(*s [(ad_simple_path_dist x y)] computes the minimum path from x to y *)
Definition ad_simple_path_dist (x y : ad) :=
ad_simple_path_dist_1 x y nil (MapCard _ cg).

Lemma ad_simple_path_dist_1_complete_1 :
forall (n : nat) (x y : ad) (prefix : list ad) (d' : D),
CG_path x d' (rev (x :: prefix)) ->
ad_list_stutters (x :: prefix) = false ->
forall d0 : D,
ad_simple_path_dist_1 x y prefix n = Some d0 ->
{l : list ad &
and_sp (CG_path y (Dplus d0 d') (rev (x :: prefix) ++ l))
(ad_list_stutters (rev (x :: prefix) ++ l) = false /\ length l <= n)}.