Loogle!

Result

Found 5 declarations mentioning Ideal.primeCompl_le_nonZeroDivisors.

• Ideal.primeCompl_le_nonZeroDivisors 📋 Mathlib.RingTheory.Ideal.Operations
  {R : Type u_1} [CommSemiring R] [NoZeroDivisors R] (P : Ideal R) [P.IsPrime] : P.primeCompl ≤ nonZeroDivisors R
• PrimeSpectrum.iInf_localization_eq_bot 📋 Mathlib.RingTheory.Spectrum.Maximal.Localization
  (R : Type u_4) [CommRing R] [IsDomain R] (K : Type u_5) [Field K] [Algebra R K] [IsFractionRing R K] : ⨅ v, Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯ = ⊥
• MaximalSpectrum.iInf_localization_eq_bot 📋 Mathlib.RingTheory.Spectrum.Maximal.Localization
  (R : Type u_4) [CommRing R] [IsDomain R] (K : Type u_5) [Field K] [Algebra R K] [IsFractionRing R K] : ⨅ v, Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯ = ⊥
• IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_bot 📋 Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
  (R : Type u_1) (K : Type u_3) [CommRing R] [Field K] [IsDedekindDomain R] [Algebra R K] [hK : IsFractionRing R K] : ⨅ v, Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯ = ⊥
• IsIntegrallyClosed.of_localization 📋 Mathlib.RingTheory.LocalProperties.IntegrallyClosed
  {R : Type u_1} [CommRing R] [IsDomain R] (S : Set (PrimeSpectrum R)) (h : ∀ p ∈ S, IsIntegrallyClosed (Localization.AtPrime p.asIdeal)) (hs : ⨅ p ∈ S, Localization.subalgebra (FractionRing R) p.asIdeal.primeCompl ⋯ = ⊥) : IsIntegrallyClosed R

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 1c119a3