simple induction 1; intros; auto with pts.
apply le_type_trans with y0; auto with pts.
Qed.

(* Specifications: *)
Theorem red_in_env :
forall (e : env) (u v t T : term),
typ the_PTS (Ax u :: e) t T ->
forall s : sort,
typ the_PTS e v (Srt s) -> red R e u v -> typ the_PTS (Ax v :: e) t T.