Recognizer for nat-val-alistp.
(nat-val-alistp-p x) → *
Function:
(defun nat-val-alistp-p (x) (declare (xargs :guard t)) (let ((__function__ 'nat-val-alistp-p)) (declare (ignorable __function__)) (if (atom x) (eq x nil) (and (consp (car x)) (natp (cdar x)) (nat-val-alistp-p (cdr x))))))
Theorem:
(defthm nat-val-alistp-p-of-repeat (iff (nat-val-alistp-p (acl2::repeat acl2::n x)) (or (and (consp x) (natp (cdr x))) (zp acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-butlast (implies (nat-val-alistp-p (double-rewrite x)) (nat-val-alistp-p (butlast x acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-rev (equal (nat-val-alistp-p (acl2::rev x)) (nat-val-alistp-p (acl2::list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-append (equal (nat-val-alistp-p (append acl2::a acl2::b)) (and (nat-val-alistp-p (acl2::list-fix acl2::a)) (nat-val-alistp-p acl2::b))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-list-fix (implies (nat-val-alistp-p x) (nat-val-alistp-p (acl2::list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm true-listp-when-nat-val-alistp-p-compound-recognizer (implies (nat-val-alistp-p x) (true-listp x)) :rule-classes :compound-recognizer)
Theorem:
(defthm nat-val-alistp-p-when-not-consp (implies (not (consp x)) (equal (nat-val-alistp-p x) (not x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-cdr-when-nat-val-alistp-p (implies (nat-val-alistp-p (double-rewrite x)) (nat-val-alistp-p (cdr x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-cons (equal (nat-val-alistp-p (cons acl2::a x)) (and (and (consp acl2::a) (natp (cdr acl2::a))) (nat-val-alistp-p x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-remove-assoc (implies (nat-val-alistp-p x) (nat-val-alistp-p (remove-assoc-equal acl2::name x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-put-assoc (implies (and (nat-val-alistp-p x)) (iff (nat-val-alistp-p (put-assoc-equal acl2::name acl2::val x)) (and t (natp acl2::val)))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-fast-alist-clean (implies (nat-val-alistp-p x) (nat-val-alistp-p (fast-alist-clean x))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-hons-shrink-alist (implies (and (nat-val-alistp-p x) (nat-val-alistp-p acl2::y)) (nat-val-alistp-p (hons-shrink-alist x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm nat-val-alistp-p-of-hons-acons (equal (nat-val-alistp-p (hons-acons acl2::a acl2::n x)) (and t (natp acl2::n) (nat-val-alistp-p x))) :rule-classes ((:rewrite)))
Theorem:
(defthm natp-of-cdr-of-hons-assoc-equal-when-nat-val-alistp-p (implies (nat-val-alistp-p x) (iff (natp (cdr (hons-assoc-equal acl2::k x))) (or (hons-assoc-equal acl2::k x) (natp nil)))) :rule-classes ((:rewrite)))
Theorem:
(defthm alistp-when-nat-val-alistp-p-rewrite (implies (nat-val-alistp-p x) (alistp x)) :rule-classes ((:rewrite)))
Theorem:
(defthm alistp-when-nat-val-alistp-p (implies (nat-val-alistp-p x) (alistp x)) :rule-classes :tau-system)
Theorem:
(defthm natp-of-cdar-when-nat-val-alistp-p (implies (nat-val-alistp-p x) (iff (natp (cdar x)) (or (consp x) (natp nil)))) :rule-classes ((:rewrite)))