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Hex-quad

Hex-quad-fix

Fixing function for hex-quad structures.

Signature
(hex-quad-fix x) → new-x
Arguments: x — Guard (hex-quadp x).
Returns: new-x — Type (hex-quadp new-x).

Definitions and Theorems

Function: hex-quad-fix$inline

(defun hex-quad-fix$inline (x)
  (declare (xargs :guard (hex-quadp x)))
  (let ((__function__ 'hex-quad-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((1st (hex-digit-fix (cdr (std::da-nth 0 (cdr x)))))
              (2nd (hex-digit-fix (cdr (std::da-nth 1 (cdr x)))))
              (3rd (hex-digit-fix (cdr (std::da-nth 2 (cdr x)))))
              (4th (hex-digit-fix (cdr (std::da-nth 3 (cdr x))))))
           (cons :hex-quad (list (cons '1st 1st)
                                 (cons '2nd 2nd)
                                 (cons '3rd 3rd)
                                 (cons '4th 4th))))
         :exec x)))

Theorem: hex-quadp-of-hex-quad-fix

(defthm hex-quadp-of-hex-quad-fix
  (b* ((new-x (hex-quad-fix$inline x)))
    (hex-quadp new-x))
  :rule-classes :rewrite)

Theorem: hex-quad-fix-when-hex-quadp

(defthm hex-quad-fix-when-hex-quadp
  (implies (hex-quadp x)
           (equal (hex-quad-fix x) x)))

Function: hex-quad-equiv$inline

(defun hex-quad-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (hex-quadp acl2::x)
                              (hex-quadp acl2::y))))
  (equal (hex-quad-fix acl2::x)
         (hex-quad-fix acl2::y)))

Theorem: hex-quad-equiv-is-an-equivalence

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

Theorem: hex-quad-equiv-implies-equal-hex-quad-fix-1

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

Theorem: hex-quad-fix-under-hex-quad-equiv

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

Theorem: equal-of-hex-quad-fix-1-forward-to-hex-quad-equiv

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

Theorem: equal-of-hex-quad-fix-2-forward-to-hex-quad-equiv

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

Theorem: hex-quad-equiv-of-hex-quad-fix-1-forward

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

Theorem: hex-quad-equiv-of-hex-quad-fix-2-forward

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