Fixpoint append (q : queue data) : queue data -> queue data :=
  fun q2 : queue data =>
  match q with
  | empty => q2
  | input d q' => input data d (append q' q2)
  end.

Lemma append_assoc : forall q1 q2 q3 : queue data, append q1 (append q2 q3) = append (append q1 q2) q3.