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• Backref-alist

Backref-alist-p

Recognizer for backref-alist.

Signature

(backref-alist-p x) → *

Definitions and Theorems

Function: backref-alist-p

(defun backref-alist-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'backref-alist-p))
    (declare (ignorable __function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (backref-p (cdar x))
           (backref-alist-p (cdr x))))))

Theorem: backref-alist-p-of-undup

(defthm backref-alist-p-of-undup
  (implies (backref-alist-p x)
           (backref-alist-p (undup x)))
  :rule-classes ((:rewrite)))

Theorem: backref-alist-p-of-union-equal

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

Theorem: backref-alist-p-of-intersection-equal-2

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

Theorem: backref-alist-p-of-intersection-equal-1

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

Theorem: backref-alist-p-of-set-difference-equal

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

Theorem: backref-alist-p-when-subsetp-equal

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

Theorem: backref-alist-p-of-rcons

(defthm backref-alist-p-of-rcons
  (iff (backref-alist-p (rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (backref-p (cdr acl2::a)))
            (backref-alist-p (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: backref-alist-p-of-append

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

Theorem: backref-alist-p-of-rev

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

Theorem: backref-alist-p-of-list-fix

(defthm backref-alist-p-of-list-fix
  (implies (backref-alist-p acl2::x)
           (backref-alist-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-backref-alist-p-compound-recognizer

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

Theorem: backref-alist-p-when-not-consp

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

Theorem: backref-alist-p-of-cdr-when-backref-alist-p

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

Theorem: backref-alist-p-of-cons

(defthm backref-alist-p-of-cons
  (equal (backref-alist-p (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (backref-p (cdr acl2::a)))
              (backref-alist-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: backref-alist-p-of-remove-assoc

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

Theorem: backref-alist-p-of-put-assoc

(defthm backref-alist-p-of-put-assoc
 (implies
  (and (backref-alist-p acl2::x))
  (iff
    (backref-alist-p (put-assoc-equal acl2::name acl2::val acl2::x))
    (and t (backref-p acl2::val))))
 :rule-classes ((:rewrite)))

Theorem: backref-alist-p-of-fast-alist-clean

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

Theorem: backref-alist-p-of-hons-shrink-alist

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

Theorem: backref-alist-p-of-hons-acons

(defthm backref-alist-p-of-hons-acons
  (equal (backref-alist-p (hons-acons acl2::a acl2::n acl2::x))
         (and t (backref-p acl2::n)
              (backref-alist-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: backref-p-of-cdr-of-hons-assoc-equal-when-backref-alist-p

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

Theorem: alistp-when-backref-alist-p-rewrite

(defthm alistp-when-backref-alist-p-rewrite
  (implies (backref-alist-p acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-backref-alist-p

(defthm alistp-when-backref-alist-p
  (implies (backref-alist-p acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: backref-p-of-cdar-when-backref-alist-p

(defthm backref-p-of-cdar-when-backref-alist-p
  (implies (backref-alist-p acl2::x)
           (iff (backref-p (cdar acl2::x))
                (or (consp acl2::x) (backref-p nil))))
  :rule-classes ((:rewrite)))