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• ACL2-number-listp

Std/typed-lists/ACL2-number-listp

Lemmas about ACL2-number-listp available in the std/typed-lists library.

Most of these are generated automatically with std::deflist.

Definitions and Theorems

Theorem: acl2-number-listp-of-cons

(defthm acl2-number-listp-of-cons
  (equal (acl2-number-listp (cons a x))
         (and (acl2-numberp a)
              (acl2-number-listp x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-cdr-when-acl2-number-listp

(defthm acl2-number-listp-of-cdr-when-acl2-number-listp
  (implies (acl2-number-listp (double-rewrite x))
           (acl2-number-listp (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-when-not-consp

(defthm acl2-number-listp-when-not-consp
  (implies (not (consp x))
           (equal (acl2-number-listp x) (not x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-numberp-of-car-when-acl2-number-listp

(defthm acl2-numberp-of-car-when-acl2-number-listp
  (implies (acl2-number-listp x)
           (iff (acl2-numberp (car x)) (consp x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-acl2-number-listp-compound-recognizer

(defthm true-listp-when-acl2-number-listp-compound-recognizer
  (implies (acl2-number-listp x)
           (true-listp x))
  :rule-classes :compound-recognizer)

Theorem: acl2-number-listp-of-list-fix

(defthm acl2-number-listp-of-list-fix
  (implies (acl2-number-listp x)
           (acl2-number-listp (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-sfix

(defthm acl2-number-listp-of-sfix
  (iff (acl2-number-listp (set::sfix x))
       (or (acl2-number-listp x)
           (not (set::setp x))))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-insert

(defthm acl2-number-listp-of-insert
  (iff (acl2-number-listp (set::insert a x))
       (and (acl2-number-listp (set::sfix x))
            (acl2-numberp a)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-delete

(defthm acl2-number-listp-of-delete
  (implies (acl2-number-listp x)
           (acl2-number-listp (set::delete k x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-mergesort

(defthm acl2-number-listp-of-mergesort
  (iff (acl2-number-listp (set::mergesort x))
       (acl2-number-listp (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-union

(defthm acl2-number-listp-of-union
  (iff (acl2-number-listp (set::union x y))
       (and (acl2-number-listp (set::sfix x))
            (acl2-number-listp (set::sfix y))))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-intersect-1

(defthm acl2-number-listp-of-intersect-1
  (implies (acl2-number-listp x)
           (acl2-number-listp (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-intersect-2

(defthm acl2-number-listp-of-intersect-2
  (implies (acl2-number-listp y)
           (acl2-number-listp (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-difference

(defthm acl2-number-listp-of-difference
  (implies (acl2-number-listp x)
           (acl2-number-listp (set::difference x y)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-duplicated-members

(defthm acl2-number-listp-of-duplicated-members
  (implies (acl2-number-listp x)
           (acl2-number-listp (duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-rev

(defthm acl2-number-listp-of-rev
  (equal (acl2-number-listp (rev x))
         (acl2-number-listp (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-append

(defthm acl2-number-listp-of-append
  (equal (acl2-number-listp (append a b))
         (and (acl2-number-listp (list-fix a))
              (acl2-number-listp b)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-rcons

(defthm acl2-number-listp-of-rcons
  (iff (acl2-number-listp (rcons a x))
       (and (acl2-numberp a)
            (acl2-number-listp (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: acl2-numberp-when-member-equal-of-acl2-number-listp

(defthm acl2-numberp-when-member-equal-of-acl2-number-listp
  (and (implies (and (member-equal a x)
                     (acl2-number-listp x))
                (acl2-numberp a))
       (implies (and (acl2-number-listp x)
                     (member-equal a x))
                (acl2-numberp a)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-when-subsetp-equal

(defthm acl2-number-listp-when-subsetp-equal
  (and (implies (and (subsetp-equal x y)
                     (acl2-number-listp y))
                (equal (acl2-number-listp x)
                       (true-listp x)))
       (implies (and (acl2-number-listp y)
                     (subsetp-equal x y))
                (equal (acl2-number-listp x)
                       (true-listp x))))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-set-difference-equal

(defthm acl2-number-listp-of-set-difference-equal
  (implies (acl2-number-listp x)
           (acl2-number-listp (set-difference-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-intersection-equal-1

(defthm acl2-number-listp-of-intersection-equal-1
  (implies (acl2-number-listp (double-rewrite x))
           (acl2-number-listp (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-intersection-equal-2

(defthm acl2-number-listp-of-intersection-equal-2
  (implies (acl2-number-listp (double-rewrite y))
           (acl2-number-listp (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-union-equal

(defthm acl2-number-listp-of-union-equal
  (equal (acl2-number-listp (union-equal x y))
         (and (acl2-number-listp (list-fix x))
              (acl2-number-listp (double-rewrite y))))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-take

(defthm acl2-number-listp-of-take
  (implies (acl2-number-listp (double-rewrite x))
           (iff (acl2-number-listp (take n x))
                (or (acl2-numberp nil)
                    (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-repeat

(defthm acl2-number-listp-of-repeat
  (iff (acl2-number-listp (repeat n x))
       (or (acl2-numberp x) (zp n)))
  :rule-classes ((:rewrite)))

Theorem: acl2-numberp-of-nth-when-acl2-number-listp

(defthm acl2-numberp-of-nth-when-acl2-number-listp
  (implies (acl2-number-listp x)
           (iff (acl2-numberp (nth n x))
                (< (nfix n) (len x))))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-update-nth

(defthm acl2-number-listp-of-update-nth
  (implies (acl2-number-listp (double-rewrite x))
           (iff (acl2-number-listp (update-nth n y x))
                (and (acl2-numberp y)
                     (or (<= (nfix n) (len x))
                         (acl2-numberp nil)))))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-butlast

(defthm acl2-number-listp-of-butlast
  (implies (acl2-number-listp (double-rewrite x))
           (acl2-number-listp (butlast x n)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-nthcdr

(defthm acl2-number-listp-of-nthcdr
  (implies (acl2-number-listp (double-rewrite x))
           (acl2-number-listp (nthcdr n x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-last

(defthm acl2-number-listp-of-last
  (implies (acl2-number-listp (double-rewrite x))
           (acl2-number-listp (last x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-remove

(defthm acl2-number-listp-of-remove
  (implies (acl2-number-listp x)
           (acl2-number-listp (remove a x)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-revappend

(defthm acl2-number-listp-of-revappend
  (equal (acl2-number-listp (revappend x y))
         (and (acl2-number-listp (list-fix x))
              (acl2-number-listp y)))
  :rule-classes ((:rewrite)))

Theorem: acl2-number-listp-of-remove-equal

(defthm acl2-number-listp-of-remove-equal
  (implies (acl2-number-listp x)
           (acl2-number-listp (remove-equal a x))))

Theorem: acl2-number-listp-of-make-list-ac

(defthm acl2-number-listp-of-make-list-ac
  (equal (acl2-number-listp (make-list-ac n x ac))
         (and (acl2-number-listp ac)
              (or (acl2-numberp x) (zp n)))))

Theorem: eqlable-listp-when-acl2-number-listp

(defthm eqlable-listp-when-acl2-number-listp
  (implies (acl2-number-listp x)
           (eqlable-listp x)))