(* Blocks that are not mapped by f are set to empty permission. This makes the invariant preservation easier. *)
Definition projectMap (f : memren) (pmap : access_map) : access_map :=
(pmap#1, Maps.PTree.fold (fun acc b bperm =>
match f b with
| Some b' =>
Maps.PTree.set b' bperm acc
| None =>
acc end)
pmap.2 (Maps.PTree.empty _)).

Lemma projectMap_tree_unchanged :
forall f (pmap : access_map) (b2 : positive)
(Hb1: ~ (exists b1 : block, f b1 = Some b2)),
Maps.PTree.get b2 (projectMap f pmap).2 = None.