unfold refineList.
apply in_map_iff.
exists (d :: p).
split; auto.
apply filter_In.
split; simpl; auto.
unfold eq_dir_b; destruct d; auto.
Qed.

(* Specifications: *)
Lemma refinePreservesPrefixFree : forall d l, PrefixFree l -> PrefixFree (refineList d l).
intros.