proposition: Lemma chemin_A_e1_g0 : forall (q qd qa d : Ensf) (g0 e1 e2 : Elt) (w : Word), automate q qd qa d -> ~ dans g0 q -> chemin_A (add g0 q) (union d (delta_D g0 qd)) e1 e2 w -> e2 = g0 :>Elt -> w = nil :>Word.
proof: Proof. intros q qd qa d g0 e1 e2 w H_aut H_g0 H. elim H; auto. intros e0 e e3 x w0 H0 H1 H2 H3 H4 H5. absurd (chemin_A (add g0 q) (union d (delta_D g0 qd)) e g0 w0). 2: cut (chemin_A (add g0 q) (union d (delta_D g0 qd)) e e3 w0); auto. 2: rewrite H5; auto. apply chemin_A_e1_g0_abs with qa; auto. 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. cut (dans x alph /\ dans e q). 2: apply coupl2; auto. intro Ht; elim Ht; clear Ht; intros. 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 H8. 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 H11. red in |- *; intro; apply not_dans_epsilon_alph; assumption. Qed.
imports: Ensf Max Words Dec