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• Dec-digit-char-p

Dec-digit-char

Fixtype of decimal digit characters.

This is a type introduced by fty::deffixtype.

Definitions and Theorems

Function: dec-digit-char-fix

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

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

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

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

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

Function: dec-digit-char-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Subtopics

Dec-digit-char-fix

Fixer for dec-digit-char.

Dec-digit-char-list

Fixtype of lists of decimal digit characters.