Section Prop_R_lift_subst.
Variable R : red_rule.
Hypothesis R_lifts : R_lift R.
Hypothesis R_substs : R_subst R.
Proof R_rt_lift (ctxt R) (ctxt_lift R R_lifts).
Proof R_rst_lift (ctxt R) (ctxt_lift R R_lifts).
Proof R_rt_subst (ctxt R) (ctxt_subst R R_substs).
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).