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• Expdata-input-processing

Expdata-pos-surjmap-alistp

Recognize alists from positive integers to surjective mapping records.

This is an ordinary std::defalist.

Function: expdata-pos-surjmap-alistp

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

Definitions and Theorems

Function: expdata-pos-surjmap-alistp

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

Theorem: expdata-pos-surjmap-alistp-of-revappend

(defthm expdata-pos-surjmap-alistp-of-revappend
  (equal (expdata-pos-surjmap-alistp (revappend acl2::x acl2::y))
         (and (expdata-pos-surjmap-alistp (list-fix acl2::x))
              (expdata-pos-surjmap-alistp acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-remove

(defthm expdata-pos-surjmap-alistp-of-remove
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (expdata-pos-surjmap-alistp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-last

(defthm expdata-pos-surjmap-alistp-of-last
  (implies (expdata-pos-surjmap-alistp (double-rewrite acl2::x))
           (expdata-pos-surjmap-alistp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-nthcdr

(defthm expdata-pos-surjmap-alistp-of-nthcdr
  (implies (expdata-pos-surjmap-alistp (double-rewrite acl2::x))
           (expdata-pos-surjmap-alistp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-butlast

(defthm expdata-pos-surjmap-alistp-of-butlast
  (implies (expdata-pos-surjmap-alistp (double-rewrite acl2::x))
           (expdata-pos-surjmap-alistp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-update-nth

(defthm expdata-pos-surjmap-alistp-of-update-nth
 (implies
  (expdata-pos-surjmap-alistp (double-rewrite acl2::x))
  (iff
   (expdata-pos-surjmap-alistp (update-nth acl2::n acl2::y acl2::x))
   (and (and (consp acl2::y)
             (posp (car acl2::y))
             (expdata-surjmapp (cdr acl2::y)))
        (or (<= (nfix acl2::n) (len acl2::x))
            (and (consp nil)
                 (posp (car nil))
                 (expdata-surjmapp (cdr nil)))))))
 :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-repeat

(defthm expdata-pos-surjmap-alistp-of-repeat
  (iff (expdata-pos-surjmap-alistp (repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (posp (car acl2::x))
                (expdata-surjmapp (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-take

(defthm expdata-pos-surjmap-alistp-of-take
  (implies (expdata-pos-surjmap-alistp (double-rewrite acl2::x))
           (iff (expdata-pos-surjmap-alistp (take acl2::n acl2::x))
                (or (and (consp nil)
                         (posp (car nil))
                         (expdata-surjmapp (cdr nil)))
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-union-equal

(defthm expdata-pos-surjmap-alistp-of-union-equal
 (equal (expdata-pos-surjmap-alistp (union-equal acl2::x acl2::y))
        (and (expdata-pos-surjmap-alistp (list-fix acl2::x))
             (expdata-pos-surjmap-alistp (double-rewrite acl2::y))))
 :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-intersection-equal-2

(defthm expdata-pos-surjmap-alistp-of-intersection-equal-2
 (implies
  (expdata-pos-surjmap-alistp (double-rewrite acl2::y))
  (expdata-pos-surjmap-alistp (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-intersection-equal-1

(defthm expdata-pos-surjmap-alistp-of-intersection-equal-1
 (implies
  (expdata-pos-surjmap-alistp (double-rewrite acl2::x))
  (expdata-pos-surjmap-alistp (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-set-difference-equal

(defthm expdata-pos-surjmap-alistp-of-set-difference-equal
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (expdata-pos-surjmap-alistp
                (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-when-subsetp-equal

(defthm expdata-pos-surjmap-alistp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (expdata-pos-surjmap-alistp acl2::y))
                (equal (expdata-pos-surjmap-alistp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (expdata-pos-surjmap-alistp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (expdata-pos-surjmap-alistp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-rcons

(defthm expdata-pos-surjmap-alistp-of-rcons
  (iff (expdata-pos-surjmap-alistp (rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (posp (car acl2::a))
                 (expdata-surjmapp (cdr acl2::a)))
            (expdata-pos-surjmap-alistp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-append

(defthm expdata-pos-surjmap-alistp-of-append
  (equal (expdata-pos-surjmap-alistp (append acl2::a acl2::b))
         (and (expdata-pos-surjmap-alistp (list-fix acl2::a))
              (expdata-pos-surjmap-alistp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-rev

(defthm expdata-pos-surjmap-alistp-of-rev
  (equal (expdata-pos-surjmap-alistp (rev acl2::x))
         (expdata-pos-surjmap-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-duplicated-members

(defthm expdata-pos-surjmap-alistp-of-duplicated-members
 (implies (expdata-pos-surjmap-alistp acl2::x)
          (expdata-pos-surjmap-alistp (duplicated-members acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-difference

(defthm expdata-pos-surjmap-alistp-of-difference
 (implies (expdata-pos-surjmap-alistp acl2::x)
          (expdata-pos-surjmap-alistp (difference acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-intersect-2

(defthm expdata-pos-surjmap-alistp-of-intersect-2
  (implies (expdata-pos-surjmap-alistp acl2::y)
           (expdata-pos-surjmap-alistp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-intersect-1

(defthm expdata-pos-surjmap-alistp-of-intersect-1
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (expdata-pos-surjmap-alistp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-union

(defthm expdata-pos-surjmap-alistp-of-union
  (iff (expdata-pos-surjmap-alistp (union acl2::x acl2::y))
       (and (expdata-pos-surjmap-alistp (sfix acl2::x))
            (expdata-pos-surjmap-alistp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-mergesort

(defthm expdata-pos-surjmap-alistp-of-mergesort
  (iff (expdata-pos-surjmap-alistp (mergesort acl2::x))
       (expdata-pos-surjmap-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-delete

(defthm expdata-pos-surjmap-alistp-of-delete
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (expdata-pos-surjmap-alistp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-insert

(defthm expdata-pos-surjmap-alistp-of-insert
  (iff (expdata-pos-surjmap-alistp (insert acl2::a acl2::x))
       (and (expdata-pos-surjmap-alistp (sfix acl2::x))
            (and (consp acl2::a)
                 (posp (car acl2::a))
                 (expdata-surjmapp (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-sfix

(defthm expdata-pos-surjmap-alistp-of-sfix
  (iff (expdata-pos-surjmap-alistp (sfix acl2::x))
       (or (expdata-pos-surjmap-alistp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-list-fix

(defthm expdata-pos-surjmap-alistp-of-list-fix
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (expdata-pos-surjmap-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-expdata-pos-surjmap-alistp-compound-recognizer

(defthm
     true-listp-when-expdata-pos-surjmap-alistp-compound-recognizer
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: expdata-pos-surjmap-alistp-when-not-consp

(defthm expdata-pos-surjmap-alistp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (expdata-pos-surjmap-alistp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-cdr-when-expdata-pos-surjmap-alistp

(defthm
  expdata-pos-surjmap-alistp-of-cdr-when-expdata-pos-surjmap-alistp
  (implies (expdata-pos-surjmap-alistp (double-rewrite acl2::x))
           (expdata-pos-surjmap-alistp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-cons

(defthm expdata-pos-surjmap-alistp-of-cons
  (equal (expdata-pos-surjmap-alistp (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (posp (car acl2::a))
                   (expdata-surjmapp (cdr acl2::a)))
              (expdata-pos-surjmap-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-make-fal

(defthm expdata-pos-surjmap-alistp-of-make-fal
  (implies (and (expdata-pos-surjmap-alistp acl2::x)
                (expdata-pos-surjmap-alistp acl2::y))
           (expdata-pos-surjmap-alistp (make-fal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expdata-surjmapp-of-cdr-when-member-equal-of-expdata-pos-surjmap-alistp

(defthm
 expdata-surjmapp-of-cdr-when-member-equal-of-expdata-pos-surjmap-alistp
 (and (implies (and (expdata-pos-surjmap-alistp acl2::x)
                    (member-equal acl2::a acl2::x))
               (expdata-surjmapp (cdr acl2::a)))
      (implies (and (member-equal acl2::a acl2::x)
                    (expdata-pos-surjmap-alistp acl2::x))
               (expdata-surjmapp (cdr acl2::a))))
 :rule-classes ((:rewrite)))

Theorem: posp-of-car-when-member-equal-of-expdata-pos-surjmap-alistp

(defthm posp-of-car-when-member-equal-of-expdata-pos-surjmap-alistp
  (and (implies (and (expdata-pos-surjmap-alistp acl2::x)
                     (member-equal acl2::a acl2::x))
                (posp (car acl2::a)))
       (implies (and (member-equal acl2::a acl2::x)
                     (expdata-pos-surjmap-alistp acl2::x))
                (posp (car acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-expdata-pos-surjmap-alistp

(defthm consp-when-member-equal-of-expdata-pos-surjmap-alistp
  (implies (and (expdata-pos-surjmap-alistp acl2::x)
                (member-equal acl2::a acl2::x))
           (consp acl2::a))
  :rule-classes
  ((:rewrite :backchain-limit-lst (0 0))
   (:rewrite
        :backchain-limit-lst (0 0)
        :corollary (implies (if (member-equal acl2::a acl2::x)
                                (expdata-pos-surjmap-alistp acl2::x)
                              'nil)
                            (consp acl2::a)))))

Theorem: expdata-pos-surjmap-alistp-of-remove-assoc

(defthm expdata-pos-surjmap-alistp-of-remove-assoc
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (expdata-pos-surjmap-alistp
                (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-put-assoc

(defthm expdata-pos-surjmap-alistp-of-put-assoc
  (implies (and (expdata-pos-surjmap-alistp acl2::x))
           (iff (expdata-pos-surjmap-alistp
                     (put-assoc-equal acl2::name acl2::val acl2::x))
                (and (posp acl2::name)
                     (expdata-surjmapp acl2::val))))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-fast-alist-clean

(defthm expdata-pos-surjmap-alistp-of-fast-alist-clean
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (expdata-pos-surjmap-alistp (fast-alist-clean acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-hons-shrink-alist

(defthm expdata-pos-surjmap-alistp-of-hons-shrink-alist
 (implies
   (and (expdata-pos-surjmap-alistp acl2::x)
        (expdata-pos-surjmap-alistp acl2::y))
   (expdata-pos-surjmap-alistp (hons-shrink-alist acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: expdata-pos-surjmap-alistp-of-hons-acons

(defthm expdata-pos-surjmap-alistp-of-hons-acons
 (equal
   (expdata-pos-surjmap-alistp (hons-acons acl2::a acl2::n acl2::x))
   (and (posp acl2::a)
        (expdata-surjmapp acl2::n)
        (expdata-pos-surjmap-alistp acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: expdata-surjmapp-of-cdr-of-hons-assoc-equal-when-expdata-pos-surjmap-alistp

(defthm
 expdata-surjmapp-of-cdr-of-hons-assoc-equal-when-expdata-pos-surjmap-alistp
 (implies
    (expdata-pos-surjmap-alistp acl2::x)
    (iff (expdata-surjmapp (cdr (hons-assoc-equal acl2::k acl2::x)))
         (hons-assoc-equal acl2::k acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: alistp-when-expdata-pos-surjmap-alistp-rewrite

(defthm alistp-when-expdata-pos-surjmap-alistp-rewrite
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-expdata-pos-surjmap-alistp

(defthm alistp-when-expdata-pos-surjmap-alistp
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: expdata-surjmapp-of-cdar-when-expdata-pos-surjmap-alistp

(defthm expdata-surjmapp-of-cdar-when-expdata-pos-surjmap-alistp
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (iff (expdata-surjmapp (cdar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: posp-of-caar-when-expdata-pos-surjmap-alistp

(defthm posp-of-caar-when-expdata-pos-surjmap-alistp
  (implies (expdata-pos-surjmap-alistp acl2::x)
           (iff (posp (caar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expdata-posp-of-key-of-pos-surjmap-alist

(defthm expdata-posp-of-key-of-pos-surjmap-alist
  (implies (and (expdata-pos-surjmap-alistp x)
                (consp (assoc-equal k x)))
           (posp (car (assoc-equal k x)))))

Theorem: expdata-surjmapp-of-val-of-pos-surjmap-alist

(defthm expdata-surjmapp-of-val-of-pos-surjmap-alist
  (implies (and (expdata-pos-surjmap-alistp x)
                (consp (assoc-equal k x)))
           (expdata-surjmapp (cdr (assoc-equal k x)))))