Loogle!

Result

Found 11 declarations mentioning Polynomial.C, Polynomial.roots, Multiset.map and Multiset.prod.

• Polynomial.roots_multiset_prod_X_sub_C 📋 Mathlib.Algebra.Polynomial.Roots
  {R : Type u} [CommRing R] [IsDomain R] (s : Multiset R) : (Multiset.map (fun a => Polynomial.X - Polynomial.C a) s).prod.roots = s
• Polynomial.prod_multiset_X_sub_C_of_monic_of_roots_card_eq 📋 Mathlib.Algebra.Polynomial.Roots
  {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp : p.Monic) (hroots : p.roots.card = p.natDegree) : (Multiset.map (fun a => Polynomial.X - Polynomial.C a) p.roots).prod = p
• Polynomial.prod_multiset_X_sub_C_dvd 📋 Mathlib.Algebra.Polynomial.Roots
  {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) : (Multiset.map (fun a => Polynomial.X - Polynomial.C a) p.roots).prod ∣ p
• 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.exists_prod_multiset_X_sub_C_mul 📋 Mathlib.Algebra.Polynomial.Roots
  {R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) : ∃ q, (Multiset.map (fun a => Polynomial.X - Polynomial.C a) p.roots).prod * q = p ∧ p.roots.card + q.natDegree = p.natDegree ∧ q.roots = 0
• Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_C 📋 Mathlib.Algebra.Polynomial.Roots
  {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hroots : p.roots.card = p.natDegree) : Polynomial.C p.leadingCoeff * (Multiset.map (fun a => Polynomial.X - Polynomial.C a) p.roots).prod = p
• Polynomial.prod_multiset_root_eq_finset_root 📋 Mathlib.Algebra.Polynomial.Roots
  {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} [DecidableEq R] : (Multiset.map (fun a => Polynomial.X - Polynomial.C a) p.roots).prod = ∏ a ∈ p.roots.toFinset, (Polynomial.X - Polynomial.C a) ^ Polynomial.rootMultiplicity a p
• Polynomial.eq_prod_roots_of_monic_of_splits_id 📋 Mathlib.Algebra.Polynomial.Splits
  {K : Type v} [Field K] {p : Polynomial K} (m : p.Monic) (hsplit : Polynomial.Splits (RingHom.id K) p) : p = (Multiset.map (fun a => Polynomial.X - Polynomial.C a) p.roots).prod
• Polynomial.eq_prod_roots_of_splits_id 📋 Mathlib.Algebra.Polynomial.Splits
  {K : Type v} [Field K] {p : Polynomial K} (hsplit : Polynomial.Splits (RingHom.id K) p) : p = Polynomial.C p.leadingCoeff * (Multiset.map (fun a => Polynomial.X - Polynomial.C a) p.roots).prod
• Polynomial.eq_prod_roots_of_splits 📋 Mathlib.Algebra.Polynomial.Splits
  {K : Type v} {L : Type w} [Field K] [Field L] {p : Polynomial K} {i : K →+* L} (hsplit : Polynomial.Splits i p) : Polynomial.map i p = Polynomial.C (i p.leadingCoeff) * (Multiset.map (fun a => Polynomial.X - Polynomial.C a) (Polynomial.map i p).roots).prod
• spectrum.exists_mem_of_not_isUnit_aeval_prod 📋 Mathlib.FieldTheory.IsAlgClosed.Spectrum
  {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] [IsDomain R] {p : Polynomial R} {a : A} (h : ¬IsUnit ((Polynomial.aeval a) (Multiset.map (fun x => Polynomial.X - Polynomial.C x) p.roots).prod)) : ∃ k ∈ spectrum R a, Polynomial.eval k p = 0

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 19971e9 serving mathlib revision bce1d65