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