Section FreeGroups.
Variable generators: Type.
Hypothesis generator_eq_dec: forall g1 g2:generators,
{ g1=g2 } + { g1<>g2 }.
Inductive alphabet: Type :=
| intro_gen: generators -> alphabet
| intro_gen_inv: generators -> alphabet.
Definition group_str := list alphabet.
Definition opposite (letter:alphabet) :=
match letter with
| intro_gen a => intro_gen_inv a
| intro_gen_inv a => intro_gen a
end.
Inductive non_reduced: group_str -> Prop :=
| intro_letter_inverse: forall (S T: group_str) (a: alphabet),
non_reduced (S ++ (opposite a) :: a :: T).
Definition reduced (S: group_str) : Prop :=
not (non_reduced S).

Lemma cons_to_nonreduced_gives_nonreduced:
forall (S:group_str) (a:alphabet),
non_reduced S -> non_reduced (a :: S).