Notation typ2 := (typ the_PTS) (only parsing).Notation wf_type2 := (wf_type the_PTS) (only parsing).Definition rule_sound :=
    forall (e : env) (x y : term),
    R e x y -> forall T : term, typ the_PTS e x T -> typ the_PTS e y T.

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.