		Search-engine friendly clone of the ACL2 documentation.

Zp-listp

Zp-listp-basics

Basic theorems about zp-listp, generated by std::deflist.

Definitions and Theorems

Theorem: zp-listp-of-cons

(defthm zp-listp-of-cons
  (equal (zp-listp (cons a x))
         (and (not (posp a)) (zp-listp x)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-cdr-when-zp-listp

(defthm zp-listp-of-cdr-when-zp-listp
  (implies (zp-listp (double-rewrite x))
           (zp-listp (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-when-not-consp

(defthm zp-listp-when-not-consp
  (implies (not (consp x))
           (equal (zp-listp x) (not x)))
  :rule-classes ((:rewrite)))

Theorem: posp-of-car-when-zp-listp

(defthm posp-of-car-when-zp-listp
  (implies (zp-listp x)
           (not (posp (car x))))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-zp-listp-compound-recognizer

(defthm true-listp-when-zp-listp-compound-recognizer
  (implies (zp-listp x) (true-listp x))
  :rule-classes :compound-recognizer)

Theorem: zp-listp-of-list-fix

(defthm zp-listp-of-list-fix
  (implies (zp-listp x)
           (zp-listp (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-sfix

(defthm zp-listp-of-sfix
  (iff (zp-listp (set::sfix x))
       (or (zp-listp x) (not (set::setp x))))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-insert

(defthm zp-listp-of-insert
  (iff (zp-listp (set::insert a x))
       (and (zp-listp (set::sfix x))
            (not (posp a))))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-delete

(defthm zp-listp-of-delete
  (implies (zp-listp x)
           (zp-listp (set::delete k x)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-mergesort

(defthm zp-listp-of-mergesort
  (iff (zp-listp (set::mergesort x))
       (zp-listp (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-union

(defthm zp-listp-of-union
  (iff (zp-listp (set::union x y))
       (and (zp-listp (set::sfix x))
            (zp-listp (set::sfix y))))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-intersect-1

(defthm zp-listp-of-intersect-1
  (implies (zp-listp x)
           (zp-listp (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-intersect-2

(defthm zp-listp-of-intersect-2
  (implies (zp-listp y)
           (zp-listp (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-difference

(defthm zp-listp-of-difference
  (implies (zp-listp x)
           (zp-listp (set::difference x y)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-duplicated-members

(defthm zp-listp-of-duplicated-members
  (implies (zp-listp x)
           (zp-listp (duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-rev

(defthm zp-listp-of-rev
  (equal (zp-listp (rev x))
         (zp-listp (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-append

(defthm zp-listp-of-append
  (equal (zp-listp (append a b))
         (and (zp-listp (list-fix a))
              (zp-listp b)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-rcons

(defthm zp-listp-of-rcons
  (iff (zp-listp (rcons a x))
       (and (not (posp a))
            (zp-listp (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: posp-when-member-equal-of-zp-listp

(defthm posp-when-member-equal-of-zp-listp
  (and (implies (and (member-equal a x) (zp-listp x))
                (not (posp a)))
       (implies (and (zp-listp x) (member-equal a x))
                (not (posp a))))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-when-subsetp-equal

(defthm zp-listp-when-subsetp-equal
  (and (implies (and (subsetp-equal x y) (zp-listp y))
                (equal (zp-listp x) (true-listp x)))
       (implies (and (zp-listp y) (subsetp-equal x y))
                (equal (zp-listp x) (true-listp x))))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-set-difference-equal

(defthm zp-listp-of-set-difference-equal
  (implies (zp-listp x)
           (zp-listp (set-difference-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-intersection-equal-1

(defthm zp-listp-of-intersection-equal-1
  (implies (zp-listp (double-rewrite x))
           (zp-listp (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-intersection-equal-2

(defthm zp-listp-of-intersection-equal-2
  (implies (zp-listp (double-rewrite y))
           (zp-listp (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-union-equal

(defthm zp-listp-of-union-equal
  (equal (zp-listp (union-equal x y))
         (and (zp-listp (list-fix x))
              (zp-listp (double-rewrite y))))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-take

(defthm zp-listp-of-take
  (implies (zp-listp (double-rewrite x))
           (iff (zp-listp (take n x))
                (or (not (posp nil))
                    (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-repeat

(defthm zp-listp-of-repeat
  (iff (zp-listp (repeat n x))
       (or (not (posp x)) (zp n)))
  :rule-classes ((:rewrite)))

Theorem: posp-of-nth-when-zp-listp

(defthm posp-of-nth-when-zp-listp
  (implies (zp-listp x)
           (not (posp (nth n x))))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-update-nth

(defthm zp-listp-of-update-nth
  (implies (zp-listp (double-rewrite x))
           (iff (zp-listp (update-nth n y x))
                (and (not (posp y))
                     (or (<= (nfix n) (len x))
                         (not (posp nil))))))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-butlast

(defthm zp-listp-of-butlast
  (implies (zp-listp (double-rewrite x))
           (zp-listp (butlast x n)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-nthcdr

(defthm zp-listp-of-nthcdr
  (implies (zp-listp (double-rewrite x))
           (zp-listp (nthcdr n x)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-last

(defthm zp-listp-of-last
  (implies (zp-listp (double-rewrite x))
           (zp-listp (last x)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-remove

(defthm zp-listp-of-remove
  (implies (zp-listp x)
           (zp-listp (remove a x)))
  :rule-classes ((:rewrite)))

Theorem: zp-listp-of-revappend

(defthm zp-listp-of-revappend
  (equal (zp-listp (revappend x y))
         (and (zp-listp (list-fix x))
              (zp-listp y)))
  :rule-classes ((:rewrite)))