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String-set

String-setp

Recognizer for string-set.

Signature
(string-setp x) → *

Definitions and Theorems

Function: string-setp

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

Theorem: booleanp-ofstring-setp

(defthm booleanp-ofstring-setp
  (booleanp (string-setp x)))

Theorem: setp-when-string-setp

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

Theorem: stringp-of-head-when-string-setp

(defthm stringp-of-head-when-string-setp
  (implies (string-setp x)
           (equal (stringp (set::head x))
                  (not (set::emptyp x)))))

Theorem: string-setp-of-tail-when-string-setp

(defthm string-setp-of-tail-when-string-setp
  (implies (string-setp x)
           (string-setp (set::tail x))))

Theorem: string-setp-of-insert

(defthm string-setp-of-insert
  (equal (string-setp (set::insert a x))
         (and (stringp a)
              (string-setp (set::sfix x)))))

Theorem: stringp-when-in-string-setp-binds-free-x

(defthm stringp-when-in-string-setp-binds-free-x
  (implies (and (set::in a x) (string-setp x))
           (stringp a)))

Theorem: not-in-string-setp-when-not-stringp

(defthm not-in-string-setp-when-not-stringp
  (implies (and (string-setp x) (not (stringp a)))
           (not (set::in a x))))

Theorem: string-setp-of-union

(defthm string-setp-of-union
  (equal (string-setp (set::union x y))
         (and (string-setp (set::sfix x))
              (string-setp (set::sfix y)))))

Theorem: string-setp-of-intersect

(defthm string-setp-of-intersect
  (implies (and (string-setp x) (string-setp y))
           (string-setp (set::intersect x y))))

Theorem: string-setp-of-difference

(defthm string-setp-of-difference
  (implies (string-setp x)
           (string-setp (set::difference x y))))

Theorem: string-setp-of-delete

(defthm string-setp-of-delete
  (implies (string-setp x)
           (string-setp (set::delete a x))))