(** 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 remove_o : forall m x y, find y (remove x m) = if eq_dec x y then None else find y m.