		Search-engine friendly clone of the ACL2 documentation.

Defbyte-standard-instances

Sbyte2

Fixtype of signed bytes of size 2.

Definitions and Theorems

Function: sbyte2-equiv$inline

(defun sbyte2-equiv$inline (x y)
  (declare (xargs :guard (and (sbyte2p x) (sbyte2p y))))
  (equal (sbyte2-fix x) (sbyte2-fix y)))

Theorem: sbyte2-equiv-is-an-equivalence

(defthm sbyte2-equiv-is-an-equivalence
  (and (booleanp (sbyte2-equiv x y))
       (sbyte2-equiv x x)
       (implies (sbyte2-equiv x y)
                (sbyte2-equiv y x))
       (implies (and (sbyte2-equiv x y)
                     (sbyte2-equiv y z))
                (sbyte2-equiv x z)))
  :rule-classes (:equivalence))

Theorem: sbyte2-equiv-implies-equal-sbyte2-fix-1

(defthm sbyte2-equiv-implies-equal-sbyte2-fix-1
  (implies (sbyte2-equiv x x-equiv)
           (equal (sbyte2-fix x)
                  (sbyte2-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sbyte2-fix-under-sbyte2-equiv

(defthm sbyte2-fix-under-sbyte2-equiv
  (sbyte2-equiv (sbyte2-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sbyte2-fix-1-forward-to-sbyte2-equiv

(defthm equal-of-sbyte2-fix-1-forward-to-sbyte2-equiv
  (implies (equal (sbyte2-fix x) y)
           (sbyte2-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sbyte2-fix-2-forward-to-sbyte2-equiv

(defthm equal-of-sbyte2-fix-2-forward-to-sbyte2-equiv
  (implies (equal x (sbyte2-fix y))
           (sbyte2-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sbyte2-equiv-of-sbyte2-fix-1-forward

(defthm sbyte2-equiv-of-sbyte2-fix-1-forward
  (implies (sbyte2-equiv (sbyte2-fix x) y)
           (sbyte2-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sbyte2-equiv-of-sbyte2-fix-2-forward

(defthm sbyte2-equiv-of-sbyte2-fix-2-forward
  (implies (sbyte2-equiv x (sbyte2-fix y))
           (sbyte2-equiv x y))
  :rule-classes :forward-chaining)

Subtopics

Sbyte2p: Recognizer for sbyte2.
Sbyte2-fix: Fixer for sbyte2.