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Unsigned-saturate

Unsigned-saturate-fn

Logical definition of unsigned-saturate, and also its executable implementation in the general case.

Signature
(unsigned-saturate-fn n x) → saturated
Arguments: n — Guard (posp n).; x — Guard (integerp x).
Returns: saturated — Type (natp saturated).

Definitions and Theorems

Function: unsigned-saturate-fn

(defun unsigned-saturate-fn (n x)
  (declare (xargs :guard (and (posp n) (integerp x))))
  (let ((__function__ 'unsigned-saturate-fn))
    (declare (ignorable __function__))
    (b* ((n (lnfix n))
         (x (lifix x))
         (|2^N| (ash 1 n))
         (max (+ -1 |2^N|))
         ((when (>= x max)) max)
         ((when (<= x 0)) 0))
      x)))

Theorem: natp-of-unsigned-saturate-fn

(defthm acl2::natp-of-unsigned-saturate-fn
  (b* ((saturated (unsigned-saturate-fn n x)))
    (natp saturated))
  :rule-classes :type-prescription)

Theorem: unsigned-byte-p-of-unsigned-saturate-fn

(defthm unsigned-byte-p-of-unsigned-saturate-fn
  (implies (natp n)
           (unsigned-byte-p n (unsigned-saturate-fn n x))))

Theorem: nat-equiv-implies-equal-unsigned-saturate-fn-1

(defthm nat-equiv-implies-equal-unsigned-saturate-fn-1
  (implies (nat-equiv n n-equiv)
           (equal (unsigned-saturate-fn n x)
                  (unsigned-saturate-fn n-equiv x)))
  :rule-classes (:congruence))

Theorem: int-equiv-implies-equal-unsigned-saturate-fn-2

(defthm int-equiv-implies-equal-unsigned-saturate-fn-2
  (implies (int-equiv x x-equiv)
           (equal (unsigned-saturate-fn n x)
                  (unsigned-saturate-fn n x-equiv)))
  :rule-classes (:congruence))