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• Vl-primalist

Vl-primalist-fix

(vl-primalist-fix x) is an ACL2::fty alist fixing function that follows the fix-keys strategy.

Signature

(vl-primalist-fix x) → fty::newx

Arguments

x — Guard (vl-primalist-p x).

Returns

fty::newx — Type (vl-primalist-p fty::newx).

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

Definitions and Theorems

Function: vl-primalist-fix$inline

(defun vl-primalist-fix$inline (x)
  (declare (xargs :guard (vl-primalist-p x)))
  (let ((__function__ 'vl-primalist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             x
           (if (consp (car x))
               (cons (cons (acl2::symbol-fix (caar x))
                           (vl-module-fix (cdar x)))
                     (vl-primalist-fix (cdr x)))
             (vl-primalist-fix (cdr x))))
         :exec x)))

Theorem: vl-primalist-p-of-vl-primalist-fix

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

Theorem: vl-primalist-fix-when-vl-primalist-p

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

Function: vl-primalist-equiv$inline

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

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

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

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

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

Theorem: vl-primalist-fix-under-vl-primalist-equiv

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

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

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

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

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

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

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

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

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

Theorem: cons-of-symbol-fix-k-under-vl-primalist-equiv

(defthm cons-of-symbol-fix-k-under-vl-primalist-equiv
  (vl-primalist-equiv (cons (cons (acl2::symbol-fix acl2::k)
                                  acl2::v)
                            acl2::x)
                      (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-symbol-equiv-congruence-on-k-under-vl-primalist-equiv

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

Theorem: cons-of-vl-module-fix-v-under-vl-primalist-equiv

(defthm cons-of-vl-module-fix-v-under-vl-primalist-equiv
  (vl-primalist-equiv (cons (cons acl2::k (vl-module-fix acl2::v))
                            acl2::x)
                      (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-vl-module-equiv-congruence-on-v-under-vl-primalist-equiv

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

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

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

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

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

Theorem: vl-primalist-fix-of-acons

(defthm vl-primalist-fix-of-acons
  (equal (vl-primalist-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (acl2::symbol-fix acl2::a)
                     (vl-module-fix acl2::b))
               (vl-primalist-fix x))))

Theorem: vl-primalist-fix-of-append

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

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

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