intros. unfold empty_map, Maps.PMap.get.
rewrite Maps.PTree.gempty; reflexivity.
Qed.
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 perm_coh_lower:
forall p1 p2 p3 p4
(Hpu: perm_coh p1 p2)
(Hperm2: Mem.perm_order'' p2 p4)
(Hperm1: Mem.perm_order'' p1 p3),
perm_coh p3 p4.