Recognizer for valid-ord-scope.
(valid-ord-scopep x) → *
Function:
(defun valid-ord-scopep (x) (declare (xargs :guard t)) (let ((__function__ 'valid-ord-scopep)) (declare (ignorable __function__)) (if (atom x) (eq x nil) (and (consp (car x)) (identp (caar x)) (valid-ord-infop (cdar x)) (valid-ord-scopep (cdr x))))))
Theorem:
(defthm valid-ord-scopep-of-revappend (equal (valid-ord-scopep (revappend acl2::x acl2::y)) (and (valid-ord-scopep (list-fix acl2::x)) (valid-ord-scopep acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-remove (implies (valid-ord-scopep acl2::x) (valid-ord-scopep (remove acl2::a acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-last (implies (valid-ord-scopep (double-rewrite acl2::x)) (valid-ord-scopep (last acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-nthcdr (implies (valid-ord-scopep (double-rewrite acl2::x)) (valid-ord-scopep (nthcdr acl2::n acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-butlast (implies (valid-ord-scopep (double-rewrite acl2::x)) (valid-ord-scopep (butlast acl2::x acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-update-nth (implies (valid-ord-scopep (double-rewrite acl2::x)) (iff (valid-ord-scopep (update-nth acl2::n acl2::y acl2::x)) (and (and (consp acl2::y) (identp (car acl2::y)) (valid-ord-infop (cdr acl2::y))) (or (<= (nfix acl2::n) (len acl2::x)) (and (consp nil) (identp (car nil)) (valid-ord-infop (cdr nil))))))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-repeat (iff (valid-ord-scopep (repeat acl2::n acl2::x)) (or (and (consp acl2::x) (identp (car acl2::x)) (valid-ord-infop (cdr acl2::x))) (zp acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-take (implies (valid-ord-scopep (double-rewrite acl2::x)) (iff (valid-ord-scopep (take acl2::n acl2::x)) (or (and (consp nil) (identp (car nil)) (valid-ord-infop (cdr nil))) (<= (nfix acl2::n) (len acl2::x))))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-union-equal (equal (valid-ord-scopep (union-equal acl2::x acl2::y)) (and (valid-ord-scopep (list-fix acl2::x)) (valid-ord-scopep (double-rewrite acl2::y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-intersection-equal-2 (implies (valid-ord-scopep (double-rewrite acl2::y)) (valid-ord-scopep (intersection-equal acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-intersection-equal-1 (implies (valid-ord-scopep (double-rewrite acl2::x)) (valid-ord-scopep (intersection-equal acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-set-difference-equal (implies (valid-ord-scopep acl2::x) (valid-ord-scopep (set-difference-equal acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-when-subsetp-equal (and (implies (and (subsetp-equal acl2::x acl2::y) (valid-ord-scopep acl2::y)) (equal (valid-ord-scopep acl2::x) (true-listp acl2::x))) (implies (and (valid-ord-scopep acl2::y) (subsetp-equal acl2::x acl2::y)) (equal (valid-ord-scopep acl2::x) (true-listp acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-rcons (iff (valid-ord-scopep (rcons acl2::a acl2::x)) (and (and (consp acl2::a) (identp (car acl2::a)) (valid-ord-infop (cdr acl2::a))) (valid-ord-scopep (list-fix acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-append (equal (valid-ord-scopep (append acl2::a acl2::b)) (and (valid-ord-scopep (list-fix acl2::a)) (valid-ord-scopep acl2::b))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-rev (equal (valid-ord-scopep (rev acl2::x)) (valid-ord-scopep (list-fix acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-duplicated-members (implies (valid-ord-scopep acl2::x) (valid-ord-scopep (duplicated-members acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-difference (implies (valid-ord-scopep acl2::x) (valid-ord-scopep (difference acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-intersect-2 (implies (valid-ord-scopep acl2::y) (valid-ord-scopep (intersect acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-intersect-1 (implies (valid-ord-scopep acl2::x) (valid-ord-scopep (intersect acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-union (iff (valid-ord-scopep (union acl2::x acl2::y)) (and (valid-ord-scopep (sfix acl2::x)) (valid-ord-scopep (sfix acl2::y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-mergesort (iff (valid-ord-scopep (mergesort acl2::x)) (valid-ord-scopep (list-fix acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-delete (implies (valid-ord-scopep acl2::x) (valid-ord-scopep (delete acl2::k acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-insert (iff (valid-ord-scopep (insert acl2::a acl2::x)) (and (valid-ord-scopep (sfix acl2::x)) (and (consp acl2::a) (identp (car acl2::a)) (valid-ord-infop (cdr acl2::a))))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-sfix (iff (valid-ord-scopep (sfix acl2::x)) (or (valid-ord-scopep acl2::x) (not (setp acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-list-fix (implies (valid-ord-scopep acl2::x) (valid-ord-scopep (list-fix acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm true-listp-when-valid-ord-scopep-compound-recognizer (implies (valid-ord-scopep acl2::x) (true-listp acl2::x)) :rule-classes :compound-recognizer)
Theorem:
(defthm valid-ord-scopep-when-not-consp (implies (not (consp acl2::x)) (equal (valid-ord-scopep acl2::x) (not acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-cdr-when-valid-ord-scopep (implies (valid-ord-scopep (double-rewrite acl2::x)) (valid-ord-scopep (cdr acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-cons (equal (valid-ord-scopep (cons acl2::a acl2::x)) (and (and (consp acl2::a) (identp (car acl2::a)) (valid-ord-infop (cdr acl2::a))) (valid-ord-scopep acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-remove-assoc (implies (valid-ord-scopep acl2::x) (valid-ord-scopep (remove-assoc-equal acl2::name acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-put-assoc (implies (and (valid-ord-scopep acl2::x)) (iff (valid-ord-scopep (put-assoc-equal acl2::name acl2::val acl2::x)) (and (identp acl2::name) (valid-ord-infop acl2::val)))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-fast-alist-clean (implies (valid-ord-scopep acl2::x) (valid-ord-scopep (fast-alist-clean acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-hons-shrink-alist (implies (and (valid-ord-scopep acl2::x) (valid-ord-scopep acl2::y)) (valid-ord-scopep (hons-shrink-alist acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-scopep-of-hons-acons (equal (valid-ord-scopep (hons-acons acl2::a acl2::n acl2::x)) (and (identp acl2::a) (valid-ord-infop acl2::n) (valid-ord-scopep acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm valid-ord-infop-of-cdr-of-hons-assoc-equal-when-valid-ord-scopep (implies (valid-ord-scopep acl2::x) (iff (valid-ord-infop (cdr (hons-assoc-equal acl2::k acl2::x))) (hons-assoc-equal acl2::k acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm alistp-when-valid-ord-scopep-rewrite (implies (valid-ord-scopep acl2::x) (alistp acl2::x)) :rule-classes ((:rewrite)))
Theorem:
(defthm alistp-when-valid-ord-scopep (implies (valid-ord-scopep acl2::x) (alistp acl2::x)) :rule-classes :tau-system)
Theorem:
(defthm valid-ord-infop-of-cdar-when-valid-ord-scopep (implies (valid-ord-scopep acl2::x) (iff (valid-ord-infop (cdar acl2::x)) (consp acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm identp-of-caar-when-valid-ord-scopep (implies (valid-ord-scopep acl2::x) (iff (identp (caar acl2::x)) (consp acl2::x))) :rule-classes ((:rewrite)))