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Fixtype of signed bytes of size 8.
Function:
(defun sbyte8-equiv$inline (x y) (declare (xargs :guard (and (sbyte8p x) (sbyte8p y)))) (equal (sbyte8-fix x) (sbyte8-fix y)))
Theorem:
(defthm sbyte8-equiv-is-an-equivalence (and (booleanp (sbyte8-equiv x y)) (sbyte8-equiv x x) (implies (sbyte8-equiv x y) (sbyte8-equiv y x)) (implies (and (sbyte8-equiv x y) (sbyte8-equiv y z)) (sbyte8-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm sbyte8-equiv-implies-equal-sbyte8-fix-1 (implies (sbyte8-equiv x x-equiv) (equal (sbyte8-fix x) (sbyte8-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm sbyte8-fix-under-sbyte8-equiv (sbyte8-equiv (sbyte8-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-sbyte8-fix-1-forward-to-sbyte8-equiv (implies (equal (sbyte8-fix x) y) (sbyte8-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-sbyte8-fix-2-forward-to-sbyte8-equiv (implies (equal x (sbyte8-fix y)) (sbyte8-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm sbyte8-equiv-of-sbyte8-fix-1-forward (implies (sbyte8-equiv (sbyte8-fix x) y) (sbyte8-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm sbyte8-equiv-of-sbyte8-fix-2-forward (implies (sbyte8-equiv x (sbyte8-fix y)) (sbyte8-equiv x y)) :rule-classes :forward-chaining)