Loogle!

Result

Found 4 definitions mentioning Algebra.adjoin and Subring.closure.

• Algebra.adjoin_int 📋 Mathlib.RingTheory.Adjoin.Basic
  {R : Type u_1} [Ring R] (s : Set R) : Algebra.adjoin ℤ s = subalgebraOfSubring (Subring.closure s)
• Algebra.adjoin_eq_ring_closure 📋 Mathlib.RingTheory.Adjoin.Basic
  {R : Type uR} {A : Type uA} [CommRing R] [Ring A] [Algebra R A] (s : Set A) : (Algebra.adjoin R s).toSubring = Subring.closure (Set.range ⇑(algebraMap R A) ∪ s)
• Algebra.mem_adjoin_iff 📋 Mathlib.RingTheory.Adjoin.Basic
  {R : Type uR} {A : Type uA} [CommRing R] [Ring A] [Algebra R A] {s : Set A} {x : A} : x ∈ Algebra.adjoin R s ↔ x ∈ Subring.closure (Set.range ⇑(algebraMap R A) ∪ s)
• Subring.closureEquivAdjoinInt 📋 Mathlib.RingTheory.Adjoin.Basic
  {R : Type u_1} [Ring R] (s : Set R) : ↥(Subring.closure s) ≃ₐ[ℤ] ↥(Algebra.adjoin ℤ 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, _ * _, _ ^ _, |- _ < _ → _
woould 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 4e1aab0 serving mathlib revision 2d53f5f