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