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• Truth

Negative-cofactor

Signature

(negative-cofactor n truth numvars) → cofactor

Arguments

n — Guard (natp n).

truth — Guard (integerp truth).

numvars — Guard (natp numvars).

Returns

cofactor — Type (integerp cofactor).

Definitions and Theorems

Function: negative-cofactor

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

Theorem: integerp-of-negative-cofactor

(defthm integerp-of-negative-cofactor
  (b* ((cofactor (negative-cofactor n truth numvars)))
    (integerp cofactor))
  :rule-classes :type-prescription)

Theorem: negative-cofactor-correct

(defthm negative-cofactor-correct
  (b* ((?cofactor (negative-cofactor n truth numvars)))
    (implies (< (nfix n) (nfix numvars))
             (equal (truth-eval cofactor env numvars)
                    (truth-eval truth (env-update n nil env)
                                numvars)))))

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

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

Theorem: negative-cofactor-size-basic

(defthm negative-cofactor-size-basic
  (b* ((?cofactor (negative-cofactor n truth numvars)))
    (implies (and (< (nfix n) numvars)
                  (natp numvars))
             (unsigned-byte-p (ash 1 numvars)
                              cofactor))))

Theorem: negative-cofactor-size

(defthm negative-cofactor-size
  (b* ((?cofactor (negative-cofactor n truth numvars)))
    (implies (and (natp m)
                  (<= (ash 1 numvars) m)
                  (< (nfix n) numvars)
                  (natp numvars))
             (unsigned-byte-p m cofactor))))

Theorem: negative-cofactor-of-truth-norm

(defthm negative-cofactor-of-truth-norm
  (equal (negative-cofactor n (truth-norm truth numvars)
                            numvars)
         (negative-cofactor n truth numvars)))

Theorem: truth-norm-of-negative-cofactor

(defthm truth-norm-of-negative-cofactor
  (implies (< (nfix n) (nfix numvars))
           (equal (truth-norm (negative-cofactor n truth numvars)
                              numvars)
                  (negative-cofactor n truth numvars))))

Theorem: negative-cofactor-of-nfix-n

(defthm negative-cofactor-of-nfix-n
  (equal (negative-cofactor (nfix n)
                            truth numvars)
         (negative-cofactor n truth numvars)))

Theorem: negative-cofactor-nat-equiv-congruence-on-n

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

Theorem: negative-cofactor-of-ifix-truth

(defthm negative-cofactor-of-ifix-truth
  (equal (negative-cofactor n (ifix truth)
                            numvars)
         (negative-cofactor n truth numvars)))

Theorem: negative-cofactor-int-equiv-congruence-on-truth

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

Theorem: negative-cofactor-of-nfix-numvars

(defthm negative-cofactor-of-nfix-numvars
  (equal (negative-cofactor n truth (nfix numvars))
         (negative-cofactor n truth numvars)))

Theorem: negative-cofactor-nat-equiv-congruence-on-numvars

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