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.

(* Specifications: *)
(* amenes a disparaitre... *)
Lemma red_lift : R_lift (red R).
Proof R_rt_lift (ctxt R) (ctxt_lift R R_lifts).
Lemma conv_lift : R_lift (conv R).
Proof R_rst_lift (ctxt R) (ctxt_lift R R_lifts).
Lemma red_subst : R_subst (red R).
Proof R_rt_subst (ctxt R) (ctxt_subst R R_substs).
Lemma conv_subst : R_subst (conv R).
Proof R_rst_subst (ctxt R) (ctxt_subst R R_substs).
Lemma red_subst_l :
forall (g : env) (u v : term),
ctxt R g u v ->
forall (t : term) (n : nat) (e : env),
trunc n e g -> red R e (subst_rec u t n) (subst_rec v t n).