mathlib documentation

algebra.free_monoid

Free monoid over a given alphabet #

Main definitions #

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]
@[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
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) :
theorem free_add_monoid.of_def {α : Type u_1} (x : α) :
def free_monoid.rec_on {α : Type u_1} {C : free_monoid αSort u_2} (xs : free_monoid α) (h0 : C 1) (ih : Π (x : α) (xs : free_monoid α), 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 : free_add_monoid αSort u_2} (xs : free_add_monoid α) (h0 : C 0) (ih : Π (x : α) (xs : free_add_monoid α), C xsC (free_add_monoid.of x + 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 : free_monoid α →* M⦄ (h : ∀ (x : α), f (free_monoid.of x) = g (free_monoid.of x)) :
f = g
@[ext]
theorem free_add_monoid.hom_eq {α : Type u_1} {M : Type u_4} [add_monoid M] ⦃f g : free_add_monoid α →+ M⦄ (h : ∀ (x : α), f (free_add_monoid.of x) = g (free_add_monoid.of x)) :
f = g
def free_add_monoid.lift {α : Type u_1} {M : Type u_4} [add_monoid M] :
(α → M) (free_add_monoid α →+ 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) (free_monoid α →* M)

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

Equations
@[simp]
@[simp]
theorem free_monoid.lift_symm_apply {α : Type u_1} {M : Type u_4} [monoid M] (f : free_monoid α →* M) :
theorem free_monoid.lift_apply {α : Type u_1} {M : Type u_4} [monoid M] (f : α → M) (l : free_monoid α) :
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) :
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) :
@[simp]
theorem free_monoid.lift_eval_of {α : Type u_1} {M : Type u_4} [monoid M] (f : α → M) (x : α) :
@[simp]
theorem free_monoid.lift_restrict {α : Type u_1} {M : Type u_4} [monoid M] (f : free_monoid α →* M) :
@[simp]
theorem free_add_monoid.lift_restrict {α : Type u_1} {M : Type u_4} [add_monoid M] (f : free_add_monoid α →+ 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) :
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) :
@[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