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Binary-digits

Bin-digit

Fixtype of Java binary digits [JLS14:3.10.1].

A Java binary digit is one of the Java ASCII characters `0' and `1'. See the grammar rule binary-digit.

This is a type introduced by fty::deffixtype.

Definitions and Theorems

Function: bin-digitp

(defun bin-digitp (x)
  (declare (xargs :guard t))
  (let ((__function__ 'bin-digitp))
    (declare (ignorable __function__))
    (or (eql x (char-code #\0))
        (eql x (char-code #\1)))))

Theorem: booleanp-of-bin-digitp

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

Function: bin-digit-fix

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

Theorem: bin-digitp-of-bin-digit-fix

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

Theorem: bin-digit-fix-when-bin-digitp

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

Function: bin-digit-equiv$inline

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

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

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

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

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

Theorem: bin-digit-fix-under-bin-digit-equiv

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

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

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

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

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

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

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

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

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

Subtopics

Bin-digitp: Recognizer for bin-digit.
Bin-digit-fix: Fixer for bin-digit.