Definition REACH_closed m (X: Values.block -> bool) : Prop :=
  (forall b, REACH m X b = true -> X b = true).Definition mapped (j:meminj) b : bool :=
  match j b with None => false | Some _ => true end.

Lemma restrict_mapped_closed: forall j m X (RC: REACH_closed m (mapped j)) (RX: REACH_closed m X), REACH_closed m (mapped (restrict j X)).