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Rebiasing Exponents

Rebiasing Exponents

Definitions and Theorems

Function: rebias

(defun rebias (expo old new)
  (declare (xargs :guard (and (integerp expo)
                              (posp old)
                              (posp new))))
  (+ expo
     (- (expt 2 (1- new))
        (expt 2 (1- old)))))

Theorem: natp-rebias-up

(defthm natp-rebias-up
  (implies (and (natp n)
                (natp m)
                (< 0 m)
                (<= m n)
                (bvecp x m))
           (natp (rebias x m n))))

Theorem: natp-rebias-down

(defthm natp-rebias-down
  (implies (and (natp n)
                (natp m)
                (< 0 m)
                (<= m n)
                (bvecp x n)
                (< x (+ (expt 2 (1- n)) (expt 2 (1- m))))
                (>= x (- (expt 2 (1- n)) (expt 2 (1- m)))))
           (natp (rebias x n m))))

Theorem: bvecp-rebias-up

(defthm bvecp-rebias-up
  (implies (and (natp n)
                (natp m)
                (< 0 m)
                (<= m n)
                (bvecp x m))
           (bvecp (rebias x m n) n)))

Theorem: bvecp-rebias-down

(defthm bvecp-rebias-down
  (implies (and (natp n)
                (natp m)
                (< 0 m)
                (<= m n)
                (bvecp x n)
                (< x (+ (expt 2 (1- n)) (expt 2 (1- m))))
                (>= x (- (expt 2 (1- n)) (expt 2 (1- m)))))
           (bvecp (rebias x n m) m)))

Theorem: rebias-lower

(defthm rebias-lower
  (implies (and (natp n)
                (natp m)
                (> n m)
                (> m 1)
                (bvecp x n)
                (< x (+ (expt 2 (1- n)) (expt 2 (1- m))))
                (>= x (- (expt 2 (1- n)) (expt 2 (1- m)))))
           (equal (rebias x n m)
                  (cat (bitn x (1- n))
                       1 (bits x (- m 2) 0)
                       (1- m)))))

Theorem: rebias-higher

(defthm rebias-higher
  (implies (and (natp n)
                (natp m)
                (> n m)
                (> m 1)
                (bvecp x m))
           (equal (rebias x m n)
                  (cat (cat (bitn x (1- m))
                            1
                            (mulcat 1 (- n m)
                                    (bitn (lognot x) (1- m)))
                            (- n m))
                       (1+ (- n m))
                       (bits x (- m 2) 0)
                       (1- m)))))