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] *)

Lemma inter_add_2: mem x s'=false ->
equal (inter (add x s) s') (inter s s')=true.