Fixing function for hexdig/uscore structures.
(hexdig/uscore-fix x) → new-x
Function:
(defun hexdig/uscore-fix$inline (x) (declare (xargs :guard (hexdig/uscore-p x))) (let ((__function__ 'hexdig/uscore-fix)) (declare (ignorable __function__)) (mbe :logic (case (hexdig/uscore-kind x) (:digit (b* ((get (hex-digit-fix (std::da-nth 0 (cdr x))))) (cons :digit (list get)))) (:underscore (cons :underscore (list)))) :exec x)))
Theorem:
(defthm hexdig/uscore-p-of-hexdig/uscore-fix (b* ((new-x (hexdig/uscore-fix$inline x))) (hexdig/uscore-p new-x)) :rule-classes :rewrite)
Theorem:
(defthm hexdig/uscore-fix-when-hexdig/uscore-p (implies (hexdig/uscore-p x) (equal (hexdig/uscore-fix x) x)))
Function:
(defun hexdig/uscore-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (hexdig/uscore-p acl2::x) (hexdig/uscore-p acl2::y)))) (equal (hexdig/uscore-fix acl2::x) (hexdig/uscore-fix acl2::y)))
Theorem:
(defthm hexdig/uscore-equiv-is-an-equivalence (and (booleanp (hexdig/uscore-equiv x y)) (hexdig/uscore-equiv x x) (implies (hexdig/uscore-equiv x y) (hexdig/uscore-equiv y x)) (implies (and (hexdig/uscore-equiv x y) (hexdig/uscore-equiv y z)) (hexdig/uscore-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm hexdig/uscore-equiv-implies-equal-hexdig/uscore-fix-1 (implies (hexdig/uscore-equiv acl2::x x-equiv) (equal (hexdig/uscore-fix acl2::x) (hexdig/uscore-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm hexdig/uscore-fix-under-hexdig/uscore-equiv (hexdig/uscore-equiv (hexdig/uscore-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-hexdig/uscore-fix-1-forward-to-hexdig/uscore-equiv (implies (equal (hexdig/uscore-fix acl2::x) acl2::y) (hexdig/uscore-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-hexdig/uscore-fix-2-forward-to-hexdig/uscore-equiv (implies (equal acl2::x (hexdig/uscore-fix acl2::y)) (hexdig/uscore-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm hexdig/uscore-equiv-of-hexdig/uscore-fix-1-forward (implies (hexdig/uscore-equiv (hexdig/uscore-fix acl2::x) acl2::y) (hexdig/uscore-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm hexdig/uscore-equiv-of-hexdig/uscore-fix-2-forward (implies (hexdig/uscore-equiv acl2::x (hexdig/uscore-fix acl2::y)) (hexdig/uscore-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm hexdig/uscore-kind$inline-of-hexdig/uscore-fix-x (equal (hexdig/uscore-kind$inline (hexdig/uscore-fix x)) (hexdig/uscore-kind$inline x)))
Theorem:
(defthm hexdig/uscore-kind$inline-hexdig/uscore-equiv-congruence-on-x (implies (hexdig/uscore-equiv x x-equiv) (equal (hexdig/uscore-kind$inline x) (hexdig/uscore-kind$inline x-equiv))) :rule-classes :congruence)
Theorem:
(defthm consp-of-hexdig/uscore-fix (consp (hexdig/uscore-fix x)) :rule-classes :type-prescription)