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• Stringp

Stringp-basics

Basic theorems about stringp, generated by std::deflist.

Definitions and Theorems

Theorem: stringp-of-cons

(defthm stringp-of-cons
  (equal (stringp (cons acl2::a acl2::x))
         (and (symbolp acl2::a)
              (stringp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-cdr-when-stringp

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

Theorem: stringp-when-not-consp

(defthm stringp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (stringp acl2::x) (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: symbolp-of-car-when-stringp

(defthm symbolp-of-car-when-stringp
  (implies (stringp acl2::x)
           (iff (symbolp (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-stringp-compound-recognizer

(defthm true-listp-when-stringp-compound-recognizer
  (implies (stringp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: stringp-of-list-fix

(defthm stringp-of-list-fix
  (implies (stringp acl2::x)
           (stringp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-sfix

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

Theorem: stringp-of-insert

(defthm stringp-of-insert
  (iff (stringp (insert acl2::a acl2::x))
       (and (stringp (sfix acl2::x))
            (symbolp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-delete

(defthm stringp-of-delete
  (implies (stringp acl2::x)
           (stringp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-mergesort

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

Theorem: stringp-of-union

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

Theorem: stringp-of-intersect-1

(defthm stringp-of-intersect-1
  (implies (stringp acl2::x)
           (stringp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-intersect-2

(defthm stringp-of-intersect-2
  (implies (stringp acl2::y)
           (stringp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-difference

(defthm stringp-of-difference
  (implies (stringp acl2::x)
           (stringp (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-duplicated-members

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

Theorem: stringp-of-rev

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

Theorem: stringp-of-append

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

Theorem: stringp-of-rcons

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

Theorem: symbolp-when-member-equal-of-stringp

(defthm symbolp-when-member-equal-of-stringp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (stringp acl2::x))
                (symbolp acl2::a))
       (implies (and (stringp acl2::x)
                     (member-equal acl2::a acl2::x))
                (symbolp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: stringp-when-subsetp-equal

(defthm stringp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (stringp acl2::y))
                (equal (stringp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (stringp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (stringp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-set-difference-equal

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

Theorem: stringp-of-intersection-equal-1

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

Theorem: stringp-of-intersection-equal-2

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

Theorem: stringp-of-union-equal

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

Theorem: stringp-of-take

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

Theorem: stringp-of-repeat

(defthm stringp-of-repeat
  (iff (stringp (repeat acl2::n acl2::x))
       (or (symbolp acl2::x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: symbolp-of-nth-when-stringp

(defthm symbolp-of-nth-when-stringp
  (implies (stringp acl2::x)
           (iff (symbolp (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-update-nth

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

Theorem: stringp-of-butlast

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

Theorem: stringp-of-nthcdr

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

Theorem: stringp-of-last

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

Theorem: stringp-of-remove

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

Theorem: stringp-of-revappend

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