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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)))