intros x y x' y' s s' Der.
change
((fun a a' : Word * Conf => inmonoid X (fst a) -> inmonoid X (fst a'))
(s, (x, y)) (s', (x', y'))) in |- *.
elim Der; simpl in |- *.
intros.
apply inmonoid_Append; auto.
auto.
Qed.

(* Specifications: *)
Lemma Derivestar_P_A_3_imp_Derivegstar :
forall x y x' y' s s' : Word,
Derivestar_P_A_3 (s, (x, y)) (s', (x', y')) ->
inmonoid X s -> Derivegstar X R (Append s x) (Append s' x').