Global Notation "a # b" := (Maps.PMap.get b a) (at level 1).

Lemma permission_at_alloc_3: forall m m' lo sz nb ofs (Halloc: Mem.alloc m lo sz = (m', nb)) (Hofs: (ofs < lo \/ ofs >= sz)%Z), forall k, permissions.permission_at m' nb ofs k = None.