Section Sound_context.
Variable the_rule : Basic_rule.
Let R := Rule the_rule.
Let Rlifts : R_lift R := R_lifts the_rule.
Hint Resolve Rlifts: pts.
Hypothesis
incl_R_le_type :
forall (e : env) (x y : term), ctxt R e x y -> le_type e y x.
simple induction 1; intros; auto with pts.
apply le_type_trans with y0; auto with pts.
Qed.

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.