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• Svar-size-alist

Svar-size-alist-p

Recognizer for svar-size-alist.

Signature

(svar-size-alist-p x) → *

Definitions and Theorems

Function: svar-size-alist-p

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

Theorem: svar-size-alist-p-of-revappend

(defthm svar-size-alist-p-of-revappend
  (equal (svar-size-alist-p (revappend x y))
         (and (svar-size-alist-p (list-fix x))
              (svar-size-alist-p y)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-remove

(defthm svar-size-alist-p-of-remove
  (implies (svar-size-alist-p x)
           (svar-size-alist-p (remove acl2::a x)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-last

(defthm svar-size-alist-p-of-last
  (implies (svar-size-alist-p (double-rewrite x))
           (svar-size-alist-p (last x)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-nthcdr

(defthm svar-size-alist-p-of-nthcdr
  (implies (svar-size-alist-p (double-rewrite x))
           (svar-size-alist-p (nthcdr acl2::n x)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-butlast

(defthm svar-size-alist-p-of-butlast
  (implies (svar-size-alist-p (double-rewrite x))
           (svar-size-alist-p (butlast x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-update-nth

(defthm svar-size-alist-p-of-update-nth
  (implies (svar-size-alist-p (double-rewrite x))
           (iff (svar-size-alist-p (update-nth acl2::n y x))
                (and (and (consp y)
                          (svar-p (car y))
                          (natp (cdr y)))
                     (or (<= (nfix acl2::n) (len x))
                         (and (consp nil)
                              (svar-p (car nil))
                              (natp (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-repeat

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

Theorem: svar-size-alist-p-of-take

(defthm svar-size-alist-p-of-take
  (implies (svar-size-alist-p (double-rewrite x))
           (iff (svar-size-alist-p (take acl2::n x))
                (or (and (consp nil)
                         (svar-p (car nil))
                         (natp (cdr nil)))
                    (<= (nfix acl2::n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-union-equal

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

Theorem: svar-size-alist-p-of-intersection-equal-2

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

Theorem: svar-size-alist-p-of-intersection-equal-1

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

Theorem: svar-size-alist-p-of-set-difference-equal

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

Theorem: svar-size-alist-p-set-equiv-congruence

(defthm svar-size-alist-p-set-equiv-congruence
  (implies (set-equiv x y)
           (equal (svar-size-alist-p x)
                  (svar-size-alist-p y)))
  :rule-classes :congruence)

Theorem: svar-size-alist-p-when-subsetp-equal

(defthm svar-size-alist-p-when-subsetp-equal
  (and (implies (and (subsetp-equal x y)
                     (svar-size-alist-p y))
                (svar-size-alist-p x))
       (implies (and (svar-size-alist-p y)
                     (subsetp-equal x y))
                (svar-size-alist-p x)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-rcons

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

Theorem: svar-size-alist-p-of-rev

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

Theorem: svar-size-alist-p-of-duplicated-members

(defthm svar-size-alist-p-of-duplicated-members
  (implies (svar-size-alist-p x)
           (svar-size-alist-p (duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-difference

(defthm svar-size-alist-p-of-difference
  (implies (svar-size-alist-p x)
           (svar-size-alist-p (difference x y)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-intersect-2

(defthm svar-size-alist-p-of-intersect-2
  (implies (svar-size-alist-p y)
           (svar-size-alist-p (intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-intersect-1

(defthm svar-size-alist-p-of-intersect-1
  (implies (svar-size-alist-p x)
           (svar-size-alist-p (intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-union

(defthm svar-size-alist-p-of-union
  (iff (svar-size-alist-p (union x y))
       (and (svar-size-alist-p (sfix x))
            (svar-size-alist-p (sfix y))))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-mergesort

(defthm svar-size-alist-p-of-mergesort
  (iff (svar-size-alist-p (mergesort x))
       (svar-size-alist-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-delete

(defthm svar-size-alist-p-of-delete
  (implies (svar-size-alist-p x)
           (svar-size-alist-p (delete acl2::k x)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-insert

(defthm svar-size-alist-p-of-insert
  (iff (svar-size-alist-p (insert acl2::a x))
       (and (svar-size-alist-p (sfix x))
            (and (consp acl2::a)
                 (svar-p (car acl2::a))
                 (natp (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-sfix

(defthm svar-size-alist-p-of-sfix
  (iff (svar-size-alist-p (sfix x))
       (or (svar-size-alist-p x)
           (not (setp x))))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-list-fix

(defthm svar-size-alist-p-of-list-fix
  (equal (svar-size-alist-p (list-fix x))
         (svar-size-alist-p x))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-append

(defthm svar-size-alist-p-of-append
  (equal (svar-size-alist-p (append acl2::a acl2::b))
         (and (svar-size-alist-p acl2::a)
              (svar-size-alist-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-when-not-consp

(defthm svar-size-alist-p-when-not-consp
  (implies (not (consp x))
           (svar-size-alist-p x))
  :rule-classes ((:rewrite)))

Theorem: svar-size-alist-p-of-cdr-when-svar-size-alist-p

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

Theorem: svar-size-alist-p-of-cons

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

Theorem: svar-size-alist-p-of-make-fal

(defthm svar-size-alist-p-of-make-fal
  (implies (and (svar-size-alist-p x)
                (svar-size-alist-p y))
           (svar-size-alist-p (make-fal x y)))
  :rule-classes ((:rewrite)))

Theorem: natp-of-cdr-when-member-equal-of-svar-size-alist-p

(defthm natp-of-cdr-when-member-equal-of-svar-size-alist-p
  (and (implies (and (svar-size-alist-p x)
                     (member-equal acl2::a x))
                (natp (cdr acl2::a)))
       (implies (and (member-equal acl2::a x)
                     (svar-size-alist-p x))
                (natp (cdr acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: svar-p-of-car-when-member-equal-of-svar-size-alist-p

(defthm svar-p-of-car-when-member-equal-of-svar-size-alist-p
  (and (implies (and (svar-size-alist-p x)
                     (member-equal acl2::a x))
                (svar-p (car acl2::a)))
       (implies (and (member-equal acl2::a x)
                     (svar-size-alist-p x))
                (svar-p (car acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-svar-size-alist-p

(defthm consp-when-member-equal-of-svar-size-alist-p
  (implies (and (svar-size-alist-p x)
                (member-equal acl2::a x))
           (consp acl2::a))
  :rule-classes
  ((:rewrite :backchain-limit-lst (0 0))
   (:rewrite :backchain-limit-lst (0 0)
             :corollary (implies (if (member-equal acl2::a x)
                                     (svar-size-alist-p x)
                                   'nil)
                                 (consp acl2::a)))))

Theorem: svar-size-alist-p-of-fast-alist-clean

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

Theorem: svar-size-alist-p-of-hons-shrink-alist

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

Theorem: svar-size-alist-p-of-hons-acons

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

Theorem: natp-of-cdr-of-hons-assoc-equal-when-svar-size-alist-p

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

Theorem: natp-of-cdar-when-svar-size-alist-p

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

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

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