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