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• Repetition/element/rulename/charstring-listp

Repetition/element/rulename/charstring-listp-basics

Basic theorems about repetition/element/rulename/charstring-listp, generated by std::deflist.

Definitions and Theorems

Theorem: repetition/element/rulename/charstring-listp-of-cons

(defthm repetition/element/rulename/charstring-listp-of-cons
  (equal
       (repetition/element/rulename/charstring-listp
            (cons acl2::a acl2::x))
       (and (repetition/element/rulename/charstring-p acl2::a)
            (repetition/element/rulename/charstring-listp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-cdr-when-repetition/element/rulename/charstring-listp

(defthm
 repetition/element/rulename/charstring-listp-of-cdr-when-repetition/element/rulename/charstring-listp
 (implies
      (repetition/element/rulename/charstring-listp
           (double-rewrite acl2::x))
      (repetition/element/rulename/charstring-listp (cdr acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-when-not-consp

(defthm repetition/element/rulename/charstring-listp-when-not-consp
  (implies
       (not (consp acl2::x))
       (equal (repetition/element/rulename/charstring-listp acl2::x)
              (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-p-of-car-when-repetition/element/rulename/charstring-listp

(defthm
 repetition/element/rulename/charstring-p-of-car-when-repetition/element/rulename/charstring-listp
 (implies
      (repetition/element/rulename/charstring-listp acl2::x)
      (iff (repetition/element/rulename/charstring-p (car acl2::x))
           (consp acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: true-listp-when-repetition/element/rulename/charstring-listp-compound-recognizer

(defthm
 true-listp-when-repetition/element/rulename/charstring-listp-compound-recognizer
 (implies (repetition/element/rulename/charstring-listp acl2::x)
          (true-listp acl2::x))
 :rule-classes :compound-recognizer)

Theorem: repetition/element/rulename/charstring-listp-of-list-fix

(defthm repetition/element/rulename/charstring-listp-of-list-fix
 (implies
  (repetition/element/rulename/charstring-listp acl2::x)
  (repetition/element/rulename/charstring-listp (list-fix acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-sfix

(defthm repetition/element/rulename/charstring-listp-of-sfix
  (iff (repetition/element/rulename/charstring-listp (sfix acl2::x))
       (or (repetition/element/rulename/charstring-listp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-insert

(defthm repetition/element/rulename/charstring-listp-of-insert
 (iff
  (repetition/element/rulename/charstring-listp
       (insert acl2::a acl2::x))
  (and (repetition/element/rulename/charstring-listp (sfix acl2::x))
       (repetition/element/rulename/charstring-p acl2::a)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-delete

(defthm repetition/element/rulename/charstring-listp-of-delete
  (implies (repetition/element/rulename/charstring-listp acl2::x)
           (repetition/element/rulename/charstring-listp
                (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-mergesort

(defthm repetition/element/rulename/charstring-listp-of-mergesort
 (iff
  (repetition/element/rulename/charstring-listp (mergesort acl2::x))
  (repetition/element/rulename/charstring-listp (list-fix acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-union

(defthm repetition/element/rulename/charstring-listp-of-union
 (iff
  (repetition/element/rulename/charstring-listp
       (union acl2::x acl2::y))
  (and
     (repetition/element/rulename/charstring-listp (sfix acl2::x))
     (repetition/element/rulename/charstring-listp (sfix acl2::y))))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-intersect-1

(defthm repetition/element/rulename/charstring-listp-of-intersect-1
  (implies (repetition/element/rulename/charstring-listp acl2::x)
           (repetition/element/rulename/charstring-listp
                (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-intersect-2

(defthm repetition/element/rulename/charstring-listp-of-intersect-2
  (implies (repetition/element/rulename/charstring-listp acl2::y)
           (repetition/element/rulename/charstring-listp
                (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-difference

(defthm repetition/element/rulename/charstring-listp-of-difference
  (implies (repetition/element/rulename/charstring-listp acl2::x)
           (repetition/element/rulename/charstring-listp
                (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-duplicated-members

(defthm
 repetition/element/rulename/charstring-listp-of-duplicated-members
 (implies (repetition/element/rulename/charstring-listp acl2::x)
          (repetition/element/rulename/charstring-listp
               (duplicated-members acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-rev

(defthm repetition/element/rulename/charstring-listp-of-rev
 (equal
  (repetition/element/rulename/charstring-listp (rev acl2::x))
  (repetition/element/rulename/charstring-listp (list-fix acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-append

(defthm repetition/element/rulename/charstring-listp-of-append
 (equal
  (repetition/element/rulename/charstring-listp
       (append acl2::a acl2::b))
  (and
   (repetition/element/rulename/charstring-listp (list-fix acl2::a))
   (repetition/element/rulename/charstring-listp acl2::b)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-rcons

(defthm repetition/element/rulename/charstring-listp-of-rcons
  (iff (repetition/element/rulename/charstring-listp
            (rcons acl2::a acl2::x))
       (and (repetition/element/rulename/charstring-p acl2::a)
            (repetition/element/rulename/charstring-listp
                 (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-p-when-member-equal-of-repetition/element/rulename/charstring-listp

(defthm
 repetition/element/rulename/charstring-p-when-member-equal-of-repetition/element/rulename/charstring-listp
 (and
   (implies
        (and (member-equal acl2::a acl2::x)
             (repetition/element/rulename/charstring-listp acl2::x))
        (repetition/element/rulename/charstring-p acl2::a))
   (implies
        (and (repetition/element/rulename/charstring-listp acl2::x)
             (member-equal acl2::a acl2::x))
        (repetition/element/rulename/charstring-p acl2::a)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-when-subsetp-equal

(defthm
    repetition/element/rulename/charstring-listp-when-subsetp-equal
 (and
  (implies
       (and (subsetp-equal acl2::x acl2::y)
            (repetition/element/rulename/charstring-listp acl2::y))
       (equal (repetition/element/rulename/charstring-listp acl2::x)
              (true-listp acl2::x)))
  (implies
       (and (repetition/element/rulename/charstring-listp acl2::y)
            (subsetp-equal acl2::x acl2::y))
       (equal (repetition/element/rulename/charstring-listp acl2::x)
              (true-listp acl2::x))))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-set-difference-equal

(defthm
 repetition/element/rulename/charstring-listp-of-set-difference-equal
 (implies (repetition/element/rulename/charstring-listp acl2::x)
          (repetition/element/rulename/charstring-listp
               (set-difference-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-intersection-equal-1

(defthm
 repetition/element/rulename/charstring-listp-of-intersection-equal-1
 (implies (repetition/element/rulename/charstring-listp
               (double-rewrite acl2::x))
          (repetition/element/rulename/charstring-listp
               (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-intersection-equal-2

(defthm
 repetition/element/rulename/charstring-listp-of-intersection-equal-2
 (implies (repetition/element/rulename/charstring-listp
               (double-rewrite acl2::y))
          (repetition/element/rulename/charstring-listp
               (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-union-equal

(defthm repetition/element/rulename/charstring-listp-of-union-equal
 (equal
  (repetition/element/rulename/charstring-listp
       (union-equal acl2::x acl2::y))
  (and
   (repetition/element/rulename/charstring-listp (list-fix acl2::x))
   (repetition/element/rulename/charstring-listp
        (double-rewrite acl2::y))))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-take

(defthm repetition/element/rulename/charstring-listp-of-take
  (implies (repetition/element/rulename/charstring-listp
                (double-rewrite acl2::x))
           (iff (repetition/element/rulename/charstring-listp
                     (take acl2::n acl2::x))
                (or (repetition/element/rulename/charstring-p nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-repeat

(defthm repetition/element/rulename/charstring-listp-of-repeat
  (iff (repetition/element/rulename/charstring-listp
            (repeat acl2::n acl2::x))
       (or (repetition/element/rulename/charstring-p acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-p-of-nth-when-repetition/element/rulename/charstring-listp

(defthm
 repetition/element/rulename/charstring-p-of-nth-when-repetition/element/rulename/charstring-listp
 (implies
  (repetition/element/rulename/charstring-listp acl2::x)
  (iff
    (repetition/element/rulename/charstring-p (nth acl2::n acl2::x))
    (< (nfix acl2::n) (len acl2::x))))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-update-nth

(defthm repetition/element/rulename/charstring-listp-of-update-nth
 (implies
    (repetition/element/rulename/charstring-listp
         (double-rewrite acl2::x))
    (iff (repetition/element/rulename/charstring-listp
              (update-nth acl2::n acl2::y acl2::x))
         (and (repetition/element/rulename/charstring-p acl2::y)
              (or (<= (nfix acl2::n) (len acl2::x))
                  (repetition/element/rulename/charstring-p nil)))))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-butlast

(defthm repetition/element/rulename/charstring-listp-of-butlast
  (implies (repetition/element/rulename/charstring-listp
                (double-rewrite acl2::x))
           (repetition/element/rulename/charstring-listp
                (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-nthcdr

(defthm repetition/element/rulename/charstring-listp-of-nthcdr
  (implies (repetition/element/rulename/charstring-listp
                (double-rewrite acl2::x))
           (repetition/element/rulename/charstring-listp
                (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-last

(defthm repetition/element/rulename/charstring-listp-of-last
 (implies
      (repetition/element/rulename/charstring-listp
           (double-rewrite acl2::x))
      (repetition/element/rulename/charstring-listp (last acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-remove

(defthm repetition/element/rulename/charstring-listp-of-remove
  (implies (repetition/element/rulename/charstring-listp acl2::x)
           (repetition/element/rulename/charstring-listp
                (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition/element/rulename/charstring-listp-of-revappend

(defthm repetition/element/rulename/charstring-listp-of-revappend
 (equal
  (repetition/element/rulename/charstring-listp
       (revappend acl2::x acl2::y))
  (and
   (repetition/element/rulename/charstring-listp (list-fix acl2::x))
   (repetition/element/rulename/charstring-listp acl2::y)))
 :rule-classes ((:rewrite)))