simple induction 1; intros; auto with pts.
apply rst_step; auto with pts.
apply R_stable with R'0 e; auto with pts.
apply rst_sym; auto with pts.
apply rst_trans with y0; auto with pts.
Qed.
End Prop_R_stable_env.
End Properties.
Hint Resolve ctxt_lift R_rt_lift R_rst_lift ctxt_subst R_rt_subst
R_rst_subst: pts.
Section Prop_R_lift_subst.
Variable R : red_rule.
Hypothesis R_lifts : R_lift R.
Hypothesis R_substs : R_subst R.

Lemma conv_lift : R_lift (conv R).
Proof R_rst_lift (ctxt R) (ctxt_lift R R_lifts).