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Regaccess

Regaccess-fix

Fixing function for regaccess structures.

Signature
(regaccess-fix x) → new-x
Arguments: x — Guard (regaccessp x).
Returns: new-x — Type (regaccessp new-x).

Definitions and Theorems

Function: regaccess-fix$inline

(defun regaccess-fix$inline (x)
 (declare (xargs :guard (regaccessp x)))
 (let ((__function__ 'regaccess-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (b* ((base (register-fix (cdr (std::da-nth 0 (cdr x)))))
        (comps (identifier-list-fix (cdr (std::da-nth 1 (cdr x))))))
     (cons :regaccess (list (cons 'base base)
                            (cons 'comps comps))))
   :exec x)))

Theorem: regaccessp-of-regaccess-fix

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

Theorem: regaccess-fix-when-regaccessp

(defthm regaccess-fix-when-regaccessp
  (implies (regaccessp x)
           (equal (regaccess-fix x) x)))

Function: regaccess-equiv$inline

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

Theorem: regaccess-equiv-is-an-equivalence

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

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

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

Theorem: regaccess-fix-under-regaccess-equiv

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

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

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

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

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

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

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

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

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