proposition: Lemma compare_lt_not_gt (a b: A): compare a b = Lt -> ~ compare a b = Gt.
proof: Proof. intros Hlt Hgt. apply compare_lt_iff in Hlt. apply compare_gt_iff in Hgt. now apply lt_not_gt in Hgt. Qed.
imports: Coq(Relations RelationClasses Setoid)