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Bitops/extra-defs

Nth-slice2

Extract the nth 2-bit slice of the integer x.

Signature
(nth-slice2 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-slice2$inline

(defun acl2::nth-slice2$inline (n x)
  (declare (xargs :guard (and (natp n) (integerp x))))
  (let ((__function__ 'nth-slice2))
    (declare (ignorable __function__))
    (mbe :logic (logand (ash (ifix x) (* (nfix n) -2))
                        (1- (expt 2 2)))
         :exec (the (unsigned-byte 2)
                    (logand (ash x (the (integer * 0) (* n -2)))
                            3)))))

Theorem: natp-of-nth-slice2

(defthm acl2::natp-of-nth-slice2
  (b* ((slice (acl2::nth-slice2$inline n x)))
    (natp slice))
  :rule-classes :type-prescription)

Theorem: nat-equiv-implies-equal-nth-slice2-1

(defthm nat-equiv-implies-equal-nth-slice2-1
  (implies (nat-equiv n n-equiv)
           (equal (nth-slice2 n x)
                  (nth-slice2 n-equiv x)))
  :rule-classes (:congruence))

Theorem: int-equiv-implies-equal-nth-slice2-2

(defthm int-equiv-implies-equal-nth-slice2-2
  (implies (int-equiv x x-equiv)
           (equal (nth-slice2 n x)
                  (nth-slice2 n x-equiv)))
  :rule-classes (:congruence))

Theorem: unsigned-byte-p-2-of-nth-slice2

(defthm unsigned-byte-p-2-of-nth-slice2
  (unsigned-byte-p 2 (nth-slice2 n x)))

Theorem: nth-slice2-is-nth-slice

(defthm nth-slice2-is-nth-slice
  (equal (nth-slice2 n x)
         (nth-slice 2 n x)))