Section CGAdd.
Variable cg1 : CGraph1.
Variable x y : ad.
Variable d : D.
Definition cg2 := CG_add cg1 x y d.
(*s If there is no cycle of negative weight in the old graph [cg1], and the distance from [y] to [x] is $\geq -d$, then there is no simple cycle of negative weight in the new graph  [cg2] either: *)
End CGAdd.
(*s Properties of the range of the graph *)
Definition cg_range := DMerge D.
(*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.