(** Useful tactic for simplifying expressions like [In y (add x e (remove z m))] *)

Ltac map_iff :=
 repeat (progress (
  rewrite add_mapsto_iff || rewrite add_in_iff ||
  rewrite remove_mapsto_iff || rewrite remove_in_iff ||
  rewrite empty_mapsto_iff || rewrite empty_in_iff ||
  rewrite map_mapsto_iff || rewrite map_in_iff ||
  rewrite mapi_in_iff)).

Lemma add_b : forall m x y e, mem y (add x e m) = eqb x y || mem y m.