# mathlibdocumentation

algebra.free_monoid

# Free monoid over a given alphabet #

## Main definitions #

• free_monoid α: free monoid over alphabet α; defined as a synonym for list α with multiplication given by (++).
• free_monoid.of: embedding α → free_monoid α sending each element x to [x];
• free_monoid.lift: natural equivalence between α → M and free_monoid α →* M
• free_monoid.map: embedding of α → β into free_monoid α →* free_monoid β given by list.map.
def free_add_monoid (α : Type u_1) :
Type u_1

Free nonabelian additive monoid over a given alphabet

def free_monoid (α : Type u_1) :
Type u_1

Free monoid over a given alphabet.

Equations
@[instance]
@[instance]
def free_monoid.monoid {α : Type u_1} :
Equations
@[instance]
def free_add_monoid.inhabited {α : Type u_1} :
@[instance]
def free_monoid.inhabited {α : Type u_1} :
Equations
theorem free_monoid.one_def {α : Type u_1} :
theorem free_add_monoid.zero_def {α : Type u_1} :
theorem free_add_monoid.add_def {α : Type u_1} (xs ys : list α) :
xs + ys = xs ++ ys
theorem free_monoid.mul_def {α : Type u_1} (xs ys : list α) :
xs * ys = xs ++ ys
def free_monoid.of {α : Type u_1} (x : α) :

Embeds an element of α into free_monoid α as a singleton list.

Equations
• = [x]
def free_add_monoid.of {α : Type u_1} (x : α) :

Embeds an element of α into free_add_monoid α as a singleton list.

theorem free_monoid.of_def {α : Type u_1} (x : α) :
= [x]
theorem free_add_monoid.of_def {α : Type u_1} (x : α) :
= [x]
theorem free_add_monoid.of_injective {α : Type u_1} :
theorem free_monoid.of_injective {α : Type u_1} :
def free_monoid.rec_on {α : Type u_1} {C : Sort u_2} (xs : free_monoid α) (h0 : C 1) (ih : Π (x : α) (xs : , C xsC ((free_monoid.of x) * xs)) :
C xs

Recursor for free_monoid using 1 and of x * xs instead of [] and x :: xs.

Equations
• xs.rec_on h0 ih = list.rec_on xs h0 ih
def free_add_monoid.rec_on {α : Type u_1} {C : Sort u_2} (xs : free_add_monoid α) (h0 : C 0) (ih : Π (x : α) (xs : , C xsC + xs)) :
C xs

Recursor for free_add_monoid using 0 and of x + xs instead of [] and x :: xs.

@[ext]
theorem free_monoid.hom_eq {α : Type u_1} {M : Type u_4} [monoid M] ⦃f g : →* M⦄ (h : ∀ (x : α), f = g ) :
f = g
@[ext]
theorem free_add_monoid.hom_eq {α : Type u_1} {M : Type u_4} [add_monoid M] ⦃f g : →+ M⦄ (h : ∀ (x : α), f = g ) :
f = g
def free_add_monoid.lift {α : Type u_1} {M : Type u_4} [add_monoid M] :
(α → M) →+ M)

Equivalence between maps α → A and additive monoid homomorphisms free_add_monoid α →+ A.

def free_monoid.lift {α : Type u_1} {M : Type u_4} [monoid M] :
(α → M) →* M)

Equivalence between maps α → M and monoid homomorphisms free_monoid α →* M.

Equations
@[simp]
theorem free_add_monoid.lift_symm_apply {α : Type u_1} {M : Type u_4} [add_monoid M] (f : →+ M) :
@[simp]
theorem free_monoid.lift_symm_apply {α : Type u_1} {M : Type u_4} [monoid M] (f : →* M) :
theorem free_monoid.lift_apply {α : Type u_1} {M : Type u_4} [monoid M] (f : α → M) (l : free_monoid α) :
l = (list.map f l).prod
theorem free_add_monoid.lift_apply {α : Type u_1} {M : Type u_4} [add_monoid M] (f : α → M) (l : free_add_monoid α) :
= (list.map f l).sum
theorem free_add_monoid.lift_comp_of {α : Type u_1} {M : Type u_4} [add_monoid M] (f : α → M) :
theorem free_monoid.lift_comp_of {α : Type u_1} {M : Type u_4} [monoid M] (f : α → M) :
@[simp]
theorem free_add_monoid.lift_eval_of {α : Type u_1} {M : Type u_4} [add_monoid M] (f : α → M) (x : α) :
= f x
@[simp]
theorem free_monoid.lift_eval_of {α : Type u_1} {M : Type u_4} [monoid M] (f : α → M) (x : α) :
= f x
@[simp]
theorem free_monoid.lift_restrict {α : Type u_1} {M : Type u_4} [monoid M] (f : →* M) :
@[simp]
theorem free_add_monoid.lift_restrict {α : Type u_1} {M : Type u_4} [add_monoid M] (f : →+ M) :
theorem free_monoid.comp_lift {α : Type u_1} {M : Type u_4} [monoid M] {N : Type u_5} [monoid N] (g : M →* N) (f : α → M) :
theorem free_add_monoid.comp_lift {α : Type u_1} {M : Type u_4} [add_monoid M] {N : Type u_5} [add_monoid N] (g : M →+ N) (f : α → M) :
theorem free_monoid.hom_map_lift {α : Type u_1} {M : Type u_4} [monoid M] {N : Type u_5} [monoid N] (g : M →* N) (f : α → M) (x : free_monoid α) :
theorem free_add_monoid.hom_map_lift {α : Type u_1} {M : Type u_4} [add_monoid M] {N : Type u_5} [add_monoid N] (g : M →+ N) (f : α → M) (x : free_add_monoid α) :
def free_add_monoid.map {α : Type u_1} {β : Type u_2} (f : α → β) :

The unique additive monoid homomorphism free_add_monoid α →+ free_add_monoid β that sends each of x to of (f x).

def free_monoid.map {α : Type u_1} {β : Type u_2} (f : α → β) :

The unique monoid homomorphism free_monoid α →* free_monoid β that sends each of x to of (f x).

Equations
@[simp]
theorem free_add_monoid.map_of {α : Type u_1} {β : Type u_2} (f : α → β) (x : α) :
@[simp]
theorem free_monoid.map_of {α : Type u_1} {β : Type u_2} (f : α → β) (x : α) :
theorem free_monoid.lift_of_comp_eq_map {α : Type u_1} {β : Type u_2} (f : α → β) :
free_monoid.lift (λ (x : α), free_monoid.of (f x)) =
theorem free_add_monoid.lift_of_comp_eq_map {α : Type u_1} {β : Type u_2} (f : α → β) :
theorem free_add_monoid.map_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} (g : β → γ) (f : α → β) :
theorem free_monoid.map_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} (g : β → γ) (f : α → β) :
@[instance]
def free_monoid.star_monoid {α : Type u_1} :
Equations
@[simp]
theorem free_monoid.star_of {α : Type u_1} (x : α) :
@[simp]
theorem free_monoid.star_one {α : Type u_1} :
star 1 = 1

Note that star_one is already a global simp lemma, but this one works with dsimp too