Lemmas about nat-listp available in the std/typed-lists library.
Most of these are generated automatically with std::deflist.
BOZO some additional lemmas are found in
Theorem:
(defthm natp-of-car-when-nat-listp (implies (nat-listp x) (iff (natp (car x)) (consp x))) :rule-classes ((:rewrite)))
Theorem:
(defthm true-listp-when-nat-listp-compound-recognizer (implies (nat-listp x) (true-listp x)) :rule-classes :compound-recognizer)
Theorem:
(defthm nat-listp-of-list-fix (implies (nat-listp x) (nat-listp (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-sfix (iff (nat-listp (set::sfix x)) (or (nat-listp x) (not (set::setp x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-insert (iff (nat-listp (set::insert a x)) (and (nat-listp (set::sfix x)) (natp a))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-delete (implies (nat-listp x) (nat-listp (set::delete k x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-mergesort (iff (nat-listp (set::mergesort x)) (nat-listp (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-union (iff (nat-listp (set::union x y)) (and (nat-listp (set::sfix x)) (nat-listp (set::sfix y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-intersect-1 (implies (nat-listp x) (nat-listp (set::intersect x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-intersect-2 (implies (nat-listp y) (nat-listp (set::intersect x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-difference (implies (nat-listp x) (nat-listp (set::difference x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-duplicated-members (implies (nat-listp x) (nat-listp (duplicated-members x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-rev (equal (nat-listp (rev x)) (nat-listp (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-append (equal (nat-listp (append a b)) (and (nat-listp (list-fix a)) (nat-listp b))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-rcons (iff (nat-listp (rcons a x)) (and (natp a) (nat-listp (list-fix x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm natp-when-member-equal-of-nat-listp (and (implies (and (member-equal a x) (nat-listp x)) (natp a)) (implies (and (nat-listp x) (member-equal a x)) (natp a))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-when-subsetp-equal (and (implies (and (subsetp-equal x y) (nat-listp y)) (equal (nat-listp x) (true-listp x))) (implies (and (nat-listp y) (subsetp-equal x y)) (equal (nat-listp x) (true-listp x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-set-difference-equal (implies (nat-listp x) (nat-listp (set-difference-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-intersection-equal-1 (implies (nat-listp (double-rewrite x)) (nat-listp (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-intersection-equal-2 (implies (nat-listp (double-rewrite y)) (nat-listp (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-union-equal (equal (nat-listp (union-equal x y)) (and (nat-listp (list-fix x)) (nat-listp (double-rewrite y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-take (implies (nat-listp (double-rewrite x)) (iff (nat-listp (take n x)) (or (natp nil) (<= (nfix n) (len x))))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-repeat (iff (nat-listp (repeat n x)) (or (natp x) (zp n))) :rule-classes ((:rewrite)))
Theorem:
(defthm natp-of-nth-when-nat-listp (implies (nat-listp x) (iff (natp (nth n x)) (< (nfix n) (len x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-update-nth (implies (nat-listp (double-rewrite x)) (iff (nat-listp (update-nth n y x)) (and (natp y) (or (<= (nfix n) (len x)) (natp nil))))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-butlast (implies (nat-listp (double-rewrite x)) (nat-listp (butlast x n))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-nthcdr (implies (nat-listp (double-rewrite x)) (nat-listp (nthcdr n x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-last (implies (nat-listp (double-rewrite x)) (nat-listp (last x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-remove (implies (nat-listp x) (nat-listp (remove a x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-listp-of-revappend (equal (nat-listp (revappend x y)) (and (nat-listp (list-fix x)) (nat-listp y))) :rule-classes ((:rewrite)))
Theorem:
(defthm integerp-of-car-when-nat-listp (implies (nat-listp x) (equal (integerp (car x)) (consp x))))
Theorem:
(defthm lower-bound-of-car-when-nat-listp (implies (nat-listp x) (<= 0 (car x))) :rule-classes ((:rewrite) (:linear)))
Theorem:
(defthm nat-listp-of-remove-equal (implies (nat-listp x) (nat-listp (remove-equal a x))))
Theorem:
(defthm nat-listp-of-make-list-ac (equal (nat-listp (make-list-ac n x ac)) (and (nat-listp ac) (or (natp x) (zp n)))))
Theorem:
(defthm eqlable-listp-when-nat-listp (implies (nat-listp x) (eqlable-listp x)))