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• Chars-in-charset-p

Chars-in-charset-p-basics

Basic theorems about chars-in-charset-p, generated by std::deflist.

Definitions and Theorems

Theorem: chars-in-charset-p-of-cons

(defthm chars-in-charset-p-of-cons
  (equal (chars-in-charset-p (cons a x) set)
         (and (char-in-charset-p a set)
              (chars-in-charset-p x set)))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-cdr-when-chars-in-charset-p

(defthm chars-in-charset-p-of-cdr-when-chars-in-charset-p
  (implies (chars-in-charset-p (double-rewrite x)
                               set)
           (chars-in-charset-p (cdr x) set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-when-not-consp

(defthm chars-in-charset-p-when-not-consp
  (implies (not (consp x))
           (chars-in-charset-p x set))
  :rule-classes ((:rewrite)))

Theorem: char-in-charset-p-of-car-when-chars-in-charset-p

(defthm char-in-charset-p-of-car-when-chars-in-charset-p
  (implies (chars-in-charset-p x set)
           (iff (char-in-charset-p (car x) set)
                (or (consp x)
                    (char-in-charset-p nil set))))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-append

(defthm chars-in-charset-p-of-append
  (equal (chars-in-charset-p (append a b) set)
         (and (chars-in-charset-p a set)
              (chars-in-charset-p b set)))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-list-fix

(defthm chars-in-charset-p-of-list-fix
  (equal (chars-in-charset-p (list-fix x) set)
         (chars-in-charset-p x set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-sfix

(defthm chars-in-charset-p-of-sfix
  (iff (chars-in-charset-p (set::sfix x) set)
       (or (chars-in-charset-p x set)
           (not (set::setp x))))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-insert

(defthm chars-in-charset-p-of-insert
  (iff (chars-in-charset-p (set::insert a x)
                           set)
       (and (chars-in-charset-p (set::sfix x) set)
            (char-in-charset-p a set)))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-delete

(defthm chars-in-charset-p-of-delete
  (implies (chars-in-charset-p x set)
           (chars-in-charset-p (set::delete k x)
                               set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-mergesort

(defthm chars-in-charset-p-of-mergesort
  (iff (chars-in-charset-p (set::mergesort x)
                           set)
       (chars-in-charset-p (list-fix x) set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-union

(defthm chars-in-charset-p-of-union
  (iff (chars-in-charset-p (set::union x y)
                           set)
       (and (chars-in-charset-p (set::sfix x) set)
            (chars-in-charset-p (set::sfix y) set)))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-intersect-1

(defthm chars-in-charset-p-of-intersect-1
  (implies (chars-in-charset-p x set)
           (chars-in-charset-p (set::intersect x y)
                               set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-intersect-2

(defthm chars-in-charset-p-of-intersect-2
  (implies (chars-in-charset-p y set)
           (chars-in-charset-p (set::intersect x y)
                               set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-difference

(defthm chars-in-charset-p-of-difference
  (implies (chars-in-charset-p x set)
           (chars-in-charset-p (set::difference x y)
                               set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-duplicated-members

(defthm chars-in-charset-p-of-duplicated-members
  (implies (chars-in-charset-p x set)
           (chars-in-charset-p (acl2::duplicated-members x)
                               set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-rev

(defthm chars-in-charset-p-of-rev
  (equal (chars-in-charset-p (rev x) set)
         (chars-in-charset-p (list-fix x) set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-rcons

(defthm chars-in-charset-p-of-rcons
  (iff (chars-in-charset-p (acl2::rcons a x)
                           set)
       (and (char-in-charset-p a set)
            (chars-in-charset-p (list-fix x) set)))
  :rule-classes ((:rewrite)))

Theorem: char-in-charset-p-when-member-equal-of-chars-in-charset-p

(defthm char-in-charset-p-when-member-equal-of-chars-in-charset-p
  (and (implies (and (member-equal a x)
                     (chars-in-charset-p x set))
                (char-in-charset-p a set))
       (implies (and (chars-in-charset-p x set)
                     (member-equal a x))
                (char-in-charset-p a set)))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-when-subsetp-equal

(defthm chars-in-charset-p-when-subsetp-equal
  (and (implies (and (subsetp-equal x y)
                     (chars-in-charset-p y set))
                (chars-in-charset-p x set))
       (implies (and (chars-in-charset-p y set)
                     (subsetp-equal x y))
                (chars-in-charset-p x set)))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-set-equiv-congruence

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

Theorem: chars-in-charset-p-of-set-difference-equal

(defthm chars-in-charset-p-of-set-difference-equal
  (implies (chars-in-charset-p x set)
           (chars-in-charset-p (set-difference-equal x y)
                               set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-intersection-equal-1

(defthm chars-in-charset-p-of-intersection-equal-1
  (implies (chars-in-charset-p (double-rewrite x)
                               set)
           (chars-in-charset-p (intersection-equal x y)
                               set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-intersection-equal-2

(defthm chars-in-charset-p-of-intersection-equal-2
  (implies (chars-in-charset-p (double-rewrite y)
                               set)
           (chars-in-charset-p (intersection-equal x y)
                               set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-union-equal

(defthm chars-in-charset-p-of-union-equal
  (equal (chars-in-charset-p (union-equal x y)
                             set)
         (and (chars-in-charset-p (list-fix x) set)
              (chars-in-charset-p (double-rewrite y)
                                  set)))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-take

(defthm chars-in-charset-p-of-take
  (implies (chars-in-charset-p (double-rewrite x)
                               set)
           (iff (chars-in-charset-p (take n x) set)
                (or (char-in-charset-p nil set)
                    (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-repeat

(defthm chars-in-charset-p-of-repeat
  (iff (chars-in-charset-p (repeat n x) set)
       (or (char-in-charset-p x set) (zp n)))
  :rule-classes ((:rewrite)))

Theorem: char-in-charset-p-of-nth-when-chars-in-charset-p

(defthm char-in-charset-p-of-nth-when-chars-in-charset-p
  (implies (and (chars-in-charset-p x set)
                (< (nfix n) (len x)))
           (char-in-charset-p (nth n x) set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-update-nth

(defthm chars-in-charset-p-of-update-nth
  (implies (chars-in-charset-p (double-rewrite x)
                               set)
           (iff (chars-in-charset-p (update-nth n y x)
                                    set)
                (and (char-in-charset-p y set)
                     (or (<= (nfix n) (len x))
                         (char-in-charset-p nil set)))))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-butlast

(defthm chars-in-charset-p-of-butlast
  (implies (chars-in-charset-p (double-rewrite x)
                               set)
           (chars-in-charset-p (butlast x n) set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-nthcdr

(defthm chars-in-charset-p-of-nthcdr
  (implies (chars-in-charset-p (double-rewrite x)
                               set)
           (chars-in-charset-p (nthcdr n x) set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-last

(defthm chars-in-charset-p-of-last
  (implies (chars-in-charset-p (double-rewrite x)
                               set)
           (chars-in-charset-p (last x) set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-remove

(defthm chars-in-charset-p-of-remove
  (implies (chars-in-charset-p x set)
           (chars-in-charset-p (remove a x) set))
  :rule-classes ((:rewrite)))

Theorem: chars-in-charset-p-of-revappend

(defthm chars-in-charset-p-of-revappend
  (equal (chars-in-charset-p (revappend x y) set)
         (and (chars-in-charset-p (list-fix x) set)
              (chars-in-charset-p y set)))
  :rule-classes ((:rewrite)))