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• Certificate-listp

Certificate-listp-basics

Basic theorems about certificate-listp, generated by std::deflist.

Definitions and Theorems

Theorem: certificate-listp-of-cons

(defthm certificate-listp-of-cons
  (equal (certificate-listp (cons acl2::a acl2::x))
         (and (certificatep acl2::a)
              (certificate-listp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-cdr-when-certificate-listp

(defthm certificate-listp-of-cdr-when-certificate-listp
  (implies (certificate-listp (double-rewrite acl2::x))
           (certificate-listp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-when-not-consp

(defthm certificate-listp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (certificate-listp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificatep-of-car-when-certificate-listp

(defthm certificatep-of-car-when-certificate-listp
  (implies (certificate-listp acl2::x)
           (iff (certificatep (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-certificate-listp-compound-recognizer

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

Theorem: certificate-listp-of-list-fix

(defthm certificate-listp-of-list-fix
  (implies (certificate-listp acl2::x)
           (certificate-listp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-sfix

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

Theorem: certificate-listp-of-insert

(defthm certificate-listp-of-insert
  (iff (certificate-listp (insert acl2::a acl2::x))
       (and (certificate-listp (sfix acl2::x))
            (certificatep acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-delete

(defthm certificate-listp-of-delete
  (implies (certificate-listp acl2::x)
           (certificate-listp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-mergesort

(defthm certificate-listp-of-mergesort
  (iff (certificate-listp (mergesort acl2::x))
       (certificate-listp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-union

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

Theorem: certificate-listp-of-intersect-1

(defthm certificate-listp-of-intersect-1
  (implies (certificate-listp acl2::x)
           (certificate-listp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-intersect-2

(defthm certificate-listp-of-intersect-2
  (implies (certificate-listp acl2::y)
           (certificate-listp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-difference

(defthm certificate-listp-of-difference
  (implies (certificate-listp acl2::x)
           (certificate-listp (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-duplicated-members

(defthm certificate-listp-of-duplicated-members
  (implies (certificate-listp acl2::x)
           (certificate-listp (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-rev

(defthm certificate-listp-of-rev
  (equal (certificate-listp (rev acl2::x))
         (certificate-listp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-append

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

Theorem: certificate-listp-of-rcons

(defthm certificate-listp-of-rcons
  (iff (certificate-listp (rcons acl2::a acl2::x))
       (and (certificatep acl2::a)
            (certificate-listp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: certificatep-when-member-equal-of-certificate-listp

(defthm certificatep-when-member-equal-of-certificate-listp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (certificate-listp acl2::x))
                (certificatep acl2::a))
       (implies (and (certificate-listp acl2::x)
                     (member-equal acl2::a acl2::x))
                (certificatep acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-when-subsetp-equal

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

Theorem: certificate-listp-of-set-difference-equal

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

Theorem: certificate-listp-of-intersection-equal-1

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

Theorem: certificate-listp-of-intersection-equal-2

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

Theorem: certificate-listp-of-union-equal

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

Theorem: certificate-listp-of-take

(defthm certificate-listp-of-take
  (implies (certificate-listp (double-rewrite acl2::x))
           (iff (certificate-listp (take acl2::n acl2::x))
                (or (certificatep nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-repeat

(defthm certificate-listp-of-repeat
  (iff (certificate-listp (repeat acl2::n acl2::x))
       (or (certificatep acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: certificatep-of-nth-when-certificate-listp

(defthm certificatep-of-nth-when-certificate-listp
  (implies (certificate-listp acl2::x)
           (iff (certificatep (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-update-nth

(defthm certificate-listp-of-update-nth
  (implies
       (certificate-listp (double-rewrite acl2::x))
       (iff (certificate-listp (update-nth acl2::n acl2::y acl2::x))
            (and (certificatep acl2::y)
                 (or (<= (nfix acl2::n) (len acl2::x))
                     (certificatep nil)))))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-butlast

(defthm certificate-listp-of-butlast
  (implies (certificate-listp (double-rewrite acl2::x))
           (certificate-listp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-nthcdr

(defthm certificate-listp-of-nthcdr
  (implies (certificate-listp (double-rewrite acl2::x))
           (certificate-listp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-last

(defthm certificate-listp-of-last
  (implies (certificate-listp (double-rewrite acl2::x))
           (certificate-listp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-remove

(defthm certificate-listp-of-remove
  (implies (certificate-listp acl2::x)
           (certificate-listp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: certificate-listp-of-revappend

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