simple induction 1; simpl in |- *; auto with pts; intros.
apply ctx_rule.
apply R_lifts with (2 := H1); auto with pts.
apply ctx_abs_r.
apply H1.
replace (Ax (lift_rec 1 N k)) with (lift_decl 1 (Ax N) k); auto with pts.
apply ctx_prod_r.
apply H1.
replace (Ax (lift_rec 1 M1 k)) with (lift_decl 1 (Ax M1) k); auto with pts.
Qed.

Lemma R_rt_lift : R_lift (R_rt R).
unfold R_lift, R_rt in |- *.