Basic theorems about cons-listp, generated by std::deflist.
Theorem:
(defthm cons-listp-of-cons (equal (cons-listp (cons a x)) (and (consp a) (cons-listp x))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-cdr-when-cons-listp (implies (cons-listp (double-rewrite x)) (cons-listp (cdr x))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-when-not-consp (implies (not (consp x)) (cons-listp x)) :rule-classes ((:rewrite)))
Theorem:
(defthm consp-of-car-when-cons-listp (implies (cons-listp x) (iff (consp (car x)) (consp x))) :rule-classes ((:rewrite :backchain-limit-lst 0)))
Theorem:
(defthm cons-listp-of-append (equal (cons-listp (append a b)) (and (cons-listp a) (cons-listp b))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-list-fix (equal (cons-listp (list-fix x)) (cons-listp x)) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-sfix (iff (cons-listp (set::sfix x)) (or (cons-listp x) (not (set::setp x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-insert (iff (cons-listp (set::insert a x)) (and (cons-listp (set::sfix x)) (consp a))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-delete (implies (cons-listp x) (cons-listp (set::delete k x))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-mergesort (iff (cons-listp (set::mergesort x)) (cons-listp (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-union (iff (cons-listp (set::union x y)) (and (cons-listp (set::sfix x)) (cons-listp (set::sfix y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-intersect-1 (implies (cons-listp x) (cons-listp (set::intersect x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-intersect-2 (implies (cons-listp y) (cons-listp (set::intersect x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-difference (implies (cons-listp x) (cons-listp (set::difference x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-duplicated-members (implies (cons-listp x) (cons-listp (duplicated-members x))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-rev (equal (cons-listp (rev x)) (cons-listp (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-rcons (iff (cons-listp (rcons a x)) (and (consp a) (cons-listp (list-fix x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm consp-when-member-equal-of-cons-listp (and (implies (and (member-equal a x) (cons-listp x)) (consp a)) (implies (and (cons-listp x) (member-equal a x)) (consp a))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-when-subsetp-equal (and (implies (and (subsetp-equal x y) (cons-listp y)) (cons-listp x)) (implies (and (cons-listp y) (subsetp-equal x y)) (cons-listp x))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-set-equiv-congruence (implies (set-equiv x y) (equal (cons-listp x) (cons-listp y))) :rule-classes :congruence)
Theorem:
(defthm cons-listp-of-set-difference-equal (implies (cons-listp x) (cons-listp (set-difference-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-intersection-equal-1 (implies (cons-listp (double-rewrite x)) (cons-listp (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-intersection-equal-2 (implies (cons-listp (double-rewrite y)) (cons-listp (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-union-equal (equal (cons-listp (union-equal x y)) (and (cons-listp (list-fix x)) (cons-listp (double-rewrite y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-take (implies (cons-listp (double-rewrite x)) (iff (cons-listp (take n x)) (or (consp nil) (<= (nfix n) (len x))))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-repeat (iff (cons-listp (repeat n x)) (or (consp x) (zp n))) :rule-classes ((:rewrite)))
Theorem:
(defthm consp-of-nth-when-cons-listp (implies (cons-listp x) (iff (consp (nth n x)) (< (nfix n) (len x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-update-nth (implies (cons-listp (double-rewrite x)) (iff (cons-listp (update-nth n y x)) (and (consp y) (or (<= (nfix n) (len x)) (consp nil))))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-butlast (implies (cons-listp (double-rewrite x)) (cons-listp (butlast x n))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-nthcdr (implies (cons-listp (double-rewrite x)) (cons-listp (nthcdr n x))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-last (implies (cons-listp (double-rewrite x)) (cons-listp (last x))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-remove (implies (cons-listp x) (cons-listp (remove a x))) :rule-classes ((:rewrite)))
Theorem:
(defthm cons-listp-of-revappend (equal (cons-listp (revappend x y)) (and (cons-listp (list-fix x)) (cons-listp y))) :rule-classes ((:rewrite)))