intros x dans_x_d.
elimtype (dans x (map f_R_d R) \/ dans x (map f_X_d X)).
intros dans_x.
elimtype (exists y : Elt, dans y R /\ x = f_R_d y).
intros y temp; elim temp; clear temp.
unfold f_R_d in |- *.
intros dans_y_R eg_x_f_y.
exists (cons (first y) nil).
apply inmonoid_cons.
trivial.
elim (Regles_X_V_R y dans_y_R).
intros f_y dans_f_y_V temp; elim temp; clear temp.
intros u eg_y inmono_u.
rewrite eg_y.
unfold first in |- *.
unfold P in |- *.
auto.
exists (eps X).
auto.
exists (word_inv (second y)).
elim (Regles_X_V_R y dans_y_R).
intros f_y dans_f_y_V temp; elim temp; clear temp.
intros u eg_y inmono_u.
rewrite eg_y.
unfold second in |- *.
unfold word_inv in |- *.
assumption.
elim (Regles_X_V_R y dans_y_R).
intros f_y dans_f_y_V temp; elim temp; clear temp.
intros u eg_y inmono_u.
replace (word (word_inv (second y))) with (second y).
assumption.
replace (second y) with (word u).
apply refl_equal.
rewrite eg_y.
apply refl_equal.
apply dans_map.
assumption.
intros dans_x.
elimtype (exists y : Elt, dans y X /\ x = f_X_d y).
intros y temp; elim temp; clear temp.
unfold f_X_d in |- *.
intros dans_y_X eg_x_f_y.
exists (cons y nil).
apply inmonoid_cons.
trivial.
unfold P in |- *.
auto.
exists y.
auto.
exists nil.
trivial.
assumption.
apply dans_map.
assumption.
auto.
Qed.

Lemma X_P_wd_wa_d : P_automata X P wd wa d.
red in |- *.