Loogle!

Result

Found 6 declarations mentioning Polynomial, RatFunc, and Inv.inv. Of these, 6 match your pattern(s).

• RatFunc.inv_def 📋 Mathlib.FieldTheory.RatFunc.Basic
  {K : Type u_1} [CommRing K] [IsDomain K] (x✝ : RatFunc K) : x✝.inv = match x✝ with | { toFractionRing := p } => { toFractionRing := p⁻¹ }
• RatFunc.ofFractionRing_inv 📋 Mathlib.FieldTheory.RatFunc.Basic
  {K : Type u} [CommRing K] [IsDomain K] (p : FractionRing (Polynomial K)) : { toFractionRing := p⁻¹ } = { toFractionRing := p }⁻¹
• RatFunc.numDenom.eq_1 📋 Mathlib.FieldTheory.RatFunc.Basic
  {K : Type u} [Field K] (x : RatFunc K) : x.numDenom = x.liftOn' (fun p q => if q = 0 then (0, 1) else have r := gcd p q; (Polynomial.C (q / r).leadingCoeff⁻¹ * (p / r), Polynomial.C (q / r).leadingCoeff⁻¹ * (q / r))) ⋯
• RatFunc.num_div 📋 Mathlib.FieldTheory.RatFunc.Basic
  {K : Type u} [Field K] (p q : Polynomial K) : ((algebraMap (Polynomial K) (RatFunc K)) p / (algebraMap (Polynomial K) (RatFunc K)) q).num = Polynomial.C (q / gcd p q).leadingCoeff⁻¹ * (p / gcd p q)
• RatFunc.denom_div 📋 Mathlib.FieldTheory.RatFunc.Basic
  {K : Type u} [Field K] (p : Polynomial K) {q : Polynomial K} (hq : q ≠ 0) : ((algebraMap (Polynomial K) (RatFunc K)) p / (algebraMap (Polynomial K) (RatFunc K)) q).denom = Polynomial.C (q / gcd p q).leadingCoeff⁻¹ * (q / gcd p q)
• RatFunc.numDenom_div 📋 Mathlib.FieldTheory.RatFunc.Basic
  {K : Type u} [Field K] (p : Polynomial K) {q : Polynomial K} (hq : q ≠ 0) : ((algebraMap (Polynomial K) (RatFunc K)) p / (algebraMap (Polynomial K) (RatFunc K)) q).numDenom = (Polynomial.C (q / gcd p q).leadingCoeff⁻¹ * (p / gcd p q), Polynomial.C (q / gcd p q).leadingCoeff⁻¹ * (q / gcd p q))

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 83a7f42