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• Truth

Depends-on-witness

Signature

(depends-on-witness n truth numvars) → diff-env

Arguments

n — Guard (natp n).

truth — Guard (integerp truth).

numvars — Guard (natp numvars).

Returns

diff-env — Type (natp diff-env).

Definitions and Theorems

Function: depends-on-witness

(defun depends-on-witness (n truth numvars)
  (declare (xargs :guard (and (natp n)
                              (integerp truth)
                              (natp numvars))))
  (declare (xargs :guard (< n numvars)))
  (let ((__function__ 'depends-on-witness))
    (declare (ignorable __function__))
    (b* ((var (var n numvars))
         (truth (truth-norm truth numvars))
         (diff (logxor (logand (lognot var) truth)
                       (ash (logand var truth)
                            (- (ash 1 (lnfix n)))))))
      (bitops::trailing-0-count diff))))

Theorem: natp-of-depends-on-witness

(defthm natp-of-depends-on-witness
  (b* ((diff-env (depends-on-witness n truth numvars)))
    (natp diff-env))
  :rule-classes :type-prescription)

Theorem: depends-on-witness-correct

(defthm depends-on-witness-correct
  (b* ((?diff-env (depends-on-witness n truth numvars)))
    (implies (and (natp numvars)
                  (< (nfix n) numvars)
                  (depends-on n truth numvars))
             (not (equal (truth-eval truth (env-update n t diff-env)
                                     numvars)
                         (truth-eval truth diff-env numvars))))))

Theorem: depends-on-witness-of-nfix-n

(defthm depends-on-witness-of-nfix-n
  (equal (depends-on-witness (nfix n)
                             truth numvars)
         (depends-on-witness n truth numvars)))

Theorem: depends-on-witness-nat-equiv-congruence-on-n

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

Theorem: depends-on-witness-of-ifix-truth

(defthm depends-on-witness-of-ifix-truth
  (equal (depends-on-witness n (ifix truth)
                             numvars)
         (depends-on-witness n truth numvars)))

Theorem: depends-on-witness-int-equiv-congruence-on-truth

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

Theorem: depends-on-witness-of-nfix-numvars

(defthm depends-on-witness-of-nfix-numvars
  (equal (depends-on-witness n truth (nfix numvars))
         (depends-on-witness n truth numvars)))

Theorem: depends-on-witness-nat-equiv-congruence-on-numvars

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