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.

Lemma cg_out_of_range_2 :
forall (cg : CGraph1) (y : ad),
in_dom D y (cg_range cg) = false ->
forall x : ad, N.eqb x y = false -> ad_simple_path_dist cg x y = None.