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Sig-table

Sig-table-p

Recognizer for sig-table.

Signature
(sig-table-p x) → *

Definitions and Theorems

Function: sig-table-p

(defun sig-table-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'sig-table-p))
    (declare (ignorable __function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (fgl-objectlist-p (caar x))
           (fgl-object-bindingslist-p (cdar x))
           (sig-table-p (cdr x))))))

Theorem: sig-table-p-of-revappend

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

Theorem: sig-table-p-of-remove

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

Theorem: sig-table-p-of-last

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

Theorem: sig-table-p-of-nthcdr

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

Theorem: sig-table-p-of-butlast

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

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

(defthm sig-table-p-of-update-nth
  (implies
       (sig-table-p (double-rewrite x))
       (iff (sig-table-p (update-nth n y x))
            (and (and (consp y)
                      (fgl-objectlist-p (car y))
                      (fgl-object-bindingslist-p (cdr y)))
                 (or (<= (nfix n) (len x))
                     (and (consp nil)
                          (fgl-objectlist-p (car nil))
                          (fgl-object-bindingslist-p (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: sig-table-p-of-repeat

(defthm sig-table-p-of-repeat
  (iff (sig-table-p (acl2::repeat n x))
       (or (and (consp x)
                (fgl-objectlist-p (car x))
                (fgl-object-bindingslist-p (cdr x)))
           (zp n)))
  :rule-classes ((:rewrite)))

Theorem: sig-table-p-of-take

(defthm sig-table-p-of-take
  (implies (sig-table-p (double-rewrite x))
           (iff (sig-table-p (take n x))
                (or (and (consp nil)
                         (fgl-objectlist-p (car nil))
                         (fgl-object-bindingslist-p (cdr nil)))
                    (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

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

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

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

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

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

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

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

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

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

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

Theorem: sig-table-p-of-rcons

(defthm sig-table-p-of-rcons
  (iff (sig-table-p (acl2::rcons a x))
       (and (and (consp a)
                 (fgl-objectlist-p (car a))
                 (fgl-object-bindingslist-p (cdr a)))
            (sig-table-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: sig-table-p-of-append

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

Theorem: sig-table-p-of-rev

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

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

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

Theorem: sig-table-p-of-difference

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

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

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

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

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

Theorem: sig-table-p-of-union

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

Theorem: sig-table-p-of-mergesort

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

Theorem: sig-table-p-of-delete

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

Theorem: sig-table-p-of-insert

(defthm sig-table-p-of-insert
  (iff (sig-table-p (set::insert a x))
       (and (sig-table-p (set::sfix x))
            (and (consp a)
                 (fgl-objectlist-p (car a))
                 (fgl-object-bindingslist-p (cdr a)))))
  :rule-classes ((:rewrite)))

Theorem: sig-table-p-of-sfix

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

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

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

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

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

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

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

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

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

Theorem: sig-table-p-of-cons

(defthm sig-table-p-of-cons
  (equal (sig-table-p (cons a x))
         (and (and (consp a)
                   (fgl-objectlist-p (car a))
                   (fgl-object-bindingslist-p (cdr a)))
              (sig-table-p x)))
  :rule-classes ((:rewrite)))

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

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

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

(defthm sig-table-p-of-put-assoc
  (implies (and (sig-table-p x))
           (iff (sig-table-p (put-assoc-equal name acl2::val x))
                (and (fgl-objectlist-p name)
                     (fgl-object-bindingslist-p acl2::val))))
  :rule-classes ((:rewrite)))

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

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

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

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

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

(defthm sig-table-p-of-hons-acons
  (equal (sig-table-p (hons-acons a n x))
         (and (fgl-objectlist-p a)
              (fgl-object-bindingslist-p n)
              (sig-table-p x)))
  :rule-classes ((:rewrite)))

Theorem: fgl-object-bindingslist-p-of-cdr-of-hons-assoc-equal-when-sig-table-p

(defthm
 fgl-object-bindingslist-p-of-cdr-of-hons-assoc-equal-when-sig-table-p
 (implies
      (sig-table-p x)
      (iff (fgl-object-bindingslist-p (cdr (hons-assoc-equal k x)))
           (or (hons-assoc-equal k x)
               (fgl-object-bindingslist-p nil))))
 :rule-classes ((:rewrite)))

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

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

Theorem: alistp-when-sig-table-p

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

Theorem: fgl-object-bindingslist-p-of-cdar-when-sig-table-p

(defthm fgl-object-bindingslist-p-of-cdar-when-sig-table-p
  (implies (sig-table-p x)
           (iff (fgl-object-bindingslist-p (cdar x))
                (or (consp x)
                    (fgl-object-bindingslist-p nil))))
  :rule-classes ((:rewrite)))

Theorem: fgl-objectlist-p-of-caar-when-sig-table-p

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