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