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• Plexeme-list-tokenp

Plexeme-list-tokenp-basics

Basic theorems about plexeme-list-tokenp, generated by std::deflist.

Definitions and Theorems

Theorem: plexeme-list-tokenp-of-cons

(defthm plexeme-list-tokenp-of-cons
  (equal (plexeme-list-tokenp (cons acl2::a acl2::x))
         (and (plexeme-tokenp acl2::a)
              (plexeme-list-tokenp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-cdr-when-plexeme-list-tokenp

(defthm plexeme-list-tokenp-of-cdr-when-plexeme-list-tokenp
  (implies (plexeme-list-tokenp (double-rewrite acl2::x))
           (plexeme-list-tokenp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-when-not-consp

(defthm plexeme-list-tokenp-when-not-consp
  (implies (not (consp acl2::x))
           (plexeme-list-tokenp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: plexeme-tokenp-of-car-when-plexeme-list-tokenp

(defthm plexeme-tokenp-of-car-when-plexeme-list-tokenp
  (implies (plexeme-list-tokenp acl2::x)
           (plexeme-tokenp (car acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-append

(defthm plexeme-list-tokenp-of-append
  (equal (plexeme-list-tokenp (append acl2::a acl2::b))
         (and (plexeme-list-tokenp acl2::a)
              (plexeme-list-tokenp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-list-fix

(defthm plexeme-list-tokenp-of-list-fix
  (equal (plexeme-list-tokenp (list-fix acl2::x))
         (plexeme-list-tokenp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-sfix

(defthm plexeme-list-tokenp-of-sfix
  (iff (plexeme-list-tokenp (sfix acl2::x))
       (or (plexeme-list-tokenp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-insert

(defthm plexeme-list-tokenp-of-insert
  (iff (plexeme-list-tokenp (insert acl2::a acl2::x))
       (and (plexeme-list-tokenp (sfix acl2::x))
            (plexeme-tokenp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-delete

(defthm plexeme-list-tokenp-of-delete
  (implies (plexeme-list-tokenp acl2::x)
           (plexeme-list-tokenp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-mergesort

(defthm plexeme-list-tokenp-of-mergesort
  (iff (plexeme-list-tokenp (mergesort acl2::x))
       (plexeme-list-tokenp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-union

(defthm plexeme-list-tokenp-of-union
  (iff (plexeme-list-tokenp (union acl2::x acl2::y))
       (and (plexeme-list-tokenp (sfix acl2::x))
            (plexeme-list-tokenp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-intersect-1

(defthm plexeme-list-tokenp-of-intersect-1
  (implies (plexeme-list-tokenp acl2::x)
           (plexeme-list-tokenp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-intersect-2

(defthm plexeme-list-tokenp-of-intersect-2
  (implies (plexeme-list-tokenp acl2::y)
           (plexeme-list-tokenp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-difference

(defthm plexeme-list-tokenp-of-difference
  (implies (plexeme-list-tokenp acl2::x)
           (plexeme-list-tokenp (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-duplicated-members

(defthm plexeme-list-tokenp-of-duplicated-members
  (implies (plexeme-list-tokenp acl2::x)
           (plexeme-list-tokenp (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-rev

(defthm plexeme-list-tokenp-of-rev
  (equal (plexeme-list-tokenp (rev acl2::x))
         (plexeme-list-tokenp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-rcons

(defthm plexeme-list-tokenp-of-rcons
  (iff (plexeme-list-tokenp (rcons acl2::a acl2::x))
       (and (plexeme-tokenp acl2::a)
            (plexeme-list-tokenp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: plexeme-tokenp-when-member-equal-of-plexeme-list-tokenp

(defthm plexeme-tokenp-when-member-equal-of-plexeme-list-tokenp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (plexeme-list-tokenp acl2::x))
                (plexeme-tokenp acl2::a))
       (implies (and (plexeme-list-tokenp acl2::x)
                     (member-equal acl2::a acl2::x))
                (plexeme-tokenp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-when-subsetp-equal

(defthm plexeme-list-tokenp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (plexeme-list-tokenp acl2::y))
                (plexeme-list-tokenp acl2::x))
       (implies (and (plexeme-list-tokenp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (plexeme-list-tokenp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-set-equiv-congruence

(defthm plexeme-list-tokenp-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (plexeme-list-tokenp acl2::x)
                  (plexeme-list-tokenp acl2::y)))
  :rule-classes :congruence)

Theorem: plexeme-list-tokenp-of-set-difference-equal

(defthm plexeme-list-tokenp-of-set-difference-equal
  (implies
       (plexeme-list-tokenp acl2::x)
       (plexeme-list-tokenp (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-intersection-equal-1

(defthm plexeme-list-tokenp-of-intersection-equal-1
  (implies
       (plexeme-list-tokenp (double-rewrite acl2::x))
       (plexeme-list-tokenp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-intersection-equal-2

(defthm plexeme-list-tokenp-of-intersection-equal-2
  (implies
       (plexeme-list-tokenp (double-rewrite acl2::y))
       (plexeme-list-tokenp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-union-equal

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

Theorem: plexeme-list-tokenp-of-take

(defthm plexeme-list-tokenp-of-take
  (implies (plexeme-list-tokenp (double-rewrite acl2::x))
           (iff (plexeme-list-tokenp (take acl2::n acl2::x))
                (or (plexeme-tokenp nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-repeat

(defthm plexeme-list-tokenp-of-repeat
  (iff (plexeme-list-tokenp (repeat acl2::n acl2::x))
       (or (plexeme-tokenp acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-tokenp-of-nth-when-plexeme-list-tokenp

(defthm plexeme-tokenp-of-nth-when-plexeme-list-tokenp
  (implies (plexeme-list-tokenp acl2::x)
           (plexeme-tokenp (nth acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-update-nth

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

Theorem: plexeme-list-tokenp-of-butlast

(defthm plexeme-list-tokenp-of-butlast
  (implies (plexeme-list-tokenp (double-rewrite acl2::x))
           (plexeme-list-tokenp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-nthcdr

(defthm plexeme-list-tokenp-of-nthcdr
  (implies (plexeme-list-tokenp (double-rewrite acl2::x))
           (plexeme-list-tokenp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-last

(defthm plexeme-list-tokenp-of-last
  (implies (plexeme-list-tokenp (double-rewrite acl2::x))
           (plexeme-list-tokenp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-remove

(defthm plexeme-list-tokenp-of-remove
  (implies (plexeme-list-tokenp acl2::x)
           (plexeme-list-tokenp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: plexeme-list-tokenp-of-revappend

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