proposition: Lemma exists_mem_2: forall s, exists_ f s=false -> forall x, mem x s=true -> f x=false.
proof: Proof. intros. rewrite for_all_exists in H; auto. rewrite negb_false_iff in H. rewrite <- negb_true_iff. apply for_all_mem_2 with (2:=H); auto. Qed.
imports: MSetProperties Zerob Sumbool Lia DecidableTypeEx