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Dimb-scope

Dimb-scopep

Recognizer for dimb-scope.

Signature
(dimb-scopep x) → *

Definitions and Theorems

Function: dimb-scopep

(defun dimb-scopep (x)
  (declare (xargs :guard t))
  (let ((__function__ 'dimb-scopep))
    (declare (ignorable __function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (identp (caar x))
           (dimb-kindp (cdar x))
           (dimb-scopep (cdr x))))))

Theorem: dimb-scopep-of-revappend

(defthm dimb-scopep-of-revappend
  (equal (dimb-scopep (revappend acl2::x acl2::y))
         (and (dimb-scopep (list-fix acl2::x))
              (dimb-scopep acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-remove

(defthm dimb-scopep-of-remove
  (implies (dimb-scopep acl2::x)
           (dimb-scopep (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-last

(defthm dimb-scopep-of-last
  (implies (dimb-scopep (double-rewrite acl2::x))
           (dimb-scopep (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-nthcdr

(defthm dimb-scopep-of-nthcdr
  (implies (dimb-scopep (double-rewrite acl2::x))
           (dimb-scopep (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-butlast

(defthm dimb-scopep-of-butlast
  (implies (dimb-scopep (double-rewrite acl2::x))
           (dimb-scopep (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-update-nth

(defthm dimb-scopep-of-update-nth
  (implies (dimb-scopep (double-rewrite acl2::x))
           (iff (dimb-scopep (update-nth acl2::n acl2::y acl2::x))
                (and (and (consp acl2::y)
                          (identp (car acl2::y))
                          (dimb-kindp (cdr acl2::y)))
                     (or (<= (nfix acl2::n) (len acl2::x))
                         (and (consp nil)
                              (identp (car nil))
                              (dimb-kindp (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-repeat

(defthm dimb-scopep-of-repeat
  (iff (dimb-scopep (repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (identp (car acl2::x))
                (dimb-kindp (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-take

(defthm dimb-scopep-of-take
  (implies (dimb-scopep (double-rewrite acl2::x))
           (iff (dimb-scopep (take acl2::n acl2::x))
                (or (and (consp nil)
                         (identp (car nil))
                         (dimb-kindp (cdr nil)))
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-union-equal

(defthm dimb-scopep-of-union-equal
  (equal (dimb-scopep (union-equal acl2::x acl2::y))
         (and (dimb-scopep (list-fix acl2::x))
              (dimb-scopep (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-intersection-equal-2

(defthm dimb-scopep-of-intersection-equal-2
  (implies (dimb-scopep (double-rewrite acl2::y))
           (dimb-scopep (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-intersection-equal-1

(defthm dimb-scopep-of-intersection-equal-1
  (implies (dimb-scopep (double-rewrite acl2::x))
           (dimb-scopep (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-set-difference-equal

(defthm dimb-scopep-of-set-difference-equal
  (implies (dimb-scopep acl2::x)
           (dimb-scopep (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-when-subsetp-equal

(defthm dimb-scopep-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (dimb-scopep acl2::y))
                (equal (dimb-scopep acl2::x)
                       (true-listp acl2::x)))
       (implies (and (dimb-scopep acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (dimb-scopep acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-rcons

(defthm dimb-scopep-of-rcons
  (iff (dimb-scopep (rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (identp (car acl2::a))
                 (dimb-kindp (cdr acl2::a)))
            (dimb-scopep (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-append

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

Theorem: dimb-scopep-of-rev

(defthm dimb-scopep-of-rev
  (equal (dimb-scopep (rev acl2::x))
         (dimb-scopep (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-duplicated-members

(defthm dimb-scopep-of-duplicated-members
  (implies (dimb-scopep acl2::x)
           (dimb-scopep (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-difference

(defthm dimb-scopep-of-difference
  (implies (dimb-scopep acl2::x)
           (dimb-scopep (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-intersect-2

(defthm dimb-scopep-of-intersect-2
  (implies (dimb-scopep acl2::y)
           (dimb-scopep (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-intersect-1

(defthm dimb-scopep-of-intersect-1
  (implies (dimb-scopep acl2::x)
           (dimb-scopep (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-union

(defthm dimb-scopep-of-union
  (iff (dimb-scopep (union acl2::x acl2::y))
       (and (dimb-scopep (sfix acl2::x))
            (dimb-scopep (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-mergesort

(defthm dimb-scopep-of-mergesort
  (iff (dimb-scopep (mergesort acl2::x))
       (dimb-scopep (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-delete

(defthm dimb-scopep-of-delete
  (implies (dimb-scopep acl2::x)
           (dimb-scopep (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-insert

(defthm dimb-scopep-of-insert
  (iff (dimb-scopep (insert acl2::a acl2::x))
       (and (dimb-scopep (sfix acl2::x))
            (and (consp acl2::a)
                 (identp (car acl2::a))
                 (dimb-kindp (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-sfix

(defthm dimb-scopep-of-sfix
  (iff (dimb-scopep (sfix acl2::x))
       (or (dimb-scopep acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-list-fix

(defthm dimb-scopep-of-list-fix
  (implies (dimb-scopep acl2::x)
           (dimb-scopep (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-dimb-scopep-compound-recognizer

(defthm true-listp-when-dimb-scopep-compound-recognizer
  (implies (dimb-scopep acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: dimb-scopep-when-not-consp

(defthm dimb-scopep-when-not-consp
  (implies (not (consp acl2::x))
           (equal (dimb-scopep acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-cdr-when-dimb-scopep

(defthm dimb-scopep-of-cdr-when-dimb-scopep
  (implies (dimb-scopep (double-rewrite acl2::x))
           (dimb-scopep (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-cons

(defthm dimb-scopep-of-cons
  (equal (dimb-scopep (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (identp (car acl2::a))
                   (dimb-kindp (cdr acl2::a)))
              (dimb-scopep acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-remove-assoc

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

Theorem: dimb-scopep-of-put-assoc

(defthm dimb-scopep-of-put-assoc
 (implies
   (and (dimb-scopep acl2::x))
   (iff (dimb-scopep (put-assoc-equal acl2::name acl2::val acl2::x))
        (and (identp acl2::name)
             (dimb-kindp acl2::val))))
 :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-fast-alist-clean

(defthm dimb-scopep-of-fast-alist-clean
  (implies (dimb-scopep acl2::x)
           (dimb-scopep (fast-alist-clean acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-hons-shrink-alist

(defthm dimb-scopep-of-hons-shrink-alist
  (implies (and (dimb-scopep acl2::x)
                (dimb-scopep acl2::y))
           (dimb-scopep (hons-shrink-alist acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: dimb-scopep-of-hons-acons

(defthm dimb-scopep-of-hons-acons
  (equal (dimb-scopep (hons-acons acl2::a acl2::n acl2::x))
         (and (identp acl2::a)
              (dimb-kindp acl2::n)
              (dimb-scopep acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: dimb-kindp-of-cdr-of-hons-assoc-equal-when-dimb-scopep

(defthm dimb-kindp-of-cdr-of-hons-assoc-equal-when-dimb-scopep
 (implies (dimb-scopep acl2::x)
          (iff (dimb-kindp (cdr (hons-assoc-equal acl2::k acl2::x)))
               (hons-assoc-equal acl2::k acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: alistp-when-dimb-scopep-rewrite

(defthm alistp-when-dimb-scopep-rewrite
  (implies (dimb-scopep acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-dimb-scopep

(defthm alistp-when-dimb-scopep
  (implies (dimb-scopep acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: dimb-kindp-of-cdar-when-dimb-scopep

(defthm dimb-kindp-of-cdar-when-dimb-scopep
  (implies (dimb-scopep acl2::x)
           (iff (dimb-kindp (cdar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identp-of-caar-when-dimb-scopep

(defthm identp-of-caar-when-dimb-scopep
  (implies (dimb-scopep acl2::x)
           (iff (identp (caar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))