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Hexadecimal-digits

Hex-digit

Fixtype of Java hexadecimal digits [JLS14:3.10.1].

A Java hexadecimal digit is a Java ASCII character that is either a decimal digit (i.e. between `0' and `9') or an (uppercase or lowercase) letter between `A' and `F' or between `a' and `f'. See the grammar rule hex-digit (also in [JLS14:3.3]).

This is a type introduced by fty::deffixtype.

Definitions and Theorems

Function: hex-digitp

(defun hex-digitp (x)
  (declare (xargs :guard t))
  (let ((__function__ 'hex-digitp))
    (declare (ignorable __function__))
    (and (integerp x)
         (or (and (<= (char-code #\0) x)
                  (<= x (char-code #\9)))
             (and (<= (char-code #\A) x)
                  (<= x (char-code #\F)))
             (and (<= (char-code #\a) x)
                  (<= x (char-code #\f)))))))

Theorem: booleanp-of-hex-digitp

(defthm booleanp-of-hex-digitp
  (b* ((yes/no (hex-digitp x)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Function: hex-digit-fix

(defun hex-digit-fix (x)
  (declare (xargs :guard (hex-digitp x)))
  (mbe :logic (if (hex-digitp x) x (char-code #\0))
       :exec x))

Theorem: hex-digitp-of-hex-digit-fix

(defthm hex-digitp-of-hex-digit-fix
  (b* ((fixed-x (hex-digit-fix x)))
    (hex-digitp fixed-x))
  :rule-classes :rewrite)

Theorem: hex-digit-fix-when-hex-digitp

(defthm hex-digit-fix-when-hex-digitp
  (implies (hex-digitp x)
           (equal (hex-digit-fix x) x)))

Function: hex-digit-equiv$inline

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

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

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

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

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

Theorem: hex-digit-fix-under-hex-digit-equiv

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

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

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

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

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

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

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

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

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

Subtopics

Hex-digitp: Recognizer for hex-digit.
Hex-digit-fix: Fixer for hex-digit.