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