intros i u v u' v' H H0; try assumption.
apply transitive with (plus (cons u i) (cons i v)).
apply symetrique.
apply Chasles.
apply transitive with (plus (cons i u') (cons v' i)).
apply compatible.
apply R_permute.
apply symetrique; auto.
apply symetrique; auto.
apply transitive with (plus (cons v' i) (cons i u')).
apply plus_sym.
apply Chasles.
Qed.

(* Specifications: *)
Lemma calcul2 :
forall a b c d : AV,
R (plus (plus a (plus b c)) d) (plus (plus a (plus b d)) c).