Recognizer for rsh-of-concat-alist.
(rsh-of-concat-alist-p x) → *
Function:
(defun rsh-of-concat-alist-p (x) (declare (xargs :guard t)) (let ((__function__ 'rsh-of-concat-alist-p)) (declare (ignorable __function__)) (if (atom x) t (and (consp (car x)) (natp (caar x)) (svex-p (cdar x)) (rsh-of-concat-alist-p (cdr x))))))
Theorem:
(defthm rsh-of-concat-alist-p-of-union-equal (equal (rsh-of-concat-alist-p (union-equal x y)) (and (rsh-of-concat-alist-p (list-fix x)) (rsh-of-concat-alist-p (double-rewrite y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-intersection-equal-2 (implies (rsh-of-concat-alist-p (double-rewrite y)) (rsh-of-concat-alist-p (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-intersection-equal-1 (implies (rsh-of-concat-alist-p (double-rewrite x)) (rsh-of-concat-alist-p (intersection-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-set-difference-equal (implies (rsh-of-concat-alist-p x) (rsh-of-concat-alist-p (set-difference-equal x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-set-equiv-congruence (implies (set-equiv x y) (equal (rsh-of-concat-alist-p x) (rsh-of-concat-alist-p y))) :rule-classes :congruence)
Theorem:
(defthm rsh-of-concat-alist-p-when-subsetp-equal (and (implies (and (subsetp-equal x y) (rsh-of-concat-alist-p y)) (rsh-of-concat-alist-p x)) (implies (and (rsh-of-concat-alist-p y) (subsetp-equal x y)) (rsh-of-concat-alist-p x))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-rcons (iff (rsh-of-concat-alist-p (acl2::rcons acl2::a x)) (and (and (consp acl2::a) (natp (car acl2::a)) (svex-p (cdr acl2::a))) (rsh-of-concat-alist-p (list-fix x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-repeat (iff (rsh-of-concat-alist-p (repeat acl2::n x)) (or (and (consp x) (natp (car x)) (svex-p (cdr x))) (zp acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-rev (equal (rsh-of-concat-alist-p (rev x)) (rsh-of-concat-alist-p (list-fix x))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-list-fix (equal (rsh-of-concat-alist-p (list-fix x)) (rsh-of-concat-alist-p x)) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-append (equal (rsh-of-concat-alist-p (append acl2::a acl2::b)) (and (rsh-of-concat-alist-p acl2::a) (rsh-of-concat-alist-p acl2::b))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-when-not-consp (implies (not (consp x)) (rsh-of-concat-alist-p x)) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-cdr-when-rsh-of-concat-alist-p (implies (rsh-of-concat-alist-p (double-rewrite x)) (rsh-of-concat-alist-p (cdr x))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-cons (equal (rsh-of-concat-alist-p (cons acl2::a x)) (and (and (consp acl2::a) (natp (car acl2::a)) (svex-p (cdr acl2::a))) (rsh-of-concat-alist-p x))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-fast-alist-clean (implies (rsh-of-concat-alist-p x) (rsh-of-concat-alist-p (fast-alist-clean x))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-hons-shrink-alist (implies (and (rsh-of-concat-alist-p x) (rsh-of-concat-alist-p y)) (rsh-of-concat-alist-p (hons-shrink-alist x y))) :rule-classes ((:rewrite)))
Theorem:
(defthm rsh-of-concat-alist-p-of-hons-acons (equal (rsh-of-concat-alist-p (hons-acons acl2::a acl2::n x)) (and (natp acl2::a) (svex-p acl2::n) (rsh-of-concat-alist-p x))) :rule-classes ((:rewrite)))
Theorem:
(defthm svex-p-of-cdr-of-hons-assoc-equal-when-rsh-of-concat-alist-p (implies (rsh-of-concat-alist-p x) (iff (svex-p (cdr (hons-assoc-equal acl2::k x))) (or (hons-assoc-equal acl2::k x) (svex-p nil)))) :rule-classes ((:rewrite)))
Theorem:
(defthm svex-p-of-cdar-when-rsh-of-concat-alist-p (implies (rsh-of-concat-alist-p x) (iff (svex-p (cdar x)) (or (consp x) (svex-p nil)))) :rule-classes ((:rewrite)))
Theorem:
(defthm natp-of-caar-when-rsh-of-concat-alist-p (implies (rsh-of-concat-alist-p x) (iff (natp (caar x)) (or (consp x) (natp nil)))) :rule-classes ((:rewrite)))