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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)