Loogle!

Result

Found 14 declarations mentioning FreeGroup.map.

• FreeGroup.map 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u} {β : Type v} (f : α → β) : FreeGroup α →* FreeGroup β
• FreeGroup.map.id 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u} (x : FreeGroup α) : (FreeGroup.map id) x = x
• FreeGroup.map.id' 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u} (x : FreeGroup α) : (FreeGroup.map fun z => z) x = x
• FreeGroup.map.of 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u} {β : Type v} {f : α → β} {x : α} : (FreeGroup.map f) (FreeGroup.of x) = FreeGroup.of (f x)
• FreeGroup.map.eq_1 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u} {β : Type v} (f : α → β) : FreeGroup.map f = MonoidHom.mk' (Quot.map (List.map fun x => (f x.1, x.2)) ⋯) ⋯
• FreeGroup.map.mk 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u} {L : List (α × Bool)} {β : Type v} {f : α → β} : (FreeGroup.map f) (FreeGroup.mk L) = FreeGroup.mk (List.map (fun x => (f x.1, x.2)) L)
• FreeGroup.freeGroupCongr_apply 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u_1} {β : Type u_2} (e : α ≃ β) (a : FreeGroup α) : (FreeGroup.freeGroupCongr e) a = (FreeGroup.map ⇑e) a
• FreeGroup.freeGroupCongr.eq_1 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u_1} {β : Type u_2} (e : α ≃ β) : FreeGroup.freeGroupCongr e = { toFun := ⇑(FreeGroup.map ⇑e), invFun := ⇑(FreeGroup.map ⇑e.symm), left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ }
• FreeGroup.map.comp 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u} {β : Type v} {γ : Type w} (f : α → β) (g : β → γ) (x : FreeGroup α) : (FreeGroup.map g) ((FreeGroup.map f) x) = (FreeGroup.map (g ∘ f)) x
• FreeGroup.map.unique 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u} {β : Type v} {f : α → β} (g : FreeGroup α →* FreeGroup β) (hg : ∀ (x : α), g (FreeGroup.of x) = FreeGroup.of (f x)) {x : FreeGroup α} : g x = (FreeGroup.map f) x
• FreeGroup.map_eq_lift 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u} {β : Type v} {f : α → β} {x : FreeGroup α} : (FreeGroup.map f) x = (FreeGroup.lift (FreeGroup.of ∘ f)) x
• FreeGroup.lift_eq_prod_map 📋 Mathlib.GroupTheory.FreeGroup.Basic
  {α : Type u} {β : Type v} [Group β] {f : α → β} {x : FreeGroup α} : (FreeGroup.lift f) x = FreeGroup.prod ((FreeGroup.map f) x)
• FreeGroup.instMonad.eq_1 📋 Mathlib.GroupTheory.FreeGroup.Basic
  : FreeGroup.instMonad = { map := fun {_α _β} {f} => ⇑(FreeGroup.map f), pure := fun {_α} => FreeGroup.of, seq := fun {α β} f x {_α _β} {x} {f} => (FreeGroup.lift f) x, seqLeft := fun {α β} x y {_α _β} {x} {f} => (FreeGroup.lift f) x, seqRight := fun {α β} x y {_α _β} {x} {f} => (FreeGroup.lift f) x, bind := fun {_α _β} {x} {f} => (FreeGroup.lift f) x }
• FreeGroup.freeGroupUnitEquivInt.eq_1 📋 Mathlib.GroupTheory.CoprodI
  : FreeGroup.freeGroupUnitEquivInt = { toFun := fun x => ((FreeGroup.map fun x => 1) x).sum, invFun := fun x => FreeGroup.of () ^ x, left_inv := FreeGroup.freeGroupUnitEquivInt._proof_63, right_inv := FreeGroup.freeGroupUnitEquivInt._proof_64 }

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