(* Some None represents the empty permission. None is used for permissions that conflict/race. *)
Definition perm_union (p1 p2 : option permission) : option (option permission) :=
match p1,p2 with
| None, _ => Some p2
| _, None => Some p1
| Some p1', Some p2' =>
match p1', p2' with
| Freeable, _ => None
| _, Freeable => None
| Nonempty, _ => Some p2
| _, Nonempty => Some p1
| Writable, _ => None
| _, Writable => None
| Readable, Readable => Some (Some Readable)
end
end.

Lemma perm_union_comm :
forall p1 p2,
perm_union p1 p2 = perm_union p2 p1.