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Decimal-digits

Dec-digit

Fixtype of Java decimal digits [JLS14:3.10.1].

A Java decimal digit is a Java ASCII character between `0' and `9'. See the grammar rule digit.

This is a type introduced by fty::deffixtype.

Definitions and Theorems

Function: dec-digitp

(defun dec-digitp (x)
  (declare (xargs :guard t))
  (let ((__function__ 'dec-digitp))
    (declare (ignorable __function__))
    (and (integerp x)
         (<= (char-code #\0) x)
         (<= x (char-code #\9)))))

Theorem: booleanp-of-dec-digitp

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

Function: dec-digit-fix

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

Theorem: dec-digitp-of-dec-digit-fix

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

Theorem: dec-digit-fix-when-dec-digitp

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

Function: dec-digit-equiv$inline

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

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

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

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

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

Theorem: dec-digit-fix-under-dec-digit-equiv

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

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

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

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

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

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

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

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

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

Subtopics

Dec-digitp: Recognizer for dec-digit.
Dec-digit-fix: Fixer for dec-digit.