intros x y x' y' s s' Der.
change
((fun a a' : Word * Conf =>
inmonoid X (fst a) ->
Derivegstar X R (Append (fst a) (fst (snd a)))
(Append (fst a') (fst (snd a')))) (s, (x, y))
(s', (x', y'))) in |- *.
unfold Derivegstar in |- *.
elim Der; simpl in |- *.
intros w u s0 x0 dans_x0_X inmon_s.
replace (Append (Append s0 (cons x0 nil)) w) with
(Append s0 (Append (cons x0 nil) w)).
apply Rstar_reflexive.
auto.
intros v w u s0 x0 dans_couple_x0_u_R inmon_s.
apply Rstar_R with (Append s0 (Append u w)).
apply Deriveg_imp_Deriveg_App_2.
assumption.
apply Deriveg1.
assumption.
apply Rstar_reflexive.
Qed.

Lemma Derive_P_A_3_conserve_inmonoid_s :
forall x y x' y' s s' : Word,
Derive_P_A_3 (s, (x, y)) (s', (x', y')) -> inmonoid X s -> inmonoid X s'.