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