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Address+pos-set

Address+pos-setp

Recognizer for address+pos-set.

Signature
(address+pos-setp x) → *

Definitions and Theorems

Function: address+pos-setp

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

Theorem: booleanp-ofaddress+pos-setp

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

Theorem: setp-when-address+pos-setp

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

Theorem: address+pos-p-of-head-when-address+pos-setp

(defthm address+pos-p-of-head-when-address+pos-setp
  (implies (address+pos-setp x)
           (equal (address+pos-p (head x))
                  (not (emptyp x)))))

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

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

Theorem: address+pos-setp-of-insert

(defthm address+pos-setp-of-insert
  (equal (address+pos-setp (insert a x))
         (and (address+pos-p a)
              (address+pos-setp (sfix x)))))

Theorem: address+pos-p-when-in-address+pos-setp-binds-free-x

(defthm address+pos-p-when-in-address+pos-setp-binds-free-x
  (implies (and (in a x) (address+pos-setp x))
           (address+pos-p a)))

Theorem: not-in-address+pos-setp-when-not-address+pos-p

(defthm not-in-address+pos-setp-when-not-address+pos-p
  (implies (and (address+pos-setp x)
                (not (address+pos-p a)))
           (not (in a x))))

Theorem: address+pos-setp-of-union

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

Theorem: address+pos-setp-of-intersect

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

Theorem: address+pos-setp-of-difference

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

Theorem: address+pos-setp-of-delete

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