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Symbol-set

Symbol-setp

Recognizer for symbol-set.

Signature
(symbol-setp x) → *

Definitions and Theorems

Function: symbol-setp

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

Theorem: booleanp-ofsymbol-setp

(defthm booleanp-ofsymbol-setp
  (booleanp (symbol-setp x)))

Theorem: setp-when-symbol-setp

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

Theorem: symbolp-of-head-when-symbol-setp

(defthm symbolp-of-head-when-symbol-setp
  (implies (symbol-setp x)
           (symbolp (set::head x))))

Theorem: symbol-setp-of-tail-when-symbol-setp

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

Theorem: symbol-setp-of-insert

(defthm symbol-setp-of-insert
  (equal (symbol-setp (set::insert a x))
         (and (symbolp a)
              (symbol-setp (set::sfix x)))))

Theorem: symbolp-when-in-symbol-setp-binds-free-x

(defthm symbolp-when-in-symbol-setp-binds-free-x
  (implies (and (set::in a x) (symbol-setp x))
           (symbolp a)))

Theorem: not-in-symbol-setp-when-not-symbolp

(defthm not-in-symbol-setp-when-not-symbolp
  (implies (and (symbol-setp x) (not (symbolp a)))
           (not (set::in a x))))

Theorem: symbol-setp-of-union

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

Theorem: symbol-setp-of-intersect

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

Theorem: symbol-setp-of-difference

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

Theorem: symbol-setp-of-delete

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