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