Definition pures_sub (phi phi' : rmap) :=
  forall adr,
  match resource_at phi adr with
    | PURE k pp => resource_at phi' adr
                 = PURE k (preds_fmap (approx (level phi')) (approx (level phi')) pp)
    | _ => True
  end.

(* Specifications: *)
Lemma pures_sub_trans phi1 phi2 phi3 : (level phi3 <= level phi2)%nat -> pures_sub phi1 phi2 -> pures_sub phi2 phi3 -> pures_sub phi1 phi3.
Lemma pures_sub_refl phi : pures_sub phi phi.