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