Inductive pl_path_true : pl_path -> Map bool -> Prop :=
| plp_true_nil : forall m : Map bool, pl_path_true pl_path_nil m
| plp_true_cons :
forall (m : Map bool) (a : ad) (pl : pl_path),
pl_path_true pl m ->
MapGet bool m a = Some true -> pl_path_true (pl_path_cons a pl) m.
Definition pl_non_empty_path_true_def_0 (pl : pl_path)
(p : prec_list) : Prop :=
forall m : Map bool,
pl_path_incl pl p -> pl_path_true pl m -> pl_non_empty m p = true.

(* Specifications: *)
Lemma pl_non_empty_path_true_0 :
pl_non_empty_path_true_def_0 pl_path_nil prec_empty.
Lemma pl_non_empty_path_true_1 :
forall (plp : pl_path) (a : ad) (la ls : prec_list),
pl_path_incl plp la ->
pl_non_empty_path_true_def_0 plp la ->
pl_non_empty_path_true_def_0 (pl_path_cons a plp) (prec_cons a la ls).
Lemma pl_non_empty_path_true_2 :
forall (plp : pl_path) (a : ad) (la ls : prec_list),
pl_path_incl plp ls ->
pl_non_empty_path_true_def_0 plp ls ->
plp <> pl_path_nil -> pl_non_empty_path_true_def_0 plp (prec_cons a la ls).
Lemma pl_non_empty_path_true :
forall (pl : pl_path) (p : prec_list) (m : Map bool),
pl_path_incl pl p -> pl_path_true pl m -> pl_non_empty m p = true.
Lemma pl_non_empty_path_true_rev :
forall (p : prec_list) (m : Map bool),
pl_non_empty m p = true ->
exists plp : pl_path, pl_path_incl plp p /\ pl_path_true plp m.
Lemma st_non_empty_0 :
forall (m : Map bool) (s : state) (p : prec_list) (a : ad),
MapGet prec_list s a = Some p ->
pl_non_empty m p = true -> st_non_empty m s = true.
Lemma st_non_empty_1 :
forall (d : preDTA) (m r : Map bool) (a : ad) (l : state),
MapGet state d a = Some l ->
domain_equal state bool d r ->
st_non_empty m l = true ->
MapGet bool (dta_app_ne_aux d m r) a = Some true.