simple induction 2; intros.
repeat rewrite lift0; trivial with arith pts.
rewrite simpl_lift.
rewrite (simpl_lift u).
unfold lift at 1 3 in |- *.
apply H with x e0 e0; auto with arith pts.
Qed.

Lemma sub_sort_env_indep :
forall (R : Subtyping_rule) (e : env) (s1 s2 : sort),
Rule (Le_type R) e (Srt s1) (Srt s2) ->
forall f : env, Rule (Le_type R) f (Srt s1) (Srt s2).