Loogle!

Result

Found 165 declarations mentioning Equiv and Option. Of these, 28 match your pattern(s).

• Set.subtypeInsertEquivOption 📋 Mathlib.Data.Set.Insert
  {α : Type u} [DecidableEq α] {t : Set α} {x : α} (h : x ∉ t) : { i // i ∈ insert x t } ≃ Option { i // i ∈ t }
• Equiv.optionEquivSumPUnit 📋 Mathlib.Logic.Equiv.Option
  (α : Type w) : Option α ≃ α ⊕ PUnit.{v + 1}
• Equiv.optionCongr 📋 Mathlib.Logic.Equiv.Option
  {α : Type u_1} {β : Type u_2} (e : α ≃ β) : Option α ≃ Option β
• Equiv.removeNone 📋 Mathlib.Logic.Equiv.Option
  {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) : α ≃ β
• Equiv.optionIsSomeEquiv 📋 Mathlib.Logic.Equiv.Option
  (α : Type u_4) : { x // x.isSome = true } ≃ α
• Equiv.optionSubtypeNe 📋 Mathlib.Logic.Equiv.Option
  {α : Type u_1} [DecidableEq α] (a : α) : Option { b // b ≠ a } ≃ α
• Equiv.optionSubtype 📋 Mathlib.Logic.Equiv.Option
  {α : Type u_1} {β : Type u_2} [DecidableEq β] (x : β) : { e // e none = x } ≃ (α ≃ { y // y ≠ x })
• optionProdEquiv 📋 Mathlib.Logic.Equiv.Prod
  {α : Type u_9} {β : Type u_10} : Option α × β ≃ β ⊕ α × β
• Equiv.piOptionEquivProd 📋 Mathlib.Logic.Equiv.Basic
  {α : Type u_10} {β : Option α → Type u_9} : ((a : Option α) → β a) ≃ β none × ((a : α) → β (some a))
• Equiv.sigmaOptionEquivOfSome 📋 Mathlib.Logic.Equiv.Basic
  {α : Type u_9} (p : Option α → Type v) (h : ∀ (a : p none), False) : (x : Option α) × p x ≃ (x : α) × p (some x)
• Finset.subtypeInsertEquivOption 📋 Mathlib.Data.Finset.Insert
  {α : Type u_1} [DecidableEq α] {t : Finset α} {x : α} (h : x ∉ t) : { i // i ∈ insert x t } ≃ Option { i // i ∈ t }
• Function.Embedding.optionEmbeddingEquiv 📋 Mathlib.Logic.Embedding.Set
  (α : Type u_1) (β : Type u_2) : (Option α ↪ β) ≃ (f : α ↪ β) × ↑(Set.range ⇑f)ᶜ
• Part.equivOption 📋 Mathlib.Data.Part
  {α : Type u_1} : Part α ≃ Option α
• symOptionSuccEquiv 📋 Mathlib.Data.Sym.Basic
  {α : Type u_1} {n : ℕ} [DecidableEq α] : Sym (Option α) n.succ ≃ Sym (Option α) n ⊕ Sym α n.succ
• Finsupp.optionEquiv 📋 Mathlib.Data.Finsupp.Option
  {α : Type u_1} {M : Type u_2} [Zero M] : (Option α →₀ M) ≃ M × (α →₀ M)
• finSuccEquiv 📋 Mathlib.Logic.Equiv.Fin.Basic
  (n : ℕ) : Fin (n + 1) ≃ Option (Fin n)
• finSuccEquivLast 📋 Mathlib.Logic.Equiv.Fin.Basic
  {n : ℕ} : Fin (n + 1) ≃ Option (Fin n)
• finSuccEquiv' 📋 Mathlib.Logic.Equiv.Fin.Basic
  {n : ℕ} (i : Fin (n + 1)) : Fin (n + 1) ≃ Option (Fin n)
• DFinsupp.equivProdDFinsupp 📋 Mathlib.Data.DFinsupp.Defs
  {ι : Type u} {α : Option ι → Type v} [(i : Option ι) → Zero (α i)] : (Π₀ (i : Option ι), α i) ≃ α none × Π₀ (i : ι), α (some i)
• Equiv.Perm.decomposeOption 📋 Mathlib.GroupTheory.Perm.Option
  {α : Type u_1} [DecidableEq α] : Equiv.Perm (Option α) ≃ Option α × Equiv.Perm α
• CategoryTheory.Limits.WalkingParallelFamily.arrowEquiv 📋 Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers
  (J : Type w) : CategoryTheory.Arrow (CategoryTheory.Limits.WalkingParallelFamily J) ≃ Option (Option J)
• Multiset.consEquiv 📋 Mathlib.Data.Multiset.Fintype
  {α : Type u_1} [DecidableEq α] {m : Multiset α} {v : α} : (v ::ₘ m).ToType ≃ Option m.ToType
• OrderedFinpartition.extendEquiv 📋 Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
  (n : ℕ) : (c : OrderedFinpartition n) × Option (Fin c.length) ≃ OrderedFinpartition (n + 1)
• CategoryTheory.WithInitial.optionEquiv 📋 Mathlib.CategoryTheory.WithTerminal.FinCategory
  (C : Type u) : Option C ≃ CategoryTheory.WithInitial C
• CategoryTheory.WithTerminal.optionEquiv 📋 Mathlib.CategoryTheory.WithTerminal.FinCategory
  (C : Type u) : Option C ≃ CategoryTheory.WithTerminal C
• derangements.derangementsOptionEquivSigmaAtMostOneFixedPoint 📋 Mathlib.Combinatorics.Derangements.Basic
  {α : Type u_1} [DecidableEq α] : ↑(derangements (Option α)) ≃ (a : α) × ↑{f | Function.fixedPoints ⇑f ⊆ {a}}
• derangements.derangementsRecursionEquiv 📋 Mathlib.Combinatorics.Derangements.Basic
  {α : Type u_1} [DecidableEq α] : ↑(derangements (Option α)) ≃ (a : α) × (↑(derangements ↑{a}ᶜ) ⊕ ↑(derangements α))
• Finmap.keysLookupEquiv 📋 Mathlib.Data.Finmap
  {α : Type u} {β : α → Type v} [DecidableEq α] : Finmap β ≃ { f // ∀ (i : α), (f.2 i).isSome = true ↔ i ∈ f.1 }

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 519f454