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Normalize-<-/-to-*-3

Rewrite: Replace x < y/z and x > y/z with x*z < y or x*z > y, depending on the sign of z.

Definitions and Theorems

Theorem: normalize-<-/-to-*-3

(defthm normalize-<-/-to-*-3
  (implies (and (real/rationalp x)
                (real/rationalp y)
                (real/rationalp z)
                (not (equal z 0)))
           (and (equal (< x (* y (/ z)))
                       (if (< z 0)
                           (< y (* x z))
                         (< (* x z) y)))
                (equal (< x (* (/ z) y))
                       (if (< z 0)
                           (< y (* x z))
                         (< (* x z) y)))
                (equal (< (* y (/ z)) x)
                       (if (< z 0)
                           (< (* x z) y)
                         (< y (* x z))))
                (equal (< (* (/ z) y) x)
                       (if (< z 0)
                           (< (* x z) y)
                         (< y (* x z)))))))