Basic theorems about addresslist-p, generated by std::deflist.
Theorem:
(defthm addresslist-p-of-cons (equal (addresslist-p (cons acl2::a x)) (and (address-p acl2::a) (addresslist-p x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-cdr-when-addresslist-p (implies (addresslist-p (double-rewrite x)) (addresslist-p (cdr x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-when-not-consp (implies (not (consp x)) (equal (addresslist-p x) (not x))) :rule-classes ((:rewrite)))
Theorem:
(defthm address-p-of-car-when-addresslist-p (implies (addresslist-p x) (iff (address-p (car x)) (or (consp x) (address-p nil)))) :rule-classes ((:rewrite)))
Theorem:
(defthm true-listp-when-addresslist-p-compound-recognizer (implies (addresslist-p x) (true-listp x)) :rule-classes :compound-recognizer)
Theorem:
(defthm addresslist-p-of-list-fix (implies (addresslist-p x) (addresslist-p (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-sfix (iff (addresslist-p (sfix x)) (or (addresslist-p x) (not (setp x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-insert (iff (addresslist-p (insert acl2::a x)) (and (addresslist-p (sfix x)) (address-p acl2::a))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-delete (implies (addresslist-p x) (addresslist-p (delete acl2::k x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-mergesort (iff (addresslist-p (mergesort x)) (addresslist-p (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-union (iff (addresslist-p (union x y)) (and (addresslist-p (sfix x)) (addresslist-p (sfix y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-intersect-1 (implies (addresslist-p x) (addresslist-p (intersect x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-intersect-2 (implies (addresslist-p y) (addresslist-p (intersect x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-difference (implies (addresslist-p x) (addresslist-p (difference x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-duplicated-members (implies (addresslist-p x) (addresslist-p (duplicated-members x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-rev (equal (addresslist-p (rev x)) (addresslist-p (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-append (equal (addresslist-p (append acl2::a acl2::b)) (and (addresslist-p (list-fix acl2::a)) (addresslist-p acl2::b))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-rcons (iff (addresslist-p (acl2::rcons acl2::a x)) (and (address-p acl2::a) (addresslist-p (list-fix x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm address-p-when-member-equal-of-addresslist-p (and (implies (and (member-equal acl2::a x) (addresslist-p x)) (address-p acl2::a)) (implies (and (addresslist-p x) (member-equal acl2::a x)) (address-p acl2::a))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-when-subsetp-equal (and (implies (and (subsetp-equal x y) (addresslist-p y)) (equal (addresslist-p x) (true-listp x))) (implies (and (addresslist-p y) (subsetp-equal x y)) (equal (addresslist-p x) (true-listp x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-set-difference-equal (implies (addresslist-p x) (addresslist-p (set-difference-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-intersection-equal-1 (implies (addresslist-p (double-rewrite x)) (addresslist-p (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-intersection-equal-2 (implies (addresslist-p (double-rewrite y)) (addresslist-p (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-union-equal (equal (addresslist-p (union-equal x y)) (and (addresslist-p (list-fix x)) (addresslist-p (double-rewrite y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-take (implies (addresslist-p (double-rewrite x)) (iff (addresslist-p (take acl2::n x)) (or (address-p nil) (<= (nfix acl2::n) (len x))))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-repeat (iff (addresslist-p (repeat acl2::n x)) (or (address-p x) (zp acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm address-p-of-nth-when-addresslist-p (implies (and (addresslist-p x) (< (nfix acl2::n) (len x))) (address-p (nth acl2::n x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-update-nth (implies (addresslist-p (double-rewrite x)) (iff (addresslist-p (update-nth acl2::n y x)) (and (address-p y) (or (<= (nfix acl2::n) (len x)) (address-p nil))))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-butlast (implies (addresslist-p (double-rewrite x)) (addresslist-p (butlast x acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-nthcdr (implies (addresslist-p (double-rewrite x)) (addresslist-p (nthcdr acl2::n x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-last (implies (addresslist-p (double-rewrite x)) (addresslist-p (last x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-remove (implies (addresslist-p x) (addresslist-p (remove acl2::a x))) :rule-classes ((:rewrite)))
Theorem:
(defthm addresslist-p-of-revappend (equal (addresslist-p (revappend x y)) (and (addresslist-p (list-fix x)) (addresslist-p y))) :rule-classes ((:rewrite)))