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• Svarlist-addr-p

Svarlist-addr-p-basics

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

Definitions and Theorems

Theorem: svarlist-addr-p-of-cons

(defthm svarlist-addr-p-of-cons
  (equal (svarlist-addr-p (cons acl2::a x))
         (and (svar-addr-p acl2::a)
              (svarlist-addr-p x)))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-of-cdr-when-svarlist-addr-p

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

Theorem: svarlist-addr-p-when-not-consp

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

Theorem: svar-addr-p-of-car-when-svarlist-addr-p

(defthm svar-addr-p-of-car-when-svarlist-addr-p
  (implies (svarlist-addr-p x)
           (iff (svar-addr-p (car x))
                (or (consp x) (svar-addr-p nil))))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-of-append

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

Theorem: svarlist-addr-p-of-list-fix

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

Theorem: svarlist-addr-p-of-rev

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

Theorem: svarlist-addr-p-of-repeat

(defthm svarlist-addr-p-of-repeat
  (iff (svarlist-addr-p (repeat acl2::n x))
       (or (svar-addr-p x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-of-rcons

(defthm svarlist-addr-p-of-rcons
  (iff (svarlist-addr-p (acl2::rcons acl2::a x))
       (and (svar-addr-p acl2::a)
            (svarlist-addr-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: svar-addr-p-when-member-equal-of-svarlist-addr-p

(defthm svar-addr-p-when-member-equal-of-svarlist-addr-p
  (and (implies (and (member-equal acl2::a x)
                     (svarlist-addr-p x))
                (svar-addr-p acl2::a))
       (implies (and (svarlist-addr-p x)
                     (member-equal acl2::a x))
                (svar-addr-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-when-subsetp-equal

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

Theorem: svarlist-addr-p-set-equiv-congruence

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

Theorem: svarlist-addr-p-of-set-difference-equal

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

Theorem: svarlist-addr-p-of-intersection-equal-1

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

Theorem: svarlist-addr-p-of-intersection-equal-2

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

Theorem: svarlist-addr-p-of-union-equal

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

Theorem: svarlist-addr-p-of-sfix

(defthm svarlist-addr-p-of-sfix
  (iff (svarlist-addr-p (sfix x))
       (or (svarlist-addr-p x) (not (setp x))))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-of-insert

(defthm svarlist-addr-p-of-insert
  (iff (svarlist-addr-p (insert acl2::a x))
       (and (svarlist-addr-p (sfix x))
            (svar-addr-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-of-delete

(defthm svarlist-addr-p-of-delete
  (implies (svarlist-addr-p x)
           (svarlist-addr-p (delete acl2::k x)))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-of-mergesort

(defthm svarlist-addr-p-of-mergesort
  (iff (svarlist-addr-p (mergesort x))
       (svarlist-addr-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-of-union

(defthm svarlist-addr-p-of-union
  (iff (svarlist-addr-p (union x y))
       (and (svarlist-addr-p (sfix x))
            (svarlist-addr-p (sfix y))))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-of-intersect-1

(defthm svarlist-addr-p-of-intersect-1
  (implies (svarlist-addr-p x)
           (svarlist-addr-p (intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-of-intersect-2

(defthm svarlist-addr-p-of-intersect-2
  (implies (svarlist-addr-p y)
           (svarlist-addr-p (intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: svarlist-addr-p-of-difference

(defthm svarlist-addr-p-of-difference
  (implies (svarlist-addr-p x)
           (svarlist-addr-p (difference x y)))
  :rule-classes ((:rewrite)))