Definition permission_at (m : mem) (b : block) (ofs : Z) (k : perm_kind) :=
    Maps.PMap.get b (Mem.mem_access m) ofs k.(** Coherence between permissions. This is used for the relation between data
  and lock permissions*)
  (** Note: p1 should be data permission and p2 lock permission*)
  Definition perm_coh (p1 p2 : option permission) :=
    match p1 with
    | Some Freeable | Some Writable | Some Readable =>
                                      match p2 with
                                      | None => True
                                      | _ => False
                                      end
    | Some Nonempty | None =>
                      match p2 with
                      | Some Freeable => False
                      | _ => True
                      end
    end.

Lemma po_join_sub_lock: forall r1 r2 , join_sub r2 r1 -> Mem.perm_order'' (perm_of_res_lock r1) (perm_of_res_lock r2).