Fixing function for member-type structures.
(member-type-fix x) → new-x
Function:
(defun member-type-fix$inline (x) (declare (xargs :guard (member-typep x))) (let ((__function__ 'member-type-fix)) (declare (ignorable __function__)) (mbe :logic (b* ((name (ident-fix (cdr (std::da-nth 0 (cdr x))))) (type (type-fix (cdr (std::da-nth 1 (cdr x)))))) (cons :member-type (list (cons 'name name) (cons 'type type)))) :exec x)))
Theorem:
(defthm member-typep-of-member-type-fix (b* ((new-x (member-type-fix$inline x))) (member-typep new-x)) :rule-classes :rewrite)
Theorem:
(defthm member-type-fix-when-member-typep (implies (member-typep x) (equal (member-type-fix x) x)))
Function:
(defun member-type-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (member-typep acl2::x) (member-typep acl2::y)))) (equal (member-type-fix acl2::x) (member-type-fix acl2::y)))
Theorem:
(defthm member-type-equiv-is-an-equivalence (and (booleanp (member-type-equiv x y)) (member-type-equiv x x) (implies (member-type-equiv x y) (member-type-equiv y x)) (implies (and (member-type-equiv x y) (member-type-equiv y z)) (member-type-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm member-type-equiv-implies-equal-member-type-fix-1 (implies (member-type-equiv acl2::x x-equiv) (equal (member-type-fix acl2::x) (member-type-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm member-type-fix-under-member-type-equiv (member-type-equiv (member-type-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-member-type-fix-1-forward-to-member-type-equiv (implies (equal (member-type-fix acl2::x) acl2::y) (member-type-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-member-type-fix-2-forward-to-member-type-equiv (implies (equal acl2::x (member-type-fix acl2::y)) (member-type-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm member-type-equiv-of-member-type-fix-1-forward (implies (member-type-equiv (member-type-fix acl2::x) acl2::y) (member-type-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm member-type-equiv-of-member-type-fix-2-forward (implies (member-type-equiv acl2::x (member-type-fix acl2::y)) (member-type-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)