auto.
Qed.
(**************************************************************)
Inductive Derive_P_A_3 : Word * Conf -> Word * Conf -> Prop :=
| Derive3_X :
forall (w u s : Word) (x : Elt),
dans x X ->
Derive_P_A_3 (s, (cons x w, cons x u)) (Append s (cons x nil), (w, u))
| Derive3_V :
forall (v w u s : Word) (x : Elt),
dans (couple x (word u)) R ->
Derive_P_A_3 (s, (cons x w, v)) (s, (Append u w, v)).
Definition Derivestar_P_A_3 := Rstar (Word * Conf) Derive_P_A_3.

Lemma Conserve_App_s_u :
forall s1 s2 u1 u2 v1 v2 : Word,
Derive_P_A_3 (s1, (u1, v1)) (s2, (u2, v2)) -> Append s1 v1 = Append s2 v2.