Definition rel_base (b : pred T) :=
  forall x' y', ~~ b (h x') -> e (h x') (h y') = e' x' y'.

(* Specifications: *)
Lemma map_path b x' p' (Bb : rel_base b) : ~~ has (preim h b) (belast x' p') -> path e (h x') (map h p') = path e' x' p'.
Lemma mem2_map x' y' p' : mem2 (map h p') (h x') (h y') = mem2 p' x' y'.
Lemma next_map p : uniq p -> forall x, next (map h p) (h x) = h (next p x).
Lemma prev_map p : uniq p -> forall x, prev (map h p) (h x) = h (prev p x).