intros.
elim CR with e x' y'; intros.
apply conv_hn_inv_trans with x0; auto with arith pts.
apply conv_hn_inv_sym; auto with arith pts.
change (conv R e x' y') in |- *.
apply conv_trans with x; auto with arith pts.
apply conv_trans with y; auto with arith pts.
Qed.

Lemma inv_conv_conv :
forall (e : env) (x y : term), conv_hn_inv e x y -> conv R e x y.