(* terme t reconnu en l'etat s : f^(high t)=true *)

Definition dt_non_empty_def_0 (d : preDTA) (a : ad) 
  (t : term) (pr : reconnaissance d a t) :=
  forall n : nat,
  term_high t <= n ->
  MapGet bool (power (Map bool) (dta_app_ne d) (map_mini state d) n) a =
  Some true.Definition dt_non_empty_def_1 (d : preDTA) (s : state) 
  (t : term) (pr : state_reconnait d s t) :=
  forall n : nat,
  term_high t <= S n ->
  st_non_empty (power (Map bool) (dta_app_ne d) (map_mini state d) n) s =
  true.Definition dt_non_empty_def_2 (d : preDTA) (p : prec_list) 
  (t : term_list) (pr : liste_reconnait d p t) :=
  forall n : nat,
  term_high_0 t <= n ->
  pl_non_empty (power (Map bool) (dta_app_ne d) (map_mini state d) n) p =
  true.

Lemma dt_non_empty_r_0 : forall (d : preDTA) (m r : Map bool) (a : ad) (l : state), MapGet state d a = Some l -> domain_equal state bool d r -> MapGet bool (dta_app_ne_aux d m r) a = Some true -> MapGet bool r a = Some true \/ st_non_empty m l = true.