Fixpoint card (d : data) (q : queue) {struct q} : Z :=
  match q with
  | empty => 0%Z
  | input d' q' =>
      if eq_data_dec d d'
      then (card d q' + 1)%Z
      else card d q'
  end.

Lemma in_q_output : forall (d : data) (q : queue data), In_queue data d (first_out data q) -> In_queue data d q.