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• Resolving-multiple-drivers

Vl-res-sigma-p

An alist that records the fresh wires we introduce for multiply driven wires.

The basic idea is that if W is a multiply-driven wire, and we are going to rewrite the occurrences so that they drive W_1, W_2, ... instead of W, then this alist should bind

W --> (W_1 W_2 ...)

In short, this alist will end up saying which wires need to be resolved together to drive W; see vl-make-res-occs.

Definitions and Theorems

Function: vl-res-sigma-p

(defun vl-res-sigma-p (x)
       (declare (xargs :guard t))
       (if (consp x)
           (and (consp (car x))
                (vl-emodwire-p (caar x))
                (vl-emodwirelist-p (cdar x))
                (vl-res-sigma-p (cdr x)))
           t))

Theorem: vl-res-sigma-p-of-revappend

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

Theorem: vl-res-sigma-p-of-remove

(defthm vl-res-sigma-p-of-remove
        (implies (vl-res-sigma-p acl2::x)
                 (vl-res-sigma-p (remove acl2::a acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-last

(defthm vl-res-sigma-p-of-last
        (implies (vl-res-sigma-p (double-rewrite acl2::x))
                 (vl-res-sigma-p (last acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-nthcdr

(defthm vl-res-sigma-p-of-nthcdr
        (implies (vl-res-sigma-p (double-rewrite acl2::x))
                 (vl-res-sigma-p (nthcdr acl2::n acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-butlast

(defthm vl-res-sigma-p-of-butlast
        (implies (vl-res-sigma-p (double-rewrite acl2::x))
                 (vl-res-sigma-p (butlast acl2::x acl2::n)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-update-nth

(defthm
 vl-res-sigma-p-of-update-nth
 (implies (vl-res-sigma-p (double-rewrite acl2::x))
          (iff (vl-res-sigma-p (update-nth acl2::n acl2::y acl2::x))
               (and (and (consp acl2::y)
                         (vl-emodwire-p (car acl2::y))
                         (vl-emodwirelist-p (cdr acl2::y)))
                    (or (<= (nfix acl2::n) (len acl2::x))
                        (and (consp nil)
                             (vl-emodwire-p (car nil))
                             (vl-emodwirelist-p (cdr nil)))))))
 :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-repeat

(defthm vl-res-sigma-p-of-repeat
        (iff (vl-res-sigma-p (repeat acl2::n acl2::x))
             (or (and (consp acl2::x)
                      (vl-emodwire-p (car acl2::x))
                      (vl-emodwirelist-p (cdr acl2::x)))
                 (zp acl2::n)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-take

(defthm vl-res-sigma-p-of-take
        (implies (vl-res-sigma-p (double-rewrite acl2::x))
                 (iff (vl-res-sigma-p (take acl2::n acl2::x))
                      (or (and (consp nil)
                               (vl-emodwire-p (car nil))
                               (vl-emodwirelist-p (cdr nil)))
                          (<= (nfix acl2::n) (len acl2::x)))))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-union-equal

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

Theorem: vl-res-sigma-p-of-intersection-equal-2

(defthm
     vl-res-sigma-p-of-intersection-equal-2
     (implies (vl-res-sigma-p (double-rewrite acl2::y))
              (vl-res-sigma-p (intersection-equal acl2::x acl2::y)))
     :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-intersection-equal-1

(defthm
     vl-res-sigma-p-of-intersection-equal-1
     (implies (vl-res-sigma-p (double-rewrite acl2::x))
              (vl-res-sigma-p (intersection-equal acl2::x acl2::y)))
     :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-set-difference-equal

(defthm
   vl-res-sigma-p-of-set-difference-equal
   (implies (vl-res-sigma-p acl2::x)
            (vl-res-sigma-p (set-difference-equal acl2::x acl2::y)))
   :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-set-equiv-congruence

(defthm vl-res-sigma-p-set-equiv-congruence
        (implies (set-equiv acl2::x acl2::y)
                 (equal (vl-res-sigma-p acl2::x)
                        (vl-res-sigma-p acl2::y)))
        :rule-classes :congruence)

Theorem: vl-res-sigma-p-when-subsetp-equal

(defthm vl-res-sigma-p-when-subsetp-equal
        (and (implies (and (subsetp-equal acl2::x acl2::y)
                           (vl-res-sigma-p acl2::y))
                      (vl-res-sigma-p acl2::x))
             (implies (and (vl-res-sigma-p acl2::y)
                           (subsetp-equal acl2::x acl2::y))
                      (vl-res-sigma-p acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-rcons

(defthm vl-res-sigma-p-of-rcons
        (iff (vl-res-sigma-p (acl2::rcons acl2::a acl2::x))
             (and (and (consp acl2::a)
                       (vl-emodwire-p (car acl2::a))
                       (vl-emodwirelist-p (cdr acl2::a)))
                  (vl-res-sigma-p (list-fix acl2::x))))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-rev

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

Theorem: vl-res-sigma-p-of-duplicated-members

(defthm vl-res-sigma-p-of-duplicated-members
        (implies (vl-res-sigma-p acl2::x)
                 (vl-res-sigma-p (duplicated-members acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-difference

(defthm vl-res-sigma-p-of-difference
        (implies (vl-res-sigma-p acl2::x)
                 (vl-res-sigma-p (difference acl2::x acl2::y)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-intersect-2

(defthm vl-res-sigma-p-of-intersect-2
        (implies (vl-res-sigma-p acl2::y)
                 (vl-res-sigma-p (intersect acl2::x acl2::y)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-intersect-1

(defthm vl-res-sigma-p-of-intersect-1
        (implies (vl-res-sigma-p acl2::x)
                 (vl-res-sigma-p (intersect acl2::x acl2::y)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-union

(defthm vl-res-sigma-p-of-union
        (iff (vl-res-sigma-p (union acl2::x acl2::y))
             (and (vl-res-sigma-p (sfix acl2::x))
                  (vl-res-sigma-p (sfix acl2::y))))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-mergesort

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

Theorem: vl-res-sigma-p-of-delete

(defthm vl-res-sigma-p-of-delete
        (implies (vl-res-sigma-p acl2::x)
                 (vl-res-sigma-p (delete acl2::k acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-insert

(defthm vl-res-sigma-p-of-insert
        (iff (vl-res-sigma-p (insert acl2::a acl2::x))
             (and (vl-res-sigma-p (sfix acl2::x))
                  (and (consp acl2::a)
                       (vl-emodwire-p (car acl2::a))
                       (vl-emodwirelist-p (cdr acl2::a)))))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-sfix

(defthm vl-res-sigma-p-of-sfix
        (iff (vl-res-sigma-p (sfix acl2::x))
             (or (vl-res-sigma-p acl2::x)
                 (not (setp acl2::x))))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-list-fix

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

Theorem: vl-res-sigma-p-of-append

(defthm vl-res-sigma-p-of-append
        (equal (vl-res-sigma-p (append acl2::a acl2::b))
               (and (vl-res-sigma-p acl2::a)
                    (vl-res-sigma-p acl2::b)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-when-not-consp

(defthm vl-res-sigma-p-when-not-consp
        (implies (not (consp acl2::x))
                 (vl-res-sigma-p acl2::x))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-cdr-when-vl-res-sigma-p

(defthm vl-res-sigma-p-of-cdr-when-vl-res-sigma-p
        (implies (vl-res-sigma-p (double-rewrite acl2::x))
                 (vl-res-sigma-p (cdr acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-cons

(defthm vl-res-sigma-p-of-cons
        (equal (vl-res-sigma-p (cons acl2::a acl2::x))
               (and (and (consp acl2::a)
                         (vl-emodwire-p (car acl2::a))
                         (vl-emodwirelist-p (cdr acl2::a)))
                    (vl-res-sigma-p acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-make-fal

(defthm vl-res-sigma-p-of-make-fal
        (implies (and (vl-res-sigma-p acl2::x)
                      (vl-res-sigma-p acl2::y))
                 (vl-res-sigma-p (make-fal acl2::x acl2::y)))
        :rule-classes ((:rewrite)))

Theorem: vl-emodwirelist-p-of-cdr-when-member-equal-of-vl-res-sigma-p

(defthm vl-emodwirelist-p-of-cdr-when-member-equal-of-vl-res-sigma-p
        (and (implies (and (vl-res-sigma-p acl2::x)
                           (member-equal acl2::a acl2::x))
                      (vl-emodwirelist-p (cdr acl2::a)))
             (implies (and (member-equal acl2::a acl2::x)
                           (vl-res-sigma-p acl2::x))
                      (vl-emodwirelist-p (cdr acl2::a))))
        :rule-classes ((:rewrite)))

Theorem: vl-emodwire-p-of-car-when-member-equal-of-vl-res-sigma-p

(defthm vl-emodwire-p-of-car-when-member-equal-of-vl-res-sigma-p
        (and (implies (and (vl-res-sigma-p acl2::x)
                           (member-equal acl2::a acl2::x))
                      (vl-emodwire-p (car acl2::a)))
             (implies (and (member-equal acl2::a acl2::x)
                           (vl-res-sigma-p acl2::x))
                      (vl-emodwire-p (car acl2::a))))
        :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-vl-res-sigma-p

(defthm
   consp-when-member-equal-of-vl-res-sigma-p
   (implies (and (vl-res-sigma-p acl2::x)
                 (member-equal acl2::a acl2::x))
            (consp acl2::a))
   :rule-classes
   ((:rewrite :backchain-limit-lst (0 0))
    (:rewrite :backchain-limit-lst (0 0)
              :corollary (implies (if (member-equal acl2::a acl2::x)
                                      (vl-res-sigma-p acl2::x)
                                      'nil)
                                  (consp acl2::a)))))

Theorem: vl-emodwirelist-p-of-cdr-of-assoc-when-vl-res-sigma-p

(defthm
   vl-emodwirelist-p-of-cdr-of-assoc-when-vl-res-sigma-p
   (implies (vl-res-sigma-p acl2::x)
            (vl-emodwirelist-p (cdr (assoc-equal acl2::k acl2::x))))
   :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-fast-alist-clean

(defthm vl-res-sigma-p-of-fast-alist-clean
        (implies (vl-res-sigma-p acl2::x)
                 (vl-res-sigma-p (fast-alist-clean acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-hons-shrink-alist

(defthm
     vl-res-sigma-p-of-hons-shrink-alist
     (implies (and (vl-res-sigma-p acl2::x)
                   (vl-res-sigma-p acl2::y))
              (vl-res-sigma-p (hons-shrink-alist acl2::x acl2::y)))
     :rule-classes ((:rewrite)))

Theorem: vl-res-sigma-p-of-hons-acons

(defthm vl-res-sigma-p-of-hons-acons
        (equal (vl-res-sigma-p (hons-acons acl2::a acl2::n acl2::x))
               (and (vl-emodwire-p acl2::a)
                    (vl-emodwirelist-p acl2::n)
                    (vl-res-sigma-p acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-emodwirelist-p-of-cdr-of-hons-assoc-equal-when-vl-res-sigma-p

(defthm
  vl-emodwirelist-p-of-cdr-of-hons-assoc-equal-when-vl-res-sigma-p
  (implies
       (vl-res-sigma-p acl2::x)
       (vl-emodwirelist-p (cdr (hons-assoc-equal acl2::k acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vl-emodwire-p-of-caar-when-vl-res-sigma-p

(defthm vl-emodwire-p-of-caar-when-vl-res-sigma-p
        (implies (vl-res-sigma-p acl2::x)
                 (iff (vl-emodwire-p (caar acl2::x))
                      (consp acl2::x)))
        :rule-classes ((:rewrite)))

Theorem: vl-emodwirelist-p-of-cdar-when-vl-res-sigma-p

(defthm vl-emodwirelist-p-of-cdar-when-vl-res-sigma-p
        (implies (vl-res-sigma-p acl2::x)
                 (vl-emodwirelist-p (cdar acl2::x)))
        :rule-classes ((:rewrite)))