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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                                 RatReg.v                                 *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)
(************************************************************************)
(*									*)
(*            Un langage reduit a un mot est regulier.			*)
(*									*)
(************************************************************************)
(*  Le mot vide est reconnu par un automate reduit a un etat (zero)	*)
(*  et sans transition.							*)
unfold reconnait at 1 in |- *.
split.
apply inmonoid_nil.
exists zero; exists zero.
auto.
Qed.
(* Seul le mot vide est reconnu par l'automate ci-dessus.		*)
unfold reconnait at 1 in |- *.
simple induction w.
auto.
intros x w0 H H0.
clear H.
elim H0; intros; clear H0.
elim H1; intros; clear H1.
elim H0; intros; clear H0.
elim H1; intros; clear H1.
elim H2; intros; clear H2.
cut
(Chemin x0 x1 (singleton zero) (prodcart empty (prodcart alph empty))
(cons x w0)).
2: auto.
unfold Chemin in |- *; simpl in |- *.
intro H2.
absurd (dans x0 empty).
auto.
elim H2; intros; clear H2.
elim H4; intros; clear H4.
elim H5; intros; clear H5.
elim H6; intros; clear H6.
cut (dans x0 empty /\ dans (couple x x2) (prodcart alph empty)).
2: apply coupl2; assumption.
tauto.
Qed.
(*									*)
(*  Pour pouvoir construire un automate qui reconnait (cons a w) a	*)
(*  partir d'un automate qui reconnait w, il faut un resultat un peu	*)
(*  plus precis que "il existe un automate tel que...".			*)
(*									*)
(*  On precise que cet automate a un unique etat de depart.		*)
(*									*)
exists (singleton zero).
exists zero.
exists (singleton zero).
exists (prodcart empty (prodcart alph empty)).
split.
red in |- *; auto.
red in |- *; split.
red in |- *.
symmetry  in |- *; apply lwordnil_is_reg2.
assumption.
intro.
compute in H.
rewrite <- H.
apply lwordnil_is_reg1.
Qed.
(*  D'ou bien sur...							*)

Lemma lwordnil_is_reg : isregular (lword nil).