(*s Adding a lis of adresses connected to [root] with weight $0$ *)

Fixpoint CG_add_root (root : ad) (cg : CGraph1) (l : list ad) {struct l} :
 CGraph1 :=
  match l with
  | nil => cg
  | x :: l' => CG_add_root root (CG_add cg root x Dz) l'
  end.

Lemma CG_edge_dist_some : forall (cg : CGraph1) (x y : ad) (d : D), CG_edge cg x y = Some d -> {d' : D | ad_simple_path_dist cg x y = Some d'}.