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Register-fix

Fixing function for register structures.

Signature
(register-fix x) → new-x
Arguments: x — Guard (registerp x).
Returns: new-x — Type (registerp new-x).

Definitions and Theorems

Function: register-fix$inline

(defun register-fix$inline (x)
  (declare (xargs :guard (registerp x)))
  (let ((__function__ 'register-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((number (nfix (cdr (std::da-nth 0 (cdr x))))))
           (cons :register (list (cons 'number number))))
         :exec x)))

Theorem: registerp-of-register-fix

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

Theorem: register-fix-when-registerp

(defthm register-fix-when-registerp
  (implies (registerp x)
           (equal (register-fix x) x)))

Function: register-equiv$inline

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

Theorem: register-equiv-is-an-equivalence

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

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

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

Theorem: register-fix-under-register-equiv

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

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

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

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

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

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

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

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

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