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Sparseint

Sparseint-bitcount-from

Count the 1 bits in a sparseint if positive, 0 bits if negative, starting from offset.

Signature
(sparseint-bitcount-from offset x) → count
Arguments: offset — Guard (natp offset).; x — Guard (sparseint-p x).
Returns: count — Type (natp count).

Definitions and Theorems

Function: sparseint-bitcount-from

(defun sparseint-bitcount-from (offset x)
  (declare (xargs :guard (and (natp offset) (sparseint-p x))))
  (let ((__function__ 'sparseint-bitcount-from))
    (declare (ignorable __function__))
    (sparseint$-bitcount-rec offset (bool->bit (sparseint-< x 0))
                             (sparseint-fix x))))

Theorem: natp-of-sparseint-bitcount-from

(defthm natp-of-sparseint-bitcount-from
  (b* ((count (sparseint-bitcount-from offset x)))
    (natp count))
  :rule-classes :type-prescription)

Theorem: sparseint-bitcount-from-correct

(defthm sparseint-bitcount-from-correct
  (b* ((common-lisp::?count (sparseint-bitcount-from offset x)))
    (equal count
           (logcount (logtail offset (sparseint-val x))))))

Theorem: sparseint-bitcount-from-of-nfix-offset

(defthm sparseint-bitcount-from-of-nfix-offset
  (equal (sparseint-bitcount-from (nfix offset)
                                  x)
         (sparseint-bitcount-from offset x)))

Theorem: sparseint-bitcount-from-nat-equiv-congruence-on-offset

(defthm sparseint-bitcount-from-nat-equiv-congruence-on-offset
  (implies (nat-equiv offset offset-equiv)
           (equal (sparseint-bitcount-from offset x)
                  (sparseint-bitcount-from offset-equiv x)))
  :rule-classes :congruence)

Theorem: sparseint-bitcount-from-of-sparseint-fix-x

(defthm sparseint-bitcount-from-of-sparseint-fix-x
  (equal (sparseint-bitcount-from offset (sparseint-fix x))
         (sparseint-bitcount-from offset x)))

Theorem: sparseint-bitcount-from-sparseint-equiv-congruence-on-x

(defthm sparseint-bitcount-from-sparseint-equiv-congruence-on-x
  (implies (sparseint-equiv x x-equiv)
           (equal (sparseint-bitcount-from offset x)
                  (sparseint-bitcount-from offset x-equiv)))
  :rule-classes :congruence)