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Vl-scopeitem-alist

Vl-scopeitem-alist-fix

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

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

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

Definitions and Theorems

Function: vl-scopeitem-alist-fix$inline

(defun vl-scopeitem-alist-fix$inline (x)
  (declare (xargs :guard (vl-scopeitem-alist-p x)))
  (let ((__function__ 'vl-scopeitem-alist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             x
           (if (consp (car x))
               (cons (cons (str-fix (caar x))
                           (vl-scopeitem-fix (cdar x)))
                     (vl-scopeitem-alist-fix (cdr x)))
             (vl-scopeitem-alist-fix (cdr x))))
         :exec x)))

Theorem: vl-scopeitem-alist-p-of-vl-scopeitem-alist-fix

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

Theorem: vl-scopeitem-alist-fix-when-vl-scopeitem-alist-p

(defthm vl-scopeitem-alist-fix-when-vl-scopeitem-alist-p
  (implies (vl-scopeitem-alist-p x)
           (equal (vl-scopeitem-alist-fix x) x)))

Function: vl-scopeitem-alist-equiv$inline

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

Theorem: vl-scopeitem-alist-equiv-is-an-equivalence

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

Theorem: vl-scopeitem-alist-equiv-implies-equal-vl-scopeitem-alist-fix-1

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

Theorem: vl-scopeitem-alist-fix-under-vl-scopeitem-alist-equiv

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

Theorem: equal-of-vl-scopeitem-alist-fix-1-forward-to-vl-scopeitem-alist-equiv

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

Theorem: equal-of-vl-scopeitem-alist-fix-2-forward-to-vl-scopeitem-alist-equiv

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

Theorem: vl-scopeitem-alist-equiv-of-vl-scopeitem-alist-fix-1-forward

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

Theorem: vl-scopeitem-alist-equiv-of-vl-scopeitem-alist-fix-2-forward

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

Theorem: cons-of-str-fix-k-under-vl-scopeitem-alist-equiv

(defthm cons-of-str-fix-k-under-vl-scopeitem-alist-equiv
  (vl-scopeitem-alist-equiv (cons (cons (str-fix acl2::k) acl2::v)
                                  acl2::x)
                            (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-streqv-congruence-on-k-under-vl-scopeitem-alist-equiv

(defthm cons-streqv-congruence-on-k-under-vl-scopeitem-alist-equiv
 (implies
   (streqv acl2::k k-equiv)
   (vl-scopeitem-alist-equiv (cons (cons acl2::k acl2::v) acl2::x)
                             (cons (cons k-equiv acl2::v) acl2::x)))
 :rule-classes :congruence)

Theorem: cons-of-vl-scopeitem-fix-v-under-vl-scopeitem-alist-equiv

(defthm cons-of-vl-scopeitem-fix-v-under-vl-scopeitem-alist-equiv
  (vl-scopeitem-alist-equiv
       (cons (cons acl2::k (vl-scopeitem-fix acl2::v))
             acl2::x)
       (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-vl-scopeitem-equiv-congruence-on-v-under-vl-scopeitem-alist-equiv

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

Theorem: cons-of-vl-scopeitem-alist-fix-y-under-vl-scopeitem-alist-equiv

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

Theorem: cons-vl-scopeitem-alist-equiv-congruence-on-y-under-vl-scopeitem-alist-equiv

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

Theorem: vl-scopeitem-alist-fix-of-acons

(defthm vl-scopeitem-alist-fix-of-acons
  (equal (vl-scopeitem-alist-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (str-fix acl2::a)
                     (vl-scopeitem-fix acl2::b))
               (vl-scopeitem-alist-fix x))))

Theorem: vl-scopeitem-alist-fix-of-append

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

Theorem: consp-car-of-vl-scopeitem-alist-fix

(defthm consp-car-of-vl-scopeitem-alist-fix
  (equal (consp (car (vl-scopeitem-alist-fix x)))
         (consp (vl-scopeitem-alist-fix x))))