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• Svarlist-p

Svarlist-p-basics

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

Definitions and Theorems

Theorem: svarlist-p-of-cons

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

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

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

Theorem: svarlist-p-when-not-consp

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

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

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

Theorem: true-listp-when-svarlist-p-compound-recognizer

(defthm true-listp-when-svarlist-p-compound-recognizer
  (implies (svarlist-p x) (true-listp x))
  :rule-classes :compound-recognizer)

Theorem: svarlist-p-of-list-fix

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

Theorem: svarlist-p-of-rev

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

Theorem: svarlist-p-of-repeat

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

Theorem: svar-p-of-nth-when-svarlist-p

(defthm svar-p-of-nth-when-svarlist-p
  (implies (svarlist-p x)
           (iff (svar-p (nth acl2::n x))
                (< (nfix acl2::n) (len x))))
  :rule-classes ((:rewrite)))

Theorem: svarlist-p-of-append

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

Theorem: svarlist-p-of-rcons

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

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

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

Theorem: svarlist-p-when-subsetp-equal

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

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

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

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

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

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

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

Theorem: svarlist-p-of-union-equal

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