(*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 :
forall (cg : CGraph1) (root : ad),
(forall (x : ad) (d : D) (l : list ad),
CG_path cg x d (x :: l) -> Dle Dz d = true) ->
in_dom _ root cg = false ->
forall a : ad,
in_dom _ a (CG_add_root root cg (ad_list_of_dom _ cg)) = true ->
CG_leq (CG_add_root root cg (ad_list_of_dom _ cg)) root a = true.