(*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_rooted_1 :
forall (cg : CGraph1) (root a : ad) (d : D),
CG_edge cg root a = None -> CG_edge (CG_add cg root a d) root a = Some d.