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_out_of_range :
forall (l : list ad) (cg : CGraph1) (root : ad),
(forall (x0 : ad) (d0 : D) (l : list ad),
CG_path cg x0 d0 (x0 :: l) -> Dle Dz d0 = true) ->
MapSubset _ _ (Elems l) (MapDom _ cg) ->
ad_in_list root l = false ->
in_dom _ root (cg_range cg) = false ->
forall (x0 : ad) (d0 : D) (l0 : list ad),
CG_path (CG_add_root root cg l) x0 d0 (x0 :: l0) -> Dle Dz d0 = true.