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Constrel-set

Constrel-setp

Recognizer for constrel-set.

Signature
(constrel-setp x) → *

Definitions and Theorems

Function: constrel-setp

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

Theorem: booleanp-ofconstrel-setp

(defthm booleanp-ofconstrel-setp
  (booleanp (constrel-setp x)))

Theorem: setp-when-constrel-setp

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

Theorem: constrelp-of-head-when-constrel-setp

(defthm constrelp-of-head-when-constrel-setp
  (implies (constrel-setp x)
           (equal (constrelp (head x))
                  (not (emptyp x)))))

Theorem: constrel-setp-of-tail-when-constrel-setp

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

Theorem: constrel-setp-of-insert

(defthm constrel-setp-of-insert
  (equal (constrel-setp (insert a x))
         (and (constrelp a)
              (constrel-setp (sfix x)))))

Theorem: constrelp-when-in-constrel-setp-binds-free-x

(defthm constrelp-when-in-constrel-setp-binds-free-x
  (implies (and (in a x) (constrel-setp x))
           (constrelp a)))

Theorem: not-in-constrel-setp-when-not-constrelp

(defthm not-in-constrel-setp-when-not-constrelp
  (implies (and (constrel-setp x)
                (not (constrelp a)))
           (not (in a x))))

Theorem: constrel-setp-of-union

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

Theorem: constrel-setp-of-intersect

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

Theorem: constrel-setp-of-difference

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

Theorem: constrel-setp-of-delete

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