Recognize alists from keywords to fact information.
An alist of this type stores information about all the facts proved so far when running the proof.
Function:
(defun keyword-fact-info-alistp (x) (declare (xargs :guard t)) (if (consp x) (and (consp (car x)) (keywordp (caar x)) (fact-infop (cdar x)) (keyword-fact-info-alistp (cdr x))) (null x)))
Theorem:
(defthm keyword-fact-info-alistp-of-revappend (equal (keyword-fact-info-alistp (revappend acl2::x acl2::y)) (and (keyword-fact-info-alistp (list-fix acl2::x)) (keyword-fact-info-alistp acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-remove (implies (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp (remove acl2::a acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-last (implies (keyword-fact-info-alistp (double-rewrite acl2::x)) (keyword-fact-info-alistp (last acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-nthcdr (implies (keyword-fact-info-alistp (double-rewrite acl2::x)) (keyword-fact-info-alistp (nthcdr acl2::n acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-butlast (implies (keyword-fact-info-alistp (double-rewrite acl2::x)) (keyword-fact-info-alistp (butlast acl2::x acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-update-nth (implies (keyword-fact-info-alistp (double-rewrite acl2::x)) (iff (keyword-fact-info-alistp (update-nth acl2::n acl2::y acl2::x)) (and (and (consp acl2::y) (keywordp (car acl2::y)) (fact-infop (cdr acl2::y))) (or (<= (nfix acl2::n) (len acl2::x)) (and (consp nil) (keywordp (car nil)) (fact-infop (cdr nil))))))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-repeat (iff (keyword-fact-info-alistp (repeat acl2::n acl2::x)) (or (and (consp acl2::x) (keywordp (car acl2::x)) (fact-infop (cdr acl2::x))) (zp acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-take (implies (keyword-fact-info-alistp (double-rewrite acl2::x)) (iff (keyword-fact-info-alistp (take acl2::n acl2::x)) (or (and (consp nil) (keywordp (car nil)) (fact-infop (cdr nil))) (<= (nfix acl2::n) (len acl2::x))))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-union-equal (equal (keyword-fact-info-alistp (union-equal acl2::x acl2::y)) (and (keyword-fact-info-alistp (list-fix acl2::x)) (keyword-fact-info-alistp (double-rewrite acl2::y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-intersection-equal-2 (implies (keyword-fact-info-alistp (double-rewrite acl2::y)) (keyword-fact-info-alistp (intersection-equal acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-intersection-equal-1 (implies (keyword-fact-info-alistp (double-rewrite acl2::x)) (keyword-fact-info-alistp (intersection-equal acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-set-difference-equal (implies (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp (set-difference-equal acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-when-subsetp-equal (and (implies (and (subsetp-equal acl2::x acl2::y) (keyword-fact-info-alistp acl2::y)) (equal (keyword-fact-info-alistp acl2::x) (true-listp acl2::x))) (implies (and (keyword-fact-info-alistp acl2::y) (subsetp-equal acl2::x acl2::y)) (equal (keyword-fact-info-alistp acl2::x) (true-listp acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-rcons (iff (keyword-fact-info-alistp (rcons acl2::a acl2::x)) (and (and (consp acl2::a) (keywordp (car acl2::a)) (fact-infop (cdr acl2::a))) (keyword-fact-info-alistp (list-fix acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-append (equal (keyword-fact-info-alistp (append acl2::a acl2::b)) (and (keyword-fact-info-alistp (list-fix acl2::a)) (keyword-fact-info-alistp acl2::b))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-rev (equal (keyword-fact-info-alistp (rev acl2::x)) (keyword-fact-info-alistp (list-fix acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-duplicated-members (implies (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp (duplicated-members acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-difference (implies (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp (difference acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-intersect-2 (implies (keyword-fact-info-alistp acl2::y) (keyword-fact-info-alistp (intersect acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-intersect-1 (implies (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp (intersect acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-union (iff (keyword-fact-info-alistp (union acl2::x acl2::y)) (and (keyword-fact-info-alistp (sfix acl2::x)) (keyword-fact-info-alistp (sfix acl2::y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-mergesort (iff (keyword-fact-info-alistp (mergesort acl2::x)) (keyword-fact-info-alistp (list-fix acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-delete (implies (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp (delete acl2::k acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-insert (iff (keyword-fact-info-alistp (insert acl2::a acl2::x)) (and (keyword-fact-info-alistp (sfix acl2::x)) (and (consp acl2::a) (keywordp (car acl2::a)) (fact-infop (cdr acl2::a))))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-sfix (iff (keyword-fact-info-alistp (sfix acl2::x)) (or (keyword-fact-info-alistp acl2::x) (not (setp acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-list-fix (implies (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp (list-fix acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm true-listp-when-keyword-fact-info-alistp-compound-recognizer (implies (keyword-fact-info-alistp acl2::x) (true-listp acl2::x)) :rule-classes :compound-recognizer)
Theorem:
(defthm keyword-fact-info-alistp-when-not-consp (implies (not (consp acl2::x)) (equal (keyword-fact-info-alistp acl2::x) (not acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-cdr-when-keyword-fact-info-alistp (implies (keyword-fact-info-alistp (double-rewrite acl2::x)) (keyword-fact-info-alistp (cdr acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-cons (equal (keyword-fact-info-alistp (cons acl2::a acl2::x)) (and (and (consp acl2::a) (keywordp (car acl2::a)) (fact-infop (cdr acl2::a))) (keyword-fact-info-alistp acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-make-fal (implies (and (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp acl2::y)) (keyword-fact-info-alistp (make-fal acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm fact-infop-of-cdr-when-member-equal-of-keyword-fact-info-alistp (and (implies (and (keyword-fact-info-alistp acl2::x) (member-equal acl2::a acl2::x)) (fact-infop (cdr acl2::a))) (implies (and (member-equal acl2::a acl2::x) (keyword-fact-info-alistp acl2::x)) (fact-infop (cdr acl2::a)))) :rule-classes ((:rewrite)))
Theorem:
(defthm keywordp-of-car-when-member-equal-of-keyword-fact-info-alistp (and (implies (and (keyword-fact-info-alistp acl2::x) (member-equal acl2::a acl2::x)) (keywordp (car acl2::a))) (implies (and (member-equal acl2::a acl2::x) (keyword-fact-info-alistp acl2::x)) (keywordp (car acl2::a)))) :rule-classes ((:rewrite)))
Theorem:
(defthm consp-when-member-equal-of-keyword-fact-info-alistp (implies (and (keyword-fact-info-alistp acl2::x) (member-equal acl2::a acl2::x)) (consp acl2::a)) :rule-classes ((:rewrite :backchain-limit-lst (0 0)) (:rewrite :backchain-limit-lst (0 0) :corollary (implies (if (member-equal acl2::a acl2::x) (keyword-fact-info-alistp acl2::x) 'nil) (consp acl2::a)))))
Theorem:
(defthm keyword-fact-info-alistp-of-remove-assoc (implies (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp (remove-assoc-equal acl2::name acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-put-assoc (implies (and (keyword-fact-info-alistp acl2::x)) (iff (keyword-fact-info-alistp (put-assoc-equal acl2::name acl2::val acl2::x)) (and (keywordp acl2::name) (fact-infop acl2::val)))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-fast-alist-clean (implies (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp (fast-alist-clean acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-hons-shrink-alist (implies (and (keyword-fact-info-alistp acl2::x) (keyword-fact-info-alistp acl2::y)) (keyword-fact-info-alistp (hons-shrink-alist acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm keyword-fact-info-alistp-of-hons-acons (equal (keyword-fact-info-alistp (hons-acons acl2::a acl2::n acl2::x)) (and (keywordp acl2::a) (fact-infop acl2::n) (keyword-fact-info-alistp acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm fact-infop-of-cdr-of-hons-assoc-equal-when-keyword-fact-info-alistp (implies (keyword-fact-info-alistp acl2::x) (iff (fact-infop (cdr (hons-assoc-equal acl2::k acl2::x))) (or (hons-assoc-equal acl2::k acl2::x) (fact-infop nil)))) :rule-classes ((:rewrite)))
Theorem:
(defthm alistp-when-keyword-fact-info-alistp-rewrite (implies (keyword-fact-info-alistp acl2::x) (alistp acl2::x)) :rule-classes ((:rewrite)))
Theorem:
(defthm alistp-when-keyword-fact-info-alistp (implies (keyword-fact-info-alistp acl2::x) (alistp acl2::x)) :rule-classes :tau-system)
Theorem:
(defthm fact-infop-of-cdar-when-keyword-fact-info-alistp (implies (keyword-fact-info-alistp acl2::x) (iff (fact-infop (cdar acl2::x)) (or (consp acl2::x) (fact-infop nil)))) :rule-classes ((:rewrite)))
Theorem:
(defthm keywordp-of-caar-when-keyword-fact-info-alistp (implies (keyword-fact-info-alistp acl2::x) (iff (keywordp (caar acl2::x)) (or (consp acl2::x) (keywordp nil)))) :rule-classes ((:rewrite)))
Theorem:
(defthm fact-info-listp-of-strip-cdr-when-keyword-fact-info-alistp (implies (keyword-fact-info-alistp x) (fact-info-listp (strip-cdrs x))))