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• Scratchlist-p

Scratchlist-p-basics

Basic theorems about scratchlist-p, generated by deflist.

Definitions and Theorems

Theorem: scratchlist-p-of-cons

(defthm scratchlist-p-of-cons
  (equal (scratchlist-p (cons a x))
         (and (scratchobj-p a)
              (scratchlist-p x)))
  :rule-classes ((:rewrite)))

Theorem: scratchlist-p-of-cdr-when-scratchlist-p

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

Theorem: scratchlist-p-when-not-consp

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

Theorem: scratchobj-p-of-car-when-scratchlist-p

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

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

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

Theorem: scratchlist-p-of-list-fix

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

Theorem: scratchlist-p-of-sfix

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

Theorem: scratchlist-p-of-insert

(defthm scratchlist-p-of-insert
  (iff (scratchlist-p (set::insert a x))
       (and (scratchlist-p (set::sfix x))
            (scratchobj-p a)))
  :rule-classes ((:rewrite)))

Theorem: scratchlist-p-of-delete

(defthm scratchlist-p-of-delete
  (implies (scratchlist-p x)
           (scratchlist-p (set::delete k x)))
  :rule-classes ((:rewrite)))

Theorem: scratchlist-p-of-mergesort

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

Theorem: scratchlist-p-of-union

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

Theorem: scratchlist-p-of-intersect-1

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

Theorem: scratchlist-p-of-intersect-2

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

Theorem: scratchlist-p-of-difference

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

Theorem: scratchlist-p-of-duplicated-members

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

Theorem: scratchlist-p-of-rev

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

Theorem: scratchlist-p-of-append

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

Theorem: scratchlist-p-of-rcons

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

Theorem: scratchobj-p-when-member-equal-of-scratchlist-p

(defthm scratchobj-p-when-member-equal-of-scratchlist-p
  (and (implies (and (member-equal a x)
                     (scratchlist-p x))
                (scratchobj-p a))
       (implies (and (scratchlist-p x)
                     (member-equal a x))
                (scratchobj-p a)))
  :rule-classes ((:rewrite)))

Theorem: scratchlist-p-when-subsetp-equal

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

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

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

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

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

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

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

Theorem: scratchlist-p-of-union-equal

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

Theorem: scratchlist-p-of-take

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

Theorem: scratchlist-p-of-repeat

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

Theorem: scratchobj-p-of-nth-when-scratchlist-p

(defthm scratchobj-p-of-nth-when-scratchlist-p
  (implies (and (scratchlist-p x)
                (< (nfix n) (len x)))
           (scratchobj-p (nth n x)))
  :rule-classes ((:rewrite)))

Theorem: scratchlist-p-of-update-nth

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

Theorem: scratchlist-p-of-butlast

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

Theorem: scratchlist-p-of-nthcdr

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

Theorem: scratchlist-p-of-last

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

Theorem: scratchlist-p-of-remove

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

Theorem: scratchlist-p-of-revappend

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