Loogle!

Result

Found 5 definitions mentioning Filter.HasBasis and Metric.closedBall.

• Metric.nhds_basis_closedBall 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] {x : α} : (nhds x).HasBasis (fun ε => 0 < ε) (Metric.closedBall x)
• Metric.nhds_basis_closedBall_pow 📋 Mathlib.Topology.MetricSpace.Pseudo.Defs
  {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ} (h0 : 0 < r) (h1 : r < 1) : (nhds x).HasBasis (fun x => True) fun n => Metric.closedBall x (r ^ n)
• Metric.hasBasis_cobounded_compl_closedBall 📋 Mathlib.Topology.MetricSpace.Bounded
  {α : Type u} [PseudoMetricSpace α] (c : α) : (Bornology.cobounded α).HasBasis (fun x => True) fun r => (Metric.closedBall c r)ᶜ
• VitaliFamily.filterAt_basis_closedBall 📋 Mathlib.MeasureTheory.Covering.VitaliFamily
  {X : Type u_1} [PseudoMetricSpace X] {m0 : MeasurableSpace X} {μ : MeasureTheory.Measure X} (v : VitaliFamily μ) (x : X) : (v.filterAt x).HasBasis (fun x => 0 < x) fun x_1 => {t | t ∈ v.setsAt x ∧ t ⊆ Metric.closedBall x x_1}
• SmoothBumpFunction.nhdsWithin_range_basis 📋 Mathlib.Geometry.Manifold.BumpFunction
  {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} : (nhdsWithin (↑(extChartAt I c) c) (Set.range ↑I)).HasBasis (fun x => True) fun f => Metric.closedBall (↑(extChartAt I c) c) f.rOut ∩ Set.range ↑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, _ * _, _ ^ _, |- _ < _ → _
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