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• String-literalp

String-literalp-basics

Basic theorems about string-literalp, generated by std::deflist.

Definitions and Theorems

Theorem: string-literalp-of-cons

(defthm string-literalp-of-cons
  (equal (string-literalp (cons acl2::a acl2::x))
         (and (string-literal-char-p acl2::a)
              (string-literalp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: string-literalp-of-cdr-when-string-literalp

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

Theorem: string-literalp-when-not-consp

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

Theorem: string-literal-char-p-of-car-when-string-literalp

(defthm string-literal-char-p-of-car-when-string-literalp
  (implies (string-literalp acl2::x)
           (iff (string-literal-char-p (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-string-literalp-compound-recognizer

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

Theorem: string-literalp-of-list-fix

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

Theorem: string-literalp-of-sfix

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

Theorem: string-literalp-of-insert

(defthm string-literalp-of-insert
  (iff (string-literalp (insert acl2::a acl2::x))
       (and (string-literalp (sfix acl2::x))
            (string-literal-char-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: string-literalp-of-delete

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

Theorem: string-literalp-of-mergesort

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

Theorem: string-literalp-of-union

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

Theorem: string-literalp-of-intersect-1

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

Theorem: string-literalp-of-intersect-2

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

Theorem: string-literalp-of-difference

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

Theorem: string-literalp-of-duplicated-members

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

Theorem: string-literalp-of-rev

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

Theorem: string-literalp-of-append

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

Theorem: string-literalp-of-rcons

(defthm string-literalp-of-rcons
  (iff (string-literalp (rcons acl2::a acl2::x))
       (and (string-literal-char-p acl2::a)
            (string-literalp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: string-literal-char-p-when-member-equal-of-string-literalp

(defthm string-literal-char-p-when-member-equal-of-string-literalp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (string-literalp acl2::x))
                (string-literal-char-p acl2::a))
       (implies (and (string-literalp acl2::x)
                     (member-equal acl2::a acl2::x))
                (string-literal-char-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: string-literalp-when-subsetp-equal

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

Theorem: string-literalp-of-set-difference-equal

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

Theorem: string-literalp-of-intersection-equal-1

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

Theorem: string-literalp-of-intersection-equal-2

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

Theorem: string-literalp-of-union-equal

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

Theorem: string-literalp-of-take

(defthm string-literalp-of-take
  (implies (string-literalp (double-rewrite acl2::x))
           (iff (string-literalp (take acl2::n acl2::x))
                (or (string-literal-char-p nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: string-literalp-of-repeat

(defthm string-literalp-of-repeat
  (iff (string-literalp (repeat acl2::n acl2::x))
       (or (string-literal-char-p acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: string-literal-char-p-of-nth-when-string-literalp

(defthm string-literal-char-p-of-nth-when-string-literalp
  (implies (string-literalp acl2::x)
           (iff (string-literal-char-p (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: string-literalp-of-update-nth

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

Theorem: string-literalp-of-butlast

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

Theorem: string-literalp-of-nthcdr

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

Theorem: string-literalp-of-last

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

Theorem: string-literalp-of-remove

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

Theorem: string-literalp-of-revappend

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