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Bitops/merge

Merge-4-u4s

Concatenate 4 nibbles together to form an 16-bit result.

Signature
(merge-4-u4s a3 a2 a1 a0) → result
Returns: result — Type (natp result).

Definitions and Theorems

Function: merge-4-u4s$inline

(defun acl2::merge-4-u4s$inline (a3 a2 a1 a0)
  (declare (type (unsigned-byte 4) a3 a2 a1 a0))
  (declare (xargs :guard t))
  (let ((__function__ 'merge-4-u4s))
    (declare (ignorable __function__))
    (mbe :logic
         (logapp* 4 (nfix a0)
                  (nfix a1)
                  (nfix a2)
                  (nfix a3)
                  0)
         :exec
         (b* ((ans a0)
              (ans (the (unsigned-byte 16)
                        (logior (the (unsigned-byte 16)
                                     (ash a1 (* 1 4)))
                                (the (unsigned-byte 16) ans))))
              (ans (the (unsigned-byte 16)
                        (logior (the (unsigned-byte 16)
                                     (ash a2 (* 2 4)))
                                (the (unsigned-byte 16) ans)))))
           (the (unsigned-byte 16)
                (logior (the (unsigned-byte 16)
                             (ash a3 (* 3 4)))
                        (the (unsigned-byte 16) ans)))))))

Theorem: natp-of-merge-4-u4s

(defthm acl2::natp-of-merge-4-u4s
  (b* ((result (acl2::merge-4-u4s$inline a3 a2 a1 a0)))
    (natp result))
  :rule-classes :type-prescription)

Theorem: unsigned-byte-p-16-of-merge-4-u4s

(defthm unsigned-byte-p-16-of-merge-4-u4s
  (unsigned-byte-p 16 (merge-4-u4s a3 a2 a1 a0))
  :rule-classes
  ((:rewrite
        :corollary
        (implies (>= (nfix n) 16)
                 (unsigned-byte-p n (merge-4-u4s a3 a2 a1 a0)))
        :hints (("Goal" :in-theory (disable unsigned-byte-p))))))

Theorem: merge-4-u4s-is-merge-unsigneds

(defthm merge-4-u4s-is-merge-unsigneds
  (equal (merge-4-u4s a3 a2 a1 a0)
         (merge-unsigneds 4
                          (list (nfix a3)
                                (nfix a2)
                                (nfix a1)
                                (nfix a0)))))

Basic nat-equiv congruences.

Theorem: nat-equiv-implies-equal-merge-4-u4s-4

(defthm nat-equiv-implies-equal-merge-4-u4s-4
  (implies (nat-equiv a0 a0-equiv)
           (equal (merge-4-u4s a3 a2 a1 a0)
                  (merge-4-u4s a3 a2 a1 a0-equiv)))
  :rule-classes (:congruence))

Theorem: nat-equiv-implies-equal-merge-4-u4s-3

(defthm nat-equiv-implies-equal-merge-4-u4s-3
  (implies (nat-equiv a1 a1-equiv)
           (equal (merge-4-u4s a3 a2 a1 a0)
                  (merge-4-u4s a3 a2 a1-equiv a0)))
  :rule-classes (:congruence))

Theorem: nat-equiv-implies-equal-merge-4-u4s-2

(defthm nat-equiv-implies-equal-merge-4-u4s-2
  (implies (nat-equiv a2 a2-equiv)
           (equal (merge-4-u4s a3 a2 a1 a0)
                  (merge-4-u4s a3 a2-equiv a1 a0)))
  :rule-classes (:congruence))

Theorem: nat-equiv-implies-equal-merge-4-u4s-1

(defthm nat-equiv-implies-equal-merge-4-u4s-1
  (implies (nat-equiv a3 a3-equiv)
           (equal (merge-4-u4s a3 a2 a1 a0)
                  (merge-4-u4s a3-equiv a2 a1 a0)))
  :rule-classes (:congruence))