Loogle!

Result

Found 11 declarations mentioning Ultrafilter.map.

• Ultrafilter.map 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} {β : Type v} (m : α → β) (f : Ultrafilter α) : Ultrafilter β
• Ultrafilter.map_id 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} (f : Ultrafilter α) : Ultrafilter.map id f = f
• Ultrafilter.map_id' 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} (f : Ultrafilter α) : Ultrafilter.map (fun x => x) f = f
• Ultrafilter.coe_map 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} {β : Type v} (m : α → β) (f : Ultrafilter α) : ↑(Ultrafilter.map m f) = Filter.map m ↑f
• Ultrafilter.map_pure 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} {β : Type v} (m : α → β) (a : α) : Ultrafilter.map m (pure a) = pure (m a)
• Ultrafilter.map_map 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} {β : Type v} {γ : Type u_1} (f : Ultrafilter α) (m : α → β) (n : β → γ) : Ultrafilter.map n (Ultrafilter.map m f) = Ultrafilter.map (n ∘ m) f
• Ultrafilter.mem_map 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} {β : Type v} {m : α → β} {f : Ultrafilter α} {s : Set β} : s ∈ Ultrafilter.map m f ↔ m ⁻¹' s ∈ f
• Ultrafilter.ofComapInfPrincipal_eq_of_map 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} {β : Type v} {m : α → β} {s : Set α} {g : Ultrafilter β} (h : m '' s ∈ g) : Ultrafilter.map m (Ultrafilter.ofComapInfPrincipal h) = g
• ultrafilter_extend_eq_iff 📋 Mathlib.Topology.StoneCech
  {α : Type u} {γ : Type u_1} [TopologicalSpace γ] [T2Space γ] [CompactSpace γ] {f : α → γ} {b : Ultrafilter α} {c : γ} : Ultrafilter.extend f b = c ↔ ↑(Ultrafilter.map f b) ≤ nhds c
• Compactum.join_distrib 📋 Mathlib.Topology.Category.Compactum
  (X : Compactum) (uux : Ultrafilter (Ultrafilter X.A)) : X.str (X.join uux) = X.str (Ultrafilter.map X.str uux)
• Compactum.str_hom_commute 📋 Mathlib.Topology.Category.Compactum
  (X Y : Compactum) (f : X ⟶ Y) (xs : Ultrafilter X.A) : (CategoryTheory.ConcreteCategory.hom f) (X.str xs) = Y.str (Ultrafilter.map (⇑(CategoryTheory.ConcreteCategory.hom f)) xs)

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 19971e9 serving mathlib revision bce1d65