Section Prop_R_lift.
Hypothesis R_lifts : R_lift R.
Hypothesis R_lifts' : R_lift R'.
red in |- *.
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.
unfold R_lift, R_rt in |- *.
simple induction 1; intros; auto with pts.
apply rt_step.
apply R_lifts with (2 := H1); auto with pts.
apply rt_trans with (lift_rec 1 y k); auto with pts.
Qed.

Lemma R_rst_lift : R_lift (R_rst R).