(*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).

(* Specifications: *)
Lemma ad_simple_path_dist_correct_1 : forall (x y : ad) (l : list ad) (d : D), CG_path y d (x :: l) -> ad_list_stutters (x :: l) = false -> Ddle (ad_simple_path_dist x y) (Some d) = true.
Lemma ad_simple_path_dist_correct : forall (x y : ad) (l : list ad) (d : D), CG_path y d (x :: l) -> Ddle (ad_simple_path_dist x y) (Some d) = true.
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)}.
Lemma ad_simple_path_dist_1_complete : forall (n : nat) (x y : ad) (d : D), ad_simple_path_dist_1 x y nil n = Some d -> {l : list ad & and_sp (CG_path y d (x :: l)) (ad_list_stutters (x :: l) = false)}.
Lemma ad_simple_path_dist_complete : forall (x y : ad) (d : D), ad_simple_path_dist x y = Some d -> {l : list ad & and_sp (CG_path y d (x :: l)) (ad_list_stutters (x :: l) = false)}.
Lemma ad_simple_path_dist_complete_2 : forall (x y : ad) (d : D), ad_simple_path_dist x y = Some d -> {l : list ad & CG_path y d (x :: l)}.
Lemma ad_simple_path_dist_complete_3 : forall (x y : ad) (dd : option D), (forall (l : list ad) (d : D), CG_path y d (x :: l) -> Ddle dd (Some d) = true) -> Ddle dd (ad_simple_path_dist x y) = true.
Lemma ad_simple_path_dist_d_1 : forall x : ad, ad_simple_path_dist x x = Some Dz.
Lemma ad_simple_path_dist_d_2 : forall (x y z : ad) (d d' : D), ad_simple_path_dist x y = Some d -> ad_simple_path_dist y z = Some d' -> Ddle (ad_simple_path_dist x z) (Some (Dplus d' d)) = true.
Lemma ad_simple_path_dist_d_3 : forall (x y z : ad) (d d' : D), Ddle (ad_simple_path_dist x y) (Some d) = true -> Ddle (ad_simple_path_dist y z) (Some d') = true -> Ddle (ad_simple_path_dist x z) (Some (Dplus d' d)) = true.