Basic equivalence relation for sbyte4-list structures.
Function:
(defun sbyte4-list-equiv$inline (x y) (declare (xargs :guard (and (sbyte4-listp x) (sbyte4-listp y)))) (equal (sbyte4-list-fix x) (sbyte4-list-fix y)))
Theorem:
(defthm sbyte4-list-equiv-is-an-equivalence (and (booleanp (sbyte4-list-equiv x y)) (sbyte4-list-equiv x x) (implies (sbyte4-list-equiv x y) (sbyte4-list-equiv y x)) (implies (and (sbyte4-list-equiv x y) (sbyte4-list-equiv y z)) (sbyte4-list-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm sbyte4-list-equiv-implies-equal-sbyte4-list-fix-1 (implies (sbyte4-list-equiv x x-equiv) (equal (sbyte4-list-fix x) (sbyte4-list-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm sbyte4-list-fix-under-sbyte4-list-equiv (sbyte4-list-equiv (sbyte4-list-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-sbyte4-list-fix-1-forward-to-sbyte4-list-equiv (implies (equal (sbyte4-list-fix x) y) (sbyte4-list-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-sbyte4-list-fix-2-forward-to-sbyte4-list-equiv (implies (equal x (sbyte4-list-fix y)) (sbyte4-list-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm sbyte4-list-equiv-of-sbyte4-list-fix-1-forward (implies (sbyte4-list-equiv (sbyte4-list-fix x) y) (sbyte4-list-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm sbyte4-list-equiv-of-sbyte4-list-fix-2-forward (implies (sbyte4-list-equiv x (sbyte4-list-fix y)) (sbyte4-list-equiv x y)) :rule-classes :forward-chaining)