unfold Derivestar_P_A_2 in |- *.
simple induction x.
intro.
apply Rstar_reflexive.
intros x0 w Hyp inmon.
apply Rstar_R with (w, w).
apply Derive_X.
apply inmonoid_cons_inv2 with w; assumption.
apply Hyp.
apply inmonoid_cons_inv with x0; assumption.
Qed.
Hint Resolve Derivestar_P_A_2_x.

Lemma Derivegstar_imp_Derivestar_P_A_2 :
forall x y : Word,
Derivegstar X R x y -> inmonoid X y -> Derivestar_P_A_2 (x, y) (nil, nil).