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Keyword-fact-info-alistp

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.

Definitions and Theorems

Function: keyword-fact-info-alistp

(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: keyword-fact-info-alistp-of-revappend

(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: keyword-fact-info-alistp-of-remove

(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: keyword-fact-info-alistp-of-last

(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: keyword-fact-info-alistp-of-nthcdr

(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: keyword-fact-info-alistp-of-butlast

(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: keyword-fact-info-alistp-of-update-nth

(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: keyword-fact-info-alistp-of-repeat

(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: keyword-fact-info-alistp-of-take

(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: keyword-fact-info-alistp-of-union-equal

(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: keyword-fact-info-alistp-of-intersection-equal-2

(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: keyword-fact-info-alistp-of-intersection-equal-1

(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: keyword-fact-info-alistp-of-set-difference-equal

(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: keyword-fact-info-alistp-when-subsetp-equal

(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: keyword-fact-info-alistp-of-rcons

(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: keyword-fact-info-alistp-of-append

(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: keyword-fact-info-alistp-of-rev

(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: keyword-fact-info-alistp-of-duplicated-members

(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: keyword-fact-info-alistp-of-difference

(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: keyword-fact-info-alistp-of-intersect-2

(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: keyword-fact-info-alistp-of-intersect-1

(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: keyword-fact-info-alistp-of-union

(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: keyword-fact-info-alistp-of-mergesort

(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: keyword-fact-info-alistp-of-delete

(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: keyword-fact-info-alistp-of-insert

(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: keyword-fact-info-alistp-of-sfix

(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: keyword-fact-info-alistp-of-list-fix

(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: true-listp-when-keyword-fact-info-alistp-compound-recognizer

(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: keyword-fact-info-alistp-when-not-consp

(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: keyword-fact-info-alistp-of-cdr-when-keyword-fact-info-alistp

(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: keyword-fact-info-alistp-of-cons

(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: keyword-fact-info-alistp-of-make-fal

(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: fact-infop-of-cdr-when-member-equal-of-keyword-fact-info-alistp

(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: keywordp-of-car-when-member-equal-of-keyword-fact-info-alistp

(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: consp-when-member-equal-of-keyword-fact-info-alistp

(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: keyword-fact-info-alistp-of-remove-assoc

(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: keyword-fact-info-alistp-of-put-assoc

(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: keyword-fact-info-alistp-of-fast-alist-clean

(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: keyword-fact-info-alistp-of-hons-shrink-alist

(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: keyword-fact-info-alistp-of-hons-acons

(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: fact-infop-of-cdr-of-hons-assoc-equal-when-keyword-fact-info-alistp

(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: alistp-when-keyword-fact-info-alistp-rewrite

(defthm alistp-when-keyword-fact-info-alistp-rewrite
  (implies (keyword-fact-info-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-keyword-fact-info-alistp

(defthm alistp-when-keyword-fact-info-alistp
  (implies (keyword-fact-info-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: fact-infop-of-cdar-when-keyword-fact-info-alistp

(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: keywordp-of-caar-when-keyword-fact-info-alistp

(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: fact-info-listp-of-strip-cdr-when-keyword-fact-info-alistp

(defthm fact-info-listp-of-strip-cdr-when-keyword-fact-info-alistp
  (implies (keyword-fact-info-alistp x)
           (fact-info-listp (strip-cdrs x))))