(* Restrict a map by a set of keys *)
(* Fold over the set we are restricting with for better induction. *)
Definition restrict {A} (m : M.t A) s :=
S.fold (fun k m' => match M.find k m with
| Some v => M.add k v m'
| None => m'
end) s (@M.empty _).

Lemma restrict_m {A} : forall s s',
S.Equal s s' ->
forall k k' : M.t A, M.Equal k k' -> M.Equal (restrict k s) (restrict k' s').