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Special-char-alist

Special-char-alistp

Signature
(special-char-alistp x) → *

Definitions and Theorems

Function: special-char-alistp

(defun special-char-alistp (x)
  (declare (xargs :guard t))
  (let ((acl2::__function__ 'special-char-alistp))
    (declare (ignorable acl2::__function__))
    (if (atom x)
        t
      (and (consp (car x))
           (characterp (caar x))
           (character-listp (cdar x))
           (special-char-alistp (cdr x))))))

Theorem: special-char-alistp-of-revappend

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

Theorem: special-char-alistp-of-remove

(defthm special-char-alistp-of-remove
  (implies (special-char-alistp acl2::x)
           (special-char-alistp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-last

(defthm special-char-alistp-of-last
  (implies (special-char-alistp (double-rewrite acl2::x))
           (special-char-alistp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-nthcdr

(defthm special-char-alistp-of-nthcdr
  (implies (special-char-alistp (double-rewrite acl2::x))
           (special-char-alistp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-butlast

(defthm special-char-alistp-of-butlast
  (implies (special-char-alistp (double-rewrite acl2::x))
           (special-char-alistp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-update-nth

(defthm special-char-alistp-of-update-nth
 (implies
     (special-char-alistp (double-rewrite acl2::x))
     (iff (special-char-alistp (update-nth acl2::n acl2::y acl2::x))
          (and (and (consp acl2::y)
                    (characterp (car acl2::y))
                    (character-listp (cdr acl2::y)))
               (or (<= (nfix acl2::n) (len acl2::x))
                   (and (consp nil)
                        (characterp (car nil))
                        (character-listp (cdr nil)))))))
 :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-repeat

(defthm special-char-alistp-of-repeat
  (iff (special-char-alistp (acl2::repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (characterp (car acl2::x))
                (character-listp (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-take

(defthm special-char-alistp-of-take
  (implies (special-char-alistp (double-rewrite acl2::x))
           (iff (special-char-alistp (take acl2::n acl2::x))
                (or (and (consp nil)
                         (characterp (car nil))
                         (character-listp (cdr nil)))
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-union-equal

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

Theorem: special-char-alistp-of-intersection-equal-2

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

Theorem: special-char-alistp-of-intersection-equal-1

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

Theorem: special-char-alistp-of-set-difference-equal

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

Theorem: special-char-alistp-set-equiv-congruence

(defthm special-char-alistp-set-equiv-congruence
  (implies (acl2::set-equiv acl2::x acl2::y)
           (equal (special-char-alistp acl2::x)
                  (special-char-alistp acl2::y)))
  :rule-classes :congruence)

Theorem: special-char-alistp-when-subsetp-equal

(defthm special-char-alistp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (special-char-alistp acl2::y))
                (special-char-alistp acl2::x))
       (implies (and (special-char-alistp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (special-char-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-rcons

(defthm special-char-alistp-of-rcons
  (iff (special-char-alistp (acl2::rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (characterp (car acl2::a))
                 (character-listp (cdr acl2::a)))
            (special-char-alistp (acl2::list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-rev

(defthm special-char-alistp-of-rev
  (equal (special-char-alistp (acl2::rev acl2::x))
         (special-char-alistp (acl2::list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-duplicated-members

(defthm special-char-alistp-of-duplicated-members
  (implies (special-char-alistp acl2::x)
           (special-char-alistp (acl2::duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-difference

(defthm special-char-alistp-of-difference
  (implies (special-char-alistp acl2::x)
           (special-char-alistp (set::difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-intersect-2

(defthm special-char-alistp-of-intersect-2
  (implies (special-char-alistp acl2::y)
           (special-char-alistp (set::intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-intersect-1

(defthm special-char-alistp-of-intersect-1
  (implies (special-char-alistp acl2::x)
           (special-char-alistp (set::intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-union

(defthm special-char-alistp-of-union
  (iff (special-char-alistp (set::union acl2::x acl2::y))
       (and (special-char-alistp (set::sfix acl2::x))
            (special-char-alistp (set::sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-mergesort

(defthm special-char-alistp-of-mergesort
  (iff (special-char-alistp (set::mergesort acl2::x))
       (special-char-alistp (acl2::list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-delete

(defthm special-char-alistp-of-delete
  (implies (special-char-alistp acl2::x)
           (special-char-alistp (set::delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-insert

(defthm special-char-alistp-of-insert
  (iff (special-char-alistp (set::insert acl2::a acl2::x))
       (and (special-char-alistp (set::sfix acl2::x))
            (and (consp acl2::a)
                 (characterp (car acl2::a))
                 (character-listp (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-sfix

(defthm special-char-alistp-of-sfix
  (iff (special-char-alistp (set::sfix acl2::x))
       (or (special-char-alistp acl2::x)
           (not (set::setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-list-fix

(defthm special-char-alistp-of-list-fix
  (equal (special-char-alistp (acl2::list-fix acl2::x))
         (special-char-alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-append

(defthm special-char-alistp-of-append
  (equal (special-char-alistp (append acl2::a acl2::b))
         (and (special-char-alistp acl2::a)
              (special-char-alistp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-when-not-consp

(defthm special-char-alistp-when-not-consp
  (implies (not (consp acl2::x))
           (special-char-alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-cdr-when-special-char-alistp

(defthm special-char-alistp-of-cdr-when-special-char-alistp
  (implies (special-char-alistp (double-rewrite acl2::x))
           (special-char-alistp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-cons

(defthm special-char-alistp-of-cons
  (equal (special-char-alistp (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (characterp (car acl2::a))
                   (character-listp (cdr acl2::a)))
              (special-char-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-fast-alist-clean

(defthm special-char-alistp-of-fast-alist-clean
  (implies (special-char-alistp acl2::x)
           (special-char-alistp (fast-alist-clean acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-hons-shrink-alist

(defthm special-char-alistp-of-hons-shrink-alist
 (implies (and (special-char-alistp acl2::x)
               (special-char-alistp acl2::y))
          (special-char-alistp (hons-shrink-alist acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: special-char-alistp-of-hons-acons

(defthm special-char-alistp-of-hons-acons
  (equal (special-char-alistp (hons-acons acl2::a acl2::n acl2::x))
         (and (characterp acl2::a)
              (character-listp acl2::n)
              (special-char-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: character-listp-of-cdr-of-hons-assoc-equal-when-special-char-alistp

(defthm
 character-listp-of-cdr-of-hons-assoc-equal-when-special-char-alistp
 (implies
     (special-char-alistp acl2::x)
     (iff (character-listp (cdr (hons-assoc-equal acl2::k acl2::x)))
          (or (hons-assoc-equal acl2::k acl2::x)
              (character-listp nil))))
 :rule-classes ((:rewrite)))

Theorem: character-listp-of-cdar-when-special-char-alistp

(defthm character-listp-of-cdar-when-special-char-alistp
  (implies (special-char-alistp acl2::x)
           (iff (character-listp (cdar acl2::x))
                (or (consp acl2::x)
                    (character-listp nil))))
  :rule-classes ((:rewrite)))

Theorem: characterp-of-caar-when-special-char-alistp

(defthm characterp-of-caar-when-special-char-alistp
  (implies (special-char-alistp acl2::x)
           (iff (characterp (caar acl2::x))
                (or (consp acl2::x) (characterp nil))))
  :rule-classes ((:rewrite)))