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Nibble

A fixtype of (unsigned 4-bit) nibbles.

We use fty::defbyte to generate this fixtype, along with the recognizer, fixer, and equivalence. The recognizer is identical to nibblep.

Definitions and Theorems

Function: nibble-equiv$inline

(defun nibble-equiv$inline (x y)
  (declare (xargs :guard (and (nibblep x) (nibblep y))))
  (equal (nibble-fix x) (nibble-fix y)))

Theorem: nibble-equiv-is-an-equivalence

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

Theorem: nibble-equiv-implies-equal-nibble-fix-1

(defthm nibble-equiv-implies-equal-nibble-fix-1
  (implies (nibble-equiv x x-equiv)
           (equal (nibble-fix x)
                  (nibble-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: nibble-fix-under-nibble-equiv

(defthm nibble-fix-under-nibble-equiv
  (nibble-equiv (nibble-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-nibble-fix-1-forward-to-nibble-equiv

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

Theorem: equal-of-nibble-fix-2-forward-to-nibble-equiv

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

Theorem: nibble-equiv-of-nibble-fix-1-forward

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

Theorem: nibble-equiv-of-nibble-fix-2-forward

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

Subtopics

Nibble-list: A fixtype of true lists of (unsigned 4-bit) nibbles.
Nibble-fix: Fixer for nibble.
Nibblep-additional-theorems: Additional theorems about nibble.