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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

(defthm true-listp-when-bit-listp
        (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)))