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Bitops/merge

Merge-4-bits

Concatenate 4 bits together to form an 4-bit natural.

Signature
(merge-4-bits a3 a2 a1 a0) → result
Arguments: a3 — Guard (bitp a3).; a2 — Guard (bitp a2).; a1 — Guard (bitp a1).; a0 — Guard (bitp a0).
Returns: result — Type (natp result).

Definitions and Theorems

Function: merge-4-bits$inline

(defun acl2::merge-4-bits$inline (a3 a2 a1 a0)
  (declare (xargs :guard (and (bitp a3)
                              (bitp a2)
                              (bitp a1)
                              (bitp a0))))
  (declare (type bit a3 a2 a1 a0))
  (declare (xargs :split-types t))
  (let ((__function__ 'merge-4-bits))
    (declare (ignorable __function__))
    (mbe :logic
         (logapp* 1 (bfix a0)
                  (bfix a1)
                  (bfix a2)
                  (bfix a3)
                  0)
         :exec
         (b* ((ans a0)
              (ans (the (unsigned-byte 4)
                        (logior (the (unsigned-byte 4) (ash a1 1))
                                (the (unsigned-byte 4) ans))))
              (ans (the (unsigned-byte 4)
                        (logior (the (unsigned-byte 4) (ash a2 2))
                                (the (unsigned-byte 4) ans)))))
           (the (unsigned-byte 4)
                (logior (the (unsigned-byte 4) (ash a3 3))
                        (the (unsigned-byte 4) ans)))))))

Theorem: natp-of-merge-4-bits

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

Theorem: unsigned-byte-p-4-of-merge-4-bits

(defthm unsigned-byte-p-4-of-merge-4-bits
  (unsigned-byte-p 4 (merge-4-bits a3 a2 a1 a0)))

Theorem: merge-4-bits-is-merge-unsigneds

(defthm merge-4-bits-is-merge-unsigneds
  (equal (merge-4-bits a3 a2 a1 a0)
         (merge-unsigneds 1
                          (list (bfix a3)
                                (bfix a2)
                                (bfix a1)
                                (bfix a0)))))

Basic bit-equiv congruences.

Theorem: bit-equiv-implies-equal-merge-4-bits-4

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

Theorem: bit-equiv-implies-equal-merge-4-bits-3

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

Theorem: bit-equiv-implies-equal-merge-4-bits-2

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

Theorem: bit-equiv-implies-equal-merge-4-bits-1

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