(*s [(ad_simple_path_dist x y)] computes the minimum path from x to y *)
Definition ad_simple_path_dist (x y : ad) :=
  ad_simple_path_dist_1 x y nil (MapCard _ cg).

Lemma ad_simple_path_dist_correct : forall (x y : ad) (l : list ad) (d : D), CG_path y d (x :: l) -> Ddle (ad_simple_path_dist x y) (Some d) = true.