Basic equivalence relation for hexdig/uscore structures.
Function:
(defun hexdig/uscore-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (hexdig/uscore-p acl2::x) (hexdig/uscore-p acl2::y)))) (equal (hexdig/uscore-fix acl2::x) (hexdig/uscore-fix acl2::y)))
Theorem:
(defthm hexdig/uscore-equiv-is-an-equivalence (and (booleanp (hexdig/uscore-equiv x y)) (hexdig/uscore-equiv x x) (implies (hexdig/uscore-equiv x y) (hexdig/uscore-equiv y x)) (implies (and (hexdig/uscore-equiv x y) (hexdig/uscore-equiv y z)) (hexdig/uscore-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm hexdig/uscore-equiv-implies-equal-hexdig/uscore-fix-1 (implies (hexdig/uscore-equiv acl2::x x-equiv) (equal (hexdig/uscore-fix acl2::x) (hexdig/uscore-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm hexdig/uscore-fix-under-hexdig/uscore-equiv (hexdig/uscore-equiv (hexdig/uscore-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-hexdig/uscore-fix-1-forward-to-hexdig/uscore-equiv (implies (equal (hexdig/uscore-fix acl2::x) acl2::y) (hexdig/uscore-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-hexdig/uscore-fix-2-forward-to-hexdig/uscore-equiv (implies (equal acl2::x (hexdig/uscore-fix acl2::y)) (hexdig/uscore-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm hexdig/uscore-equiv-of-hexdig/uscore-fix-1-forward (implies (hexdig/uscore-equiv (hexdig/uscore-fix acl2::x) acl2::y) (hexdig/uscore-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm hexdig/uscore-equiv-of-hexdig/uscore-fix-2-forward (implies (hexdig/uscore-equiv acl2::x (hexdig/uscore-fix acl2::y)) (hexdig/uscore-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)