Loogle!

Result

Found 5 declarations mentioning Polynomial.roots, Dvd.dvd, and LE.le. Of these, 5 match your pattern(s).

• Polynomial.roots.le_of_dvd 📋 Mathlib.Algebra.Polynomial.Roots
  {R : Type u} [CommRing R] [IsDomain R] {p q : Polynomial R} (h : q ≠ 0) : p ∣ q → p.roots ≤ q.roots
• Multiset.prod_X_sub_C_dvd_iff_le_roots 📋 Mathlib.Algebra.Polynomial.Roots
  {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p ≠ 0) (s : Multiset R) : (Multiset.map (fun a => Polynomial.X - Polynomial.C a) s).prod ∣ p ↔ s ≤ p.roots
• Polynomial.Splits.dvd_of_roots_le_roots 📋 Mathlib.Algebra.Polynomial.Factors
  {R : Type u_1} [Field R] {f g : Polynomial R} (hp : f.Splits) (hp0 : f ≠ 0) (hq : f.roots ≤ g.roots) : f ∣ g
• Polynomial.Splits.dvd_iff_roots_le_roots 📋 Mathlib.Algebra.Polynomial.Factors
  {R : Type u_1} [Field R] {f g : Polynomial R} (hf : f.Splits) (hf0 : f ≠ 0) (hg0 : g ≠ 0) : f ∣ g ↔ f.roots ≤ g.roots
• IsAlgClosed.dvd_iff_roots_le_roots 📋 Mathlib.FieldTheory.IsAlgClosed.Basic
  {k : Type u} [Field k] [IsAlgClosed k] {p q : Polynomial k} (hp : p ≠ 0) (hq : q ≠ 0) : p ∣ q ↔ p.roots ≤ q.roots

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 f91c049