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Jblockp

Jblockp-basics

Basic theorems about jblockp, generated by std::deflist.

Definitions and Theorems

Theorem: jblockp-of-cons

(defthm jblockp-of-cons
  (equal (jblockp (cons acl2::a acl2::x))
         (and (jstatemp acl2::a)
              (jblockp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: jblockp-of-cdr-when-jblockp

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

Theorem: jblockp-when-not-consp

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

Theorem: jstatemp-of-car-when-jblockp

(defthm jstatemp-of-car-when-jblockp
  (implies (jblockp acl2::x)
           (iff (jstatemp (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

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

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

Theorem: jblockp-of-list-fix

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

Theorem: jblockp-of-sfix

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

Theorem: jblockp-of-insert

(defthm jblockp-of-insert
  (iff (jblockp (insert acl2::a acl2::x))
       (and (jblockp (sfix acl2::x))
            (jstatemp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: jblockp-of-delete

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

Theorem: jblockp-of-mergesort

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

Theorem: jblockp-of-union

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

Theorem: jblockp-of-intersect-1

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

Theorem: jblockp-of-intersect-2

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

Theorem: jblockp-of-difference

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

Theorem: jblockp-of-duplicated-members

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

Theorem: jblockp-of-rev

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

Theorem: jblockp-of-append

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

Theorem: jblockp-of-rcons

(defthm jblockp-of-rcons
  (iff (jblockp (rcons acl2::a acl2::x))
       (and (jstatemp acl2::a)
            (jblockp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: jstatemp-when-member-equal-of-jblockp

(defthm jstatemp-when-member-equal-of-jblockp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (jblockp acl2::x))
                (jstatemp acl2::a))
       (implies (and (jblockp acl2::x)
                     (member-equal acl2::a acl2::x))
                (jstatemp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: jblockp-when-subsetp-equal

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

Theorem: jblockp-of-set-difference-equal

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

Theorem: jblockp-of-intersection-equal-1

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

Theorem: jblockp-of-intersection-equal-2

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

Theorem: jblockp-of-union-equal

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

Theorem: jblockp-of-take

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

Theorem: jblockp-of-repeat

(defthm jblockp-of-repeat
  (iff (jblockp (repeat acl2::n acl2::x))
       (or (jstatemp acl2::x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: jstatemp-of-nth-when-jblockp

(defthm jstatemp-of-nth-when-jblockp
  (implies (jblockp acl2::x)
           (iff (jstatemp (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: jblockp-of-update-nth

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

Theorem: jblockp-of-butlast

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

Theorem: jblockp-of-nthcdr

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

Theorem: jblockp-of-last

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

Theorem: jblockp-of-remove

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

Theorem: jblockp-of-revappend

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