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• Tokenlist-p

Tokenlist-p-basics

Basic theorems about tokenlist-p, generated by deflist.

Definitions and Theorems

Theorem: tokenlist-p-of-cons

(defthm tokenlist-p-of-cons
        (equal (tokenlist-p (cons acl2::a acl2::x))
               (and (token-p acl2::a)
                    (tokenlist-p acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-cdr-when-tokenlist-p

(defthm tokenlist-p-of-cdr-when-tokenlist-p
        (implies (tokenlist-p (double-rewrite acl2::x))
                 (tokenlist-p (cdr acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-when-not-consp

(defthm tokenlist-p-when-not-consp
        (implies (not (consp acl2::x))
                 (tokenlist-p acl2::x))
        :rule-classes ((:rewrite)))

Theorem: token-p-of-car-when-tokenlist-p

(defthm token-p-of-car-when-tokenlist-p
        (implies (tokenlist-p acl2::x)
                 (iff (token-p (car acl2::x))
                      (or (consp acl2::x) (token-p nil))))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-append

(defthm tokenlist-p-of-append
        (equal (tokenlist-p (append acl2::a acl2::b))
               (and (tokenlist-p acl2::a)
                    (tokenlist-p acl2::b)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-list-fix

(defthm tokenlist-p-of-list-fix
        (equal (tokenlist-p (list-fix acl2::x))
               (tokenlist-p acl2::x))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-sfix

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

Theorem: tokenlist-p-of-insert

(defthm tokenlist-p-of-insert
        (iff (tokenlist-p (set::insert acl2::a acl2::x))
             (and (tokenlist-p (set::sfix acl2::x))
                  (token-p acl2::a)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-delete

(defthm tokenlist-p-of-delete
        (implies (tokenlist-p acl2::x)
                 (tokenlist-p (set::delete acl2::k acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-mergesort

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

Theorem: tokenlist-p-of-union

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

Theorem: tokenlist-p-of-intersect-1

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

Theorem: tokenlist-p-of-intersect-2

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

Theorem: tokenlist-p-of-difference

(defthm tokenlist-p-of-difference
        (implies (tokenlist-p acl2::x)
                 (tokenlist-p (set::difference acl2::x acl2::y)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-duplicated-members

(defthm tokenlist-p-of-duplicated-members
        (implies (tokenlist-p acl2::x)
                 (tokenlist-p (acl2::duplicated-members acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-rev

(defthm tokenlist-p-of-rev
        (equal (tokenlist-p (acl2::rev acl2::x))
               (tokenlist-p (list-fix acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-rcons

(defthm tokenlist-p-of-rcons
        (iff (tokenlist-p (acl2::rcons acl2::a acl2::x))
             (and (token-p acl2::a)
                  (tokenlist-p (list-fix acl2::x))))
        :rule-classes ((:rewrite)))

Theorem: token-p-when-member-equal-of-tokenlist-p

(defthm token-p-when-member-equal-of-tokenlist-p
        (and (implies (and (member-equal acl2::a acl2::x)
                           (tokenlist-p acl2::x))
                      (token-p acl2::a))
             (implies (and (tokenlist-p acl2::x)
                           (member-equal acl2::a acl2::x))
                      (token-p acl2::a)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-when-subsetp-equal

(defthm tokenlist-p-when-subsetp-equal
        (and (implies (and (subsetp-equal acl2::x acl2::y)
                           (tokenlist-p acl2::y))
                      (tokenlist-p acl2::x))
             (implies (and (tokenlist-p acl2::y)
                           (subsetp-equal acl2::x acl2::y))
                      (tokenlist-p acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-set-equiv-congruence

(defthm tokenlist-p-set-equiv-congruence
        (implies (acl2::set-equiv acl2::x acl2::y)
                 (equal (tokenlist-p acl2::x)
                        (tokenlist-p acl2::y)))
        :rule-classes :congruence)

Theorem: tokenlist-p-of-set-difference-equal

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

Theorem: tokenlist-p-of-intersection-equal-1

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

Theorem: tokenlist-p-of-intersection-equal-2

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

Theorem: tokenlist-p-of-union-equal

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

Theorem: tokenlist-p-of-take

(defthm tokenlist-p-of-take
        (implies (tokenlist-p (double-rewrite acl2::x))
                 (iff (tokenlist-p (take acl2::n acl2::x))
                      (or (token-p nil)
                          (<= (nfix acl2::n) (len acl2::x)))))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-repeat

(defthm tokenlist-p-of-repeat
        (iff (tokenlist-p (acl2::repeat acl2::n acl2::x))
             (or (token-p acl2::x) (zp acl2::n)))
        :rule-classes ((:rewrite)))

Theorem: token-p-of-nth-when-tokenlist-p

(defthm token-p-of-nth-when-tokenlist-p
        (implies (and (tokenlist-p acl2::x)
                      (< (nfix acl2::n) (len acl2::x)))
                 (token-p (nth acl2::n acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-update-nth

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

Theorem: tokenlist-p-of-butlast

(defthm tokenlist-p-of-butlast
        (implies (tokenlist-p (double-rewrite acl2::x))
                 (tokenlist-p (butlast acl2::x acl2::n)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-nthcdr

(defthm tokenlist-p-of-nthcdr
        (implies (tokenlist-p (double-rewrite acl2::x))
                 (tokenlist-p (nthcdr acl2::n acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-last

(defthm tokenlist-p-of-last
        (implies (tokenlist-p (double-rewrite acl2::x))
                 (tokenlist-p (last acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-remove

(defthm tokenlist-p-of-remove
        (implies (tokenlist-p acl2::x)
                 (tokenlist-p (remove acl2::a acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: tokenlist-p-of-revappend

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