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• Cons-list-listp

Cons-list-listp-basics

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

Definitions and Theorems

Theorem: cons-list-listp-of-cons

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

Theorem: cons-list-listp-of-cdr-when-cons-list-listp

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

Theorem: cons-list-listp-when-not-consp

(defthm cons-list-listp-when-not-consp
        (implies (not (consp x))
                 (cons-list-listp x))
        :rule-classes ((:rewrite)))

Theorem: cons-listp-of-car-when-cons-list-listp

(defthm cons-listp-of-car-when-cons-list-listp
        (implies (cons-list-listp x)
                 (cons-listp (car x)))
        :rule-classes ((:rewrite)))

Theorem: cons-list-listp-of-append

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

Theorem: cons-list-listp-of-list-fix

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

Theorem: cons-list-listp-of-sfix

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

Theorem: cons-list-listp-of-insert

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

Theorem: cons-list-listp-of-delete

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

Theorem: cons-list-listp-of-mergesort

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

Theorem: cons-list-listp-of-union

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

Theorem: cons-list-listp-of-intersect-1

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

Theorem: cons-list-listp-of-intersect-2

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

Theorem: cons-list-listp-of-difference

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

Theorem: cons-list-listp-of-duplicated-members

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

Theorem: cons-list-listp-of-rev

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

Theorem: cons-list-listp-of-rcons

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

Theorem: cons-listp-when-member-equal-of-cons-list-listp

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

Theorem: cons-list-listp-when-subsetp-equal

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

Theorem: cons-list-listp-set-equiv-congruence

(defthm cons-list-listp-set-equiv-congruence
        (implies (set-equiv x y)
                 (equal (cons-list-listp x)
                        (cons-list-listp y)))
        :rule-classes :congruence)

Theorem: cons-list-listp-of-set-difference-equal

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

Theorem: cons-list-listp-of-intersection-equal-1

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

Theorem: cons-list-listp-of-intersection-equal-2

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

Theorem: cons-list-listp-of-union-equal

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

Theorem: cons-list-listp-of-take

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

Theorem: cons-list-listp-of-repeat

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

Theorem: cons-listp-of-nth-when-cons-list-listp

(defthm cons-listp-of-nth-when-cons-list-listp
        (implies (cons-list-listp x)
                 (cons-listp (nth n x)))
        :rule-classes ((:rewrite)))

Theorem: cons-list-listp-of-update-nth

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

Theorem: cons-list-listp-of-butlast

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

Theorem: cons-list-listp-of-nthcdr

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

Theorem: cons-list-listp-of-last

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

Theorem: cons-list-listp-of-remove

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

Theorem: cons-list-listp-of-revappend

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