(*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_add_root_rooted_3 : forall (cg : CGraph1) (root : ad), in_dom _ root cg = false -> forall a : ad, in_dom _ a cg = true -> CG_edge (CG_add_root root cg (ad_list_of_dom _ cg)) root a = Some Dz.