Fixpoint inc_dec_in_queue (s : Site) (q : queue Message) {struct q} : Prop :=
  match q with
  | empty => False
  | input (inc_dec s2) q' =>
      if eq_site_dec s2 s then True else False
  | input _ q' => inc_dec_in_queue s q'
  end.

Lemma inc_dec_in : forall (s : Site) (q : queue Message), inc_dec_in_queue s q -> In_queue Message (inc_dec s) q.