(* insertion d'un état *)
Definition insert_state (d : preDTA) (a : ad) (s : state) : preDTA :=
MapPut state d a s.
Definition insert_main_state_0 (d : preDTA) (a : ad)
(s : state) : DTA := dta (insert_state d a s) a.
Definition insert_main_state (d : preDTA) (s : state) : DTA :=
insert_main_state_0 d (new_preDTA_ad d) s.
Definition insert_ostate (d : preDTA) (a : ad) (o : option state) : preDTA :=
match o with
| None => d
| Some s => MapPut state d a s
end.
Definition insert_main_ostate_0 (d : preDTA) (a : ad)
(o : option state) : DTA := dta (insert_ostate d a o) a.

Lemma upl_conv_1_occur_in_img :
forall (pl : prec_list) (a : ad),
prec_occur (upl_conv_1 pl) a -> exists b : ad, a = uad_conv_1 b.