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