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Num-range-set

Num-range-setp

Recognizer for num-range-set.

Signature
(num-range-setp x) → *

Definitions and Theorems

Function: num-range-setp

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

Theorem: booleanp-ofnum-range-setp

(defthm booleanp-ofnum-range-setp
  (booleanp (num-range-setp x)))

Theorem: setp-when-num-range-setp

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

Theorem: num-range-p-of-head-when-num-range-setp

(defthm num-range-p-of-head-when-num-range-setp
  (implies (num-range-setp x)
           (equal (num-range-p (head x))
                  (not (emptyp x)))))

Theorem: num-range-setp-of-tail-when-num-range-setp

(defthm num-range-setp-of-tail-when-num-range-setp
  (implies (num-range-setp x)
           (num-range-setp (tail x))))

Theorem: num-range-setp-of-insert

(defthm num-range-setp-of-insert
  (equal (num-range-setp (insert a x))
         (and (num-range-p a)
              (num-range-setp (sfix x)))))

Theorem: num-range-p-when-in-num-range-setp-binds-free-x

(defthm num-range-p-when-in-num-range-setp-binds-free-x
  (implies (and (in a x) (num-range-setp x))
           (num-range-p a)))

Theorem: not-in-num-range-setp-when-not-num-range-p

(defthm not-in-num-range-setp-when-not-num-range-p
  (implies (and (num-range-setp x)
                (not (num-range-p a)))
           (not (in a x))))

Theorem: num-range-setp-of-union

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

Theorem: num-range-setp-of-intersect

(defthm num-range-setp-of-intersect
  (implies (and (num-range-setp x)
                (num-range-setp y))
           (num-range-setp (intersect x y))))

Theorem: num-range-setp-of-difference

(defthm num-range-setp-of-difference
  (implies (num-range-setp x)
           (num-range-setp (difference x y))))

Theorem: num-range-setp-of-delete

(defthm num-range-setp-of-delete
  (implies (num-range-setp x)
           (num-range-setp (delete a x))))