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 input_append : forall (d : data) (q1 q2 : queue data), input data d (append q1 q2) = append (input data d q1) q2.