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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)))