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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)