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• Svex-alist

Svex-alist-p

Recognizer for svex-alist.

Signature

(svex-alist-p x) → *

Definitions and Theorems

Function: svex-alist-p

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

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

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

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

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

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

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

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

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

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

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

Theorem: svex-alist-p-of-rcons

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

Theorem: svex-alist-p-of-append

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

Theorem: svex-alist-p-of-repeat

(defthm svex-alist-p-of-repeat
        (iff (svex-alist-p (repeat acl2::n x))
             (or (and (consp x)
                      (svar-p (car x))
                      (svex-p (cdr x)))
                 (zp acl2::n)))
        :rule-classes ((:rewrite)))

Theorem: svex-alist-p-of-rev

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

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

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

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

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

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

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

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

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

Theorem: svex-alist-p-of-cons

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: alistp-when-svex-alist-p

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

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

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

Theorem: svar-p-of-caar-when-svex-alist-p

(defthm svar-p-of-caar-when-svex-alist-p
        (implies (svex-alist-p x)
                 (iff (svar-p (caar x))
                      (or (consp x) (svar-p nil))))
        :rule-classes ((:rewrite)))