Section BasicProperties.
(** Some old specifications written with boolean equalities. *)
Variable s s' s'': t.
Variable x y z : elt.
(** properties of [mem] *)
(** Properties of [equal] *)
(* Properties of [subset] *)
(** Properties of [choose] *)
(** Properties of [add] *)
(** Properties of [remove] *)
(** Properties of [is_empty] *)
(** Properties of [singleton] *)
(** Properties of [union] *)
(* characterisation of [union] via [subset] *)
(** Properties of [inter] *)
(* characterisation of [union] via [subset] *)
(** Properties of [diff] *)

Lemma remove_inter_singleton:
equal (remove x s) (diff s (singleton x))=true.