		Search-engine friendly clone of the ACL2 documentation.

Driverlist-p

Driverlist-p-basics

Basic theorems about driverlist-p, generated by std::deflist.

Definitions and Theorems

Theorem: driverlist-p-of-cons

(defthm driverlist-p-of-cons
  (equal (driverlist-p (cons acl2::a x))
         (and (driver-p acl2::a)
              (driverlist-p x)))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-cdr-when-driverlist-p

(defthm driverlist-p-of-cdr-when-driverlist-p
  (implies (driverlist-p (double-rewrite x))
           (driverlist-p (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-when-not-consp

(defthm driverlist-p-when-not-consp
  (implies (not (consp x))
           (driverlist-p x))
  :rule-classes ((:rewrite)))

Theorem: driver-p-of-car-when-driverlist-p

(defthm driver-p-of-car-when-driverlist-p
  (implies (driverlist-p x)
           (iff (driver-p (car x))
                (or (consp x) (driver-p nil))))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-append

(defthm driverlist-p-of-append
  (equal (driverlist-p (append acl2::a acl2::b))
         (and (driverlist-p acl2::a)
              (driverlist-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-list-fix

(defthm driverlist-p-of-list-fix
  (equal (driverlist-p (list-fix x))
         (driverlist-p x))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-rev

(defthm driverlist-p-of-rev
  (equal (driverlist-p (rev x))
         (driverlist-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-repeat

(defthm driverlist-p-of-repeat
  (iff (driverlist-p (repeat acl2::n x))
       (or (driver-p x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-rcons

(defthm driverlist-p-of-rcons
  (iff (driverlist-p (acl2::rcons acl2::a x))
       (and (driver-p acl2::a)
            (driverlist-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: driver-p-when-member-equal-of-driverlist-p

(defthm driver-p-when-member-equal-of-driverlist-p
  (and (implies (and (member-equal acl2::a x)
                     (driverlist-p x))
                (driver-p acl2::a))
       (implies (and (driverlist-p x)
                     (member-equal acl2::a x))
                (driver-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-when-subsetp-equal

(defthm driverlist-p-when-subsetp-equal
  (and (implies (and (subsetp-equal x y)
                     (driverlist-p y))
                (driverlist-p x))
       (implies (and (driverlist-p y)
                     (subsetp-equal x y))
                (driverlist-p x)))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-set-equiv-congruence

(defthm driverlist-p-set-equiv-congruence
  (implies (set-equiv x y)
           (equal (driverlist-p x)
                  (driverlist-p y)))
  :rule-classes :congruence)

Theorem: driverlist-p-of-set-difference-equal

(defthm driverlist-p-of-set-difference-equal
  (implies (driverlist-p x)
           (driverlist-p (set-difference-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-intersection-equal-1

(defthm driverlist-p-of-intersection-equal-1
  (implies (driverlist-p (double-rewrite x))
           (driverlist-p (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-intersection-equal-2

(defthm driverlist-p-of-intersection-equal-2
  (implies (driverlist-p (double-rewrite y))
           (driverlist-p (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-union-equal

(defthm driverlist-p-of-union-equal
  (equal (driverlist-p (union-equal x y))
         (and (driverlist-p (list-fix x))
              (driverlist-p (double-rewrite y))))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-update-nth

(defthm driverlist-p-of-update-nth
  (implies (driverlist-p (double-rewrite x))
           (iff (driverlist-p (update-nth acl2::n y x))
                (and (driver-p y)
                     (or (<= (nfix acl2::n) (len x))
                         (driver-p nil)))))
  :rule-classes ((:rewrite)))

Theorem: driverlist-p-of-butlast

(defthm driverlist-p-of-butlast
  (implies (driverlist-p (double-rewrite x))
           (driverlist-p (butlast x acl2::n)))
  :rule-classes ((:rewrite)))