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• Std/typed-alists

Cons-pos-alistp

Recognize alists from cons pairs to positive integers.

This is an ordinary std::defalist.

Function: cons-pos-alistp

(defun cons-pos-alistp (x)
       (declare (xargs :guard t))
       (if (consp x)
           (and (consp (car x))
                (consp (caar x))
                (posp (cdar x))
                (cons-pos-alistp (cdr x)))
           (null x)))

Definitions and Theorems

Function: cons-pos-alistp

(defun cons-pos-alistp (x)
       (declare (xargs :guard t))
       (if (consp x)
           (and (consp (car x))
                (consp (caar x))
                (posp (cdar x))
                (cons-pos-alistp (cdr x)))
           (null x)))

Theorem: cons-pos-alistp-of-revappend

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

Theorem: cons-pos-alistp-of-remove

(defthm cons-pos-alistp-of-remove
        (implies (cons-pos-alistp x)
                 (cons-pos-alistp (remove a x)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-last

(defthm cons-pos-alistp-of-last
        (implies (cons-pos-alistp (double-rewrite x))
                 (cons-pos-alistp (last x)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-nthcdr

(defthm cons-pos-alistp-of-nthcdr
        (implies (cons-pos-alistp (double-rewrite x))
                 (cons-pos-alistp (nthcdr n x)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-butlast

(defthm cons-pos-alistp-of-butlast
        (implies (cons-pos-alistp (double-rewrite x))
                 (cons-pos-alistp (butlast x n)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-update-nth

(defthm cons-pos-alistp-of-update-nth
        (implies (cons-pos-alistp (double-rewrite x))
                 (iff (cons-pos-alistp (update-nth n y x))
                      (and (and (consp y)
                                (consp (car y))
                                (posp (cdr y)))
                           (or (<= (nfix n) (len x))
                               (and (consp nil)
                                    (consp (car nil))
                                    (posp (cdr nil)))))))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-repeat

(defthm cons-pos-alistp-of-repeat
        (iff (cons-pos-alistp (repeat n x))
             (or (and (consp x)
                      (consp (car x))
                      (posp (cdr x)))
                 (zp n)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-take

(defthm cons-pos-alistp-of-take
        (implies (cons-pos-alistp (double-rewrite x))
                 (iff (cons-pos-alistp (take n x))
                      (or (and (consp nil)
                               (consp (car nil))
                               (posp (cdr nil)))
                          (<= (nfix n) (len x)))))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-union-equal

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

Theorem: cons-pos-alistp-of-intersection-equal-2

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

Theorem: cons-pos-alistp-of-intersection-equal-1

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

Theorem: cons-pos-alistp-of-set-difference-equal

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

Theorem: cons-pos-alistp-when-subsetp-equal

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

Theorem: cons-pos-alistp-of-rcons

(defthm cons-pos-alistp-of-rcons
        (iff (cons-pos-alistp (rcons a x))
             (and (and (consp a)
                       (consp (car a))
                       (posp (cdr a)))
                  (cons-pos-alistp (list-fix x))))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-append

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

Theorem: cons-pos-alistp-of-rev

(defthm cons-pos-alistp-of-rev
        (equal (cons-pos-alistp (rev x))
               (cons-pos-alistp (list-fix x)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-duplicated-members

(defthm cons-pos-alistp-of-duplicated-members
        (implies (cons-pos-alistp x)
                 (cons-pos-alistp (duplicated-members x)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-difference

(defthm cons-pos-alistp-of-difference
        (implies (cons-pos-alistp x)
                 (cons-pos-alistp (set::difference x y)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-intersect-2

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

Theorem: cons-pos-alistp-of-intersect-1

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

Theorem: cons-pos-alistp-of-union

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

Theorem: cons-pos-alistp-of-mergesort

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

Theorem: cons-pos-alistp-of-delete

(defthm cons-pos-alistp-of-delete
        (implies (cons-pos-alistp x)
                 (cons-pos-alistp (set::delete k x)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-insert

(defthm cons-pos-alistp-of-insert
        (iff (cons-pos-alistp (set::insert a x))
             (and (cons-pos-alistp (set::sfix x))
                  (and (consp a)
                       (consp (car a))
                       (posp (cdr a)))))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-sfix

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

Theorem: cons-pos-alistp-of-list-fix

(defthm cons-pos-alistp-of-list-fix
        (implies (cons-pos-alistp x)
                 (cons-pos-alistp (list-fix x)))
        :rule-classes ((:rewrite)))

Theorem: true-listp-when-cons-pos-alistp-compound-recognizer

(defthm true-listp-when-cons-pos-alistp-compound-recognizer
        (implies (cons-pos-alistp x)
                 (true-listp x))
        :rule-classes :compound-recognizer)

Theorem: cons-pos-alistp-when-not-consp

(defthm cons-pos-alistp-when-not-consp
        (implies (not (consp x))
                 (equal (cons-pos-alistp x) (not x)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-cdr-when-cons-pos-alistp

(defthm cons-pos-alistp-of-cdr-when-cons-pos-alistp
        (implies (cons-pos-alistp (double-rewrite x))
                 (cons-pos-alistp (cdr x)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-cons

(defthm cons-pos-alistp-of-cons
        (equal (cons-pos-alistp (cons a x))
               (and (and (consp a)
                         (consp (car a))
                         (posp (cdr a)))
                    (cons-pos-alistp x)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-make-fal

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

Theorem: posp-of-cdr-when-member-equal-of-cons-pos-alistp

(defthm posp-of-cdr-when-member-equal-of-cons-pos-alistp
        (and (implies (and (cons-pos-alistp x)
                           (member-equal a x))
                      (posp (cdr a)))
             (implies (and (member-equal a x)
                           (cons-pos-alistp x))
                      (posp (cdr a))))
        :rule-classes ((:rewrite)))

Theorem: consp-of-car-when-member-equal-of-cons-pos-alistp

(defthm consp-of-car-when-member-equal-of-cons-pos-alistp
        (and (implies (and (cons-pos-alistp x)
                           (member-equal a x))
                      (consp (car a)))
             (implies (and (member-equal a x)
                           (cons-pos-alistp x))
                      (consp (car a))))
        :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-cons-pos-alistp

(defthm consp-when-member-equal-of-cons-pos-alistp
        (implies (and (cons-pos-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)
                                           (cons-pos-alistp x)
                                           'nil)
                                       (consp a)))))

Theorem: cons-pos-alistp-of-remove-assoc

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

Theorem: cons-pos-alistp-of-put-assoc

(defthm cons-pos-alistp-of-put-assoc
        (implies (and (cons-pos-alistp x))
                 (iff (cons-pos-alistp (put-assoc-equal name val x))
                      (and (consp name) (posp val))))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-fast-alist-clean

(defthm cons-pos-alistp-of-fast-alist-clean
        (implies (cons-pos-alistp x)
                 (cons-pos-alistp (fast-alist-clean x)))
        :rule-classes ((:rewrite)))

Theorem: cons-pos-alistp-of-hons-shrink-alist

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

Theorem: cons-pos-alistp-of-hons-acons

(defthm cons-pos-alistp-of-hons-acons
        (equal (cons-pos-alistp (hons-acons a n x))
               (and (consp a)
                    (posp n)
                    (cons-pos-alistp x)))
        :rule-classes ((:rewrite)))

Theorem: posp-of-cdr-of-hons-assoc-equal-when-cons-pos-alistp

(defthm posp-of-cdr-of-hons-assoc-equal-when-cons-pos-alistp
        (implies (cons-pos-alistp x)
                 (iff (posp (cdr (hons-assoc-equal k x)))
                      (hons-assoc-equal k x)))
        :rule-classes ((:rewrite)))

Theorem: alistp-when-cons-pos-alistp-rewrite

(defthm alistp-when-cons-pos-alistp-rewrite
        (implies (cons-pos-alistp x) (alistp x))
        :rule-classes ((:rewrite)))

Theorem: alistp-when-cons-pos-alistp

(defthm alistp-when-cons-pos-alistp
        (implies (cons-pos-alistp x) (alistp x))
        :rule-classes :tau-system)

Theorem: posp-of-cdar-when-cons-pos-alistp

(defthm posp-of-cdar-when-cons-pos-alistp
        (implies (cons-pos-alistp x)
                 (iff (posp (cdar x)) (consp x)))
        :rule-classes ((:rewrite)))

Theorem: consp-of-caar-when-cons-pos-alistp

(defthm consp-of-caar-when-cons-pos-alistp
        (implies (cons-pos-alistp x)
                 (iff (consp (caar x)) (consp x)))
        :rule-classes ((:rewrite)))

Theorem: posp-of-cdr-of-assoc-equal-when-cons-pos-alistp

(defthm posp-of-cdr-of-assoc-equal-when-cons-pos-alistp
        (implies (cons-pos-alistp alist)
                 (iff (posp (cdr (assoc-equal key alist)))
                      (assoc-equal key alist))))