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Bitops/merge

Merge-2-u2s

Concatenate 2 2-bit numbers together to form an 4-bit result.

Signature
(merge-2-u2s a1 a0) → result
Returns: result — Type (natp result).

Definitions and Theorems

Function: merge-2-u2s$inline

(defun acl2::merge-2-u2s$inline (a1 a0)
  (declare (type (unsigned-byte 2) a1 a0))
  (declare (xargs :guard t))
  (let ((__function__ 'merge-2-u2s))
    (declare (ignorable __function__))
    (mbe :logic (logapp* 2 (nfix a0) (nfix a1) 0)
         :exec
         (b* ((ans a0))
           (the (unsigned-byte 4)
                (logior (the (unsigned-byte 4) (ash a1 (* 1 2)))
                        (the (unsigned-byte 4) ans)))))))

Theorem: natp-of-merge-2-u2s

(defthm acl2::natp-of-merge-2-u2s
  (b* ((result (acl2::merge-2-u2s$inline a1 a0)))
    (natp result))
  :rule-classes :type-prescription)

Theorem: unsigned-byte-p-4-of-merge-2-u2s

(defthm unsigned-byte-p-4-of-merge-2-u2s
  (unsigned-byte-p 4 (merge-2-u2s a1 a0))
  :rule-classes
  ((:rewrite
        :corollary (implies (>= (nfix n) 4)
                            (unsigned-byte-p n (merge-2-u2s a1 a0)))
        :hints (("Goal" :in-theory (disable unsigned-byte-p))))))

Theorem: merge-2-u2s-is-merge-unsigneds

(defthm merge-2-u2s-is-merge-unsigneds
  (equal (merge-2-u2s a1 a0)
         (merge-unsigneds 2 (list (nfix a1) (nfix a0)))))

Basic nat-equiv congruences.

Theorem: nat-equiv-implies-equal-merge-2-u2s-2

(defthm nat-equiv-implies-equal-merge-2-u2s-2
  (implies (nat-equiv a0 a0-equiv)
           (equal (merge-2-u2s a1 a0)
                  (merge-2-u2s a1 a0-equiv)))
  :rule-classes (:congruence))

Theorem: nat-equiv-implies-equal-merge-2-u2s-1

(defthm nat-equiv-implies-equal-merge-2-u2s-1
  (implies (nat-equiv a1 a1-equiv)
           (equal (merge-2-u2s a1 a0)
                  (merge-2-u2s a1-equiv a0)))
  :rule-classes (:congruence))