Definition insert_main_ostate (d : preDTA) (o : option state) : DTA :=
insert_main_ostate_0 d (new_preDTA_ad d) o.
Definition union_opt_state (o0 o1 : option state) :
option state :=
match o0, o1 with
| None, None => None
| None, Some s1 => Some s1
| Some s0, None => Some s0
| Some s0, Some s1 => Some (union_mpl s0 s1)
end.
Definition union_0 (d : preDTA) (a0 a1 : ad) : option state :=
union_opt_state (MapGet state d (uad_conv_0 a0))
(MapGet state d (uad_conv_1 a1)).
Definition union_1 (d : preDTA) (a0 a1 : ad) : DTA :=
insert_main_ostate d (union_0 d a0 a1).
Definition union (dt0 dt1 : DTA) : DTA :=
match dt0, dt1 with
| dta d0 a0, dta d1 a1 => union_1 (u_merge d0 d1) a0 a1
end.

Lemma union_semantics :
forall (d0 d1 : DTA) (sigma : signature) (t : term),
DTA_main_state_correct d0 ->
DTA_main_state_correct d1 ->
DTA_ref_ok d0 ->
DTA_ref_ok d1 ->
dta_correct_wrt_sign d0 sigma ->
dta_correct_wrt_sign d1 sigma ->
(reconnait d0 t \/ reconnait d1 t <-> reconnait (union d0 d1) t).