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Any-table

Any-table-p

Recognizer for any-table.

Signature
(any-table-p x) → *

Definitions and Theorems

Function: any-table-p

(defun any-table-p (x)
  (declare (xargs :guard t))
  (let ((acl2::__function__ 'any-table-p))
    (declare (ignorable acl2::__function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (sig-path-p (caar x))
           (booleanp (cdar x))
           (any-table-p (cdr x))))))

Theorem: any-table-p-of-revappend

(defthm any-table-p-of-revappend
  (equal (any-table-p (revappend acl2::x acl2::y))
         (and (any-table-p (acl2::list-fix acl2::x))
              (any-table-p acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-remove

(defthm any-table-p-of-remove
  (implies (any-table-p acl2::x)
           (any-table-p (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-last

(defthm any-table-p-of-last
  (implies (any-table-p (double-rewrite acl2::x))
           (any-table-p (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-nthcdr

(defthm any-table-p-of-nthcdr
  (implies (any-table-p (double-rewrite acl2::x))
           (any-table-p (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-butlast

(defthm any-table-p-of-butlast
  (implies (any-table-p (double-rewrite acl2::x))
           (any-table-p (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-update-nth

(defthm any-table-p-of-update-nth
  (implies (any-table-p (double-rewrite acl2::x))
           (iff (any-table-p (update-nth acl2::n acl2::y acl2::x))
                (and (and (consp acl2::y)
                          (sig-path-p (car acl2::y))
                          (booleanp (cdr acl2::y)))
                     (or (<= (nfix acl2::n) (len acl2::x))
                         (and (consp nil)
                              (sig-path-p (car nil))
                              (booleanp (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-repeat

(defthm any-table-p-of-repeat
  (iff (any-table-p (acl2::repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (sig-path-p (car acl2::x))
                (booleanp (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-take

(defthm any-table-p-of-take
  (implies (any-table-p (double-rewrite acl2::x))
           (iff (any-table-p (take acl2::n acl2::x))
                (or (and (consp nil)
                         (sig-path-p (car nil))
                         (booleanp (cdr nil)))
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-union-equal

(defthm any-table-p-of-union-equal
  (equal (any-table-p (union-equal acl2::x acl2::y))
         (and (any-table-p (acl2::list-fix acl2::x))
              (any-table-p (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-intersection-equal-2

(defthm any-table-p-of-intersection-equal-2
  (implies (any-table-p (double-rewrite acl2::y))
           (any-table-p (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-intersection-equal-1

(defthm any-table-p-of-intersection-equal-1
  (implies (any-table-p (double-rewrite acl2::x))
           (any-table-p (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-set-difference-equal

(defthm any-table-p-of-set-difference-equal
  (implies (any-table-p acl2::x)
           (any-table-p (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-when-subsetp-equal

(defthm any-table-p-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (any-table-p acl2::y))
                (equal (any-table-p acl2::x)
                       (true-listp acl2::x)))
       (implies (and (any-table-p acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (any-table-p acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-rcons

(defthm any-table-p-of-rcons
  (iff (any-table-p (acl2::rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (sig-path-p (car acl2::a))
                 (booleanp (cdr acl2::a)))
            (any-table-p (acl2::list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-append

(defthm any-table-p-of-append
  (equal (any-table-p (append acl2::a acl2::b))
         (and (any-table-p (acl2::list-fix acl2::a))
              (any-table-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-rev

(defthm any-table-p-of-rev
  (equal (any-table-p (acl2::rev acl2::x))
         (any-table-p (acl2::list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-duplicated-members

(defthm any-table-p-of-duplicated-members
  (implies (any-table-p acl2::x)
           (any-table-p (acl2::duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-difference

(defthm any-table-p-of-difference
  (implies (any-table-p acl2::x)
           (any-table-p (set::difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-intersect-2

(defthm any-table-p-of-intersect-2
  (implies (any-table-p acl2::y)
           (any-table-p (set::intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-intersect-1

(defthm any-table-p-of-intersect-1
  (implies (any-table-p acl2::x)
           (any-table-p (set::intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-union

(defthm any-table-p-of-union
  (iff (any-table-p (set::union acl2::x acl2::y))
       (and (any-table-p (set::sfix acl2::x))
            (any-table-p (set::sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-mergesort

(defthm any-table-p-of-mergesort
  (iff (any-table-p (set::mergesort acl2::x))
       (any-table-p (acl2::list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-delete

(defthm any-table-p-of-delete
  (implies (any-table-p acl2::x)
           (any-table-p (set::delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-insert

(defthm any-table-p-of-insert
  (iff (any-table-p (set::insert acl2::a acl2::x))
       (and (any-table-p (set::sfix acl2::x))
            (and (consp acl2::a)
                 (sig-path-p (car acl2::a))
                 (booleanp (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-sfix

(defthm any-table-p-of-sfix
  (iff (any-table-p (set::sfix acl2::x))
       (or (any-table-p acl2::x)
           (not (set::setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-list-fix

(defthm any-table-p-of-list-fix
  (implies (any-table-p acl2::x)
           (any-table-p (acl2::list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-any-table-p-compound-recognizer

(defthm true-listp-when-any-table-p-compound-recognizer
  (implies (any-table-p acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: any-table-p-when-not-consp

(defthm any-table-p-when-not-consp
  (implies (not (consp acl2::x))
           (equal (any-table-p acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-cdr-when-any-table-p

(defthm any-table-p-of-cdr-when-any-table-p
  (implies (any-table-p (double-rewrite acl2::x))
           (any-table-p (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-cons

(defthm any-table-p-of-cons
  (equal (any-table-p (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (sig-path-p (car acl2::a))
                   (booleanp (cdr acl2::a)))
              (any-table-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-make-fal

(defthm any-table-p-of-make-fal
  (implies (and (any-table-p acl2::x)
                (any-table-p acl2::y))
           (any-table-p (acl2::make-fal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: booleanp-of-cdr-when-member-equal-of-any-table-p

(defthm booleanp-of-cdr-when-member-equal-of-any-table-p
  (and (implies (and (any-table-p acl2::x)
                     (member-equal acl2::a acl2::x))
                (booleanp (cdr acl2::a)))
       (implies (and (member-equal acl2::a acl2::x)
                     (any-table-p acl2::x))
                (booleanp (cdr acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: sig-path-p-of-car-when-member-equal-of-any-table-p

(defthm sig-path-p-of-car-when-member-equal-of-any-table-p
  (and (implies (and (any-table-p acl2::x)
                     (member-equal acl2::a acl2::x))
                (sig-path-p (car acl2::a)))
       (implies (and (member-equal acl2::a acl2::x)
                     (any-table-p acl2::x))
                (sig-path-p (car acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-any-table-p

(defthm consp-when-member-equal-of-any-table-p
  (implies (and (any-table-p 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)
                                     (any-table-p acl2::x)
                                   'nil)
                                 (consp acl2::a)))))

Theorem: any-table-p-of-remove-assoc

(defthm any-table-p-of-remove-assoc
  (implies (any-table-p acl2::x)
           (any-table-p (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-put-assoc

(defthm any-table-p-of-put-assoc
 (implies
   (and (any-table-p acl2::x))
   (iff (any-table-p (put-assoc-equal acl2::name acl2::val acl2::x))
        (and (sig-path-p acl2::name)
             (booleanp acl2::val))))
 :rule-classes ((:rewrite)))

Theorem: any-table-p-of-fast-alist-clean

(defthm any-table-p-of-fast-alist-clean
  (implies (any-table-p acl2::x)
           (any-table-p (fast-alist-clean acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-hons-shrink-alist

(defthm any-table-p-of-hons-shrink-alist
  (implies (and (any-table-p acl2::x)
                (any-table-p acl2::y))
           (any-table-p (hons-shrink-alist acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: any-table-p-of-hons-acons

(defthm any-table-p-of-hons-acons
  (equal (any-table-p (hons-acons acl2::a acl2::n acl2::x))
         (and (sig-path-p acl2::a)
              (booleanp acl2::n)
              (any-table-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: booleanp-of-cdr-of-hons-assoc-equal-when-any-table-p

(defthm booleanp-of-cdr-of-hons-assoc-equal-when-any-table-p
  (implies (any-table-p acl2::x)
           (iff (booleanp (cdr (hons-assoc-equal acl2::k acl2::x)))
                (or (hons-assoc-equal acl2::k acl2::x)
                    (booleanp nil))))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-any-table-p-rewrite

(defthm alistp-when-any-table-p-rewrite
  (implies (any-table-p acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-any-table-p

(defthm alistp-when-any-table-p
  (implies (any-table-p acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: booleanp-of-cdar-when-any-table-p

(defthm booleanp-of-cdar-when-any-table-p
  (implies (any-table-p acl2::x)
           (iff (booleanp (cdar acl2::x))
                (or (consp acl2::x) (booleanp nil))))
  :rule-classes ((:rewrite)))

Theorem: sig-path-p-of-caar-when-any-table-p

(defthm sig-path-p-of-caar-when-any-table-p
  (implies (any-table-p acl2::x)
           (iff (sig-path-p (caar acl2::x))
                (or (consp acl2::x) (sig-path-p nil))))
  :rule-classes ((:rewrite)))