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Representation-of-integers

Sshort-list-from-integer-list

Lift sshort to lists.

Signature
(sshort-list-from-integer-list x) → result
Arguments: x — Guard (sshort-integer-listp x).
Returns: result — Type (sshort-listp result).

Definitions and Theorems

Function: sshort-list-from-integer-list-exec

(defun sshort-list-from-integer-list-exec (x acc)
  (declare (xargs :guard (sshort-integer-listp x)))
  (declare (xargs :guard t))
  (let ((__function__ 'sshort-list-from-integer-list-exec))
    (declare (ignorable __function__))
    (if (consp x)
        (sshort-list-from-integer-list-exec
             (cdr x)
             (cons (sshort-from-integer (car x))
                   acc))
      acc)))

Function: sshort-list-from-integer-list-nrev

(defun sshort-list-from-integer-list-nrev (x acl2::nrev)
 (declare (xargs :stobjs (acl2::nrev)))
 (declare (xargs :guard (sshort-integer-listp x)))
 (declare (xargs :guard t))
 (let ((__function__ 'sshort-list-from-integer-list-nrev))
  (declare (ignorable __function__))
  (if (atom x)
      (acl2::nrev-fix acl2::nrev)
    (let ((acl2::nrev (acl2::nrev-push (sshort-from-integer (car x))
                                       acl2::nrev)))
      (sshort-list-from-integer-list-nrev (cdr x)
                                          acl2::nrev)))))

Function: sshort-list-from-integer-list

(defun sshort-list-from-integer-list (x)
 (declare (xargs :guard (sshort-integer-listp x)))
 (declare (xargs :guard t))
 (let ((__function__ 'sshort-list-from-integer-list))
  (declare (ignorable __function__))
  (mbe :logic
       (if (consp x)
           (cons (sshort-from-integer (car x))
                 (sshort-list-from-integer-list (cdr x)))
         nil)
       :exec
       (if (atom x)
           nil
         (acl2::with-local-nrev
              (sshort-list-from-integer-list-nrev x acl2::nrev))))))

Theorem: sshort-listp-of-sshort-list-from-integer-list

(defthm sshort-listp-of-sshort-list-from-integer-list
  (b* ((result (sshort-list-from-integer-list x)))
    (sshort-listp result))
  :rule-classes :rewrite)

Theorem: sshort-list-from-integer-list-of-update-nth

(defthm sshort-list-from-integer-list-of-update-nth
  (implies
       (<= (nfix acl2::n) (len acl2::x))
       (equal (sshort-list-from-integer-list
                   (update-nth acl2::n acl2::v acl2::x))
              (update-nth acl2::n (sshort-from-integer acl2::v)
                          (sshort-list-from-integer-list acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: nth-of-sshort-list-from-integer-list

(defthm nth-of-sshort-list-from-integer-list
  (equal (nth acl2::n
              (sshort-list-from-integer-list acl2::x))
         (and (< (nfix acl2::n) (len acl2::x))
              (sshort-from-integer (nth acl2::n acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: sshort-list-from-integer-list-of-take

(defthm sshort-list-from-integer-list-of-take
  (implies
       (<= (nfix acl2::n) (len acl2::x))
       (equal (sshort-list-from-integer-list (take acl2::n acl2::x))
              (take acl2::n
                    (sshort-list-from-integer-list acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: set-equiv-congruence-over-sshort-list-from-integer-list

(defthm set-equiv-congruence-over-sshort-list-from-integer-list
  (implies (set-equiv acl2::x acl2::y)
           (set-equiv (sshort-list-from-integer-list acl2::x)
                      (sshort-list-from-integer-list acl2::y)))
  :rule-classes ((:congruence)))

Theorem: subsetp-of-sshort-list-from-integer-list-when-subsetp

(defthm subsetp-of-sshort-list-from-integer-list-when-subsetp
  (implies (subsetp acl2::x acl2::y)
           (subsetp (sshort-list-from-integer-list acl2::x)
                    (sshort-list-from-integer-list acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: member-of-sshort-from-integer-in-sshort-list-from-integer-list

(defthm
     member-of-sshort-from-integer-in-sshort-list-from-integer-list
 (implies
      (common-lisp::member acl2::k acl2::x)
      (common-lisp::member (sshort-from-integer acl2::k)
                           (sshort-list-from-integer-list acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: sshort-list-from-integer-list-nrev-removal

(defthm sshort-list-from-integer-list-nrev-removal
  (equal (sshort-list-from-integer-list-nrev acl2::x acl2::nrev)
         (append acl2::nrev
                 (sshort-list-from-integer-list acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sshort-list-from-integer-list-exec-removal

(defthm sshort-list-from-integer-list-exec-removal
  (equal (sshort-list-from-integer-list-exec acl2::x acl2::acc)
         (revappend (sshort-list-from-integer-list acl2::x)
                    acl2::acc))
  :rule-classes ((:rewrite)))

Theorem: sshort-list-from-integer-list-of-rev

(defthm sshort-list-from-integer-list-of-rev
  (equal (sshort-list-from-integer-list (rev acl2::x))
         (rev (sshort-list-from-integer-list acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: sshort-list-from-integer-list-of-list-fix

(defthm sshort-list-from-integer-list-of-list-fix
  (equal (sshort-list-from-integer-list (list-fix acl2::x))
         (sshort-list-from-integer-list acl2::x))
  :rule-classes ((:rewrite)))

Theorem: sshort-list-from-integer-list-of-append

(defthm sshort-list-from-integer-list-of-append
  (equal (sshort-list-from-integer-list (append acl2::a acl2::b))
         (append (sshort-list-from-integer-list acl2::a)
                 (sshort-list-from-integer-list acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: cdr-of-sshort-list-from-integer-list

(defthm cdr-of-sshort-list-from-integer-list
  (equal (cdr (sshort-list-from-integer-list acl2::x))
         (sshort-list-from-integer-list (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: car-of-sshort-list-from-integer-list

(defthm car-of-sshort-list-from-integer-list
  (equal (car (sshort-list-from-integer-list acl2::x))
         (and (consp acl2::x)
              (sshort-from-integer (car acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: sshort-list-from-integer-list-under-iff

(defthm sshort-list-from-integer-list-under-iff
  (iff (sshort-list-from-integer-list acl2::x)
       (consp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: consp-of-sshort-list-from-integer-list

(defthm consp-of-sshort-list-from-integer-list
  (equal (consp (sshort-list-from-integer-list acl2::x))
         (consp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: len-of-sshort-list-from-integer-list

(defthm len-of-sshort-list-from-integer-list
  (equal (len (sshort-list-from-integer-list acl2::x))
         (len acl2::x))
  :rule-classes ((:rewrite)))

Theorem: true-listp-of-sshort-list-from-integer-list

(defthm true-listp-of-sshort-list-from-integer-list
  (true-listp (sshort-list-from-integer-list acl2::x))
  :rule-classes :type-prescription)

Theorem: sshort-list-from-integer-list-when-not-consp

(defthm sshort-list-from-integer-list-when-not-consp
  (implies (not (consp acl2::x))
           (equal (sshort-list-from-integer-list acl2::x)
                  nil))
  :rule-classes ((:rewrite)))

Theorem: sshort-list-from-integer-list-of-cons

(defthm sshort-list-from-integer-list-of-cons
  (equal (sshort-list-from-integer-list (cons acl2::a acl2::b))
         (cons (sshort-from-integer acl2::a)
               (sshort-list-from-integer-list acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: sshort-list-from-integer-list-of-sshort-integer-list-fix-x

(defthm sshort-list-from-integer-list-of-sshort-integer-list-fix-x
  (equal (sshort-list-from-integer-list (sshort-integer-list-fix x))
         (sshort-list-from-integer-list x)))

Theorem: sshort-list-from-integer-list-sshort-integer-list-equiv-congruence-on-x

(defthm
 sshort-list-from-integer-list-sshort-integer-list-equiv-congruence-on-x
 (implies (sshort-integer-list-equiv x x-equiv)
          (equal (sshort-list-from-integer-list x)
                 (sshort-list-from-integer-list x-equiv)))
 :rule-classes :congruence)