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Pos-set

Pos-setp

Recognizer for pos-set.

Signature
(pos-setp x) → *

Definitions and Theorems

Function: pos-setp

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

Theorem: booleanp-ofpos-setp

(defthm booleanp-ofpos-setp
  (booleanp (pos-setp x)))

Theorem: setp-when-pos-setp

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

Theorem: posp-of-head-when-pos-setp

(defthm posp-of-head-when-pos-setp
  (implies (pos-setp x)
           (equal (posp (set::head x))
                  (not (set::emptyp x)))))

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

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

Theorem: pos-setp-of-insert

(defthm pos-setp-of-insert
  (equal (pos-setp (set::insert a x))
         (and (posp a)
              (pos-setp (set::sfix x)))))

Theorem: posp-when-in-pos-setp-binds-free-x

(defthm posp-when-in-pos-setp-binds-free-x
  (implies (and (set::in a x) (pos-setp x))
           (posp a)))

Theorem: not-in-pos-setp-when-not-posp

(defthm not-in-pos-setp-when-not-posp
  (implies (and (pos-setp x) (not (posp a)))
           (not (set::in a x))))

Theorem: pos-setp-of-union

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

Theorem: pos-setp-of-intersect

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

Theorem: pos-setp-of-difference

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

Theorem: pos-setp-of-delete

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