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• Substitution

Vl-plainarglist-subst

(vl-plainarglist-subst x sigma) maps vl-plainarg-subst across a list.

Signature

(vl-plainarglist-subst x sigma) → new-x

Arguments

x — Guard (vl-plainarglist-p x).

sigma — Guard (vl-sigma-p sigma).

Returns

new-x — Type (vl-plainarglist-p new-x).

This is an ordinary defprojection.

Definitions and Theorems

Function: vl-plainarglist-subst-exec

(defun vl-plainarglist-subst-exec (x sigma acc)
  (declare (xargs :guard (and (vl-plainarglist-p x)
                              (vl-sigma-p sigma))))
  (declare (xargs :guard t))
  (let ((__function__ 'vl-plainarglist-subst-exec))
    (declare (ignorable __function__))
    (if (consp x)
        (vl-plainarglist-subst-exec
             (cdr x)
             sigma
             (cons (vl-plainarg-subst (car x) sigma)
                   acc))
      acc)))

Function: vl-plainarglist-subst-nrev

(defun vl-plainarglist-subst-nrev (x sigma nrev)
  (declare (xargs :stobjs (nrev)))
  (declare (xargs :guard (and (vl-plainarglist-p x)
                              (vl-sigma-p sigma))))
  (declare (xargs :guard t))
  (let ((__function__ 'vl-plainarglist-subst-nrev))
    (declare (ignorable __function__))
    (if (atom x)
        (nrev-fix nrev)
      (let ((nrev (nrev-push (vl-plainarg-subst (car x) sigma)
                             nrev)))
        (vl-plainarglist-subst-nrev (cdr x)
                                    sigma nrev)))))

Function: vl-plainarglist-subst

(defun vl-plainarglist-subst (x sigma)
 (declare (xargs :guard (and (vl-plainarglist-p x)
                             (vl-sigma-p sigma))))
 (declare (xargs :guard t))
 (let ((__function__ 'vl-plainarglist-subst))
  (declare (ignorable __function__))
  (mbe
   :logic
   (if (consp x)
       (cons (vl-plainarg-subst (car x) sigma)
             (vl-plainarglist-subst (cdr x) sigma))
     nil)
   :exec
   (if (atom x)
       nil
     (with-local-nrev (vl-plainarglist-subst-nrev x sigma nrev))))))

Theorem: vl-plainarglist-p-of-vl-plainarglist-subst

(defthm vl-plainarglist-p-of-vl-plainarglist-subst
  (b* ((new-x (vl-plainarglist-subst x sigma)))
    (vl-plainarglist-p new-x))
  :rule-classes :rewrite)

Theorem: vl-plainarglist-subst-of-vl-plainarglist-fix-x

(defthm vl-plainarglist-subst-of-vl-plainarglist-fix-x
  (equal (vl-plainarglist-subst (vl-plainarglist-fix x)
                                sigma)
         (vl-plainarglist-subst x sigma)))

Theorem: vl-plainarglist-subst-vl-plainarglist-equiv-congruence-on-x

(defthm vl-plainarglist-subst-vl-plainarglist-equiv-congruence-on-x
  (implies (vl-plainarglist-equiv x x-equiv)
           (equal (vl-plainarglist-subst x sigma)
                  (vl-plainarglist-subst x-equiv sigma)))
  :rule-classes :congruence)

Theorem: vl-plainarglist-subst-of-vl-sigma-fix-sigma

(defthm vl-plainarglist-subst-of-vl-sigma-fix-sigma
  (equal (vl-plainarglist-subst x (vl-sigma-fix sigma))
         (vl-plainarglist-subst x sigma)))

Theorem: vl-plainarglist-subst-vl-sigma-equiv-congruence-on-sigma

(defthm vl-plainarglist-subst-vl-sigma-equiv-congruence-on-sigma
  (implies (vl-sigma-equiv sigma sigma-equiv)
           (equal (vl-plainarglist-subst x sigma)
                  (vl-plainarglist-subst x sigma-equiv)))
  :rule-classes :congruence)

Theorem: vl-plainarglist-subst-of-update-nth

(defthm vl-plainarglist-subst-of-update-nth
 (implies
  (<= (nfix acl2::n) (len acl2::x))
  (equal (vl-plainarglist-subst (update-nth acl2::n acl2::v acl2::x)
                                sigma)
         (update-nth acl2::n
                     (vl-plainarg-subst acl2::v sigma)
                     (vl-plainarglist-subst acl2::x sigma))))
 :rule-classes ((:rewrite)))

Theorem: vl-plainarglist-subst-of-revappend

(defthm vl-plainarglist-subst-of-revappend
  (equal (vl-plainarglist-subst (revappend acl2::x acl2::y)
                                sigma)
         (revappend (vl-plainarglist-subst acl2::x sigma)
                    (vl-plainarglist-subst acl2::y sigma)))
  :rule-classes ((:rewrite)))

Theorem: nthcdr-of-vl-plainarglist-subst

(defthm nthcdr-of-vl-plainarglist-subst
  (equal (nthcdr acl2::n
                 (vl-plainarglist-subst acl2::x sigma))
         (vl-plainarglist-subst (nthcdr acl2::n acl2::x)
                                sigma))
  :rule-classes ((:rewrite)))

Theorem: nth-of-vl-plainarglist-subst

(defthm nth-of-vl-plainarglist-subst
  (equal (nth acl2::n
              (vl-plainarglist-subst acl2::x sigma))
         (and (< (nfix acl2::n) (len acl2::x))
              (vl-plainarg-subst (nth acl2::n acl2::x)
                                 sigma)))
  :rule-classes ((:rewrite)))

Theorem: vl-plainarglist-subst-nrev-removal

(defthm vl-plainarglist-subst-nrev-removal
  (equal (vl-plainarglist-subst-nrev acl2::x sigma nrev)
         (append nrev
                 (vl-plainarglist-subst acl2::x sigma)))
  :rule-classes ((:rewrite)))

Theorem: vl-plainarglist-subst-exec-removal

(defthm vl-plainarglist-subst-exec-removal
  (equal (vl-plainarglist-subst-exec acl2::x sigma acl2::acc)
         (revappend (vl-plainarglist-subst acl2::x sigma)
                    acl2::acc))
  :rule-classes ((:rewrite)))

Theorem: vl-plainarglist-subst-of-take

(defthm vl-plainarglist-subst-of-take
  (implies (<= (nfix acl2::n) (len acl2::x))
           (equal (vl-plainarglist-subst (take acl2::n acl2::x)
                                         sigma)
                  (take acl2::n
                        (vl-plainarglist-subst acl2::x sigma))))
  :rule-classes ((:rewrite)))

Theorem: set-equiv-congruence-over-vl-plainarglist-subst

(defthm set-equiv-congruence-over-vl-plainarglist-subst
  (implies (set-equiv acl2::x acl2::y)
           (set-equiv (vl-plainarglist-subst acl2::x sigma)
                      (vl-plainarglist-subst acl2::y sigma)))
  :rule-classes ((:congruence)))

Theorem: subsetp-of-vl-plainarglist-subst-when-subsetp

(defthm subsetp-of-vl-plainarglist-subst-when-subsetp
  (implies (subsetp acl2::x acl2::y)
           (subsetp (vl-plainarglist-subst acl2::x sigma)
                    (vl-plainarglist-subst acl2::y sigma)))
  :rule-classes ((:rewrite)))

Theorem: member-of-vl-plainarg-subst-in-vl-plainarglist-subst

(defthm member-of-vl-plainarg-subst-in-vl-plainarglist-subst
  (implies (member acl2::k acl2::x)
           (member (vl-plainarg-subst acl2::k sigma)
                   (vl-plainarglist-subst acl2::x sigma)))
  :rule-classes ((:rewrite)))

Theorem: vl-plainarglist-subst-of-rev

(defthm vl-plainarglist-subst-of-rev
  (equal (vl-plainarglist-subst (rev acl2::x)
                                sigma)
         (rev (vl-plainarglist-subst acl2::x sigma)))
  :rule-classes ((:rewrite)))

Theorem: vl-plainarglist-subst-of-list-fix

(defthm vl-plainarglist-subst-of-list-fix
  (equal (vl-plainarglist-subst (list-fix acl2::x)
                                sigma)
         (vl-plainarglist-subst acl2::x sigma))
  :rule-classes ((:rewrite)))

Theorem: vl-plainarglist-subst-of-append

(defthm vl-plainarglist-subst-of-append
  (equal (vl-plainarglist-subst (append acl2::a acl2::b)
                                sigma)
         (append (vl-plainarglist-subst acl2::a sigma)
                 (vl-plainarglist-subst acl2::b sigma)))
  :rule-classes ((:rewrite)))

Theorem: cdr-of-vl-plainarglist-subst

(defthm cdr-of-vl-plainarglist-subst
  (equal (cdr (vl-plainarglist-subst acl2::x sigma))
         (vl-plainarglist-subst (cdr acl2::x)
                                sigma))
  :rule-classes ((:rewrite)))

Theorem: car-of-vl-plainarglist-subst

(defthm car-of-vl-plainarglist-subst
  (equal (car (vl-plainarglist-subst acl2::x sigma))
         (and (consp acl2::x)
              (vl-plainarg-subst (car acl2::x)
                                 sigma)))
  :rule-classes ((:rewrite)))

Theorem: vl-plainarglist-subst-under-iff

(defthm vl-plainarglist-subst-under-iff
  (iff (vl-plainarglist-subst acl2::x sigma)
       (consp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: consp-of-vl-plainarglist-subst

(defthm consp-of-vl-plainarglist-subst
  (equal (consp (vl-plainarglist-subst acl2::x sigma))
         (consp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: len-of-vl-plainarglist-subst

(defthm len-of-vl-plainarglist-subst
  (equal (len (vl-plainarglist-subst acl2::x sigma))
         (len acl2::x))
  :rule-classes ((:rewrite)))

Theorem: true-listp-of-vl-plainarglist-subst

(defthm true-listp-of-vl-plainarglist-subst
  (true-listp (vl-plainarglist-subst acl2::x sigma))
  :rule-classes :type-prescription)

Theorem: vl-plainarglist-subst-when-not-consp

(defthm vl-plainarglist-subst-when-not-consp
  (implies (not (consp acl2::x))
           (equal (vl-plainarglist-subst acl2::x sigma)
                  nil))
  :rule-classes ((:rewrite)))

Theorem: vl-plainarglist-subst-of-cons

(defthm vl-plainarglist-subst-of-cons
  (equal (vl-plainarglist-subst (cons acl2::a acl2::b)
                                sigma)
         (cons (vl-plainarg-subst acl2::a sigma)
               (vl-plainarglist-subst acl2::b sigma)))
  :rule-classes ((:rewrite)))