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.

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