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).
simple induction t; simpl in |- *; intros; auto with pts.
red in |- *; red in |- *.
elim (lt_eq_lt_dec n0 n); intros; auto with pts.
elim a; intros; auto with pts.
apply rt_step.
apply iter_R_lift with g; auto with pts.
Qed.

Lemma red_red_subst :
forall (e : env) (x y t u T : term),
red R (Ax T :: e) t u -> red R e x y -> red R e (subst x t) (subst y u).