intros.
split.
unfold eqwordset in |- *.
intro w.
split.
unfold reconnait in |- *.
intro Ht; elim Ht; clear Ht; intros H1 Ht; elim Ht; clear Ht; intros e1 Ht;
elim Ht; clear Ht; intros e2 Ht; elim Ht; clear Ht;
intros H2 Ht; elim Ht; clear Ht; intros.
unfold reconnait_A in |- *.
split.
assumption.
exists g0.
exists e2.
split; [ auto | split ].
assumption.
apply chemin_A_epsilon with e1; auto.
apply chemin_A_d1_d2 with d; auto.
apply chemin_A_q1_q2 with q; auto.
apply chemin_chemin_A; auto.
apply union_d.
unfold delta_D in |- *.
replace (couple g0 (couple epsilon e1)) with (transition_D g0 e1);
auto.
unfold reconnait_A in |- *.
intro Ht; elim Ht; clear Ht; intros H1 Ht; elim Ht; clear Ht; intros e1 Ht;
elim Ht; clear Ht; intros e2 Ht; elim Ht; clear Ht;
intros H2 Ht; elim Ht; clear Ht; intros.
unfold reconnait in |- *.
split.
assumption.
cut (e1 = g0 :>Elt); auto.
intro.
cut
(exists e : Elt,
dans e qd /\ chemin_A (add g0 q) (union d (delta_D g0 qd)) e e2 w).
2: apply chemin_A_g0_e2 with qa e1; auto.
intro Ht; elim Ht; clear Ht; intros e Ht; elim Ht; clear Ht; intros.
exists e.
exists e2.
split; [ assumption | split ].
assumption.
apply chemin_D_chemin with qd qa g0; auto.
apply dans_trans with qd; auto.
apply automate_def2 with qa d; auto.
intros.
apply chemin_A_e1_g0 with q qd qa d g0 g0 g0; auto.
Qed.
(*									*)
(*  Si l est regulier, alors l est regulier au sens de isregular_D,	*)
(*  ie il existe un automate a un seul etat de depart g0 et possedant	*)
(*  la propriete que seul le mot nil permet d'aller de g0 a g0 par 	*)
(*  un chemin qui reconnait l.						*)
(*									*)

Lemma isregular_isregular_D :
forall l : wordset, isregular l -> isregular_D l.