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Defbyte-standard-instances

Sbyte4

Fixtype of signed bytes of size 4.

Definitions and Theorems

Function: sbyte4-equiv$inline

(defun sbyte4-equiv$inline (x y)
  (declare (xargs :guard (and (sbyte4p x) (sbyte4p y))))
  (equal (sbyte4-fix x) (sbyte4-fix y)))

Theorem: sbyte4-equiv-is-an-equivalence

(defthm sbyte4-equiv-is-an-equivalence
  (and (booleanp (sbyte4-equiv x y))
       (sbyte4-equiv x x)
       (implies (sbyte4-equiv x y)
                (sbyte4-equiv y x))
       (implies (and (sbyte4-equiv x y)
                     (sbyte4-equiv y z))
                (sbyte4-equiv x z)))
  :rule-classes (:equivalence))

Theorem: sbyte4-equiv-implies-equal-sbyte4-fix-1

(defthm sbyte4-equiv-implies-equal-sbyte4-fix-1
  (implies (sbyte4-equiv x x-equiv)
           (equal (sbyte4-fix x)
                  (sbyte4-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sbyte4-fix-under-sbyte4-equiv

(defthm sbyte4-fix-under-sbyte4-equiv
  (sbyte4-equiv (sbyte4-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sbyte4-fix-1-forward-to-sbyte4-equiv

(defthm equal-of-sbyte4-fix-1-forward-to-sbyte4-equiv
  (implies (equal (sbyte4-fix x) y)
           (sbyte4-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sbyte4-fix-2-forward-to-sbyte4-equiv

(defthm equal-of-sbyte4-fix-2-forward-to-sbyte4-equiv
  (implies (equal x (sbyte4-fix y))
           (sbyte4-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sbyte4-equiv-of-sbyte4-fix-1-forward

(defthm sbyte4-equiv-of-sbyte4-fix-1-forward
  (implies (sbyte4-equiv (sbyte4-fix x) y)
           (sbyte4-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sbyte4-equiv-of-sbyte4-fix-2-forward

(defthm sbyte4-equiv-of-sbyte4-fix-2-forward
  (implies (sbyte4-equiv x (sbyte4-fix y))
           (sbyte4-equiv x y))
  :rule-classes :forward-chaining)

Subtopics

Sbyte4p: Recognizer for sbyte4.
Sbyte4-fix: Fixer for sbyte4.