(*s [(CG_add x y d)] adds the edge (x,y) of weight d to G, 
   in case an edge already exists, only the minimal one is kept
*)

Definition CG_add (x y : ad) (d : D) :=
  match MapGet _ cg x with
  | None => MapPut _ cg x (M1 _ y d)
  | Some edges =>
      match MapGet _ edges y with
      | None => MapPut _ cg x (MapPut _ edges y d)
      | Some d0 => MapPut _ cg x (MapPut _ edges y (Dmin d d0))
      end
  end.Definition cg2 := CG_add cg1 x y d.

Lemma CG_add_4_4 : forall (l : list ad) (last : ad) (d0 : D), CG_path cg2 last d0 l -> ad_in_list x l = false -> CG_path cg1 last d0 l.