cut (isregular_D l).
2: apply isregular_isregular_D; auto.
clear H; intro H; elim H; clear H.
intros q H; elim H; clear H; intros g0 H; elim H; clear H; intros qa H;
elim H; clear H; intros d H; elim H; clear H.
intros H_aut Ht; elim Ht; clear Ht; intros H_g0 H_eq.
apply isregular_A_isregular.
unfold isregular_A in |- *.
exists q.
exists (singleton g0).
exists (singleton g0).
exists (fun_d_dstar g0 qa d).
split.
red in |- *.
elim H_aut.
intros H0 H1; elim H1; clear H1; intros H1 H2.
split; [ assumption | split ].
assumption.
apply dstar_is_good; assumption.
unfold eqwordset in |- *.
intro w.
split.
intro; apply lstar_is_reg1 with q qa d g0; auto.
intro; pattern w in |- *; apply induction_star with l; auto.
simple induction n.
unfold lpuiss in |- *.
unfold lword in |- *.
intros.
rewrite <- H0.
unfold reconnait_A in |- *.
split; auto.
exists g0.
exists g0.
split; [ auto | split ].
auto.
apply chemin_A_nil; auto.
apply dans_trans with (singleton g0); auto.
apply automate_def2 with qa d; auto.
intros y H1 w0.
change
(lconc l (lpuiss y l) w0 ->
reconnait_A q (singleton g0) (singleton g0) (fun_d_dstar g0 qa d) w0)
in |- *.
unfold lconc in |- *.
intros Ht; elim Ht; clear Ht; intros w1 Ht; elim Ht; clear Ht; intros w2 Ht;
elim Ht; clear Ht; intros H2 Ht; elim Ht; clear Ht;
intros H3 H4.
cut (reconnait_A q (singleton g0) (singleton g0) (fun_d_dstar g0 qa d) w2);
auto.
intro H5.
unfold eqwordset in H_eq.
elim (H_eq w1); intros H6 H7.
cut (reconnait q (singleton g0) qa d w1); auto.
intro H8.
clear H6 H7 H1 H_eq.
elim H8; clear H8; intros H8 Ht; elim Ht; clear Ht; intros e1 Ht; elim Ht;
clear Ht; intros e2 Ht; elim Ht; clear Ht; intros H9 Ht;
elim Ht; clear Ht; intros H10 H11.
elim H5; clear H5; intros H12 Ht; elim Ht; clear Ht; intros e3 Ht; elim Ht;
clear Ht; intros e4 Ht; elim Ht; clear Ht; intros H13 Ht;
elim Ht; clear Ht; intros H14 H15.
unfold reconnait_A in |- *.
split.
rewrite H4.
apply inmonoid_Append; auto.
exists e1.
exists e4.
split; [ assumption | split ].
assumption.
rewrite H4.
apply chemin_Append with e2.
apply chemin_A_d1_d2 with d.
apply chemin_chemin_A; auto.
unfold fun_d_dstar in |- *.
apply inclus_g.
apply chemin_A_epsilon with e3; auto.
apply dans_trans with qa; auto.
apply automate_def3 with (singleton g0) d; auto.
unfold fun_d_dstar in |- *.
apply union_d.
unfold delta in |- *.
cut (e3 = g0 :>Elt); auto.
intro H16.
rewrite H16.
replace (couple e2 (couple epsilon g0)) with (transition_back g0 e2);
auto.
Qed.
(************************************************************************)
(*									*)
(*  LE RESULTAT FINAL :							*)
(*      	   	   Tout langage rationnel est regulier.		*)
(*									*)
(************************************************************************)

Lemma rat_is_reg : forall L : wordset, isrationnal L -> isregular L.
intros L H.