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      • Symbol-symbol-alistp
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Symbol-truelist-alistp

Recognize alists from symbols to true lists.

This is an ordinary std::defalist.

Function: symbol-truelist-alistp

(defun symbol-truelist-alistp (x)
  (declare (xargs :guard t))
  (if (consp x)
      (and (consp (car x))
           (symbolp (caar x))
           (true-listp (cdar x))
           (symbol-truelist-alistp (cdr x)))
    (null x)))

Definitions and Theorems

Function: symbol-truelist-alistp

(defun symbol-truelist-alistp (x)
  (declare (xargs :guard t))
  (if (consp x)
      (and (consp (car x))
           (symbolp (caar x))
           (true-listp (cdar x))
           (symbol-truelist-alistp (cdr x)))
    (null x)))

Theorem: symbol-truelist-alistp-of-revappend

(defthm symbol-truelist-alistp-of-revappend
  (equal (symbol-truelist-alistp (revappend x y))
         (and (symbol-truelist-alistp (list-fix x))
              (symbol-truelist-alistp y)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-remove

(defthm symbol-truelist-alistp-of-remove
  (implies (symbol-truelist-alistp x)
           (symbol-truelist-alistp (remove a x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-last

(defthm symbol-truelist-alistp-of-last
  (implies (symbol-truelist-alistp (double-rewrite x))
           (symbol-truelist-alistp (last x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-nthcdr

(defthm symbol-truelist-alistp-of-nthcdr
  (implies (symbol-truelist-alistp (double-rewrite x))
           (symbol-truelist-alistp (nthcdr n x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-butlast

(defthm symbol-truelist-alistp-of-butlast
  (implies (symbol-truelist-alistp (double-rewrite x))
           (symbol-truelist-alistp (butlast x n)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-update-nth

(defthm symbol-truelist-alistp-of-update-nth
  (implies (symbol-truelist-alistp (double-rewrite x))
           (iff (symbol-truelist-alistp (update-nth n y x))
                (and (and (consp y)
                          (symbolp (car y))
                          (true-listp (cdr y)))
                     (or (<= (nfix n) (len x))
                         (and (consp nil)
                              (symbolp (car nil))
                              (true-listp (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-repeat

(defthm symbol-truelist-alistp-of-repeat
  (iff (symbol-truelist-alistp (repeat n x))
       (or (and (consp x)
                (symbolp (car x))
                (true-listp (cdr x)))
           (zp n)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-take

(defthm symbol-truelist-alistp-of-take
  (implies (symbol-truelist-alistp (double-rewrite x))
           (iff (symbol-truelist-alistp (take n x))
                (or (and (consp nil)
                         (symbolp (car nil))
                         (true-listp (cdr nil)))
                    (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-union-equal

(defthm symbol-truelist-alistp-of-union-equal
  (equal (symbol-truelist-alistp (union-equal x y))
         (and (symbol-truelist-alistp (list-fix x))
              (symbol-truelist-alistp (double-rewrite y))))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-intersection-equal-2

(defthm symbol-truelist-alistp-of-intersection-equal-2
  (implies (symbol-truelist-alistp (double-rewrite y))
           (symbol-truelist-alistp (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-intersection-equal-1

(defthm symbol-truelist-alistp-of-intersection-equal-1
  (implies (symbol-truelist-alistp (double-rewrite x))
           (symbol-truelist-alistp (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-set-difference-equal

(defthm symbol-truelist-alistp-of-set-difference-equal
  (implies (symbol-truelist-alistp x)
           (symbol-truelist-alistp (set-difference-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-when-subsetp-equal

(defthm symbol-truelist-alistp-when-subsetp-equal
  (and (implies (and (subsetp-equal x y)
                     (symbol-truelist-alistp y))
                (equal (symbol-truelist-alistp x)
                       (true-listp x)))
       (implies (and (symbol-truelist-alistp y)
                     (subsetp-equal x y))
                (equal (symbol-truelist-alistp x)
                       (true-listp x))))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-rcons

(defthm symbol-truelist-alistp-of-rcons
  (iff (symbol-truelist-alistp (rcons a x))
       (and (and (consp a)
                 (symbolp (car a))
                 (true-listp (cdr a)))
            (symbol-truelist-alistp (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-append

(defthm symbol-truelist-alistp-of-append
  (equal (symbol-truelist-alistp (append a b))
         (and (symbol-truelist-alistp (list-fix a))
              (symbol-truelist-alistp b)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-rev

(defthm symbol-truelist-alistp-of-rev
  (equal (symbol-truelist-alistp (rev x))
         (symbol-truelist-alistp (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-duplicated-members

(defthm symbol-truelist-alistp-of-duplicated-members
  (implies (symbol-truelist-alistp x)
           (symbol-truelist-alistp (duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-difference

(defthm symbol-truelist-alistp-of-difference
  (implies (symbol-truelist-alistp x)
           (symbol-truelist-alistp (set::difference x y)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-intersect-2

(defthm symbol-truelist-alistp-of-intersect-2
  (implies (symbol-truelist-alistp y)
           (symbol-truelist-alistp (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-intersect-1

(defthm symbol-truelist-alistp-of-intersect-1
  (implies (symbol-truelist-alistp x)
           (symbol-truelist-alistp (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-union

(defthm symbol-truelist-alistp-of-union
  (iff (symbol-truelist-alistp (set::union x y))
       (and (symbol-truelist-alistp (set::sfix x))
            (symbol-truelist-alistp (set::sfix y))))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-mergesort

(defthm symbol-truelist-alistp-of-mergesort
  (iff (symbol-truelist-alistp (set::mergesort x))
       (symbol-truelist-alistp (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-delete

(defthm symbol-truelist-alistp-of-delete
  (implies (symbol-truelist-alistp x)
           (symbol-truelist-alistp (set::delete k x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-insert

(defthm symbol-truelist-alistp-of-insert
  (iff (symbol-truelist-alistp (set::insert a x))
       (and (symbol-truelist-alistp (set::sfix x))
            (and (consp a)
                 (symbolp (car a))
                 (true-listp (cdr a)))))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-sfix

(defthm symbol-truelist-alistp-of-sfix
  (iff (symbol-truelist-alistp (set::sfix x))
       (or (symbol-truelist-alistp x)
           (not (set::setp x))))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-list-fix

(defthm symbol-truelist-alistp-of-list-fix
  (implies (symbol-truelist-alistp x)
           (symbol-truelist-alistp (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-symbol-truelist-alistp-compound-recognizer

(defthm true-listp-when-symbol-truelist-alistp-compound-recognizer
  (implies (symbol-truelist-alistp x)
           (true-listp x))
  :rule-classes :compound-recognizer)

Theorem: symbol-truelist-alistp-when-not-consp

(defthm symbol-truelist-alistp-when-not-consp
  (implies (not (consp x))
           (equal (symbol-truelist-alistp x)
                  (not x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-cdr-when-symbol-truelist-alistp

(defthm symbol-truelist-alistp-of-cdr-when-symbol-truelist-alistp
  (implies (symbol-truelist-alistp (double-rewrite x))
           (symbol-truelist-alistp (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-cons

(defthm symbol-truelist-alistp-of-cons
  (equal (symbol-truelist-alistp (cons a x))
         (and (and (consp a)
                   (symbolp (car a))
                   (true-listp (cdr a)))
              (symbol-truelist-alistp x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-make-fal

(defthm symbol-truelist-alistp-of-make-fal
  (implies (and (symbol-truelist-alistp x)
                (symbol-truelist-alistp y))
           (symbol-truelist-alistp (make-fal x y)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-of-cdr-when-member-equal-of-symbol-truelist-alistp

(defthm
      true-listp-of-cdr-when-member-equal-of-symbol-truelist-alistp
  (and (implies (and (symbol-truelist-alistp x)
                     (member-equal a x))
                (true-listp (cdr a)))
       (implies (and (member-equal a x)
                     (symbol-truelist-alistp x))
                (true-listp (cdr a))))
  :rule-classes ((:rewrite)))

Theorem: symbolp-of-car-when-member-equal-of-symbol-truelist-alistp

(defthm symbolp-of-car-when-member-equal-of-symbol-truelist-alistp
  (and (implies (and (symbol-truelist-alistp x)
                     (member-equal a x))
                (symbolp (car a)))
       (implies (and (member-equal a x)
                     (symbol-truelist-alistp x))
                (symbolp (car a))))
  :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-symbol-truelist-alistp

(defthm consp-when-member-equal-of-symbol-truelist-alistp
  (implies (and (symbol-truelist-alistp x)
                (member-equal a x))
           (consp a))
  :rule-classes
  ((:rewrite :backchain-limit-lst (0 0))
   (:rewrite :backchain-limit-lst (0 0)
             :corollary (implies (if (member-equal a x)
                                     (symbol-truelist-alistp x)
                                   'nil)
                                 (consp a)))))

Theorem: symbol-truelist-alistp-of-remove-assoc

(defthm symbol-truelist-alistp-of-remove-assoc
  (implies (symbol-truelist-alistp x)
           (symbol-truelist-alistp (remove-assoc-equal name x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-of-cdr-of-assoc-when-symbol-truelist-alistp

(defthm true-listp-of-cdr-of-assoc-when-symbol-truelist-alistp
  (implies (symbol-truelist-alistp x)
           (true-listp (cdr (assoc-equal k x))))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-put-assoc

(defthm symbol-truelist-alistp-of-put-assoc
 (implies (and (symbol-truelist-alistp x))
          (iff (symbol-truelist-alistp (put-assoc-equal name val x))
               (and (symbolp name) (true-listp val))))
 :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-fast-alist-clean

(defthm symbol-truelist-alistp-of-fast-alist-clean
  (implies (symbol-truelist-alistp x)
           (symbol-truelist-alistp (fast-alist-clean x)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-hons-shrink-alist

(defthm symbol-truelist-alistp-of-hons-shrink-alist
  (implies (and (symbol-truelist-alistp x)
                (symbol-truelist-alistp y))
           (symbol-truelist-alistp (hons-shrink-alist x y)))
  :rule-classes ((:rewrite)))

Theorem: symbol-truelist-alistp-of-hons-acons

(defthm symbol-truelist-alistp-of-hons-acons
  (equal (symbol-truelist-alistp (hons-acons a n x))
         (and (symbolp a)
              (true-listp n)
              (symbol-truelist-alistp x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-of-cdr-of-hons-assoc-equal-when-symbol-truelist-alistp

(defthm
  true-listp-of-cdr-of-hons-assoc-equal-when-symbol-truelist-alistp
  (implies (symbol-truelist-alistp x)
           (true-listp (cdr (hons-assoc-equal k x))))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-symbol-truelist-alistp-rewrite

(defthm alistp-when-symbol-truelist-alistp-rewrite
  (implies (symbol-truelist-alistp x)
           (alistp x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-symbol-truelist-alistp

(defthm alistp-when-symbol-truelist-alistp
  (implies (symbol-truelist-alistp x)
           (alistp x))
  :rule-classes :tau-system)

Theorem: true-listp-of-cdar-when-symbol-truelist-alistp

(defthm true-listp-of-cdar-when-symbol-truelist-alistp
  (implies (symbol-truelist-alistp x)
           (true-listp (cdar x)))
  :rule-classes ((:rewrite)))

Theorem: symbolp-of-caar-when-symbol-truelist-alistp

(defthm symbolp-of-caar-when-symbol-truelist-alistp
  (implies (symbol-truelist-alistp x)
           (symbolp (caar x)))
  :rule-classes ((:rewrite)))