(omega)


(o^ (omega) 2)

(o+ (o* (o^ (omega) 2) 4) 7)

(thm (= (o* 2 (omega)) (omega)))

(thm (implies (and (o-p a)
                   (o-p b)
                   (o< b (o^ (omega) a)))
              (equal (o+ b (o^ (omega) a))
                     (o^ (omega) a))))


(o^ (o+ (o* (omega) 3) 5) 12)

(in-theory (enable (omega)))

(thm (implies (and (o-p a)
                   (o-p b))
              (iff (o< (o+ (o^ 5 (o^ (omega) (omega))) a)
                       (o+ (o^ (o+ (o^ (omega) 12) (o^ (omega) 3))
                               (o^ (o+ (o* (o^ (omega) 15) 2) (o* (o^ (omega) 5) 5)) (omega))) 
                           b))
                   (o< a b))))