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