Loogle!

Result

Found 19 declarations mentioning FreeAddMonoid.map.

• FreeAddMonoid.map 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} (f : α → β) : FreeAddMonoid α →+ FreeAddMonoid β
• FreeAddMonoid.map_id 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} : FreeAddMonoid.map id = AddMonoidHom.id (FreeAddMonoid α)
• FreeAddMonoid.map_surjective 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} {f : α → β} : Function.Surjective ⇑(FreeAddMonoid.map f) ↔ Function.Surjective f
• FreeAddMonoid.map_of 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} (f : α → β) (x : α) : (FreeAddMonoid.map f) (FreeAddMonoid.of x) = FreeAddMonoid.of (f x)
• FreeAddMonoid.map_comp 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (g : β → γ) (f : α → β) : FreeAddMonoid.map (g ∘ f) = (FreeAddMonoid.map g).comp (FreeAddMonoid.map f)
• FreeAddMonoid.mem_map 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} {f : α → β} {a : FreeAddMonoid α} {m : β} : m ∈ (FreeAddMonoid.map f) a ↔ ∃ n ∈ a, f n = m
• FreeAddMonoid.ofList_map 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} (f : α → β) (xs : List α) : FreeAddMonoid.ofList (List.map f xs) = (FreeAddMonoid.map f) (FreeAddMonoid.ofList xs)
• FreeAddMonoid.toList_map 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} (f : α → β) (xs : FreeAddMonoid α) : FreeAddMonoid.toList ((FreeAddMonoid.map f) xs) = List.map f (FreeAddMonoid.toList xs)
• FreeAddMonoid.lift_of_comp_eq_map 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} (f : α → β) : (FreeAddMonoid.lift fun x => FreeAddMonoid.of (f x)) = FreeAddMonoid.map f
• FreeAddMonoid.map.eq_1 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} (f : α → β) : FreeAddMonoid.map f = { toFun := fun l => FreeAddMonoid.ofList (List.map f (FreeAddMonoid.toList l)), map_zero' := ⋯, map_add' := ⋯ }
• FreeAddMonoid.map_apply_map_symm_eq 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} {x : FreeAddMonoid β} (e : α ≃ β) : (FreeAddMonoid.map ⇑e) ((FreeAddMonoid.map ⇑e.symm) x) = x
• FreeAddMonoid.map_symm_apply_map_eq 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} {x : FreeAddMonoid α} (e : α ≃ β) : (FreeAddMonoid.map ⇑e.symm) ((FreeAddMonoid.map ⇑e) x) = x
• FreeAddMonoid.map_map 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_1} {β : Type u_2} {f : α → β} {α₁ : Type u_6} {g : α₁ → α} {x : FreeAddMonoid α₁} : (FreeAddMonoid.map f) ((FreeAddMonoid.map g) x) = (FreeAddMonoid.map (f ∘ g)) x
• FreeAddMonoid.freeAddMonoidCongr.eq_1 📋 Mathlib.Algebra.FreeMonoid.Basic
  {α : Type u_6} {β : Type u_7} (e : α ≃ β) : FreeAddMonoid.freeAddMonoidCongr e = { toFun := ⇑(FreeAddMonoid.map ⇑e), invFun := ⇑(FreeAddMonoid.map ⇑e.symm), left_inv := ⋯, right_inv := ⋯, map_add' := ⋯ }
• AddMonCat.free.eq_1 📋 Mathlib.Algebra.Category.MonCat.Adjunctions
  : AddMonCat.free = { obj := fun α => AddMonCat.of (FreeAddMonoid α), map := fun {X Y} f => AddMonCat.ofHom (FreeAddMonoid.map f), map_id := AddMonCat.free._proof_17, map_comp := @AddMonCat.free._proof_18 }
• PiTensorProduct.lifts_smul 📋 Mathlib.LinearAlgebra.PiTensorProduct
  {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ι → Type u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {x : PiTensorProduct R fun i => s i} {p : FreeAddMonoid (R × ((i : ι) → s i))} (h : p ∈ x.lifts) (a : R) : (FreeAddMonoid.map fun y => (a * y.1, y.2)) p ∈ (a • x).lifts
• PiTensorProduct.projectiveSeminormAux_smul 📋 Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
  {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] (p : FreeAddMonoid (𝕜 × ((i : ι) → E i))) (a : 𝕜) : PiTensorProduct.projectiveSeminormAux ((FreeAddMonoid.map fun y => (a * y.1, y.2)) p) = ‖a‖ * PiTensorProduct.projectiveSeminormAux p
• AddMonoid.Coprod.swap.eq_1 📋 Mathlib.GroupTheory.Coprod.Basic
  (M : Type u_1) (N : Type u_2) [AddZeroClass M] [AddZeroClass N] : AddMonoid.Coprod.swap M N = AddMonoid.Coprod.clift (AddMonoid.Coprod.mk.comp (FreeAddMonoid.map Sum.swap)) ⋯ ⋯ ⋯ ⋯
• AddMonoid.Coprod.map.eq_1 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} {M' : Type u_3} {N' : Type u_4} [AddZeroClass M] [AddZeroClass N] [AddZeroClass M'] [AddZeroClass N'] (f : M →+ M') (g : N →+ N') : AddMonoid.Coprod.map f g = AddMonoid.Coprod.clift (AddMonoid.Coprod.mk.comp (FreeAddMonoid.map (Sum.map ⇑f ⇑g))) ⋯ ⋯ ⋯ ⋯

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