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• Truth

Swap-polarity

Signature

(swap-polarity n truth numvars) → new-truth

Arguments

n — Guard (natp n).

truth — Guard (integerp truth).

numvars — Guard (natp numvars).

Returns

new-truth — Type (integerp new-truth).

Definitions and Theorems

Function: swap-polarity

(defun swap-polarity (n truth numvars)
  (declare (xargs :guard (and (natp n)
                              (integerp truth)
                              (natp numvars))))
  (declare (xargs :guard (< n numvars)))
  (let ((__function__ 'swap-polarity))
    (declare (ignorable __function__))
    (b* ((truth (truth-norm truth numvars))
         (var (var n numvars))
         (shift (ash 1 (lnfix n))))
      (logior (ash (logand var truth) (- shift))
              (ash (logand (lognot var)
                           (loghead (ash 1 (lnfix numvars)) truth))
                   shift)))))

Theorem: integerp-of-swap-polarity

(defthm integerp-of-swap-polarity
  (b* ((new-truth (swap-polarity n truth numvars)))
    (integerp new-truth))
  :rule-classes :type-prescription)

Theorem: swap-polarity-correct

(defthm swap-polarity-correct
  (b* ((?new-truth (swap-polarity n truth numvars)))
    (implies (< (nfix n) (nfix numvars))
             (equal (truth-eval new-truth env numvars)
                    (truth-eval truth (env-swap-polarity n env)
                                numvars)))))

Theorem: size-of-logand-by-size-of-loghead-2

(defthm size-of-logand-by-size-of-loghead-2
  (implies (and (unsigned-byte-p m a)
                (unsigned-byte-p n (loghead m b)))
           (unsigned-byte-p n (logand b a))))

Theorem: size-of-logand-with-loghead

(defthm size-of-logand-with-loghead
  (implies (unsigned-byte-p n (loghead m b))
           (unsigned-byte-p n (logand b (loghead m a)))))

Theorem: swap-polarity-size-basic

(defthm swap-polarity-size-basic
  (b* ((?new-truth (swap-polarity n truth numvars)))
    (implies (< (nfix n) (nfix numvars))
             (unsigned-byte-p (ash 1 numvars)
                              new-truth))))

Theorem: swap-polarity-size

(defthm swap-polarity-size
  (b* ((?new-truth (swap-polarity n truth numvars)))
    (implies (and (natp size)
                  (<= (ash 1 (nfix numvars)) size)
                  (< (nfix n) (nfix numvars)))
             (unsigned-byte-p size new-truth))))

Theorem: swap-polarity-of-truth-norm

(defthm swap-polarity-of-truth-norm
  (equal (swap-polarity n (truth-norm truth numvars)
                        numvars)
         (swap-polarity n truth numvars)))

Theorem: swap-polarity-of-nfix-n

(defthm swap-polarity-of-nfix-n
  (equal (swap-polarity (nfix n) truth numvars)
         (swap-polarity n truth numvars)))

Theorem: swap-polarity-nat-equiv-congruence-on-n

(defthm swap-polarity-nat-equiv-congruence-on-n
  (implies (nat-equiv n n-equiv)
           (equal (swap-polarity n truth numvars)
                  (swap-polarity n-equiv truth numvars)))
  :rule-classes :congruence)

Theorem: swap-polarity-of-ifix-truth

(defthm swap-polarity-of-ifix-truth
  (equal (swap-polarity n (ifix truth) numvars)
         (swap-polarity n truth numvars)))

Theorem: swap-polarity-int-equiv-congruence-on-truth

(defthm swap-polarity-int-equiv-congruence-on-truth
  (implies (int-equiv truth truth-equiv)
           (equal (swap-polarity n truth numvars)
                  (swap-polarity n truth-equiv numvars)))
  :rule-classes :congruence)

Theorem: swap-polarity-of-nfix-numvars

(defthm swap-polarity-of-nfix-numvars
  (equal (swap-polarity n truth (nfix numvars))
         (swap-polarity n truth numvars)))

Theorem: swap-polarity-nat-equiv-congruence-on-numvars

(defthm swap-polarity-nat-equiv-congruence-on-numvars
  (implies (nat-equiv numvars numvars-equiv)
           (equal (swap-polarity n truth numvars)
                  (swap-polarity n truth numvars-equiv)))
  :rule-classes :congruence)