Fixing function for type-subset structures.
(type-subset-fix x) → new-x
Function:
(defun type-subset-fix$inline (x) (declare (xargs :guard (type-subsetp x))) (let ((__function__ 'type-subset-fix)) (declare (ignorable __function__)) (mbe :logic (b* ((supertype (type-fix (cdr (std::da-nth 0 x)))) (variable (identifier-fix (cdr (std::da-nth 1 x)))) (restriction (expression-fix (cdr (std::da-nth 2 x)))) (witness (maybe-expression-fix (cdr (std::da-nth 3 x))))) (list (cons 'supertype supertype) (cons 'variable variable) (cons 'restriction restriction) (cons 'witness witness))) :exec x)))
Theorem:
(defthm type-subsetp-of-type-subset-fix (b* ((new-x (type-subset-fix$inline x))) (type-subsetp new-x)) :rule-classes :rewrite)
Theorem:
(defthm type-subset-fix-when-type-subsetp (implies (type-subsetp x) (equal (type-subset-fix x) x)))
Function:
(defun type-subset-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (type-subsetp acl2::x) (type-subsetp acl2::y)))) (equal (type-subset-fix acl2::x) (type-subset-fix acl2::y)))
Theorem:
(defthm type-subset-equiv-is-an-equivalence (and (booleanp (type-subset-equiv x y)) (type-subset-equiv x x) (implies (type-subset-equiv x y) (type-subset-equiv y x)) (implies (and (type-subset-equiv x y) (type-subset-equiv y z)) (type-subset-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm type-subset-equiv-implies-equal-type-subset-fix-1 (implies (type-subset-equiv acl2::x x-equiv) (equal (type-subset-fix acl2::x) (type-subset-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm type-subset-fix-under-type-subset-equiv (type-subset-equiv (type-subset-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-type-subset-fix-1-forward-to-type-subset-equiv (implies (equal (type-subset-fix acl2::x) acl2::y) (type-subset-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-type-subset-fix-2-forward-to-type-subset-equiv (implies (equal acl2::x (type-subset-fix acl2::y)) (type-subset-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm type-subset-equiv-of-type-subset-fix-1-forward (implies (type-subset-equiv (type-subset-fix acl2::x) acl2::y) (type-subset-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm type-subset-equiv-of-type-subset-fix-2-forward (implies (type-subset-equiv acl2::x (type-subset-fix acl2::y)) (type-subset-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)