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 not_in_q_input : forall (data : Set) (d' d : data) (q : queue data), d <> d' -> ~ In_queue data d (input data d' q) -> ~ In_queue data d q.