Axiom
  calcul4 :
    forall a b c d : AV,
    R (plus (plus a b) (plus c d)) (plus (plus a c) (plus b d)).

(* Specifications: *)
Lemma double_permute : forall u v w t : V, R (double (cons u v)) (double (cons w t)) -> R (double (cons v u)) (double (cons t w)).
Lemma R_double : forall a b : AV, R a b -> R (double a) (double b).
Lemma double_plus : forall a b : AV, R (double (plus a b)) (plus (double a) (double b)).