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Bit-listp

Bit-listp-basics

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

Definitions and Theorems

Theorem: bit-listp-of-cons

(defthm bit-listp-of-cons
  (equal (bit-listp (cons a x))
         (and (bitp a) (bit-listp x)))
  :rule-classes ((:rewrite)))

Theorem: bit-listp-of-cdr-when-bit-listp

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

Theorem: bit-listp-when-not-consp

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

Theorem: bitp-of-car-when-bit-listp

(defthm bitp-of-car-when-bit-listp
  (implies (bit-listp x)
           (iff (bitp (car x)) (consp x)))
  :rule-classes ((:rewrite)))

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

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

Theorem: bit-listp-of-list-fix

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

Theorem: bit-listp-of-sfix

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

Theorem: bit-listp-of-insert

(defthm bit-listp-of-insert
  (iff (bit-listp (set::insert a x))
       (and (bit-listp (set::sfix x))
            (bitp a)))
  :rule-classes ((:rewrite)))

Theorem: bit-listp-of-delete

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

Theorem: bit-listp-of-mergesort

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

Theorem: bit-listp-of-union

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

Theorem: bit-listp-of-intersect-1

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

Theorem: bit-listp-of-intersect-2

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

Theorem: bit-listp-of-difference

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

Theorem: bit-listp-of-duplicated-members

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

Theorem: bit-listp-of-rev

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

Theorem: bit-listp-of-append

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

Theorem: bit-listp-of-rcons

(defthm bit-listp-of-rcons
  (iff (bit-listp (rcons a x))
       (and (bitp a) (bit-listp (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: bitp-when-member-equal-of-bit-listp

(defthm bitp-when-member-equal-of-bit-listp
  (and (implies (and (member-equal a x) (bit-listp x))
                (bitp a))
       (implies (and (bit-listp x) (member-equal a x))
                (bitp a)))
  :rule-classes ((:rewrite)))

Theorem: bit-listp-when-subsetp-equal

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

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

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

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

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

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

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

Theorem: bit-listp-of-union-equal

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

Theorem: bit-listp-of-take

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

Theorem: bit-listp-of-repeat

(defthm bit-listp-of-repeat
  (iff (bit-listp (repeat n x))
       (or (bitp x) (zp n)))
  :rule-classes ((:rewrite)))

Theorem: bitp-of-nth-when-bit-listp

(defthm bitp-of-nth-when-bit-listp
  (implies (bit-listp x)
           (iff (bitp (nth n x))
                (< (nfix n) (len x))))
  :rule-classes ((:rewrite)))

Theorem: bit-listp-of-update-nth

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

Theorem: bit-listp-of-butlast

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

Theorem: bit-listp-of-nthcdr

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

Theorem: bit-listp-of-last

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

Theorem: bit-listp-of-remove

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

Theorem: bit-listp-of-revappend

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