(*s [(CG_standard_rho root d0 others x)] computes [d0-d] when
     [root] and [x] are connected by a path of minimal 
     length [d] and [(others x)] when there are not connected 
*)

Definition CG_standard_rho (root : ad) (d0 : D) (others : ad -> D)
  (x : ad) :=
  match ad_simple_path_dist root x with
  | Some d => Dplus d0 (Dneg d)
  | None => others x (* dummy *)
  end.

(* Specifications: *)
Lemma CG_standard_rho_root : forall (root : ad) (d0 : D) (others : ad -> D), CG_standard_rho root d0 others root = d0.
Lemma CG_rooted_sat_1 : forall (root : ad) (d0 : D) (others : ad -> D), (forall x : ad, in_dom _ x cg = true -> CG_leq root x = true) -> CGsat (CG_standard_rho root d0 others).
Lemma CG_rooted_sat : forall (root : ad) (d0 : D), (forall x : ad, in_dom _ x cg = true -> CG_leq root x = true) -> {rho : ad -> D | CGsat rho /\ rho root = d0}.
Lemma CG_standard_rho_minimal : forall (root : ad) (d0 : D) (others rho : ad -> D), CGsat rho -> Dle d0 (rho root) = true -> forall x : ad, CG_leq root x = true -> Dle (CG_standard_rho root d0 others x) (rho x) = true.
Lemma CG_sat_add_1 : forall (x y : ad) (d : D) (rho : ad -> D), CGsat rho -> Dle (rho x) (Dplus (rho y) d) = true -> Ddle (Some (Dneg d)) (ad_simple_path_dist y x) = true.