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Certificate-set

Certificate-setp

Recognizer for certificate-set.

Signature
(certificate-setp x) → *

Definitions and Theorems

Function: certificate-setp

(defun certificate-setp (x)
  (declare (xargs :guard t))
  (if (atom x)
      (null x)
    (and (certificatep (car x))
         (or (null (cdr x))
             (and (consp (cdr x))
                  (acl2::fast-<< (car x) (cadr x))
                  (certificate-setp (cdr x)))))))

Theorem: booleanp-ofcertificate-setp

(defthm booleanp-ofcertificate-setp
  (booleanp (certificate-setp x)))

Theorem: setp-when-certificate-setp

(defthm setp-when-certificate-setp
  (implies (certificate-setp x) (setp x))
  :rule-classes (:rewrite))

Theorem: certificatep-of-head-when-certificate-setp

(defthm certificatep-of-head-when-certificate-setp
  (implies (certificate-setp x)
           (equal (certificatep (head x))
                  (not (emptyp x)))))

Theorem: certificate-setp-of-tail-when-certificate-setp

(defthm certificate-setp-of-tail-when-certificate-setp
  (implies (certificate-setp x)
           (certificate-setp (tail x))))

Theorem: certificate-setp-of-insert

(defthm certificate-setp-of-insert
  (equal (certificate-setp (insert a x))
         (and (certificatep a)
              (certificate-setp (sfix x)))))

Theorem: certificatep-when-in-certificate-setp-binds-free-x

(defthm certificatep-when-in-certificate-setp-binds-free-x
  (implies (and (in a x) (certificate-setp x))
           (certificatep a)))

Theorem: not-in-certificate-setp-when-not-certificatep

(defthm not-in-certificate-setp-when-not-certificatep
  (implies (and (certificate-setp x)
                (not (certificatep a)))
           (not (in a x))))

Theorem: certificate-setp-of-union

(defthm certificate-setp-of-union
  (equal (certificate-setp (union x y))
         (and (certificate-setp (sfix x))
              (certificate-setp (sfix y)))))

Theorem: certificate-setp-of-intersect

(defthm certificate-setp-of-intersect
  (implies (and (certificate-setp x)
                (certificate-setp y))
           (certificate-setp (intersect x y))))

Theorem: certificate-setp-of-difference

(defthm certificate-setp-of-difference
  (implies (certificate-setp x)
           (certificate-setp (difference x y))))

Theorem: certificate-setp-of-delete

(defthm certificate-setp-of-delete
  (implies (certificate-setp x)
           (certificate-setp (delete a x))))