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_range_4 :
forall (cg : CGraph1) (x y : ad) (d : D) (a : ad),
in_dom D a (cg_range (CG_add cg x y d)) =
N.eqb a y || in_dom D a (cg_range cg).