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4vec-operations

4vec-remainder

Integer remainder as in rem for 4vecs.

Signature
(4vec-remainder x y) → remainder
Arguments: x — Guard (4vec-p x).; y — Guard (4vec-p y).
Returns: remainder — Type (4vec-p remainder).

This is a fairly conservative definition in the style of the Verilog semantics: if either input has X or Z bits, or if you try to divide by zero, then the result is all X bits. Otherwise, we produce the integer remainder of \frac{x}{y}, with rounding toward zero as in rem.

Definitions and Theorems

Function: 4vec-remainder

(defun 4vec-remainder (x y)
  (declare (xargs :guard (and (4vec-p x) (4vec-p y))))
  (let ((__function__ '4vec-remainder))
    (declare (ignorable __function__))
    (if (and (2vec-p x)
             (2vec-p y)
             (not (eql (2vec->val y) 0)))
        (2vec (rem (the integer (2vec->val x))
                   (the integer (2vec->val y))))
      (4vec-x))))

Theorem: 4vec-p-of-4vec-remainder

(defthm 4vec-p-of-4vec-remainder
  (b* ((remainder (4vec-remainder x y)))
    (4vec-p remainder))
  :rule-classes :rewrite)

Theorem: 4vec-remainder-of-2vecx-fix-x

(defthm 4vec-remainder-of-2vecx-fix-x
  (equal (4vec-remainder (2vecx-fix x) y)
         (4vec-remainder x y)))

Theorem: 4vec-remainder-2vecx-equiv-congruence-on-x

(defthm 4vec-remainder-2vecx-equiv-congruence-on-x
  (implies (2vecx-equiv x x-equiv)
           (equal (4vec-remainder x y)
                  (4vec-remainder x-equiv y)))
  :rule-classes :congruence)

Theorem: 4vec-remainder-of-2vecx-fix-y

(defthm 4vec-remainder-of-2vecx-fix-y
  (equal (4vec-remainder x (2vecx-fix y))
         (4vec-remainder x y)))

Theorem: 4vec-remainder-2vecx-equiv-congruence-on-y

(defthm 4vec-remainder-2vecx-equiv-congruence-on-y
  (implies (2vecx-equiv y y-equiv)
           (equal (4vec-remainder x y)
                  (4vec-remainder x y-equiv)))
  :rule-classes :congruence)