanh_zero 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  : Real.artanh 0 = 0
• Real.artanh_injOn 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  : Set.InjOn Real.artanh (Set.Ioo (-1) 1)
• Real.artanh_nonneg 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x : ℝ} (hx : 0 ≤ x) : 0 ≤ Real.artanh x
• Real.artanh_nonpos 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x : ℝ} (hx : x ≤ 0) : Real.artanh x ≤ 0
• Real.artanh_bijOn 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  : Set.BijOn Real.artanh (Set.Ioo (-1) 1) Set.univ
• Real.artanh_surjOn 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  : Set.SurjOn Real.artanh (Set.Ioo (-1) 1) Set.univ
• Real.strictMonoOn_artanh 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  : StrictMonoOn Real.artanh (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
• Real.artanh_pos 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x : ℝ} (hx : x ∈ Set.Ioo 0 1) : 0 < Real.artanh x
• Real.artanh_le_artanh 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x y : ℝ} (hx : -1 < x) (hy : y < 1) (hxy : x ≤ y) : Real.artanh x ≤ Real.artanh y
• Real.artanh_lt_artanh 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x y : ℝ} (hx : -1 < x) (hy : y < 1) (hxy : x < y) : Real.artanh x < Real.artanh y
• Real.artanh_neg 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x : ℝ} (hx : x ∈ Set.Ioo (-1) 0) : Real.artanh x < 0
• Real.artanh_eq_zero_iff 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x : ℝ} : Real.artanh x = 0 ↔ x ≤ -1 ∨ x = 0 ∨ 1 ≤ x
• Real.artanh_le_artanh_iff 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x y : ℝ} (hx : x ∈ Set.Ioo (-1) 1) (hy : y ∈ Set.Ioo (-1) 1) : Real.artanh x ≤ Real.artanh y ↔ x ≤ y
• Real.artanh_lt_artanh_iff 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x y : ℝ} (hx : x ∈ Set.Ioo (-1) 1) (hy : y ∈ Set.Ioo (-1) 1) : Real.artanh x < Real.artanh y ↔ x < y
• Real.exp_artanh 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x : ℝ} (hx : x ∈ Set.Ioo (-1) 1) : Real.exp (Real.artanh x) = √((1 + x) / (1 - x))
• Real.sinh_artanh 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x : ℝ} (hx : x ∈ Set.Ioo (-1) 1) : Real.sinh (Real.artanh x) = x / √(1 - x ^ 2)
• Real.cosh_artanh 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x : ℝ} (hx : x ∈ Set.Ioo (-1) 1) : Real.cosh (Real.artanh x) = 1 / √(1 - x ^ 2)
• Real.artanh_eq_half_log 📋 Mathlib.Analysis.SpecialFunctions.Artanh
  {x : ℝ} (hx : x ∈ Set.Icc (-1) 1) : Real.artanh x = 1 / 2 * Real.log ((1 + x) / (1 - 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