intros x y Derivestar_P_A_x_y_nil_nil.
elimtype (exists s2 : Word, Derivestar_P_A_3 (nil, (x, y)) (s2, (nil, nil))).
intros s2 Derivestar_P_A_3_x_y.
apply Derivegstar_Derivestar with X.
replace x with (Append nil x).
replace y with (Append s2 nil).
apply Derivestar_P_A_3_imp_Derivegstar with y nil.
assumption.
trivial.
replace y with (Append nil y).
apply sym_equal.
apply Derisvestar_P_A_3_conserve with x nil.
assumption.
trivial.
trivial.
apply Derivestar_P_A_2_imp_Derivestar_P_A_3.
auto.
Qed.
Hint Resolve Derivestar_P_A_imp_Derivestar.
(******************************************)
(*equivalence de G et de l'automate a pile*)
(******************************************)

Theorem equiv_APD_Gram : l_egal (LA X wd wa d) (LG X V R S').
red in |- *.