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Vl-elabscopes

Vl-elabscopes-fix

(vl-elabscopes-fix x) is an fty alist fixing function that follows the fix-keys strategy.

Signature
(vl-elabscopes-fix x) → fty::newx
Arguments: x — Guard (vl-elabscopes-p x).
Returns: fty::newx — Type (vl-elabscopes-p fty::newx).

Note that in the execution this is just an inline identity function.

Definitions and Theorems

Function: vl-elabscopes-fix$inline

(defun vl-elabscopes-fix$inline (x)
  (declare (xargs :guard (vl-elabscopes-p x)))
  (let ((__function__ 'vl-elabscopes-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (if (consp (car x))
               (cons (cons (vl-maybe-elabkey-fix (caar x))
                           (vl-elabscope-fix (cdar x)))
                     (vl-elabscopes-fix (cdr x)))
             (vl-elabscopes-fix (cdr x))))
         :exec x)))

Theorem: vl-elabscopes-p-of-vl-elabscopes-fix

(defthm vl-elabscopes-p-of-vl-elabscopes-fix
  (b* ((fty::newx (vl-elabscopes-fix$inline x)))
    (vl-elabscopes-p fty::newx))
  :rule-classes :rewrite)

Theorem: vl-elabscopes-fix-when-vl-elabscopes-p

(defthm vl-elabscopes-fix-when-vl-elabscopes-p
  (implies (vl-elabscopes-p x)
           (equal (vl-elabscopes-fix x) x)))

Function: vl-elabscopes-equiv$inline

(defun vl-elabscopes-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (vl-elabscopes-p acl2::x)
                              (vl-elabscopes-p acl2::y))))
  (equal (vl-elabscopes-fix acl2::x)
         (vl-elabscopes-fix acl2::y)))

Theorem: vl-elabscopes-equiv-is-an-equivalence

(defthm vl-elabscopes-equiv-is-an-equivalence
  (and (booleanp (vl-elabscopes-equiv x y))
       (vl-elabscopes-equiv x x)
       (implies (vl-elabscopes-equiv x y)
                (vl-elabscopes-equiv y x))
       (implies (and (vl-elabscopes-equiv x y)
                     (vl-elabscopes-equiv y z))
                (vl-elabscopes-equiv x z)))
  :rule-classes (:equivalence))

Theorem: vl-elabscopes-equiv-implies-equal-vl-elabscopes-fix-1

(defthm vl-elabscopes-equiv-implies-equal-vl-elabscopes-fix-1
  (implies (vl-elabscopes-equiv acl2::x x-equiv)
           (equal (vl-elabscopes-fix acl2::x)
                  (vl-elabscopes-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: vl-elabscopes-fix-under-vl-elabscopes-equiv

(defthm vl-elabscopes-fix-under-vl-elabscopes-equiv
  (vl-elabscopes-equiv (vl-elabscopes-fix acl2::x)
                       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-vl-elabscopes-fix-1-forward-to-vl-elabscopes-equiv

(defthm equal-of-vl-elabscopes-fix-1-forward-to-vl-elabscopes-equiv
  (implies (equal (vl-elabscopes-fix acl2::x)
                  acl2::y)
           (vl-elabscopes-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-vl-elabscopes-fix-2-forward-to-vl-elabscopes-equiv

(defthm equal-of-vl-elabscopes-fix-2-forward-to-vl-elabscopes-equiv
  (implies (equal acl2::x (vl-elabscopes-fix acl2::y))
           (vl-elabscopes-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: vl-elabscopes-equiv-of-vl-elabscopes-fix-1-forward

(defthm vl-elabscopes-equiv-of-vl-elabscopes-fix-1-forward
  (implies (vl-elabscopes-equiv (vl-elabscopes-fix acl2::x)
                                acl2::y)
           (vl-elabscopes-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: vl-elabscopes-equiv-of-vl-elabscopes-fix-2-forward

(defthm vl-elabscopes-equiv-of-vl-elabscopes-fix-2-forward
  (implies (vl-elabscopes-equiv acl2::x (vl-elabscopes-fix acl2::y))
           (vl-elabscopes-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: cons-of-vl-maybe-elabkey-fix-k-under-vl-elabscopes-equiv

(defthm cons-of-vl-maybe-elabkey-fix-k-under-vl-elabscopes-equiv
  (vl-elabscopes-equiv (cons (cons (vl-maybe-elabkey-fix acl2::k)
                                   acl2::v)
                             acl2::x)
                       (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-vl-maybe-elabkey-equiv-congruence-on-k-under-vl-elabscopes-equiv

(defthm
 cons-vl-maybe-elabkey-equiv-congruence-on-k-under-vl-elabscopes-equiv
 (implies
      (vl-maybe-elabkey-equiv acl2::k k-equiv)
      (vl-elabscopes-equiv (cons (cons acl2::k acl2::v) acl2::x)
                           (cons (cons k-equiv acl2::v) acl2::x)))
 :rule-classes :congruence)

Theorem: cons-of-vl-elabscope-fix-v-under-vl-elabscopes-equiv

(defthm cons-of-vl-elabscope-fix-v-under-vl-elabscopes-equiv
  (vl-elabscopes-equiv
       (cons (cons acl2::k (vl-elabscope-fix acl2::v))
             acl2::x)
       (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-vl-elabscope-equiv-congruence-on-v-under-vl-elabscopes-equiv

(defthm
  cons-vl-elabscope-equiv-congruence-on-v-under-vl-elabscopes-equiv
  (implies
       (vl-elabscope-equiv acl2::v v-equiv)
       (vl-elabscopes-equiv (cons (cons acl2::k acl2::v) acl2::x)
                            (cons (cons acl2::k v-equiv) acl2::x)))
  :rule-classes :congruence)

Theorem: cons-of-vl-elabscopes-fix-y-under-vl-elabscopes-equiv

(defthm cons-of-vl-elabscopes-fix-y-under-vl-elabscopes-equiv
  (vl-elabscopes-equiv (cons acl2::x (vl-elabscopes-fix acl2::y))
                       (cons acl2::x acl2::y)))

Theorem: cons-vl-elabscopes-equiv-congruence-on-y-under-vl-elabscopes-equiv

(defthm
 cons-vl-elabscopes-equiv-congruence-on-y-under-vl-elabscopes-equiv
 (implies (vl-elabscopes-equiv acl2::y y-equiv)
          (vl-elabscopes-equiv (cons acl2::x acl2::y)
                               (cons acl2::x y-equiv)))
 :rule-classes :congruence)

Theorem: vl-elabscopes-fix-of-acons

(defthm vl-elabscopes-fix-of-acons
  (equal (vl-elabscopes-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (vl-maybe-elabkey-fix acl2::a)
                     (vl-elabscope-fix acl2::b))
               (vl-elabscopes-fix x))))

Theorem: vl-elabscopes-fix-of-append

(defthm vl-elabscopes-fix-of-append
  (equal (vl-elabscopes-fix (append std::a std::b))
         (append (vl-elabscopes-fix std::a)
                 (vl-elabscopes-fix std::b))))

Theorem: consp-car-of-vl-elabscopes-fix

(defthm consp-car-of-vl-elabscopes-fix
  (equal (consp (car (vl-elabscopes-fix x)))
         (consp (vl-elabscopes-fix x))))