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Truth

Var

Signature
(var n numvars) → var-enc
Arguments: n — Guard (natp n).; numvars — Guard (natp numvars).
Returns: var-enc — Type (natp var-enc).

Definitions and Theorems

Function: var

(defun var (n numvars)
       (declare (xargs :guard (and (natp n) (natp numvars))))
       (declare (xargs :guard (< n numvars)))
       (let ((__function__ 'var))
            (declare (ignorable __function__))
            (b* ((n (lnfix n))
                 (numvars (lnfix numvars))
                 (w (ash 1 n))
                 (rep (ash (logmask w) w)))
                (var-repetitions (+ 1 n) rep numvars))))

Theorem: natp-of-var

(defthm natp-of-var
        (b* ((var-enc (var n numvars)))
            (natp var-enc))
        :rule-classes :type-prescription)

Theorem: logbitp-of-var

(defthm logbitp-of-var
        (b* ((?var-enc (var n numvars)))
            (implies (< (nfix n) (nfix numvars))
                     (equal (logbitp env var-enc)
                            (and (< (nfix env) (ash 1 numvars))
                                 (logbitp n (nfix env)))))))

Theorem: eval-of-var

(defthm eval-of-var
        (b* ((?var-enc (var n numvars)))
            (implies (< (nfix n) (nfix numvars))
                     (equal (truth-eval var-enc env numvars)
                            (env-lookup n env)))))

Theorem: var-size-basic

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

Theorem: var-size

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

Theorem: var-negated-masked-size

(defthm
  var-negated-masked-size
  (b* ((?var-enc (var n numvars)))
      (implies (and (< (nfix n) numvars)
                    (natp numvars))
               (unsigned-byte-p (- (ash 1 numvars) (ash 1 (nfix n)))
                                (loghead (ash 1 numvars)
                                         (lognot var-enc))))))

Theorem: var-negated

(defthm var-negated
        (b* ((?var-enc (var n numvars)))
            (implies (and (< (nfix n) numvars)
                          (natp numvars))
                     (equal (loghead (ash 1 numvars)
                                     (lognot var-enc))
                            (logtail (ash 1 (nfix n)) var-enc)))))

Theorem: var-of-nfix-n

(defthm var-of-nfix-n
        (equal (var (nfix n) numvars)
               (var n numvars)))

Theorem: var-nat-equiv-congruence-on-n

(defthm var-nat-equiv-congruence-on-n
        (implies (nat-equiv n n-equiv)
                 (equal (var n numvars)
                        (var n-equiv numvars)))
        :rule-classes :congruence)

Theorem: var-of-nfix-numvars

(defthm var-of-nfix-numvars
        (equal (var n (nfix numvars))
               (var n numvars)))

Theorem: var-nat-equiv-congruence-on-numvars

(defthm var-nat-equiv-congruence-on-numvars
        (implies (nat-equiv numvars numvars-equiv)
                 (equal (var n numvars)
                        (var n numvars-equiv)))
        :rule-classes :congruence)