Inductive non_reduced: group_str -> Prop :=
  | intro_letter_inverse: forall (S T: group_str) (a: alphabet),
    non_reduced (S ++ (opposite a) :: a :: T).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).