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A4vec-operations

A4vec-remainder

Symbolic version of 4vec-remainder.

Signature
(a4vec-remainder x y) → res
Arguments: x — Guard (a4vec-p x).; y — Guard (a4vec-p y).
Returns: res — Type (a4vec-p res).

Definitions and Theorems

Function: a4vec-remainder

(defun a4vec-remainder (x y)
  (declare (xargs :guard (and (a4vec-p x) (a4vec-p y))))
  (let ((__function__ 'a4vec-remainder))
    (declare (ignorable __function__))
    (a4vec-ite (aig-and (a2vec-p x)
                        (a2vec-p y)
                        (aig-not (aig-=-ss (a4vec->upper y)
                                           (aig-sterm nil))))
               (b* (((a4vec x))
                    ((a4vec y))
                    (res (aig-rem-ss x.upper y.upper)))
                 (a4vec res res))
               (a4vec-x))))

Theorem: a4vec-p-of-a4vec-remainder

(defthm a4vec-p-of-a4vec-remainder
  (b* ((res (a4vec-remainder x y)))
    (a4vec-p res))
  :rule-classes :rewrite)

Theorem: a4vec-remainder-correct

(defthm a4vec-remainder-correct
  (equal (a4vec-eval (a4vec-remainder x y) env)
         (4vec-remainder (a4vec-eval x env)
                         (a4vec-eval y env))))

Theorem: a4vec-remainder-of-a4vec-fix-x

(defthm a4vec-remainder-of-a4vec-fix-x
  (equal (a4vec-remainder (a4vec-fix x) y)
         (a4vec-remainder x y)))

Theorem: a4vec-remainder-a4vec-equiv-congruence-on-x

(defthm a4vec-remainder-a4vec-equiv-congruence-on-x
  (implies (a4vec-equiv x x-equiv)
           (equal (a4vec-remainder x y)
                  (a4vec-remainder x-equiv y)))
  :rule-classes :congruence)

Theorem: a4vec-remainder-of-a4vec-fix-y

(defthm a4vec-remainder-of-a4vec-fix-y
  (equal (a4vec-remainder x (a4vec-fix y))
         (a4vec-remainder x y)))

Theorem: a4vec-remainder-a4vec-equiv-congruence-on-y

(defthm a4vec-remainder-a4vec-equiv-congruence-on-y
  (implies (a4vec-equiv y y-equiv)
           (equal (a4vec-remainder x y)
                  (a4vec-remainder x y-equiv)))
  :rule-classes :congruence)