		Search-engine friendly clone of the ACL2 documentation.

Defbyte-standard-instances

Sbyte16

Fixtype of signed bytes of size 16.

Definitions and Theorems

Function: sbyte16-equiv$inline

(defun sbyte16-equiv$inline (x y)
  (declare (xargs :guard (and (sbyte16p x) (sbyte16p y))))
  (equal (sbyte16-fix x) (sbyte16-fix y)))

Theorem: sbyte16-equiv-is-an-equivalence

(defthm sbyte16-equiv-is-an-equivalence
  (and (booleanp (sbyte16-equiv x y))
       (sbyte16-equiv x x)
       (implies (sbyte16-equiv x y)
                (sbyte16-equiv y x))
       (implies (and (sbyte16-equiv x y)
                     (sbyte16-equiv y z))
                (sbyte16-equiv x z)))
  :rule-classes (:equivalence))

Theorem: sbyte16-equiv-implies-equal-sbyte16-fix-1

(defthm sbyte16-equiv-implies-equal-sbyte16-fix-1
  (implies (sbyte16-equiv x x-equiv)
           (equal (sbyte16-fix x)
                  (sbyte16-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sbyte16-fix-under-sbyte16-equiv

(defthm sbyte16-fix-under-sbyte16-equiv
  (sbyte16-equiv (sbyte16-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sbyte16-fix-1-forward-to-sbyte16-equiv

(defthm equal-of-sbyte16-fix-1-forward-to-sbyte16-equiv
  (implies (equal (sbyte16-fix x) y)
           (sbyte16-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sbyte16-fix-2-forward-to-sbyte16-equiv

(defthm equal-of-sbyte16-fix-2-forward-to-sbyte16-equiv
  (implies (equal x (sbyte16-fix y))
           (sbyte16-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sbyte16-equiv-of-sbyte16-fix-1-forward

(defthm sbyte16-equiv-of-sbyte16-fix-1-forward
  (implies (sbyte16-equiv (sbyte16-fix x) y)
           (sbyte16-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sbyte16-equiv-of-sbyte16-fix-2-forward

(defthm sbyte16-equiv-of-sbyte16-fix-2-forward
  (implies (sbyte16-equiv x (sbyte16-fix y))
           (sbyte16-equiv x y))
  :rule-classes :forward-chaining)

Subtopics

Sbyte16p: Recognizer for sbyte16.
Sbyte16-fix: Fixer for sbyte16.
Sbyte16-ihs-theorems: Theorems about sbyte16 and IHS functions.