(nat-val-alistp-fix x) is an ACL2::fty alist fixing function that follows the drop-keys strategy.
(nat-val-alistp-fix x) → fty::newx
Note that in the execution this is just an inline identity function.
Function:
(defun nat-val-alistp-fix$inline (x) (declare (xargs :guard (nat-val-alistp-p x))) (let ((__function__ 'nat-val-alistp-fix)) (declare (ignorable __function__)) (mbe :logic (if (atom x) nil (if (consp (car x)) (cons (cons (caar x) (nfix (cdar x))) (nat-val-alistp-fix (cdr x))) (nat-val-alistp-fix (cdr x)))) :exec x)))
Theorem:
(defthm nat-val-alistp-p-of-nat-val-alistp-fix (b* ((fty::newx (nat-val-alistp-fix$inline x))) (nat-val-alistp-p fty::newx)) :rule-classes :rewrite)
Theorem:
(defthm nat-val-alistp-fix-when-nat-val-alistp-p (implies (nat-val-alistp-p x) (equal (nat-val-alistp-fix x) x)))
Function:
(defun nat-val-alistp-equiv$inline (x acl2::y) (declare (xargs :guard (and (nat-val-alistp-p x) (nat-val-alistp-p acl2::y)))) (equal (nat-val-alistp-fix x) (nat-val-alistp-fix acl2::y)))
Theorem:
(defthm nat-val-alistp-equiv-is-an-equivalence (and (booleanp (nat-val-alistp-equiv x y)) (nat-val-alistp-equiv x x) (implies (nat-val-alistp-equiv x y) (nat-val-alistp-equiv y x)) (implies (and (nat-val-alistp-equiv x y) (nat-val-alistp-equiv y z)) (nat-val-alistp-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm nat-val-alistp-equiv-implies-equal-nat-val-alistp-fix-1 (implies (nat-val-alistp-equiv x x-equiv) (equal (nat-val-alistp-fix x) (nat-val-alistp-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm nat-val-alistp-fix-under-nat-val-alistp-equiv (nat-val-alistp-equiv (nat-val-alistp-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-nat-val-alistp-fix-1-forward-to-nat-val-alistp-equiv (implies (equal (nat-val-alistp-fix x) acl2::y) (nat-val-alistp-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-nat-val-alistp-fix-2-forward-to-nat-val-alistp-equiv (implies (equal x (nat-val-alistp-fix acl2::y)) (nat-val-alistp-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm nat-val-alistp-equiv-of-nat-val-alistp-fix-1-forward (implies (nat-val-alistp-equiv (nat-val-alistp-fix x) acl2::y) (nat-val-alistp-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm nat-val-alistp-equiv-of-nat-val-alistp-fix-2-forward (implies (nat-val-alistp-equiv x (nat-val-alistp-fix acl2::y)) (nat-val-alistp-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm cons-of-nfix-v-under-nat-val-alistp-equiv (nat-val-alistp-equiv (cons (cons acl2::k (nfix acl2::v)) x) (cons (cons acl2::k acl2::v) x)))
Theorem:
(defthm cons-nat-equiv-congruence-on-v-under-nat-val-alistp-equiv (implies (nat-equiv acl2::v v-equiv) (nat-val-alistp-equiv (cons (cons acl2::k acl2::v) x) (cons (cons acl2::k v-equiv) x))) :rule-classes :congruence)
Theorem:
(defthm cons-of-nat-val-alistp-fix-y-under-nat-val-alistp-equiv (nat-val-alistp-equiv (cons x (nat-val-alistp-fix acl2::y)) (cons x acl2::y)))
Theorem:
(defthm cons-nat-val-alistp-equiv-congruence-on-y-under-nat-val-alistp-equiv (implies (nat-val-alistp-equiv acl2::y y-equiv) (nat-val-alistp-equiv (cons x acl2::y) (cons x y-equiv))) :rule-classes :congruence)
Theorem:
(defthm nat-val-alistp-fix-of-acons (equal (nat-val-alistp-fix (cons (cons acl2::a acl2::b) x)) (cons (cons acl2::a (nfix acl2::b)) (nat-val-alistp-fix x))))
Theorem:
(defthm hons-assoc-equal-of-nat-val-alistp-fix (equal (hons-assoc-equal acl2::k (nat-val-alistp-fix x)) (let ((fty::pair (hons-assoc-equal acl2::k x))) (and fty::pair (cons acl2::k (nfix (cdr fty::pair)))))))
Theorem:
(defthm nat-val-alistp-fix-of-append (equal (nat-val-alistp-fix (append std::a std::b)) (append (nat-val-alistp-fix std::a) (nat-val-alistp-fix std::b))))
Theorem:
(defthm consp-car-of-nat-val-alistp-fix (equal (consp (car (nat-val-alistp-fix x))) (consp (nat-val-alistp-fix x))))