intros q qd qa d g0 e1 e2 w H_aut H_g0 H.
elim H; clear H.
intros e0 e3 H H0 H1 H2.
absurd (dans g0 q); auto.
rewrite <- H1.
rewrite H0.
apply dans_trans with qa; auto.
apply automate_def3 with qd d; auto.
intros e0 e e3 x w0 H H0 H1 H2 H3 H4 H5.
clear H0.
cut
(dans (couple e0 (couple x e)) d \/
dans (couple e0 (couple x e)) (delta_D g0 qd)); auto.
intro Ht; elim Ht; clear Ht.
intro.
cut (dans (couple e0 (couple x e)) (prodcart q (prodcart alph q))).
2: apply dans_trans with d; auto.
2: apply automate_def1 with qd qa; auto.
intro.
cut (dans e0 q /\ dans (couple x e) (prodcart alph q)).
2: apply coupl2; auto.
intro Ht; elim Ht; clear Ht; intros.
absurd (dans g0 q); auto.
rewrite <- H4.
assumption.
unfold delta_D in |- *.
intro.
cut
(exists y : Elt,
dans y qd /\ couple e0 (couple x e) = transition_D g0 y :>Elt).
2: apply dans_map; auto.
intro Ht; elim Ht; clear Ht; intros y' Ht; elim Ht; clear Ht; intros.
unfold transition_D in H7.
cut (e0 = g0 :>Elt /\ couple x e = couple epsilon y' :>Elt).
2: apply equal_couple; auto.
intro Ht; elim Ht; clear Ht; intros.
cut (x = epsilon :>Elt /\ e = y' :>Elt).
2: apply equal_couple; auto.
intro Ht; elim Ht; clear Ht; intros.
absurd (dans x alph); auto.
rewrite H10.
red in |- *; intro; apply not_dans_epsilon_alph; assumption.
intros e0 e e3 w0 H H0 H1 H2 H3 H4.
clear H0.
exists e.
split.
2: assumption.
cut
(dans (couple e0 (couple epsilon e)) d \/
dans (couple e0 (couple epsilon e)) (delta_D g0 qd));
auto.
intro Ht; elim Ht; clear Ht.
intro.
cut (dans (couple e0 (couple epsilon e)) (prodcart q (prodcart alph q))).
2: apply dans_trans with d; auto.
2: apply automate_def1 with qd qa; auto.
intro.
cut (dans e0 q /\ dans (couple epsilon e) (prodcart alph q)).
2: apply coupl2; auto.
intro Ht; elim Ht; clear Ht; intros.
cut (dans epsilon alph /\ dans e q).
2: apply coupl2; auto.
intro Ht; elim Ht; clear Ht; intros.
absurd (dans epsilon alph); auto.
red in |- *; intro; apply not_dans_epsilon_alph; assumption.
unfold delta_D in |- *.
intro.
cut
(exists y : Elt,
dans y qd /\ couple e0 (couple epsilon e) = transition_D g0 y :>Elt).
2: apply dans_map; auto.
intro Ht; elim Ht; clear Ht; intros y' Ht; elim Ht; clear Ht; intros.
unfold transition_D in H6.
cut (e0 = g0 :>Elt /\ couple epsilon e = couple epsilon y' :>Elt).
2: apply equal_couple; auto.
intro Ht; elim Ht; clear Ht; intros.
cut (epsilon = epsilon :>Elt /\ e = y' :>Elt).
2: apply equal_couple; auto.
intro Ht; elim Ht; clear Ht; intros.
rewrite H10.
assumption.
Qed.
(*									*)
(*  On ne peut avoir de chemin de e a g0 si e est dans q.		*)
(*									*)

Lemma chemin_A_e1_g0_abs :
forall (q qd qa d : Ensf) (g0 e e2 : Elt) (w : Word),
automate q qd qa d ->
dans e q ->
~ dans g0 q ->
e2 = g0 :>Elt -> ~ chemin_A (add g0 q) (union d (delta_D g0 qd)) e e2 w.