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• Arithmetic-1

Basic-product-normalization

Trivial normalization and cancellation rules for products.

Definitions and Theorems

Theorem: commutativity-2-of-*

(defthm commutativity-2-of-*
  (equal (* x (* y z)) (* y (* x z))))

Theorem: functional-self-inversion-of-/

(defthm functional-self-inversion-of-/
  (equal (/ (/ x)) (fix x)))

Theorem: distributivity-of-/-over-*

(defthm distributivity-of-/-over-*
  (equal (/ (* x y)) (* (/ x) (/ y))))

Theorem: /-cancellation-on-right

(defthm /-cancellation-on-right
  (implies (and (fc (acl2-numberp x))
                (fc (not (equal x 0))))
           (equal (* x y (/ x)) (fix y))))

Theorem: /-cancellation-on-left

(defthm /-cancellation-on-left
  (implies (and (fc (acl2-numberp x))
                (fc (not (equal 0 x))))
           (equal (* x (/ x) y) (fix y))))

Theorem: right-cancellation-for-*

(defthm right-cancellation-for-*
  (equal (equal (* x z) (* y z))
         (or (equal 0 (fix z))
             (equal (fix x) (fix y)))))

Theorem: left-cancellation-for-*

(defthm left-cancellation-for-*
  (equal (equal (* z x) (* z y))
         (or (equal 0 (fix z))
             (equal (fix x) (fix y)))))

Theorem: zero-is-only-zero-divisor

(defthm zero-is-only-zero-divisor
  (equal (equal (* x y) 0)
         (or (equal (fix x) 0)
             (equal (fix y) 0))))

Theorem: equal-*-x-y-x

(defthm equal-*-x-y-x
  (equal (equal (* x y) x)
         (or (equal x 0)
             (and (equal y 1) (acl2-numberp x)))))

Theorem: equal-*-x-y-y

(defthm equal-*-x-y-y
  (equal (equal (* x y) y)
         (or (equal y 0)
             (and (equal x 1) (acl2-numberp y)))))

Theorem: equal-/

(defthm equal-/
  (implies (and (fc (acl2-numberp x))
                (fc (not (equal 0 x))))
           (equal (equal (/ x) y)
                  (equal 1 (* x y)))))

Theorem: numerator-nonzero-forward

(defthm numerator-nonzero-forward
 (implies (not (equal (numerator r) 0))
          (and (not (equal r 0))
               (acl2-numberp r)))
 :rule-classes ((:forward-chaining :trigger-terms ((numerator r)))))

Theorem: uniqueness-of-*-inverses

(defthm uniqueness-of-*-inverses
  (equal (equal (* x y) 1)
         (and (not (equal (fix x) 0))
              (equal y (/ x)))))

Theorem: equal-/-/

(defthm equal-/-/
  (equal (equal (/ a) (/ b))
         (equal (fix a) (fix b))))

Theorem: equal-*-/-1

(defthm equal-*-/-1
  (implies (and (acl2-numberp x) (not (equal x 0)))
           (equal (equal (* (/ x) y) z)
                  (and (acl2-numberp z)
                       (equal (fix y) (* x z))))))

Theorem: equal-*-/-2

(defthm equal-*-/-2
  (implies (and (acl2-numberp x) (not (equal x 0)))
           (equal (equal (* y (/ x)) z)
                  (and (acl2-numberp z)
                       (equal (fix y) (* z x))))))

Theorem: fold-consts-in-*

(defthm fold-consts-in-*
  (implies (and (syntaxp (quotep x))
                (syntaxp (quotep y)))
           (equal (* x (* y z)) (* (* x y) z))))

Theorem: times-zero

(defthm times-zero (equal (* 0 x) 0))