(* 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 _)).

(** Its proof of correctness under the assumption that f is injective *)
Lemma projectMap_tree:
forall (f:memren) pmap
(Hf_injective: forall b1 b1' b2,
f b1 = Some b2 ->
f b1' = Some b2 ->
b1 = b1') b b'
(Hf: f b = Some b'),
Maps.PTree.get b pmap.2 = Maps.PTree.get b' (projectMap f pmap).2.