(* 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 adj_restrict : forall g s i j,
S.In i (adj (restrict g s) j) <-> S.In i (adj g j) /\ S.In j s.