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