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