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Vl-binary-digit-p

x_digit | z_digit | 0 | 1

Definitions and Theorems

Function: vl-binary-digit-p$inline

(defun vl-binary-digit-p$inline (x)
  (declare (type character x))
  (and (mbt (characterp x))
       (or (eql x #\0)
           (eql x #\1)
           (vl-x-digit-p x)
           (vl-z-digit-p x))))

Function: vl-binary-digit-echar-p$inline

(defun vl-binary-digit-echar-p$inline (x)
  (declare (xargs :guard (vl-echar-p x)))
  (vl-binary-digit-p (vl-echar->char x)))

Function: vl-binary-digit-list-p

(defun vl-binary-digit-list-p (x)
  (declare (xargs :guard (character-listp x)))
  (let ((__function__ 'vl-binary-digit-list-p))
    (declare (ignorable __function__))
    (if (consp x)
        (and (vl-binary-digit-p (car x))
             (vl-binary-digit-list-p (cdr x)))
      t)))

Theorem: vl-binary-digit-list-p-of-cons

(defthm vl-binary-digit-list-p-of-cons
  (equal (vl-binary-digit-list-p (cons acl2::a acl2::x))
         (and (vl-binary-digit-p acl2::a)
              (vl-binary-digit-list-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-cdr-when-vl-binary-digit-list-p

(defthm vl-binary-digit-list-p-of-cdr-when-vl-binary-digit-list-p
  (implies (vl-binary-digit-list-p (double-rewrite acl2::x))
           (vl-binary-digit-list-p (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-when-not-consp

(defthm vl-binary-digit-list-p-when-not-consp
  (implies (not (consp acl2::x))
           (vl-binary-digit-list-p acl2::x))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-p-of-car-when-vl-binary-digit-list-p

(defthm vl-binary-digit-p-of-car-when-vl-binary-digit-list-p
  (implies (vl-binary-digit-list-p acl2::x)
           (iff (vl-binary-digit-p (car acl2::x))
                (or (consp acl2::x)
                    (vl-binary-digit-p nil))))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-append

(defthm vl-binary-digit-list-p-of-append
  (equal (vl-binary-digit-list-p (append acl2::a acl2::b))
         (and (vl-binary-digit-list-p acl2::a)
              (vl-binary-digit-list-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-list-fix

(defthm vl-binary-digit-list-p-of-list-fix
  (equal (vl-binary-digit-list-p (list-fix acl2::x))
         (vl-binary-digit-list-p acl2::x))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-sfix

(defthm vl-binary-digit-list-p-of-sfix
  (iff (vl-binary-digit-list-p (sfix acl2::x))
       (or (vl-binary-digit-list-p acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-insert

(defthm vl-binary-digit-list-p-of-insert
  (iff (vl-binary-digit-list-p (insert acl2::a acl2::x))
       (and (vl-binary-digit-list-p (sfix acl2::x))
            (vl-binary-digit-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-delete

(defthm vl-binary-digit-list-p-of-delete
  (implies (vl-binary-digit-list-p acl2::x)
           (vl-binary-digit-list-p (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-mergesort

(defthm vl-binary-digit-list-p-of-mergesort
  (iff (vl-binary-digit-list-p (mergesort acl2::x))
       (vl-binary-digit-list-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-union

(defthm vl-binary-digit-list-p-of-union
  (iff (vl-binary-digit-list-p (union acl2::x acl2::y))
       (and (vl-binary-digit-list-p (sfix acl2::x))
            (vl-binary-digit-list-p (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-intersect-1

(defthm vl-binary-digit-list-p-of-intersect-1
  (implies (vl-binary-digit-list-p acl2::x)
           (vl-binary-digit-list-p (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-intersect-2

(defthm vl-binary-digit-list-p-of-intersect-2
  (implies (vl-binary-digit-list-p acl2::y)
           (vl-binary-digit-list-p (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-difference

(defthm vl-binary-digit-list-p-of-difference
  (implies (vl-binary-digit-list-p acl2::x)
           (vl-binary-digit-list-p (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-duplicated-members

(defthm vl-binary-digit-list-p-of-duplicated-members
  (implies (vl-binary-digit-list-p acl2::x)
           (vl-binary-digit-list-p (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-rev

(defthm vl-binary-digit-list-p-of-rev
  (equal (vl-binary-digit-list-p (rev acl2::x))
         (vl-binary-digit-list-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-rcons

(defthm vl-binary-digit-list-p-of-rcons
  (iff (vl-binary-digit-list-p (acl2::rcons acl2::a acl2::x))
       (and (vl-binary-digit-p acl2::a)
            (vl-binary-digit-list-p (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-p-when-member-equal-of-vl-binary-digit-list-p

(defthm
      vl-binary-digit-p-when-member-equal-of-vl-binary-digit-list-p
  (and (implies (and (member-equal acl2::a acl2::x)
                     (vl-binary-digit-list-p acl2::x))
                (vl-binary-digit-p acl2::a))
       (implies (and (vl-binary-digit-list-p acl2::x)
                     (member-equal acl2::a acl2::x))
                (vl-binary-digit-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-when-subsetp-equal

(defthm vl-binary-digit-list-p-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (vl-binary-digit-list-p acl2::y))
                (vl-binary-digit-list-p acl2::x))
       (implies (and (vl-binary-digit-list-p acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (vl-binary-digit-list-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-set-equiv-congruence

(defthm vl-binary-digit-list-p-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (vl-binary-digit-list-p acl2::x)
                  (vl-binary-digit-list-p acl2::y)))
  :rule-classes :congruence)

Theorem: vl-binary-digit-list-p-of-set-difference-equal

(defthm vl-binary-digit-list-p-of-set-difference-equal
 (implies
    (vl-binary-digit-list-p acl2::x)
    (vl-binary-digit-list-p (set-difference-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-intersection-equal-1

(defthm vl-binary-digit-list-p-of-intersection-equal-1
 (implies
      (vl-binary-digit-list-p (double-rewrite acl2::x))
      (vl-binary-digit-list-p (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-intersection-equal-2

(defthm vl-binary-digit-list-p-of-intersection-equal-2
 (implies
      (vl-binary-digit-list-p (double-rewrite acl2::y))
      (vl-binary-digit-list-p (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-union-equal

(defthm vl-binary-digit-list-p-of-union-equal
  (equal (vl-binary-digit-list-p (union-equal acl2::x acl2::y))
         (and (vl-binary-digit-list-p (list-fix acl2::x))
              (vl-binary-digit-list-p (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-take

(defthm vl-binary-digit-list-p-of-take
  (implies (vl-binary-digit-list-p (double-rewrite acl2::x))
           (iff (vl-binary-digit-list-p (take acl2::n acl2::x))
                (or (vl-binary-digit-p nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-repeat

(defthm vl-binary-digit-list-p-of-repeat
  (iff (vl-binary-digit-list-p (repeat acl2::n acl2::x))
       (or (vl-binary-digit-p acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-p-of-nth-when-vl-binary-digit-list-p

(defthm vl-binary-digit-p-of-nth-when-vl-binary-digit-list-p
  (implies (and (vl-binary-digit-list-p acl2::x)
                (< (nfix acl2::n) (len acl2::x)))
           (vl-binary-digit-p (nth acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-update-nth

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

Theorem: vl-binary-digit-list-p-of-butlast

(defthm vl-binary-digit-list-p-of-butlast
  (implies (vl-binary-digit-list-p (double-rewrite acl2::x))
           (vl-binary-digit-list-p (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-nthcdr

(defthm vl-binary-digit-list-p-of-nthcdr
  (implies (vl-binary-digit-list-p (double-rewrite acl2::x))
           (vl-binary-digit-list-p (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-last

(defthm vl-binary-digit-list-p-of-last
  (implies (vl-binary-digit-list-p (double-rewrite acl2::x))
           (vl-binary-digit-list-p (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-remove

(defthm vl-binary-digit-list-p-of-remove
  (implies (vl-binary-digit-list-p acl2::x)
           (vl-binary-digit-list-p (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-binary-digit-list-p-of-revappend

(defthm vl-binary-digit-list-p-of-revappend
  (equal (vl-binary-digit-list-p (revappend acl2::x acl2::y))
         (and (vl-binary-digit-list-p (list-fix acl2::x))
              (vl-binary-digit-list-p acl2::y)))
  :rule-classes ((:rewrite)))

Function: vl-read-while-binary-digit-impl

(defun vl-read-while-binary-digit-impl (echars acc)
  (declare (xargs :guard (vl-echarlist-p echars)))
  (cond ((atom echars) (mv acc echars))
        ((vl-binary-digit-p (vl-echar->char (car echars)))
         (vl-read-while-binary-digit-impl (cdr echars)
                                          (cons (car echars) acc)))
        (t (mv acc echars))))

Function: vl-read-while-binary-digit$inline

(defun vl-read-while-binary-digit$inline (echars)
  (declare (xargs :guard (vl-echarlist-p echars)))
  (mbe :logic
       (cond ((atom echars) (mv nil echars))
             ((vl-binary-digit-p (vl-echar->char (car echars)))
              (mv-let (prefix remainder)
                      (vl-read-while-binary-digit (cdr echars))
                (mv (cons (car echars) prefix)
                    remainder)))
             (t (mv nil echars)))
       :exec
       (mv-let (prefix-rev remainder)
               (vl-read-while-binary-digit-impl echars nil)
         (mv (reverse prefix-rev) remainder))))

Theorem: prefix-of-vl-read-while-binary-digit

(defthm prefix-of-vl-read-while-binary-digit
 (and
  (true-listp (mv-nth 0 (vl-read-while-binary-digit echars)))
  (implies
     (force (vl-echarlist-p echars))
     (vl-echarlist-p (mv-nth 0
                             (vl-read-while-binary-digit echars)))))
 :rule-classes
 ((:rewrite)
  (:type-prescription
       :corollary
       (true-listp (mv-nth 0
                           (vl-read-while-binary-digit echars))))))

Theorem: remainder-of-vl-read-while-binary-digit

(defthm remainder-of-vl-read-while-binary-digit
 (and
  (equal (true-listp (mv-nth 1 (vl-read-while-binary-digit echars)))
         (true-listp echars))
  (implies
     (vl-echarlist-p echars)
     (vl-echarlist-p (mv-nth 1
                             (vl-read-while-binary-digit echars)))))
 :rule-classes
 ((:rewrite)
  (:type-prescription
   :corollary
   (implies
       (true-listp echars)
       (true-listp (mv-nth 1
                           (vl-read-while-binary-digit echars)))))))

Theorem: prefix-of-vl-read-while-binary-digit-when-vl-binary-digit-p

(defthm prefix-of-vl-read-while-binary-digit-when-vl-binary-digit-p
  (implies (vl-binary-digit-p (vl-echar->char (car echars)))
           (iff (mv-nth 0 (vl-read-while-binary-digit echars))
                (consp echars))))

Theorem: vl-read-while-binary-digit-sound

(defthm vl-read-while-binary-digit-sound
  (vl-binary-digit-list-p
       (vl-echarlist->chars
            (mv-nth 0
                    (vl-read-while-binary-digit echars)))))

Theorem: vl-read-while-binary-digit-complete

(defthm vl-read-while-binary-digit-complete
 (equal
      (vl-binary-digit-p
           (vl-echar->char
                (car (mv-nth 1
                             (vl-read-while-binary-digit echars)))))
      (if (consp (mv-nth 1 (vl-read-while-binary-digit echars)))
          nil
        (vl-binary-digit-p (vl-echar->char nil)))))

Theorem: append-of-vl-read-while-binary-digit

(defthm append-of-vl-read-while-binary-digit
  (equal (append (mv-nth 0 (vl-read-while-binary-digit echars))
                 (mv-nth 1 (vl-read-while-binary-digit echars)))
         echars))

Theorem: no-change-loser-of-vl-read-while-binary-digit

(defthm no-change-loser-of-vl-read-while-binary-digit
  (implies (not (mv-nth 0 (vl-read-while-binary-digit echars)))
           (equal (mv-nth 1 (vl-read-while-binary-digit echars))
                  echars)))

Theorem: acl2-count-of-vl-read-while-binary-digit-weak

(defthm acl2-count-of-vl-read-while-binary-digit-weak
  (<= (acl2-count (mv-nth 1 (vl-read-while-binary-digit echars)))
      (acl2-count echars))
  :rule-classes ((:rewrite) (:linear)))

Theorem: acl2-count-of-vl-read-while-binary-digit-strong

(defthm acl2-count-of-vl-read-while-binary-digit-strong
 (implies
      (mv-nth 0 (vl-read-while-binary-digit echars))
      (< (acl2-count (mv-nth 1 (vl-read-while-binary-digit echars)))
         (acl2-count echars)))
 :rule-classes ((:rewrite) (:linear)))