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• Comp-db

Comp-dbp

Recognizer for comp-db.

Signature

(comp-dbp x) → *

Definitions and Theorems

Function: comp-dbp

(defun comp-dbp (x)
  (declare (xargs :guard t))
  (if (atom x)
      (eq x nil)
    (and (consp (car x))
         (stringp (caar x))
         (comp-db-entryp (cdar x))
         (comp-dbp (cdr x)))))

Theorem: comp-dbp-of-revappend

(defthm comp-dbp-of-revappend
  (equal (comp-dbp (revappend acl2::x acl2::y))
         (and (comp-dbp (list-fix acl2::x))
              (comp-dbp acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-remove

(defthm comp-dbp-of-remove
  (implies (comp-dbp acl2::x)
           (comp-dbp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-last

(defthm comp-dbp-of-last
  (implies (comp-dbp (double-rewrite acl2::x))
           (comp-dbp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-nthcdr

(defthm comp-dbp-of-nthcdr
  (implies (comp-dbp (double-rewrite acl2::x))
           (comp-dbp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-butlast

(defthm comp-dbp-of-butlast
  (implies (comp-dbp (double-rewrite acl2::x))
           (comp-dbp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-update-nth

(defthm comp-dbp-of-update-nth
  (implies (comp-dbp (double-rewrite acl2::x))
           (iff (comp-dbp (update-nth acl2::n acl2::y acl2::x))
                (and (and (consp acl2::y)
                          (stringp (car acl2::y))
                          (comp-db-entryp (cdr acl2::y)))
                     (or (<= (nfix acl2::n) (len acl2::x))
                         (and (consp nil)
                              (stringp (car nil))
                              (comp-db-entryp (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-repeat

(defthm comp-dbp-of-repeat
  (iff (comp-dbp (repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (stringp (car acl2::x))
                (comp-db-entryp (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-take

(defthm comp-dbp-of-take
  (implies (comp-dbp (double-rewrite acl2::x))
           (iff (comp-dbp (take acl2::n acl2::x))
                (or (and (consp nil)
                         (stringp (car nil))
                         (comp-db-entryp (cdr nil)))
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-union-equal

(defthm comp-dbp-of-union-equal
  (equal (comp-dbp (union-equal acl2::x acl2::y))
         (and (comp-dbp (list-fix acl2::x))
              (comp-dbp (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-intersection-equal-2

(defthm comp-dbp-of-intersection-equal-2
  (implies (comp-dbp (double-rewrite acl2::y))
           (comp-dbp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-intersection-equal-1

(defthm comp-dbp-of-intersection-equal-1
  (implies (comp-dbp (double-rewrite acl2::x))
           (comp-dbp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-set-difference-equal

(defthm comp-dbp-of-set-difference-equal
  (implies (comp-dbp acl2::x)
           (comp-dbp (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-when-subsetp-equal

(defthm comp-dbp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (comp-dbp acl2::y))
                (equal (comp-dbp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (comp-dbp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (comp-dbp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-rcons

(defthm comp-dbp-of-rcons
  (iff (comp-dbp (rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (stringp (car acl2::a))
                 (comp-db-entryp (cdr acl2::a)))
            (comp-dbp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-append

(defthm comp-dbp-of-append
  (equal (comp-dbp (append acl2::a acl2::b))
         (and (comp-dbp (list-fix acl2::a))
              (comp-dbp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-rev

(defthm comp-dbp-of-rev
  (equal (comp-dbp (rev acl2::x))
         (comp-dbp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-duplicated-members

(defthm comp-dbp-of-duplicated-members
  (implies (comp-dbp acl2::x)
           (comp-dbp (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-difference

(defthm comp-dbp-of-difference
  (implies (comp-dbp acl2::x)
           (comp-dbp (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-intersect-2

(defthm comp-dbp-of-intersect-2
  (implies (comp-dbp acl2::y)
           (comp-dbp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-intersect-1

(defthm comp-dbp-of-intersect-1
  (implies (comp-dbp acl2::x)
           (comp-dbp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-union

(defthm comp-dbp-of-union
  (iff (comp-dbp (union acl2::x acl2::y))
       (and (comp-dbp (sfix acl2::x))
            (comp-dbp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-mergesort

(defthm comp-dbp-of-mergesort
  (iff (comp-dbp (mergesort acl2::x))
       (comp-dbp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-delete

(defthm comp-dbp-of-delete
  (implies (comp-dbp acl2::x)
           (comp-dbp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-insert

(defthm comp-dbp-of-insert
  (iff (comp-dbp (insert acl2::a acl2::x))
       (and (comp-dbp (sfix acl2::x))
            (and (consp acl2::a)
                 (stringp (car acl2::a))
                 (comp-db-entryp (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-sfix

(defthm comp-dbp-of-sfix
  (iff (comp-dbp (sfix acl2::x))
       (or (comp-dbp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-list-fix

(defthm comp-dbp-of-list-fix
  (implies (comp-dbp acl2::x)
           (comp-dbp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-comp-dbp-compound-recognizer

(defthm true-listp-when-comp-dbp-compound-recognizer
  (implies (comp-dbp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: comp-dbp-when-not-consp

(defthm comp-dbp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (comp-dbp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-cdr-when-comp-dbp

(defthm comp-dbp-of-cdr-when-comp-dbp
  (implies (comp-dbp (double-rewrite acl2::x))
           (comp-dbp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-cons

(defthm comp-dbp-of-cons
  (equal (comp-dbp (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (stringp (car acl2::a))
                   (comp-db-entryp (cdr acl2::a)))
              (comp-dbp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-make-fal

(defthm comp-dbp-of-make-fal
  (implies (and (comp-dbp acl2::x)
                (comp-dbp acl2::y))
           (comp-dbp (make-fal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: comp-db-entryp-of-cdr-when-member-equal-of-comp-dbp

(defthm comp-db-entryp-of-cdr-when-member-equal-of-comp-dbp
  (and (implies (and (comp-dbp acl2::x)
                     (member-equal acl2::a acl2::x))
                (comp-db-entryp (cdr acl2::a)))
       (implies (and (member-equal acl2::a acl2::x)
                     (comp-dbp acl2::x))
                (comp-db-entryp (cdr acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-car-when-member-equal-of-comp-dbp

(defthm stringp-of-car-when-member-equal-of-comp-dbp
  (and (implies (and (comp-dbp acl2::x)
                     (member-equal acl2::a acl2::x))
                (stringp (car acl2::a)))
       (implies (and (member-equal acl2::a acl2::x)
                     (comp-dbp acl2::x))
                (stringp (car acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-comp-dbp

(defthm consp-when-member-equal-of-comp-dbp
  (implies (and (comp-dbp 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)
                                     (comp-dbp acl2::x)
                                   'nil)
                                 (consp acl2::a)))))

Theorem: comp-dbp-of-remove-assoc

(defthm comp-dbp-of-remove-assoc
  (implies (comp-dbp acl2::x)
           (comp-dbp (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-put-assoc

(defthm comp-dbp-of-put-assoc
 (implies
      (and (comp-dbp acl2::x))
      (iff (comp-dbp (put-assoc-equal acl2::name acl2::val acl2::x))
           (and (stringp acl2::name)
                (comp-db-entryp acl2::val))))
 :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-fast-alist-clean

(defthm comp-dbp-of-fast-alist-clean
  (implies (comp-dbp acl2::x)
           (comp-dbp (fast-alist-clean acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-hons-shrink-alist

(defthm comp-dbp-of-hons-shrink-alist
  (implies (and (comp-dbp acl2::x)
                (comp-dbp acl2::y))
           (comp-dbp (hons-shrink-alist acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: comp-dbp-of-hons-acons

(defthm comp-dbp-of-hons-acons
  (equal (comp-dbp (hons-acons acl2::a acl2::n acl2::x))
         (and (stringp acl2::a)
              (comp-db-entryp acl2::n)
              (comp-dbp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: comp-db-entryp-of-cdr-of-hons-assoc-equal-when-comp-dbp

(defthm comp-db-entryp-of-cdr-of-hons-assoc-equal-when-comp-dbp
 (implies
      (comp-dbp acl2::x)
      (iff (comp-db-entryp (cdr (hons-assoc-equal acl2::k acl2::x)))
           (or (hons-assoc-equal acl2::k acl2::x)
               (comp-db-entryp nil))))
 :rule-classes ((:rewrite)))

Theorem: alistp-when-comp-dbp-rewrite

(defthm alistp-when-comp-dbp-rewrite
  (implies (comp-dbp acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-comp-dbp

(defthm alistp-when-comp-dbp
  (implies (comp-dbp acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: comp-db-entryp-of-cdar-when-comp-dbp

(defthm comp-db-entryp-of-cdar-when-comp-dbp
  (implies (comp-dbp acl2::x)
           (iff (comp-db-entryp (cdar acl2::x))
                (or (consp acl2::x)
                    (comp-db-entryp nil))))
  :rule-classes ((:rewrite)))

Theorem: stringp-of-caar-when-comp-dbp

(defthm stringp-of-caar-when-comp-dbp
  (implies (comp-dbp acl2::x)
           (iff (stringp (caar acl2::x))
                (or (consp acl2::x) (stringp nil))))
  :rule-classes ((:rewrite)))