Basic theorems about pathlist-p, generated by std::deflist.
Theorem:
(defthm pathlist-p-of-cons (equal (pathlist-p (cons acl2::a x)) (and (path-p acl2::a) (pathlist-p x))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-cdr-when-pathlist-p (implies (pathlist-p (double-rewrite x)) (pathlist-p (cdr x))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-when-not-consp (implies (not (consp x)) (equal (pathlist-p x) (not x))) :rule-classes ((:rewrite)))
Theorem:
(defthm path-p-of-car-when-pathlist-p (implies (pathlist-p x) (iff (path-p (car x)) (or (consp x) (path-p nil)))) :rule-classes ((:rewrite)))
Theorem:
(defthm true-listp-when-pathlist-p-compound-recognizer (implies (pathlist-p x) (true-listp x)) :rule-classes :compound-recognizer)
Theorem:
(defthm pathlist-p-of-list-fix (implies (pathlist-p x) (pathlist-p (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-rev (equal (pathlist-p (rev x)) (pathlist-p (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-repeat (iff (pathlist-p (repeat acl2::n x)) (or (path-p x) (zp acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-append (equal (pathlist-p (append acl2::a acl2::b)) (and (pathlist-p (list-fix acl2::a)) (pathlist-p acl2::b))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-update-nth (implies (pathlist-p (double-rewrite x)) (iff (pathlist-p (update-nth acl2::n y x)) (and (path-p y) (or (<= (nfix acl2::n) (len x)) (path-p nil))))) :rule-classes ((:rewrite)))
Theorem:
(defthm path-p-of-nth-when-pathlist-p (implies (and (pathlist-p x) (< (nfix acl2::n) (len x))) (path-p (nth acl2::n x))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-take (implies (pathlist-p (double-rewrite x)) (iff (pathlist-p (take acl2::n x)) (or (path-p nil) (<= (nfix acl2::n) (len x))))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-rcons (iff (pathlist-p (acl2::rcons acl2::a x)) (and (path-p acl2::a) (pathlist-p (list-fix x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm path-p-when-member-equal-of-pathlist-p (and (implies (and (member-equal acl2::a x) (pathlist-p x)) (path-p acl2::a)) (implies (and (pathlist-p x) (member-equal acl2::a x)) (path-p acl2::a))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-when-subsetp-equal (and (implies (and (subsetp-equal x y) (pathlist-p y)) (equal (pathlist-p x) (true-listp x))) (implies (and (pathlist-p y) (subsetp-equal x y)) (equal (pathlist-p x) (true-listp x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-set-difference-equal (implies (pathlist-p x) (pathlist-p (set-difference-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-intersection-equal-1 (implies (pathlist-p (double-rewrite x)) (pathlist-p (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-intersection-equal-2 (implies (pathlist-p (double-rewrite y)) (pathlist-p (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm pathlist-p-of-union-equal (equal (pathlist-p (union-equal x y)) (and (pathlist-p (list-fix x)) (pathlist-p (double-rewrite y)))) :rule-classes ((:rewrite)))