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Hex-pair

Hex-pair-fix

Fixing function for hex-pair structures.

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

Definitions and Theorems

Function: hex-pair-fix$inline

(defun hex-pair-fix$inline (x)
  (declare (xargs :guard (hex-pairp x)))
  (let ((__function__ 'hex-pair-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))))))
           (cons :hex-pair (list (cons '1st 1st) (cons '2nd 2nd))))
         :exec x)))

Theorem: hex-pairp-of-hex-pair-fix

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

Theorem: hex-pair-fix-when-hex-pairp

(defthm hex-pair-fix-when-hex-pairp
  (implies (hex-pairp x)
           (equal (hex-pair-fix x) x)))

Function: hex-pair-equiv$inline

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

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

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

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

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

Theorem: hex-pair-fix-under-hex-pair-equiv

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

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

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

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

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

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

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

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

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