Basic theorems about scratchlist-p, generated by deflist.
Theorem:
(defthm scratchlist-p-of-cons (equal (scratchlist-p (cons a x)) (and (scratchobj-p a) (scratchlist-p x))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-cdr-when-scratchlist-p (implies (scratchlist-p (double-rewrite x)) (scratchlist-p (cdr x))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-when-not-consp (implies (not (consp x)) (equal (scratchlist-p x) (not x))) :rule-classes ((:rewrite)))
Theorem:
(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:
(defthm true-listp-when-scratchlist-p-compound-recognizer (implies (scratchlist-p x) (true-listp x)) :rule-classes :compound-recognizer)
Theorem:
(defthm scratchlist-p-of-list-fix (implies (scratchlist-p x) (scratchlist-p (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-sfix (iff (scratchlist-p (set::sfix x)) (or (scratchlist-p x) (not (set::setp x)))) :rule-classes ((:rewrite)))
Theorem:
(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:
(defthm scratchlist-p-of-delete (implies (scratchlist-p x) (scratchlist-p (set::delete k x))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-mergesort (iff (scratchlist-p (set::mergesort x)) (scratchlist-p (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(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:
(defthm scratchlist-p-of-intersect-1 (implies (scratchlist-p x) (scratchlist-p (set::intersect x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-intersect-2 (implies (scratchlist-p y) (scratchlist-p (set::intersect x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-difference (implies (scratchlist-p x) (scratchlist-p (set::difference x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-duplicated-members (implies (scratchlist-p x) (scratchlist-p (acl2::duplicated-members x))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-rev (equal (scratchlist-p (rev x)) (scratchlist-p (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-append (equal (scratchlist-p (append a b)) (and (scratchlist-p (list-fix a)) (scratchlist-p b))) :rule-classes ((:rewrite)))
Theorem:
(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:
(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:
(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:
(defthm scratchlist-p-of-set-difference-equal (implies (scratchlist-p x) (scratchlist-p (set-difference-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-intersection-equal-1 (implies (scratchlist-p (double-rewrite x)) (scratchlist-p (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-intersection-equal-2 (implies (scratchlist-p (double-rewrite y)) (scratchlist-p (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(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:
(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:
(defthm scratchlist-p-of-repeat (iff (scratchlist-p (acl2::repeat n x)) (or (scratchobj-p x) (zp n))) :rule-classes ((:rewrite)))
Theorem:
(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:
(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:
(defthm scratchlist-p-of-butlast (implies (scratchlist-p (double-rewrite x)) (scratchlist-p (butlast x n))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-nthcdr (implies (scratchlist-p (double-rewrite x)) (scratchlist-p (nthcdr n x))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-last (implies (scratchlist-p (double-rewrite x)) (scratchlist-p (last x))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-remove (implies (scratchlist-p x) (scratchlist-p (remove a x))) :rule-classes ((:rewrite)))
Theorem:
(defthm scratchlist-p-of-revappend (equal (scratchlist-p (revappend x y)) (and (scratchlist-p (list-fix x)) (scratchlist-p y))) :rule-classes ((:rewrite)))