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 not_inqueue_append : forall (m : data) (q1 q2 : queue data), ~ In_queue data m q1 /\ ~ In_queue data m q2 -> ~ In_queue data m (append data q1 q2).