# mathlibdocumentation

group_theory.free_group

# Free groups #

This file defines free groups over a type. Furthermore, it is shown that the free group construction is an instance of a monad. For the result that free_group is the left adjoint to the forgetful functor from groups to types, see algebra/category/Group/adjunctions.

## Main definitions #

• free_group: the free group associated to a type α defined as the words over a : α × bool modulo the relation a * x * x⁻¹ * b = a * b.
• free_group.mk: the canonical quotient map list (α × bool) → free_group α.
• free_group.of: the canoical injection α → free_group α.
• free_group.lift f: the canonical group homomorphism free_group α →* G given a group G and a function f : α → G.

## Main statements #

• free_group.church_rosser: The Church-Rosser theorem for word reduction (also known as Newman's diamond lemma).
• free_group.free_group_unit_equiv_int: The free group over the one-point type is isomorphic to the integers.
• The free group construction is an instance of a monad.

## Implementation details #

First we introduce the one step reduction relation free_group.red.step: w * x * x⁻¹ * v ~> w * v, its reflexive transitive closure free_group.red.trans and prove that its join is an equivalence relation. Then we introduce free_group α as a quotient over free_group.red.step.

## Tags #

free group, Newman's diamond lemma, Church-Rosser theorem

inductive free_group.red.step {α : Type u} :
list × bool)list × bool) → Prop

Reduction step: w * x * x⁻¹ * v ~> w * v

def free_group.red {α : Type u} :
list × bool)list × bool) → Prop

Reflexive-transitive closure of red.step

Equations
Instances for free_group.red
@[refl]
theorem free_group.red.refl {α : Type u} {L : list × bool)} :
@[trans]
theorem free_group.red.trans {α : Type u} {L₁ L₂ L₃ : list × bool)} :
L₂ L₃ L₃
theorem free_group.red.step.length {α : Type u} {L₁ L₂ : list × bool)} :
L₂L₂.length + 2 = L₁.length

Predicate asserting that word w₁ can be reduced to w₂ in one step, i.e. there are words w₃ w₄ and letter x such that w₁ = w₃xx⁻¹w₄ and w₂ = w₃w₄

@[simp]
theorem free_group.red.step.bnot_rev {α : Type u} {L₁ L₂ : list × bool)} {x : α} {b : bool} :
free_group.red.step (L₁ ++ (x, !b) :: (x, b) :: L₂) (L₁ ++ L₂)
@[simp]
theorem free_group.red.step.cons_bnot {α : Type u} {L : list × bool)} {x : α} {b : bool} :
free_group.red.step ((x, b) :: (x, !b) :: L) L
@[simp]
theorem free_group.red.step.cons_bnot_rev {α : Type u} {L : list × bool)} {x : α} {b : bool} :
free_group.red.step ((x, !b) :: (x, b) :: L) L
theorem free_group.red.step.append_left {α : Type u} {L₁ L₂ L₃ : list × bool)} :
L₃free_group.red.step (L₁ ++ L₂) (L₁ ++ L₃)
theorem free_group.red.step.cons {α : Type u} {L₁ L₂ : list × bool)} {x : α × bool} (H : L₂) :
free_group.red.step (x :: L₁) (x :: L₂)
theorem free_group.red.step.append_right {α : Type u} {L₁ L₂ L₃ : list × bool)} :
L₂free_group.red.step (L₁ ++ L₃) (L₂ ++ L₃)
theorem free_group.red.not_step_nil {α : Type u} {L : list × bool)} :
theorem free_group.red.step.cons_left_iff {α : Type u} {L₁ L₂ : list × bool)} {a : α} {b : bool} :
free_group.red.step ((a, b) :: L₁) L₂ (∃ (L : list × bool)), L₂ = (a, b) :: L) L₁ = (a, !b) :: L₂
theorem free_group.red.not_step_singleton {α : Type u} {L : list × bool)} {p : α × bool} :
¬ L
theorem free_group.red.step.cons_cons_iff {α : Type u} {L₁ L₂ : list × bool)} {p : α × bool} :
free_group.red.step (p :: L₁) (p :: L₂) L₂
theorem free_group.red.step.append_left_iff {α : Type u} {L₁ L₂ : list × bool)} (L : list × bool)) :
free_group.red.step (L ++ L₁) (L ++ L₂) L₂
theorem free_group.red.step.diamond {α : Type u} {L₁ L₂ L₃ L₄ : list × bool)} :
L₃ L₄L₁ = L₂(L₃ = L₄ ∃ (L₅ : list × bool)), L₅ L₅)
theorem free_group.red.step.to_red {α : Type u} {L₁ L₂ : list × bool)} :
L₂ L₂
theorem free_group.red.church_rosser {α : Type u} {L₁ L₂ L₃ : list × bool)} :
L₂ L₃

Church-Rosser theorem for word reduction: If w1 w2 w3 are words such that w1 reduces to w2 and w3 respectively, then there is a word w4 such that w2 and w3 reduce to w4 respectively. This is also known as Newman's diamond lemma.

theorem free_group.red.cons_cons {α : Type u} {L₁ L₂ : list × bool)} {p : α × bool} :
L₂free_group.red (p :: L₁) (p :: L₂)
theorem free_group.red.cons_cons_iff {α : Type u} {L₁ L₂ : list × bool)} (p : α × bool) :
free_group.red (p :: L₁) (p :: L₂) L₂
theorem free_group.red.append_append_left_iff {α : Type u} {L₁ L₂ : list × bool)} (L : list × bool)) :
free_group.red (L ++ L₁) (L ++ L₂) L₂
theorem free_group.red.append_append {α : Type u} {L₁ L₂ L₃ L₄ : list × bool)} (h₁ : L₃) (h₂ : L₄) :
free_group.red (L₁ ++ L₂) (L₃ ++ L₄)
theorem free_group.red.to_append_iff {α : Type u} {L L₁ L₂ : list × bool)} :
(L₁ ++ L₂) ∃ (L₃ L₄ : list × bool)), L = L₃ ++ L₄ L₁ L₂
theorem free_group.red.nil_iff {α : Type u} {L : list × bool)} :

The empty word [] only reduces to itself.

theorem free_group.red.singleton_iff {α : Type u} {L₁ : list × bool)} {x : α × bool} :
L₁ L₁ = [x]

A letter only reduces to itself.

theorem free_group.red.cons_nil_iff_singleton {α : Type u} {L : list × bool)} {x : α} {b : bool} :
free_group.red ((x, b) :: L) list.nil [(x, !b)]

If x is a letter and w is a word such that xw reduces to the empty word, then w reduces to x⁻¹

theorem free_group.red.red_iff_irreducible {α : Type u} {L : list × bool)} {x1 : α} {b1 : bool} {x2 : α} {b2 : bool} (h : (x1, b1) (x2, b2)) :
free_group.red [(x1, !b1), (x2, b2)] L L = [(x1, !b1), (x2, b2)]
theorem free_group.red.inv_of_red_of_ne {α : Type u} {L₁ L₂ : list × bool)} {x1 : α} {b1 : bool} {x2 : α} {b2 : bool} (H1 : (x1, b1) (x2, b2)) (H2 : free_group.red ((x1, b1) :: L₁) ((x2, b2) :: L₂)) :
((x1, !b1) :: (x2, b2) :: L₂)

If x and y are distinct letters and w₁ w₂ are words such that xw₁ reduces to yw₂, then w₁ reduces to x⁻¹yw₂.

theorem free_group.red.step.sublist {α : Type u} {L₁ L₂ : list × bool)} (H : L₂) :
L₂ <+ L₁
@[protected]
theorem free_group.red.sublist {α : Type u} {L₁ L₂ : list × bool)} :
L₂L₂ <+ L₁

If w₁ w₂ are words such that w₁ reduces to w₂, then w₂ is a sublist of w₁.

theorem free_group.red.length_le {α : Type u} {L₁ L₂ : list × bool)} (h : L₂) :
L₂.length L₁.length
theorem free_group.red.sizeof_of_step {α : Type u} {L₁ L₂ : list × bool)} :
L₂L₂.sizeof < L₁.sizeof
theorem free_group.red.length {α : Type u} {L₁ L₂ : list × bool)} (h : L₂) :
∃ (n : ), L₁.length = L₂.length + 2 * n
theorem free_group.red.antisymm {α : Type u} {L₁ L₂ : list × bool)} (h₁₂ : L₂) (h₂₁ : L₁) :
L₁ = L₂
theorem free_group.equivalence_join_red {α : Type u} :
theorem free_group.join_red_of_step {α : Type u} {L₁ L₂ : list × bool)} (h : L₂) :
theorem free_group.eqv_gen_step_iff_join_red {α : Type u} {L₁ L₂ : list × bool)} :
def free_group (α : Type u) :
Type u

The free group over a type, i.e. the words formed by the elements of the type and their formal inverses, quotient by one step reduction.

Equations
Instances for free_group
def free_group.mk {α : Type u} (L : list × bool)) :

The canonical map from list (α × bool) to the free group on α.

Equations
@[simp]
theorem free_group.quot_mk_eq_mk {α : Type u} {L : list × bool)} :
@[simp]
theorem free_group.quot_lift_mk {α : Type u} {L : list × bool)} (β : Type v) (f : list × bool) → β) (H : ∀ (L₁ L₂ : list × bool)), L₂f L₁ = f L₂) :
quot.lift f H = f L
@[simp]
theorem free_group.quot_lift_on_mk {α : Type u} {L : list × bool)} (β : Type v) (f : list × bool) → β) (H : ∀ (L₁ L₂ : list × bool)), L₂f L₁ = f L₂) :
H = f L
@[simp]
theorem free_group.quot_map_mk {α : Type u} {L : list × bool)} (β : Type v) (f : list × bool)list × bool)) (H : f) :
H = free_group.mk (f L)
@[protected, instance]
def free_group.has_one {α : Type u} :
Equations
theorem free_group.one_eq_mk {α : Type u} :
@[protected, instance]
def free_group.inhabited {α : Type u} :
Equations
@[protected, instance]
def free_group.has_mul {α : Type u} :
Equations
@[simp]
theorem free_group.mul_mk {α : Type u} {L₁ L₂ : list × bool)} :
= free_group.mk (L₁ ++ L₂)
def free_group.inv_rev {α : Type u} (w : list × bool)) :
list × bool)

Transform a word representing a free group element into a word representing its inverse. -

Equations
@[simp]
theorem free_group.inv_rev_length {α : Type u} {L₁ : list × bool)} :
@[simp]
theorem free_group.inv_rev_inv_rev {α : Type u} {L₁ : list × bool)} :
@[simp]
theorem free_group.inv_rev_empty {α : Type u} :
theorem free_group.inv_rev_involutive {α : Type u} :
theorem free_group.inv_rev_injective {α : Type u} :
theorem free_group.inv_rev_surjective {α : Type u} :
theorem free_group.inv_rev_bijective {α : Type u} :
@[protected, instance]
def free_group.has_inv {α : Type u} :
Equations
@[simp]
theorem free_group.inv_mk {α : Type u} {L : list × bool)} :
theorem free_group.red.step.inv_rev {α : Type u} {L₁ L₂ : list × bool)} (h : L₂) :
theorem free_group.red.inv_rev {α : Type u} {L₁ L₂ : list × bool)} (h : L₂) :
@[simp]
theorem free_group.red.step_inv_rev_iff {α : Type u} {L₁ L₂ : list × bool)} :
@[simp]
theorem free_group.red_inv_rev_iff {α : Type u} {L₁ L₂ : list × bool)} :
L₂
@[protected, instance]
def free_group.group {α : Type u} :
Equations
def free_group.of {α : Type u} (x : α) :

of is the canonical injection from the type to the free group over that type by sending each element to the equivalence class of the letter that is the element.

Equations
theorem free_group.red.exact {α : Type u} {L₁ L₂ : list × bool)} :
theorem free_group.of_injective {α : Type u} :

The canonical injection from the type to the free group is an injection.

def free_group.lift.aux {α : Type u} {β : Type v} [group β] (f : α → β) :
list × bool) → β

Given f : α → β with β a group, the canonical map list (α × bool) → β

Equations
theorem free_group.red.step.lift {α : Type u} {L₁ L₂ : list × bool)} {β : Type v} [group β] {f : α → β} (H : L₂) :
=
@[simp]
theorem free_group.lift_symm_apply {α : Type u} {β : Type v} [group β] (g : →* β) (ᾰ : α) :
def free_group.lift {α : Type u} {β : Type v} [group β] :
(α → β) →* β)

If β is a group, then any function from α to β extends uniquely to a group homomorphism from the free group over α to β

Equations
@[simp]
theorem free_group.lift.mk {α : Type u} {L : list × bool)} {β : Type v} [group β] {f : α → β} :
= (list.map (λ (x : α × bool), cond x.snd (f x.fst) (f x.fst)⁻¹) L).prod
@[simp]
theorem free_group.lift.of {α : Type u} {β : Type v} [group β] {f : α → β} {x : α} :
= f x
theorem free_group.lift.unique {α : Type u} {β : Type v} [group β] {f : α → β} (g : →* β) (hg : ∀ (x : α), g = f x) {x : free_group α} :
g x = x
@[ext]
theorem free_group.ext_hom {α : Type u} {G : Type u_1} [group G] (f g : →* G) (h : ∀ (a : α), f = g ) :
f = g

Two homomorphisms out of a free group are equal if they are equal on generators.

theorem free_group.lift.of_eq {α : Type u} (x : free_group α) :
theorem free_group.lift.range_le {α : Type u} {β : Type v} [group β] {f : α → β} {s : subgroup β} (H : s) :
s
theorem free_group.lift.range_eq_closure {α : Type u} {β : Type v} [group β] {f : α → β} :
def free_group.map {α : Type u} {β : Type v} (f : α → β) :

Any function from α to β extends uniquely to a group homomorphism from the free group ver α to the free group over β.

Equations
@[simp]
theorem free_group.map.mk {α : Type u} {L : list × bool)} {β : Type v} {f : α → β} :
= free_group.mk (list.map (λ (x : α × bool), (f x.fst, x.snd)) L)
@[simp]
theorem free_group.map.id {α : Type u} (x : free_group α) :
= x
@[simp]
theorem free_group.map.id' {α : Type u} (x : free_group α) :
(free_group.map (λ (z : α), z)) x = x
theorem free_group.map.comp {α : Type u} {β : Type v} {γ : Type w} (f : α → β) (g : β → γ) (x : free_group α) :
( x) = (free_group.map (g f)) x
@[simp]
theorem free_group.map.of {α : Type u} {β : Type v} {f : α → β} {x : α} :
theorem free_group.map.unique {α : Type u} {β : Type v} {f : α → β} (g : →* ) (hg : ∀ (x : α), g = free_group.of (f x)) {x : free_group α} :
g x = x
theorem free_group.map_eq_lift {α : Type u} {β : Type v} {f : α → β} {x : free_group α} :
x = x
@[simp]
theorem free_group.free_group_congr_apply {α : Type u_1} {β : Type u_2} (e : α β) (ᾰ : free_group α) :
=
def free_group.free_group_congr {α : Type u_1} {β : Type u_2} (e : α β) :

Equivalent types give rise to multiplicatively equivalent free groups.

The converse can be found in group_theory.free_abelian_group_finsupp, as equiv.of_free_group_equiv

Equations
@[simp]
theorem free_group.free_group_congr_refl {α : Type u} :
@[simp]
theorem free_group.free_group_congr_symm {α : Type u_1} {β : Type u_2} (e : α β) :
theorem free_group.free_group_congr_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : α β) (f : β γ) :
def free_group.prod {α : Type u} [group α] :
→* α

If α is a group, then any function from α to α extends uniquely to a homomorphism from the free group over α to α. This is the multiplicative version of sum.

Equations
@[simp]
theorem free_group.prod_mk {α : Type u} {L : list × bool)} [group α] :
= (list.map (λ (x : α × bool), cond x.snd x.fst (x.fst)⁻¹) L).prod
@[simp]
theorem free_group.prod.of {α : Type u} [group α] {x : α} :
theorem free_group.prod.unique {α : Type u} [group α] (g : →* α) (hg : ∀ (x : α), g = x) {x : free_group α} :
g x =
theorem free_group.lift_eq_prod_map {α : Type u} {β : Type v} [group β] {f : α → β} {x : free_group α} :
def free_group.sum {α : Type u} [add_group α] (x : free_group α) :
α

If α is a group, then any function from α to α extends uniquely to a homomorphism from the free group over α to α. This is the additive version of prod.

Equations
@[simp]
theorem free_group.sum_mk {α : Type u} {L : list × bool)} [add_group α] :
.sum = (list.map (λ (x : α × bool), cond x.snd x.fst (-x.fst)) L).sum
@[simp]
theorem free_group.sum.of {α : Type u} [add_group α] {x : α} :
.sum = x
@[simp]
theorem free_group.sum.map_mul {α : Type u} [add_group α] {x y : free_group α} :
(x * y).sum = x.sum + y.sum
@[simp]
theorem free_group.sum.map_one {α : Type u} [add_group α] :
1.sum = 0
@[simp]
theorem free_group.sum.map_inv {α : Type u} [add_group α] {x : free_group α} :

The bijection between the free group on the empty type, and a type with one element.

Equations

The bijection between the free group on a singleton, and the integers.

Equations
@[protected, instance]
Equations
@[protected]
theorem free_group.induction_on {α : Type u} {C : → Prop} (z : free_group α) (C1 : C 1) (Cp : ∀ (x : α), C ) (Ci : ∀ (x : α), C C ⁻¹) (Cm : ∀ (x y : , C xC yC (x * y)) :
C z
@[simp]
theorem free_group.map_pure {α β : Type u} (f : α → β) (x : α) :
@[simp]
theorem free_group.map_one {α β : Type u} (f : α → β) :
f <$> 1 = 1 @[simp] theorem free_group.map_mul {α β : Type u} (f : α → β) (x y : free_group α) : f <$> (x * y) = f <$> x * f <$> y
@[simp]
theorem free_group.map_inv {α β : Type u} (f : α → β) (x : free_group α) :
f <$> x⁻¹ = (f <$> x)⁻¹
@[simp]
theorem free_group.pure_bind {α β : Type u} (f : α → ) (x : α) :
= f x
@[simp]
theorem free_group.one_bind {α β : Type u} (f : α → ) :
1 >>= f = 1
@[simp]
theorem free_group.mul_bind {α β : Type u} (f : α → ) (x y : free_group α) :
x * y >>= f = (x >>= f) * (y >>= f)
@[simp]
theorem free_group.inv_bind {α β : Type u} (f : α → ) (x : free_group α) :
x⁻¹ >>= f = (x >>= f)⁻¹
@[protected, instance]
def free_group.reduce {α : Type u} [decidable_eq α] (L : list × bool)) :
list × bool)

The maximal reduction of a word. It is computable iff α has decidable equality.

Equations
@[simp]
theorem free_group.reduce.cons {α : Type u} {L : list × bool)} [decidable_eq α] (x : α × bool) :
free_group.reduce (x :: L) = .cases_on [x] (λ (hd : α × bool) (tl : list × bool)), ite (x.fst = hd.fst x.snd = !hd.snd) tl (x :: hd :: tl))
theorem free_group.reduce.red {α : Type u} {L : list × bool)} [decidable_eq α] :

The first theorem that characterises the function reduce: a word reduces to its maximal reduction.

theorem free_group.reduce.not {α : Type u} [decidable_eq α] {p : Prop} {L₁ L₂ L₃ : list × bool)} {x : α} {b : bool} :
= L₂ ++ (x, b) :: (x, !b) :: L₃ → p
theorem free_group.reduce.min {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (H : L₂) :
= L₂

The second theorem that characterises the function reduce: the maximal reduction of a word only reduces to itself.

@[simp]
theorem free_group.reduce.idem {α : Type u} {L : list × bool)} [decidable_eq α] :

reduce is idempotent, i.e. the maximal reduction of the maximal reduction of a word is the maximal reduction of the word.

theorem free_group.reduce.step.eq {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (H : L₂) :
theorem free_group.reduce.eq_of_red {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (H : L₂) :

If a word reduces to another word, then they have a common maximal reduction.

theorem free_group.red.reduce_eq {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (H : L₂) :

Alias of free_group.reduce.eq_of_red.

theorem free_group.red.reduce_right {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (h : L₂) :
theorem free_group.red.reduce_left {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (h : L₂) :
theorem free_group.reduce.sound {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (H : = ) :

If two words correspond to the same element in the free group, then they have a common maximal reduction. This is the proof that the function that sends an element of the free group to its maximal reduction is well-defined.

theorem free_group.reduce.exact {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (H : = ) :

If two words have a common maximal reduction, then they correspond to the same element in the free group.

theorem free_group.reduce.self {α : Type u} {L : list × bool)} [decidable_eq α] :

A word and its maximal reduction correspond to the same element of the free group.

theorem free_group.reduce.rev {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (H : L₂) :

If words w₁ w₂ are such that w₁ reduces to w₂, then w₂ reduces to the maximal reduction of w₁.

def free_group.to_word {α : Type u} [decidable_eq α] :
list × bool)

The function that sends an element of the free group to its maximal reduction.

Equations
theorem free_group.mk_to_word {α : Type u} [decidable_eq α] {x : free_group α} :
theorem free_group.to_word_injective {α : Type u} [decidable_eq α] :
@[simp]
theorem free_group.to_word_inj {α : Type u} [decidable_eq α] {x y : free_group α} :
@[simp]
theorem free_group.to_word_mk {α : Type u} {L₁ : list × bool)} [decidable_eq α] :
@[simp]
theorem free_group.reduce_to_word {α : Type u} [decidable_eq α] (x : free_group α) :
@[simp]
theorem free_group.to_word_one {α : Type u} [decidable_eq α] :
@[simp]
theorem free_group.to_word_eq_nil_iff {α : Type u} [decidable_eq α] {x : free_group α} :
x = 1
theorem free_group.reduce_inv_rev {α : Type u} [decidable_eq α] {w : list × bool)} :
theorem free_group.to_word_inv {α : Type u} [decidable_eq α] {x : free_group α} :
def free_group.reduce.church_rosser {α : Type u} {L₁ L₂ L₃ : list × bool)} [decidable_eq α] (H12 : L₂) (H13 : L₃) :
{L₄ // L₄ L₄}

Constructive Church-Rosser theorem (compare church_rosser).

Equations
@[protected, instance]
def free_group.decidable_eq {α : Type u} [decidable_eq α] :
Equations
@[protected, instance]
def free_group.red.decidable_rel {α : Type u} [decidable_eq α] :
Equations
• free_group.red.decidable_rel ((x1, b1) :: tl1) ((x2, b2) :: tl2) = dite ((x1, b1) = (x2, b2)) (λ (h : (x1, b1) = (x2, b2)), free_group.red.decidable_rel._match_2 x1 b1 tl1 x2 b2 tl2 h tl2)) (λ (h : ¬(x1, b1) = (x2, b2)), free_group.red.decidable_rel._match_3 x1 b1 tl1 x2 b2 tl2 h ((x1, !b1) :: (x2, b2) :: tl2)))
• free_group.red.decidable_rel ((x, b) :: tl) list.nil = free_group.red.decidable_rel._match_1 x b tl [(x, !b)])
• = decidable.is_true free_group.red.decidable_rel._main._proof_1
• free_group.red.decidable_rel._match_2 x1 b1 tl1 x2 b2 tl2 h =
• free_group.red.decidable_rel._match_2 x1 b1 tl1 x2 b2 tl2 h =
• free_group.red.decidable_rel._match_3 x1 b1 tl1 x2 b2 tl2 h =
• free_group.red.decidable_rel._match_3 x1 b1 tl1 x2 b2 tl2 h =
• free_group.red.decidable_rel._match_1 x b tl =
• free_group.red.decidable_rel._match_1 x b tl =
def free_group.red.enum {α : Type u} [decidable_eq α] (L₁ : list × bool)) :
list (list × bool))

A list containing every word that w₁ reduces to.

Equations
theorem free_group.red.enum.sound {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (H : L₂ ) :
L₂
theorem free_group.red.enum.complete {α : Type u} {L₁ L₂ : list × bool)} [decidable_eq α] (H : L₂) :
L₂
@[protected, instance]
def free_group.subtype.fintype {α : Type u} {L₁ : list × bool)} [decidable_eq α] :
fintype {L₂ // L₂}
Equations
def free_group.norm {α : Type u} [decidable_eq α] (x : free_group α) :

The length of reduced words provides a norm on a free group. -

Equations
@[simp]
theorem free_group.norm_inv_eq {α : Type u} [decidable_eq α] {x : free_group α} :
@[simp]
theorem free_group.norm_eq_zero {α : Type u} [decidable_eq α] {x : free_group α} :
x.norm = 0 x = 1
@[simp]
theorem free_group.norm_one {α : Type u} [decidable_eq α] :
1.norm = 0
theorem free_group.norm_mk_le {α : Type u} {L₁ : list × bool)} [decidable_eq α] :
theorem free_group.norm_mul_le {α : Type u} [decidable_eq α] (x y : free_group α) :
(x * y).norm x.norm + y.norm