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