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Label

Label-fix

Fixing function for label structures.

Signature
(label-fix x) → new-x
Arguments: x — Guard (labelp x).
Returns: new-x — Type (labelp new-x).

Definitions and Theorems

Function: label-fix$inline

(defun label-fix$inline (x)
  (declare (xargs :guard (labelp x)))
  (let ((__function__ 'label-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (label-kind x)
           (:name (b* ((get (ident-fix (std::da-nth 0 (cdr x)))))
                    (cons :name (list get))))
           (:cas (b* ((get (expr-fix (std::da-nth 0 (cdr x)))))
                   (cons :cas (list get))))
           (:default (cons :default (list))))
         :exec x)))

Theorem: labelp-of-label-fix

(defthm labelp-of-label-fix
  (b* ((new-x (label-fix$inline x)))
    (labelp new-x))
  :rule-classes :rewrite)

Theorem: label-fix-when-labelp

(defthm label-fix-when-labelp
  (implies (labelp x)
           (equal (label-fix x) x)))

Function: label-equiv$inline

(defun label-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (labelp acl2::x)
                              (labelp acl2::y))))
  (equal (label-fix acl2::x)
         (label-fix acl2::y)))

Theorem: label-equiv-is-an-equivalence

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

Theorem: label-equiv-implies-equal-label-fix-1

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

Theorem: label-fix-under-label-equiv

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

Theorem: equal-of-label-fix-1-forward-to-label-equiv

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

Theorem: equal-of-label-fix-2-forward-to-label-equiv

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

Theorem: label-equiv-of-label-fix-1-forward

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

Theorem: label-equiv-of-label-fix-2-forward

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

Theorem: label-kind$inline-of-label-fix-x

(defthm label-kind$inline-of-label-fix-x
  (equal (label-kind$inline (label-fix x))
         (label-kind$inline x)))

Theorem: label-kind$inline-label-equiv-congruence-on-x

(defthm label-kind$inline-label-equiv-congruence-on-x
  (implies (label-equiv x x-equiv)
           (equal (label-kind$inline x)
                  (label-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-label-fix

(defthm consp-of-label-fix
  (consp (label-fix x))
  :rule-classes :type-prescription)