Fixpoint group_str_action (S T:group_str) :=
  match S with
  | nil => T
  | a :: S0 => letter_action a (group_str_action S0 T)
  end.

(* Specifications: *)
Lemma reduced_closed_under_str_action: forall (S T:group_str), reduced T -> reduced (group_str_action S T).