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• Bin-digit-char-p

Bin-digit-char

Fixtype of binary digit characters.

This is a type introduced by fty::deffixtype.

Definitions and Theorems

Function: bin-digit-char-fix

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

Theorem: bin-digit-char-p-of-bin-digit-char-fix

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

Theorem: bin-digit-char-fix-when-bin-digit-char-p

(defthm bin-digit-char-fix-when-bin-digit-char-p
  (implies (bin-digit-char-p x)
           (equal (bin-digit-char-fix x) x)))

Function: bin-digit-char-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Subtopics

Bin-digit-char-fix

Fixer for bin-digit-char.

Bin-digit-char-list

Fixtype of lists of binary digit characters.