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 case_first_value : forall (E : Set) (x y : E) (d : data) (q : queue data), first data q = value data d -> match q with | empty => x | input _ _ => y end = y.