intros u v w t s H H0; try assumption.
apply transitive with (plus (cons v u) (cons u t)).
apply symetrique; auto.
apply Chasles.
apply transitive with (plus (cons w u) (cons u s)).
apply compatible.
apply R_permute.
try exact H.
try exact H0.
apply Chasles.
Qed.

Lemma regulier :
forall a b c d : AV, R a c -> R (plus a b) (plus c d) -> R b d.