(*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_circuit_complete_1 : forall cg : CGraph1, (forall (x : ad) (d : D) (l : list ad), CG_path cg x d (x :: l) -> Dle Dz d = true) -> forall root : ad, root = ad_alloc_opt unit (FSetUnion (MapDom _ cg) (MapDom _ (cg_range cg))) -> forall cg' : CGraph1, cg' = CG_add_root root cg (ad_list_of_dom _ cg) -> CGconsistent cg.