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• Deftreeops-numrange-info-alist

Deftreeops-numrange-info-alistp

Recognizer for deftreeops-numrange-info-alist.

Signature

(deftreeops-numrange-info-alistp x) → *

Definitions and Theorems

Function: deftreeops-numrange-info-alistp

(defun deftreeops-numrange-info-alistp (x)
  (declare (xargs :guard t))
  (let ((__function__ 'deftreeops-numrange-info-alistp))
    (declare (ignorable __function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (num-range-p (caar x))
           (deftreeops-numrange-infop (cdar x))
           (deftreeops-numrange-info-alistp (cdr x))))))

Theorem: deftreeops-numrange-info-alistp-of-revappend

(defthm deftreeops-numrange-info-alistp-of-revappend
  (equal
       (deftreeops-numrange-info-alistp (revappend acl2::x acl2::y))
       (and (deftreeops-numrange-info-alistp (list-fix acl2::x))
            (deftreeops-numrange-info-alistp acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-remove

(defthm deftreeops-numrange-info-alistp-of-remove
  (implies
       (deftreeops-numrange-info-alistp acl2::x)
       (deftreeops-numrange-info-alistp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-last

(defthm deftreeops-numrange-info-alistp-of-last
 (implies (deftreeops-numrange-info-alistp (double-rewrite acl2::x))
          (deftreeops-numrange-info-alistp (last acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-nthcdr

(defthm deftreeops-numrange-info-alistp-of-nthcdr
  (implies
       (deftreeops-numrange-info-alistp (double-rewrite acl2::x))
       (deftreeops-numrange-info-alistp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-butlast

(defthm deftreeops-numrange-info-alistp-of-butlast
  (implies
       (deftreeops-numrange-info-alistp (double-rewrite acl2::x))
       (deftreeops-numrange-info-alistp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-update-nth

(defthm deftreeops-numrange-info-alistp-of-update-nth
  (implies
       (deftreeops-numrange-info-alistp (double-rewrite acl2::x))
       (iff (deftreeops-numrange-info-alistp
                 (update-nth acl2::n acl2::y acl2::x))
            (and (and (consp acl2::y)
                      (num-range-p (car acl2::y))
                      (deftreeops-numrange-infop (cdr acl2::y)))
                 (or (<= (nfix acl2::n) (len acl2::x))
                     (and (consp nil)
                          (num-range-p (car nil))
                          (deftreeops-numrange-infop (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-repeat

(defthm deftreeops-numrange-info-alistp-of-repeat
  (iff (deftreeops-numrange-info-alistp (repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (num-range-p (car acl2::x))
                (deftreeops-numrange-infop (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-take

(defthm deftreeops-numrange-info-alistp-of-take
  (implies
       (deftreeops-numrange-info-alistp (double-rewrite acl2::x))
       (iff (deftreeops-numrange-info-alistp (take acl2::n acl2::x))
            (or (and (consp nil)
                     (num-range-p (car nil))
                     (deftreeops-numrange-infop (cdr nil)))
                (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-union-equal

(defthm deftreeops-numrange-info-alistp-of-union-equal
 (equal
   (deftreeops-numrange-info-alistp (union-equal acl2::x acl2::y))
   (and (deftreeops-numrange-info-alistp (list-fix acl2::x))
        (deftreeops-numrange-info-alistp (double-rewrite acl2::y))))
 :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-intersection-equal-2

(defthm deftreeops-numrange-info-alistp-of-intersection-equal-2
 (implies (deftreeops-numrange-info-alistp (double-rewrite acl2::y))
          (deftreeops-numrange-info-alistp
               (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-intersection-equal-1

(defthm deftreeops-numrange-info-alistp-of-intersection-equal-1
 (implies (deftreeops-numrange-info-alistp (double-rewrite acl2::x))
          (deftreeops-numrange-info-alistp
               (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-set-difference-equal

(defthm deftreeops-numrange-info-alistp-of-set-difference-equal
  (implies (deftreeops-numrange-info-alistp acl2::x)
           (deftreeops-numrange-info-alistp
                (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-when-subsetp-equal

(defthm deftreeops-numrange-info-alistp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (deftreeops-numrange-info-alistp acl2::y))
                (equal (deftreeops-numrange-info-alistp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (deftreeops-numrange-info-alistp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (deftreeops-numrange-info-alistp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-rcons

(defthm deftreeops-numrange-info-alistp-of-rcons
  (iff (deftreeops-numrange-info-alistp (rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (num-range-p (car acl2::a))
                 (deftreeops-numrange-infop (cdr acl2::a)))
            (deftreeops-numrange-info-alistp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-append

(defthm deftreeops-numrange-info-alistp-of-append
  (equal (deftreeops-numrange-info-alistp (append acl2::a acl2::b))
         (and (deftreeops-numrange-info-alistp (list-fix acl2::a))
              (deftreeops-numrange-info-alistp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-rev

(defthm deftreeops-numrange-info-alistp-of-rev
  (equal (deftreeops-numrange-info-alistp (rev acl2::x))
         (deftreeops-numrange-info-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-duplicated-members

(defthm deftreeops-numrange-info-alistp-of-duplicated-members
 (implies
     (deftreeops-numrange-info-alistp acl2::x)
     (deftreeops-numrange-info-alistp (duplicated-members acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-difference

(defthm deftreeops-numrange-info-alistp-of-difference
 (implies
     (deftreeops-numrange-info-alistp acl2::x)
     (deftreeops-numrange-info-alistp (difference acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-intersect-2

(defthm deftreeops-numrange-info-alistp-of-intersect-2
 (implies
      (deftreeops-numrange-info-alistp acl2::y)
      (deftreeops-numrange-info-alistp (intersect acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-intersect-1

(defthm deftreeops-numrange-info-alistp-of-intersect-1
 (implies
      (deftreeops-numrange-info-alistp acl2::x)
      (deftreeops-numrange-info-alistp (intersect acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-union

(defthm deftreeops-numrange-info-alistp-of-union
  (iff (deftreeops-numrange-info-alistp (union acl2::x acl2::y))
       (and (deftreeops-numrange-info-alistp (sfix acl2::x))
            (deftreeops-numrange-info-alistp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-mergesort

(defthm deftreeops-numrange-info-alistp-of-mergesort
  (iff (deftreeops-numrange-info-alistp (mergesort acl2::x))
       (deftreeops-numrange-info-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-delete

(defthm deftreeops-numrange-info-alistp-of-delete
  (implies
       (deftreeops-numrange-info-alistp acl2::x)
       (deftreeops-numrange-info-alistp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-insert

(defthm deftreeops-numrange-info-alistp-of-insert
  (iff (deftreeops-numrange-info-alistp (insert acl2::a acl2::x))
       (and (deftreeops-numrange-info-alistp (sfix acl2::x))
            (and (consp acl2::a)
                 (num-range-p (car acl2::a))
                 (deftreeops-numrange-infop (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-sfix

(defthm deftreeops-numrange-info-alistp-of-sfix
  (iff (deftreeops-numrange-info-alistp (sfix acl2::x))
       (or (deftreeops-numrange-info-alistp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-list-fix

(defthm deftreeops-numrange-info-alistp-of-list-fix
  (implies (deftreeops-numrange-info-alistp acl2::x)
           (deftreeops-numrange-info-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-deftreeops-numrange-info-alistp-compound-recognizer

(defthm
 true-listp-when-deftreeops-numrange-info-alistp-compound-recognizer
 (implies (deftreeops-numrange-info-alistp acl2::x)
          (true-listp acl2::x))
 :rule-classes :compound-recognizer)

Theorem: deftreeops-numrange-info-alistp-when-not-consp

(defthm deftreeops-numrange-info-alistp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (deftreeops-numrange-info-alistp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-cdr-when-deftreeops-numrange-info-alistp

(defthm
 deftreeops-numrange-info-alistp-of-cdr-when-deftreeops-numrange-info-alistp
 (implies (deftreeops-numrange-info-alistp (double-rewrite acl2::x))
          (deftreeops-numrange-info-alistp (cdr acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-cons

(defthm deftreeops-numrange-info-alistp-of-cons
  (equal (deftreeops-numrange-info-alistp (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (num-range-p (car acl2::a))
                   (deftreeops-numrange-infop (cdr acl2::a)))
              (deftreeops-numrange-info-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-make-fal

(defthm deftreeops-numrange-info-alistp-of-make-fal
  (implies
       (and (deftreeops-numrange-info-alistp acl2::x)
            (deftreeops-numrange-info-alistp acl2::y))
       (deftreeops-numrange-info-alistp (make-fal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-infop-of-cdr-when-member-equal-of-deftreeops-numrange-info-alistp

(defthm
 deftreeops-numrange-infop-of-cdr-when-member-equal-of-deftreeops-numrange-info-alistp
 (and (implies (and (deftreeops-numrange-info-alistp acl2::x)
                    (member-equal acl2::a acl2::x))
               (deftreeops-numrange-infop (cdr acl2::a)))
      (implies (and (member-equal acl2::a acl2::x)
                    (deftreeops-numrange-info-alistp acl2::x))
               (deftreeops-numrange-infop (cdr acl2::a))))
 :rule-classes ((:rewrite)))

Theorem: num-range-p-of-car-when-member-equal-of-deftreeops-numrange-info-alistp

(defthm
 num-range-p-of-car-when-member-equal-of-deftreeops-numrange-info-alistp
 (and (implies (and (deftreeops-numrange-info-alistp acl2::x)
                    (member-equal acl2::a acl2::x))
               (num-range-p (car acl2::a)))
      (implies (and (member-equal acl2::a acl2::x)
                    (deftreeops-numrange-info-alistp acl2::x))
               (num-range-p (car acl2::a))))
 :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-deftreeops-numrange-info-alistp

(defthm consp-when-member-equal-of-deftreeops-numrange-info-alistp
 (implies (and (deftreeops-numrange-info-alistp acl2::x)
               (member-equal acl2::a acl2::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 acl2::x)
                           (deftreeops-numrange-info-alistp acl2::x)
                         'nil)
                       (consp acl2::a)))))

Theorem: deftreeops-numrange-info-alistp-of-remove-assoc

(defthm deftreeops-numrange-info-alistp-of-remove-assoc
  (implies (deftreeops-numrange-info-alistp acl2::x)
           (deftreeops-numrange-info-alistp
                (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-put-assoc

(defthm deftreeops-numrange-info-alistp-of-put-assoc
  (implies (and (deftreeops-numrange-info-alistp acl2::x))
           (iff (deftreeops-numrange-info-alistp
                     (put-assoc-equal acl2::name acl2::val acl2::x))
                (and (num-range-p acl2::name)
                     (deftreeops-numrange-infop acl2::val))))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-fast-alist-clean

(defthm deftreeops-numrange-info-alistp-of-fast-alist-clean
  (implies
       (deftreeops-numrange-info-alistp acl2::x)
       (deftreeops-numrange-info-alistp (fast-alist-clean acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-hons-shrink-alist

(defthm deftreeops-numrange-info-alistp-of-hons-shrink-alist
  (implies (and (deftreeops-numrange-info-alistp acl2::x)
                (deftreeops-numrange-info-alistp acl2::y))
           (deftreeops-numrange-info-alistp
                (hons-shrink-alist acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-info-alistp-of-hons-acons

(defthm deftreeops-numrange-info-alistp-of-hons-acons
  (equal (deftreeops-numrange-info-alistp
              (hons-acons acl2::a acl2::n acl2::x))
         (and (num-range-p acl2::a)
              (deftreeops-numrange-infop acl2::n)
              (deftreeops-numrange-info-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: deftreeops-numrange-infop-of-cdr-of-hons-assoc-equal-when-deftreeops-numrange-info-alistp

(defthm
 deftreeops-numrange-infop-of-cdr-of-hons-assoc-equal-when-deftreeops-numrange-info-alistp
 (implies (deftreeops-numrange-info-alistp acl2::x)
          (iff (deftreeops-numrange-infop
                    (cdr (hons-assoc-equal acl2::k acl2::x)))
               (hons-assoc-equal acl2::k acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: alistp-when-deftreeops-numrange-info-alistp-rewrite

(defthm alistp-when-deftreeops-numrange-info-alistp-rewrite
  (implies (deftreeops-numrange-info-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-deftreeops-numrange-info-alistp

(defthm alistp-when-deftreeops-numrange-info-alistp
  (implies (deftreeops-numrange-info-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: deftreeops-numrange-infop-of-cdar-when-deftreeops-numrange-info-alistp

(defthm
 deftreeops-numrange-infop-of-cdar-when-deftreeops-numrange-info-alistp
 (implies (deftreeops-numrange-info-alistp acl2::x)
          (iff (deftreeops-numrange-infop (cdar acl2::x))
               (consp acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: num-range-p-of-caar-when-deftreeops-numrange-info-alistp

(defthm num-range-p-of-caar-when-deftreeops-numrange-info-alistp
  (implies (deftreeops-numrange-info-alistp acl2::x)
           (iff (num-range-p (caar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))