Loogle!

Result

Found 52 declarations mentioning PartialOrder.toPreorder, Preorder.toLE, nonZeroDivisors, LE.le, MulZeroOneClass.toMulOneClass, MonoidWithZero.toMulZeroOneClass, Submonoid.instPartialOrder, and Submonoid. Of these, 6 match your pattern(s).

• comap_nonZeroDivisors_le_of_injective 📋 Mathlib.Algebra.GroupWithZero.NonZeroDivisors
  {F : Type u_1} {M₀ : Type u_2} {M₀' : Type u_3} [MonoidWithZero M₀] [MonoidWithZero M₀'] [FunLike F M₀ M₀'] [MonoidWithZeroHomClass F M₀ M₀'] {f : F} (hf : Function.Injective ⇑f) : Submonoid.comap f (nonZeroDivisors M₀') ≤ nonZeroDivisors M₀
• le_nonZeroDivisors_of_noZeroDivisors 📋 Mathlib.Algebra.GroupWithZero.NonZeroDivisors
  {M₀ : Type u_2} [MonoidWithZero M₀] [NoZeroDivisors M₀] {S : Submonoid M₀} (hS : 0 ∉ S) : S ≤ nonZeroDivisors M₀
• map_le_nonZeroDivisors_of_injective 📋 Mathlib.Algebra.GroupWithZero.NonZeroDivisors
  {F : Type u_1} {M₀ : Type u_2} {M₀' : Type u_3} [MonoidWithZero M₀] [MonoidWithZero M₀'] [FunLike F M₀ M₀'] [NoZeroDivisors M₀'] [MonoidWithZeroHomClass F M₀ M₀'] (f : F) (hf : Function.Injective ⇑f) {S : Submonoid M₀} (hS : S ≤ nonZeroDivisors M₀) : Submonoid.map f S ≤ nonZeroDivisors M₀'
• Submonoid.LocalizationMap.map_nonZeroDivisors_le 📋 Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero
  {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) : Submonoid.map f (nonZeroDivisors M) ≤ nonZeroDivisors N
• OreLocalization.nontrivial_of_nonZeroDivisors 📋 Mathlib.RingTheory.OreLocalization.NonZeroDivisors
  {R : Type u_1} [MonoidWithZero R] {S : Submonoid R} [OreLocalization.OreSet S] [Nontrivial R] (hS : S ≤ nonZeroDivisors R) : Nontrivial (OreLocalization S R)
• IsLocalization.map_nonZeroDivisors_le 📋 Mathlib.RingTheory.Localization.Defs
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] : Submonoid.map (algebraMap R S) (nonZeroDivisors R) ≤ nonZeroDivisors S

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 36960b0 serving mathlib revision 4166dc3