(** Properties of [remove] *)

(* Specifications: *)
Lemma remove_mem_3:
mem y (remove x s)=true -> mem y s=true.
Lemma remove_equal:
mem x s=false -> equal (remove x s) s=true.
Lemma add_remove:
mem x s=true -> equal (add x (remove x s)) s=true.
Lemma remove_add:
mem x s=false -> equal (remove x (add x s)) s=true.
Lemma remove_mem_3:
mem y (remove x s)=true -> mem y s=true.
Lemma remove_equal:
mem x s=false -> equal (remove x s) s=true.
Lemma add_remove:
mem x s=true -> equal (add x (remove x s)) s=true.
Lemma remove_add:
mem x s=false -> equal (remove x (add x s)) s=true.