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Address-set

Address-setp

Recognizer for address-set.

Signature
(address-setp x) → *

Definitions and Theorems

Function: address-setp

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

Theorem: booleanp-ofaddress-setp

(defthm booleanp-ofaddress-setp
  (booleanp (address-setp x)))

Theorem: setp-when-address-setp

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

Theorem: addressp-of-head-when-address-setp

(defthm addressp-of-head-when-address-setp
  (implies (address-setp x)
           (equal (addressp (head x))
                  (not (emptyp x)))))

Theorem: address-setp-of-tail-when-address-setp

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

Theorem: address-setp-of-insert

(defthm address-setp-of-insert
  (equal (address-setp (insert a x))
         (and (addressp a)
              (address-setp (sfix x)))))

Theorem: addressp-when-in-address-setp-binds-free-x

(defthm addressp-when-in-address-setp-binds-free-x
  (implies (and (in a x) (address-setp x))
           (addressp a)))

Theorem: not-in-address-setp-when-not-addressp

(defthm not-in-address-setp-when-not-addressp
  (implies (and (address-setp x)
                (not (addressp a)))
           (not (in a x))))

Theorem: address-setp-of-union

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

Theorem: address-setp-of-intersect

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

Theorem: address-setp-of-difference

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

Theorem: address-setp-of-delete

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