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• Arithmetic-1

Inequalities-of-sums

Basic normalization to move negative terms to the other side of an inequality.

Definitions and Theorems

Theorem: <-0-minus

(defthm <-0-minus
  (equal (< 0 (- x)) (< x 0)))

Theorem: <-minus-zero

(defthm <-minus-zero
  (equal (< (- x) 0) (< 0 x)))

Theorem: <-0-+-negative-1

(defthm <-0-+-negative-1
  (equal (< 0 (+ (- y) x)) (< y x)))

Theorem: <-0-+-negative-2

(defthm <-0-+-negative-2
  (equal (< 0 (+ x (- y))) (< y x)))

Theorem: <-+-negative-0-1

(defthm <-+-negative-0-1
  (equal (< (+ (- y) x) 0) (< x y)))

Theorem: <-+-negative-0-2

(defthm <-+-negative-0-2
  (equal (< (+ x (- y)) 0) (< x y)))