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• Rational-listp

Std/typed-lists/rational-listp

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

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

Definitions and Theorems

Theorem: rational-listp-of-cons

(defthm rational-listp-of-cons
        (equal (rational-listp (cons a x))
               (and (rationalp a) (rational-listp x)))
        :rule-classes ((:rewrite)))

Theorem: rational-listp-of-cdr-when-rational-listp

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

Theorem: rational-listp-when-not-consp

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

Theorem: rationalp-of-car-when-rational-listp

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

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

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

Theorem: rational-listp-of-list-fix

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

Theorem: rational-listp-of-sfix

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

Theorem: rational-listp-of-insert

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

Theorem: rational-listp-of-delete

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

Theorem: rational-listp-of-mergesort

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

Theorem: rational-listp-of-union

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

Theorem: rational-listp-of-intersect-1

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

Theorem: rational-listp-of-intersect-2

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

Theorem: rational-listp-of-difference

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

Theorem: rational-listp-of-duplicated-members

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

Theorem: rational-listp-of-rev

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

Theorem: rational-listp-of-append

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

Theorem: rational-listp-of-rcons

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

Theorem: rationalp-when-member-equal-of-rational-listp

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

Theorem: rational-listp-when-subsetp-equal

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

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

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

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

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

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

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

Theorem: rational-listp-of-union-equal

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

Theorem: rational-listp-of-take

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

Theorem: rational-listp-of-repeat

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

Theorem: rationalp-of-nth-when-rational-listp

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

Theorem: rational-listp-of-update-nth

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

Theorem: rational-listp-of-butlast

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

Theorem: rational-listp-of-nthcdr

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

Theorem: rational-listp-of-last

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

Theorem: rational-listp-of-remove

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

Theorem: rational-listp-of-revappend

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

Theorem: rational-listp-of-remove-equal

(defthm rational-listp-of-remove-equal
        (implies (rational-listp x)
                 (rational-listp (remove-equal a x))))

Theorem: rational-listp-of-make-list-ac

(defthm rational-listp-of-make-list-ac
        (equal (rational-listp (make-list-ac n x ac))
               (and (rational-listp ac)
                    (or (rationalp x) (zp n)))))

Theorem: eqlable-listp-when-rational-listp

(defthm eqlable-listp-when-rational-listp
        (implies (rational-listp x)
                 (eqlable-listp x)))