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Rangemap

Rangemap-p

Recognizer for rangemap.

Signature
(rangemap-p x) → *

Definitions and Theorems

Function: rangemap-p

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

Theorem: rangemap-p-of-revappend

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

Theorem: rangemap-p-of-remove

(defthm rangemap-p-of-remove
  (implies (rangemap-p x)
           (rangemap-p (remove acl2::a x)))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-last

(defthm rangemap-p-of-last
  (implies (rangemap-p (double-rewrite x))
           (rangemap-p (last x)))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-nthcdr

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

Theorem: rangemap-p-of-butlast

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

Theorem: rangemap-p-of-update-nth

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

Theorem: rangemap-p-of-repeat

(defthm rangemap-p-of-repeat
  (iff (rangemap-p (repeat acl2::n x))
       (or (and (consp x)
                (address-p (car x))
                (rangelist-p (cdr x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-take

(defthm rangemap-p-of-take
  (implies (rangemap-p (double-rewrite x))
           (iff (rangemap-p (take acl2::n x))
                (or (and (consp nil)
                         (address-p (car nil))
                         (rangelist-p (cdr nil)))
                    (<= (nfix acl2::n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-union-equal

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

Theorem: rangemap-p-of-intersection-equal-2

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

Theorem: rangemap-p-of-intersection-equal-1

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

Theorem: rangemap-p-of-set-difference-equal

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

Theorem: rangemap-p-when-subsetp-equal

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

Theorem: rangemap-p-of-rcons

(defthm rangemap-p-of-rcons
  (iff (rangemap-p (acl2::rcons acl2::a x))
       (and (and (consp acl2::a)
                 (address-p (car acl2::a))
                 (rangelist-p (cdr acl2::a)))
            (rangemap-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-append

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

Theorem: rangemap-p-of-rev

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

Theorem: rangemap-p-of-duplicated-members

(defthm rangemap-p-of-duplicated-members
  (implies (rangemap-p x)
           (rangemap-p (duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-difference

(defthm rangemap-p-of-difference
  (implies (rangemap-p x)
           (rangemap-p (difference x y)))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-intersect-2

(defthm rangemap-p-of-intersect-2
  (implies (rangemap-p y)
           (rangemap-p (intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-intersect-1

(defthm rangemap-p-of-intersect-1
  (implies (rangemap-p x)
           (rangemap-p (intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-union

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

Theorem: rangemap-p-of-mergesort

(defthm rangemap-p-of-mergesort
  (iff (rangemap-p (mergesort x))
       (rangemap-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-delete

(defthm rangemap-p-of-delete
  (implies (rangemap-p x)
           (rangemap-p (delete acl2::k x)))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-insert

(defthm rangemap-p-of-insert
  (iff (rangemap-p (insert acl2::a x))
       (and (rangemap-p (sfix x))
            (and (consp acl2::a)
                 (address-p (car acl2::a))
                 (rangelist-p (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-sfix

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

Theorem: rangemap-p-of-list-fix

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

Theorem: true-listp-when-rangemap-p-compound-recognizer

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

Theorem: rangemap-p-when-not-consp

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

Theorem: rangemap-p-of-cdr-when-rangemap-p

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

Theorem: rangemap-p-of-cons

(defthm rangemap-p-of-cons
  (equal (rangemap-p (cons acl2::a x))
         (and (and (consp acl2::a)
                   (address-p (car acl2::a))
                   (rangelist-p (cdr acl2::a)))
              (rangemap-p x)))
  :rule-classes ((:rewrite)))

Theorem: rangemap-p-of-make-fal

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

Theorem: rangelist-p-of-cdr-when-member-equal-of-rangemap-p

(defthm rangelist-p-of-cdr-when-member-equal-of-rangemap-p
  (and (implies (and (rangemap-p x)
                     (member-equal acl2::a x))
                (rangelist-p (cdr acl2::a)))
       (implies (and (member-equal acl2::a x)
                     (rangemap-p x))
                (rangelist-p (cdr acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: address-p-of-car-when-member-equal-of-rangemap-p

(defthm address-p-of-car-when-member-equal-of-rangemap-p
  (and (implies (and (rangemap-p x)
                     (member-equal acl2::a x))
                (address-p (car acl2::a)))
       (implies (and (member-equal acl2::a x)
                     (rangemap-p x))
                (address-p (car acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-rangemap-p

(defthm consp-when-member-equal-of-rangemap-p
  (implies (and (rangemap-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)
                                     (rangemap-p x)
                                   'nil)
                                 (consp acl2::a)))))

Theorem: rangemap-p-of-remove-assoc

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

Theorem: rangemap-p-of-put-assoc

(defthm rangemap-p-of-put-assoc
 (implies (and (rangemap-p x))
          (iff (rangemap-p (put-assoc-equal acl2::name acl2::val x))
               (and (address-p acl2::name)
                    (rangelist-p acl2::val))))
 :rule-classes ((:rewrite)))

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

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

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

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

Theorem: rangemap-p-of-hons-acons

(defthm rangemap-p-of-hons-acons
  (equal (rangemap-p (hons-acons acl2::a acl2::n x))
         (and (address-p acl2::a)
              (rangelist-p acl2::n)
              (rangemap-p x)))
  :rule-classes ((:rewrite)))

Theorem: rangelist-p-of-cdr-of-hons-assoc-equal-when-rangemap-p

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

Theorem: alistp-when-rangemap-p-rewrite

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

Theorem: alistp-when-rangemap-p

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

Theorem: rangelist-p-of-cdar-when-rangemap-p

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

Theorem: address-p-of-caar-when-rangemap-p

(defthm address-p-of-caar-when-rangemap-p
  (implies (rangemap-p x)
           (iff (address-p (caar x))
                (or (consp x) (address-p nil))))
  :rule-classes ((:rewrite)))