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Sinteger-bit-role-listp

Sinteger-bit-role-listp-basics

Basic theorems about sinteger-bit-role-listp, generated by std::deflist.

Definitions and Theorems

Theorem: sinteger-bit-role-listp-of-cons

(defthm sinteger-bit-role-listp-of-cons
  (equal (sinteger-bit-role-listp (cons acl2::a acl2::x))
         (and (sinteger-bit-rolep acl2::a)
              (sinteger-bit-role-listp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-cdr-when-sinteger-bit-role-listp

(defthm sinteger-bit-role-listp-of-cdr-when-sinteger-bit-role-listp
  (implies (sinteger-bit-role-listp (double-rewrite acl2::x))
           (sinteger-bit-role-listp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-when-not-consp

(defthm sinteger-bit-role-listp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (sinteger-bit-role-listp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-rolep-of-car-when-sinteger-bit-role-listp

(defthm sinteger-bit-rolep-of-car-when-sinteger-bit-role-listp
  (implies (sinteger-bit-role-listp acl2::x)
           (iff (sinteger-bit-rolep (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-sinteger-bit-role-listp-compound-recognizer

(defthm true-listp-when-sinteger-bit-role-listp-compound-recognizer
  (implies (sinteger-bit-role-listp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: sinteger-bit-role-listp-of-list-fix

(defthm sinteger-bit-role-listp-of-list-fix
  (implies (sinteger-bit-role-listp acl2::x)
           (sinteger-bit-role-listp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-rev

(defthm sinteger-bit-role-listp-of-rev
  (equal (sinteger-bit-role-listp (rev acl2::x))
         (sinteger-bit-role-listp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-append

(defthm sinteger-bit-role-listp-of-append
  (equal (sinteger-bit-role-listp (append acl2::a acl2::b))
         (and (sinteger-bit-role-listp (list-fix acl2::a))
              (sinteger-bit-role-listp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-last

(defthm sinteger-bit-role-listp-of-last
  (implies (sinteger-bit-role-listp (double-rewrite acl2::x))
           (sinteger-bit-role-listp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-rolep-of-nth-when-sinteger-bit-role-listp

(defthm sinteger-bit-rolep-of-nth-when-sinteger-bit-role-listp
  (implies (sinteger-bit-role-listp acl2::x)
           (iff (sinteger-bit-rolep (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-nthcdr

(defthm sinteger-bit-role-listp-of-nthcdr
  (implies (sinteger-bit-role-listp (double-rewrite acl2::x))
           (sinteger-bit-role-listp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-remove

(defthm sinteger-bit-role-listp-of-remove
  (implies (sinteger-bit-role-listp acl2::x)
           (sinteger-bit-role-listp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-repeat

(defthm sinteger-bit-role-listp-of-repeat
  (iff (sinteger-bit-role-listp (repeat acl2::n acl2::x))
       (or (sinteger-bit-rolep acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-revappend

(defthm sinteger-bit-role-listp-of-revappend
  (equal (sinteger-bit-role-listp (revappend acl2::x acl2::y))
         (and (sinteger-bit-role-listp (list-fix acl2::x))
              (sinteger-bit-role-listp acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-rcons

(defthm sinteger-bit-role-listp-of-rcons
  (iff (sinteger-bit-role-listp (rcons acl2::a acl2::x))
       (and (sinteger-bit-rolep acl2::a)
            (sinteger-bit-role-listp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-rolep-when-member-equal-of-sinteger-bit-role-listp

(defthm
    sinteger-bit-rolep-when-member-equal-of-sinteger-bit-role-listp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (sinteger-bit-role-listp acl2::x))
                (sinteger-bit-rolep acl2::a))
       (implies (and (sinteger-bit-role-listp acl2::x)
                     (member-equal acl2::a acl2::x))
                (sinteger-bit-rolep acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-when-subsetp-equal

(defthm sinteger-bit-role-listp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (sinteger-bit-role-listp acl2::y))
                (equal (sinteger-bit-role-listp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (sinteger-bit-role-listp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (sinteger-bit-role-listp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-set-difference-equal

(defthm sinteger-bit-role-listp-of-set-difference-equal
 (implies
   (sinteger-bit-role-listp acl2::x)
   (sinteger-bit-role-listp (set-difference-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-intersection-equal-1

(defthm sinteger-bit-role-listp-of-intersection-equal-1
 (implies
     (sinteger-bit-role-listp (double-rewrite acl2::x))
     (sinteger-bit-role-listp (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-intersection-equal-2

(defthm sinteger-bit-role-listp-of-intersection-equal-2
 (implies
     (sinteger-bit-role-listp (double-rewrite acl2::y))
     (sinteger-bit-role-listp (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-union-equal

(defthm sinteger-bit-role-listp-of-union-equal
  (equal (sinteger-bit-role-listp (union-equal acl2::x acl2::y))
         (and (sinteger-bit-role-listp (list-fix acl2::x))
              (sinteger-bit-role-listp (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-take

(defthm sinteger-bit-role-listp-of-take
  (implies (sinteger-bit-role-listp (double-rewrite acl2::x))
           (iff (sinteger-bit-role-listp (take acl2::n acl2::x))
                (or (sinteger-bit-rolep nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: sinteger-bit-role-listp-of-update-nth

(defthm sinteger-bit-role-listp-of-update-nth
 (implies
  (sinteger-bit-role-listp (double-rewrite acl2::x))
  (iff
      (sinteger-bit-role-listp (update-nth acl2::n acl2::y acl2::x))
      (and (sinteger-bit-rolep acl2::y)
           (or (<= (nfix acl2::n) (len acl2::x))
               (sinteger-bit-rolep nil)))))
 :rule-classes ((:rewrite)))