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A4veclistlist-p

A4veclistlist-p-basics

Basic theorems about a4veclistlist-p, generated by std::deflist.

Definitions and Theorems

Theorem: a4veclistlist-p-of-cons

(defthm a4veclistlist-p-of-cons
  (equal (a4veclistlist-p (cons acl2::a x))
         (and (a4veclist-p acl2::a)
              (a4veclistlist-p x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-cdr-when-a4veclistlist-p

(defthm a4veclistlist-p-of-cdr-when-a4veclistlist-p
  (implies (a4veclistlist-p (double-rewrite x))
           (a4veclistlist-p (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-when-not-consp

(defthm a4veclistlist-p-when-not-consp
  (implies (not (consp x))
           (equal (a4veclistlist-p x) (not x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclist-p-of-car-when-a4veclistlist-p

(defthm a4veclist-p-of-car-when-a4veclistlist-p
  (implies (a4veclistlist-p x)
           (iff (a4veclist-p (car x))
                (or (consp x) (a4veclist-p nil))))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-a4veclistlist-p-compound-recognizer

(defthm true-listp-when-a4veclistlist-p-compound-recognizer
  (implies (a4veclistlist-p x)
           (true-listp x))
  :rule-classes :compound-recognizer)

Theorem: a4veclistlist-p-of-list-fix

(defthm a4veclistlist-p-of-list-fix
  (implies (a4veclistlist-p x)
           (a4veclistlist-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-sfix

(defthm a4veclistlist-p-of-sfix
  (iff (a4veclistlist-p (sfix x))
       (or (a4veclistlist-p x) (not (setp x))))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-insert

(defthm a4veclistlist-p-of-insert
  (iff (a4veclistlist-p (insert acl2::a x))
       (and (a4veclistlist-p (sfix x))
            (a4veclist-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-delete

(defthm a4veclistlist-p-of-delete
  (implies (a4veclistlist-p x)
           (a4veclistlist-p (delete acl2::k x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-mergesort

(defthm a4veclistlist-p-of-mergesort
  (iff (a4veclistlist-p (mergesort x))
       (a4veclistlist-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-union

(defthm a4veclistlist-p-of-union
  (iff (a4veclistlist-p (union x y))
       (and (a4veclistlist-p (sfix x))
            (a4veclistlist-p (sfix y))))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-intersect-1

(defthm a4veclistlist-p-of-intersect-1
  (implies (a4veclistlist-p x)
           (a4veclistlist-p (intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-intersect-2

(defthm a4veclistlist-p-of-intersect-2
  (implies (a4veclistlist-p y)
           (a4veclistlist-p (intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-difference

(defthm a4veclistlist-p-of-difference
  (implies (a4veclistlist-p x)
           (a4veclistlist-p (difference x y)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-duplicated-members

(defthm a4veclistlist-p-of-duplicated-members
  (implies (a4veclistlist-p x)
           (a4veclistlist-p (duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-rev

(defthm a4veclistlist-p-of-rev
  (equal (a4veclistlist-p (rev x))
         (a4veclistlist-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-append

(defthm a4veclistlist-p-of-append
  (equal (a4veclistlist-p (append acl2::a acl2::b))
         (and (a4veclistlist-p (list-fix acl2::a))
              (a4veclistlist-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-rcons

(defthm a4veclistlist-p-of-rcons
  (iff (a4veclistlist-p (acl2::rcons acl2::a x))
       (and (a4veclist-p acl2::a)
            (a4veclistlist-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: a4veclist-p-when-member-equal-of-a4veclistlist-p

(defthm a4veclist-p-when-member-equal-of-a4veclistlist-p
  (and (implies (and (member-equal acl2::a x)
                     (a4veclistlist-p x))
                (a4veclist-p acl2::a))
       (implies (and (a4veclistlist-p x)
                     (member-equal acl2::a x))
                (a4veclist-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-when-subsetp-equal

(defthm a4veclistlist-p-when-subsetp-equal
  (and (implies (and (subsetp-equal x y)
                     (a4veclistlist-p y))
                (equal (a4veclistlist-p x)
                       (true-listp x)))
       (implies (and (a4veclistlist-p y)
                     (subsetp-equal x y))
                (equal (a4veclistlist-p x)
                       (true-listp x))))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-set-difference-equal

(defthm a4veclistlist-p-of-set-difference-equal
  (implies (a4veclistlist-p x)
           (a4veclistlist-p (set-difference-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-intersection-equal-1

(defthm a4veclistlist-p-of-intersection-equal-1
  (implies (a4veclistlist-p (double-rewrite x))
           (a4veclistlist-p (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-intersection-equal-2

(defthm a4veclistlist-p-of-intersection-equal-2
  (implies (a4veclistlist-p (double-rewrite y))
           (a4veclistlist-p (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-union-equal

(defthm a4veclistlist-p-of-union-equal
  (equal (a4veclistlist-p (union-equal x y))
         (and (a4veclistlist-p (list-fix x))
              (a4veclistlist-p (double-rewrite y))))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-take

(defthm a4veclistlist-p-of-take
  (implies (a4veclistlist-p (double-rewrite x))
           (iff (a4veclistlist-p (take acl2::n x))
                (or (a4veclist-p nil)
                    (<= (nfix acl2::n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-repeat

(defthm a4veclistlist-p-of-repeat
  (iff (a4veclistlist-p (repeat acl2::n x))
       (or (a4veclist-p x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: a4veclist-p-of-nth-when-a4veclistlist-p

(defthm a4veclist-p-of-nth-when-a4veclistlist-p
  (implies (and (a4veclistlist-p x)
                (< (nfix acl2::n) (len x)))
           (a4veclist-p (nth acl2::n x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-update-nth

(defthm a4veclistlist-p-of-update-nth
  (implies (a4veclistlist-p (double-rewrite x))
           (iff (a4veclistlist-p (update-nth acl2::n y x))
                (and (a4veclist-p y)
                     (or (<= (nfix acl2::n) (len x))
                         (a4veclist-p nil)))))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-butlast

(defthm a4veclistlist-p-of-butlast
  (implies (a4veclistlist-p (double-rewrite x))
           (a4veclistlist-p (butlast x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-nthcdr

(defthm a4veclistlist-p-of-nthcdr
  (implies (a4veclistlist-p (double-rewrite x))
           (a4veclistlist-p (nthcdr acl2::n x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-last

(defthm a4veclistlist-p-of-last
  (implies (a4veclistlist-p (double-rewrite x))
           (a4veclistlist-p (last x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-remove

(defthm a4veclistlist-p-of-remove
  (implies (a4veclistlist-p x)
           (a4veclistlist-p (remove acl2::a x)))
  :rule-classes ((:rewrite)))

Theorem: a4veclistlist-p-of-revappend

(defthm a4veclistlist-p-of-revappend
  (equal (a4veclistlist-p (revappend x y))
         (and (a4veclistlist-p (list-fix x))
              (a4veclistlist-p y)))
  :rule-classes ((:rewrite)))