Loogle!

Result

Found 336 declarations mentioning MvPolynomial.X. Of these, 7 have a name containing "dvd".

• MvPolynomial.X_dvd_X 📋 Mathlib.Algebra.MvPolynomial.Division
  {σ : Type u_1} {R : Type u_2} [CommSemiring R] [Nontrivial R] {i j : σ} : MvPolynomial.X i ∣ MvPolynomial.X j ↔ i = j
• MvPolynomial.X_dvd_iff_modMonomial_eq_zero 📋 Mathlib.Algebra.MvPolynomial.Division
  {σ : Type u_1} {R : Type u_2} [CommSemiring R] {i : σ} {x : MvPolynomial σ R} : MvPolynomial.X i ∣ x ↔ (x.modMonomial fun₀ | i => 1) = 0
• MvPolynomial.X_dvd_monomial 📋 Mathlib.Algebra.MvPolynomial.Division
  {σ : Type u_1} {R : Type u_2} [CommSemiring R] {i : σ} {j : σ →₀ ℕ} {r : R} : MvPolynomial.X i ∣ (MvPolynomial.monomial j) r ↔ r = 0 ∨ j i ≠ 0
• MvPolynomial.X_dvd_mul_iff 📋 Mathlib.Algebra.MvPolynomial.Division
  {σ : Type u_1} {R : Type u_3} [CommRing R] {i : σ} {p q : MvPolynomial σ R} [IsCancelMulZero R] : MvPolynomial.X i ∣ p * q ↔ MvPolynomial.X i ∣ p ∨ MvPolynomial.X i ∣ q
• MvPolynomial.dvd_X_mul_iff 📋 Mathlib.Algebra.MvPolynomial.Division
  {σ : Type u_1} {R : Type u_3} [CommRing R] {i : σ} {p q : MvPolynomial σ R} [IsCancelMulZero R] : p ∣ MvPolynomial.X i * q ↔ p ∣ q ∨ MvPolynomial.X i ∣ p ∧ (p.divMonomial fun₀ | i => 1) ∣ q
• MvPolynomial.dvd_X_iff_exists 📋 Mathlib.Algebra.MvPolynomial.NoZeroDivisors
  {R : Type u_1} {σ : Type u_2} [CommRing R] [NoZeroDivisors R] {p : MvPolynomial σ R} {i : σ} : p ∣ MvPolynomial.X i ↔ ∃ r, IsUnit r ∧ (p = MvPolynomial.C r ∨ p = r • MvPolynomial.X i)
• MvPolynomial.dvd_smul_X_iff_exists 📋 Mathlib.Algebra.MvPolynomial.NoZeroDivisors
  {R : Type u_1} {σ : Type u_2} [CommRing R] [NoZeroDivisors R] {p : MvPolynomial σ R} {i : σ} {r : R} (hr : r ≠ 0) : p ∣ r • MvPolynomial.X i ↔ ∃ s, s ∣ r ∧ (p = MvPolynomial.C s ∨ p = s • MvPolynomial.X i)

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 76f94b4