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Defbyte-standard-instances

Sbyte1

Fixtype of signed bytes of size 1.

Definitions and Theorems

Function: sbyte1-equiv$inline

(defun sbyte1-equiv$inline (x y)
  (declare (xargs :guard (and (sbyte1p x) (sbyte1p y))))
  (equal (sbyte1-fix x) (sbyte1-fix y)))

Theorem: sbyte1-equiv-is-an-equivalence

(defthm sbyte1-equiv-is-an-equivalence
  (and (booleanp (sbyte1-equiv x y))
       (sbyte1-equiv x x)
       (implies (sbyte1-equiv x y)
                (sbyte1-equiv y x))
       (implies (and (sbyte1-equiv x y)
                     (sbyte1-equiv y z))
                (sbyte1-equiv x z)))
  :rule-classes (:equivalence))

Theorem: sbyte1-equiv-implies-equal-sbyte1-fix-1

(defthm sbyte1-equiv-implies-equal-sbyte1-fix-1
  (implies (sbyte1-equiv x x-equiv)
           (equal (sbyte1-fix x)
                  (sbyte1-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sbyte1-fix-under-sbyte1-equiv

(defthm sbyte1-fix-under-sbyte1-equiv
  (sbyte1-equiv (sbyte1-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sbyte1-fix-1-forward-to-sbyte1-equiv

(defthm equal-of-sbyte1-fix-1-forward-to-sbyte1-equiv
  (implies (equal (sbyte1-fix x) y)
           (sbyte1-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sbyte1-fix-2-forward-to-sbyte1-equiv

(defthm equal-of-sbyte1-fix-2-forward-to-sbyte1-equiv
  (implies (equal x (sbyte1-fix y))
           (sbyte1-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sbyte1-equiv-of-sbyte1-fix-1-forward

(defthm sbyte1-equiv-of-sbyte1-fix-1-forward
  (implies (sbyte1-equiv (sbyte1-fix x) y)
           (sbyte1-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sbyte1-equiv-of-sbyte1-fix-2-forward

(defthm sbyte1-equiv-of-sbyte1-fix-2-forward
  (implies (sbyte1-equiv x (sbyte1-fix y))
           (sbyte1-equiv x y))
  :rule-classes :forward-chaining)

Subtopics

Sbyte1p: Recognizer for sbyte1.
Sbyte1-fix: Fixer for sbyte1.