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Bitops/extra-defs

Nth-slice4

Extract the nth 4-bit slice of the integer x.

Signature
(nth-slice4 n x) → slice
Arguments: n — Guard (natp n).; x — Guard (integerp x).
Returns: slice — Type (natp slice).

We leave this enabled; we would usually not expect to try to reason about it.

Definitions and Theorems

Function: nth-slice4$inline

(defun acl2::nth-slice4$inline (n x)
  (declare (xargs :guard (and (natp n) (integerp x))))
  (let ((__function__ 'nth-slice4))
    (declare (ignorable __function__))
    (mbe :logic (logand (ash (ifix x) (* (nfix n) -4))
                        (1- (expt 2 4)))
         :exec (the (unsigned-byte 4)
                    (logand (ash x (the (integer * 0) (* n -4)))
                            15)))))

Theorem: natp-of-nth-slice4

(defthm acl2::natp-of-nth-slice4
  (b* ((slice (acl2::nth-slice4$inline n x)))
    (natp slice))
  :rule-classes :type-prescription)

Theorem: nat-equiv-implies-equal-nth-slice4-1

(defthm nat-equiv-implies-equal-nth-slice4-1
  (implies (nat-equiv n n-equiv)
           (equal (nth-slice4 n x)
                  (nth-slice4 n-equiv x)))
  :rule-classes (:congruence))

Theorem: int-equiv-implies-equal-nth-slice4-2

(defthm int-equiv-implies-equal-nth-slice4-2
  (implies (int-equiv x x-equiv)
           (equal (nth-slice4 n x)
                  (nth-slice4 n x-equiv)))
  :rule-classes (:congruence))

Theorem: unsigned-byte-p-4-of-nth-slice4

(defthm unsigned-byte-p-4-of-nth-slice4
  (unsigned-byte-p 4 (nth-slice4 n x)))

Theorem: nth-slice4-is-nth-slice

(defthm nth-slice4-is-nth-slice
  (equal (nth-slice4 n x)
         (nth-slice 4 n x)))