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Sym-nat-alist

Sym-nat-alistp

Signature
(sym-nat-alistp x) → *

Definitions and Theorems

Function: sym-nat-alistp

(defun sym-nat-alistp (x)
  (declare (xargs :guard t))
  (let ((acl2::__function__ 'sym-nat-alistp))
    (declare (ignorable acl2::__function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (symbolp (caar x))
           (natp (cdar x))
           (sym-nat-alistp (cdr x))))))

Theorem: sym-nat-alistp-of-revappend

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

Theorem: sym-nat-alistp-of-remove

(defthm sym-nat-alistp-of-remove
  (implies (sym-nat-alistp acl2::x)
           (sym-nat-alistp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-last

(defthm sym-nat-alistp-of-last
  (implies (sym-nat-alistp (double-rewrite acl2::x))
           (sym-nat-alistp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-nthcdr

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

Theorem: sym-nat-alistp-of-butlast

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

Theorem: sym-nat-alistp-of-update-nth

(defthm sym-nat-alistp-of-update-nth
 (implies (sym-nat-alistp (double-rewrite acl2::x))
          (iff (sym-nat-alistp (update-nth acl2::n acl2::y acl2::x))
               (and (and (consp acl2::y)
                         (symbolp (car acl2::y))
                         (natp (cdr acl2::y)))
                    (or (<= (nfix acl2::n) (len acl2::x))
                        (and (consp nil)
                             (symbolp (car nil))
                             (natp (cdr nil)))))))
 :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-repeat

(defthm sym-nat-alistp-of-repeat
  (iff (sym-nat-alistp (acl2::repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (symbolp (car acl2::x))
                (natp (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-take

(defthm sym-nat-alistp-of-take
  (implies (sym-nat-alistp (double-rewrite acl2::x))
           (iff (sym-nat-alistp (take acl2::n acl2::x))
                (or (and (consp nil)
                         (symbolp (car nil))
                         (natp (cdr nil)))
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-union-equal

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

Theorem: sym-nat-alistp-of-intersection-equal-2

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

Theorem: sym-nat-alistp-of-intersection-equal-1

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

Theorem: sym-nat-alistp-of-set-difference-equal

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

Theorem: sym-nat-alistp-when-subsetp-equal

(defthm sym-nat-alistp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (sym-nat-alistp acl2::y))
                (equal (sym-nat-alistp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (sym-nat-alistp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (sym-nat-alistp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-rcons

(defthm sym-nat-alistp-of-rcons
  (iff (sym-nat-alistp (acl2::rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (symbolp (car acl2::a))
                 (natp (cdr acl2::a)))
            (sym-nat-alistp (acl2::list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-append

(defthm sym-nat-alistp-of-append
  (equal (sym-nat-alistp (append acl2::a acl2::b))
         (and (sym-nat-alistp (acl2::list-fix acl2::a))
              (sym-nat-alistp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-rev

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

Theorem: sym-nat-alistp-of-duplicated-members

(defthm sym-nat-alistp-of-duplicated-members
  (implies (sym-nat-alistp acl2::x)
           (sym-nat-alistp (acl2::duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-difference

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

Theorem: sym-nat-alistp-of-intersect-2

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

Theorem: sym-nat-alistp-of-intersect-1

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

Theorem: sym-nat-alistp-of-union

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

Theorem: sym-nat-alistp-of-mergesort

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

Theorem: sym-nat-alistp-of-delete

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

Theorem: sym-nat-alistp-of-insert

(defthm sym-nat-alistp-of-insert
  (iff (sym-nat-alistp (set::insert acl2::a acl2::x))
       (and (sym-nat-alistp (set::sfix acl2::x))
            (and (consp acl2::a)
                 (symbolp (car acl2::a))
                 (natp (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-sfix

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

Theorem: sym-nat-alistp-of-list-fix

(defthm sym-nat-alistp-of-list-fix
  (implies (sym-nat-alistp acl2::x)
           (sym-nat-alistp (acl2::list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-sym-nat-alistp-compound-recognizer

(defthm true-listp-when-sym-nat-alistp-compound-recognizer
  (implies (sym-nat-alistp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: sym-nat-alistp-when-not-consp

(defthm sym-nat-alistp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (sym-nat-alistp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-cdr-when-sym-nat-alistp

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

Theorem: sym-nat-alistp-of-cons

(defthm sym-nat-alistp-of-cons
  (equal (sym-nat-alistp (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (symbolp (car acl2::a))
                   (natp (cdr acl2::a)))
              (sym-nat-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-make-fal

(defthm sym-nat-alistp-of-make-fal
  (implies (and (sym-nat-alistp acl2::x)
                (sym-nat-alistp acl2::y))
           (sym-nat-alistp (acl2::make-fal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: natp-of-cdr-when-member-equal-of-sym-nat-alistp

(defthm natp-of-cdr-when-member-equal-of-sym-nat-alistp
  (and (implies (and (sym-nat-alistp acl2::x)
                     (member-equal acl2::a acl2::x))
                (natp (cdr acl2::a)))
       (implies (and (member-equal acl2::a acl2::x)
                     (sym-nat-alistp acl2::x))
                (natp (cdr acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: symbolp-of-car-when-member-equal-of-sym-nat-alistp

(defthm symbolp-of-car-when-member-equal-of-sym-nat-alistp
  (and (implies (and (sym-nat-alistp acl2::x)
                     (member-equal acl2::a acl2::x))
                (symbolp (car acl2::a)))
       (implies (and (member-equal acl2::a acl2::x)
                     (sym-nat-alistp acl2::x))
                (symbolp (car acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-sym-nat-alistp

(defthm consp-when-member-equal-of-sym-nat-alistp
  (implies (and (sym-nat-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)
                                     (sym-nat-alistp acl2::x)
                                   'nil)
                                 (consp acl2::a)))))

Theorem: sym-nat-alistp-of-remove-assoc

(defthm sym-nat-alistp-of-remove-assoc
  (implies (sym-nat-alistp acl2::x)
           (sym-nat-alistp (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-put-assoc

(defthm sym-nat-alistp-of-put-assoc
 (implies
  (and (sym-nat-alistp acl2::x))
  (iff
     (sym-nat-alistp (put-assoc-equal acl2::name acl2::val acl2::x))
     (and (symbolp acl2::name)
          (natp acl2::val))))
 :rule-classes ((:rewrite)))

Theorem: sym-nat-alistp-of-fast-alist-clean

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

Theorem: sym-nat-alistp-of-hons-shrink-alist

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

Theorem: sym-nat-alistp-of-hons-acons

(defthm sym-nat-alistp-of-hons-acons
  (equal (sym-nat-alistp (hons-acons acl2::a acl2::n acl2::x))
         (and (symbolp acl2::a)
              (natp acl2::n)
              (sym-nat-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: natp-of-cdr-of-hons-assoc-equal-when-sym-nat-alistp

(defthm natp-of-cdr-of-hons-assoc-equal-when-sym-nat-alistp
  (implies (sym-nat-alistp acl2::x)
           (iff (natp (cdr (hons-assoc-equal acl2::k acl2::x)))
                (or (hons-assoc-equal acl2::k acl2::x)
                    (natp nil))))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-sym-nat-alistp-rewrite

(defthm alistp-when-sym-nat-alistp-rewrite
  (implies (sym-nat-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-sym-nat-alistp

(defthm alistp-when-sym-nat-alistp
  (implies (sym-nat-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: natp-of-cdar-when-sym-nat-alistp

(defthm natp-of-cdar-when-sym-nat-alistp
  (implies (sym-nat-alistp acl2::x)
           (iff (natp (cdar acl2::x))
                (or (consp acl2::x) (natp nil))))
  :rule-classes ((:rewrite)))

Theorem: symbolp-of-caar-when-sym-nat-alistp

(defthm symbolp-of-caar-when-sym-nat-alistp
  (implies (sym-nat-alistp acl2::x)
           (iff (symbolp (caar acl2::x))
                (or (consp acl2::x) (symbolp nil))))
  :rule-classes ((:rewrite)))