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