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Constraint-db

Constraint-db-p

Signature
(constraint-db-p x) → *

Definitions and Theorems

Function: constraint-db-p

(defun constraint-db-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'constraint-db-p))
    (declare (ignorable __function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (pseudo-fnsym-p (caar x))
           (constraint-tuplelist-p (cdar x))
           (constraint-db-p (cdr x))))))

Theorem: constraint-db-p-of-revappend

(defthm constraint-db-p-of-revappend
  (equal (constraint-db-p (revappend x y))
         (and (constraint-db-p (list-fix x))
              (constraint-db-p y)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-remove

(defthm constraint-db-p-of-remove
  (implies (constraint-db-p x)
           (constraint-db-p (remove a x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-last

(defthm constraint-db-p-of-last
  (implies (constraint-db-p (double-rewrite x))
           (constraint-db-p (last x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-nthcdr

(defthm constraint-db-p-of-nthcdr
  (implies (constraint-db-p (double-rewrite x))
           (constraint-db-p (nthcdr n x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-butlast

(defthm constraint-db-p-of-butlast
  (implies (constraint-db-p (double-rewrite x))
           (constraint-db-p (butlast x n)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-update-nth

(defthm constraint-db-p-of-update-nth
 (implies (constraint-db-p (double-rewrite x))
          (iff (constraint-db-p (update-nth n y x))
               (and (and (consp y)
                         (pseudo-fnsym-p (car y))
                         (constraint-tuplelist-p (cdr y)))
                    (or (<= (nfix n) (len x))
                        (and (consp nil)
                             (pseudo-fnsym-p (car nil))
                             (constraint-tuplelist-p (cdr nil)))))))
 :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-repeat

(defthm constraint-db-p-of-repeat
  (iff (constraint-db-p (acl2::repeat n x))
       (or (and (consp x)
                (pseudo-fnsym-p (car x))
                (constraint-tuplelist-p (cdr x)))
           (zp n)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-take

(defthm constraint-db-p-of-take
  (implies (constraint-db-p (double-rewrite x))
           (iff (constraint-db-p (take n x))
                (or (and (consp nil)
                         (pseudo-fnsym-p (car nil))
                         (constraint-tuplelist-p (cdr nil)))
                    (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-union-equal

(defthm constraint-db-p-of-union-equal
  (equal (constraint-db-p (union-equal x y))
         (and (constraint-db-p (list-fix x))
              (constraint-db-p (double-rewrite y))))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-intersection-equal-2

(defthm constraint-db-p-of-intersection-equal-2
  (implies (constraint-db-p (double-rewrite y))
           (constraint-db-p (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-intersection-equal-1

(defthm constraint-db-p-of-intersection-equal-1
  (implies (constraint-db-p (double-rewrite x))
           (constraint-db-p (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-set-difference-equal

(defthm constraint-db-p-of-set-difference-equal
  (implies (constraint-db-p x)
           (constraint-db-p (set-difference-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-when-subsetp-equal

(defthm constraint-db-p-when-subsetp-equal
  (and (implies (and (subsetp-equal x y)
                     (constraint-db-p y))
                (equal (constraint-db-p x)
                       (true-listp x)))
       (implies (and (constraint-db-p y)
                     (subsetp-equal x y))
                (equal (constraint-db-p x)
                       (true-listp x))))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-rcons

(defthm constraint-db-p-of-rcons
  (iff (constraint-db-p (acl2::rcons a x))
       (and (and (consp a)
                 (pseudo-fnsym-p (car a))
                 (constraint-tuplelist-p (cdr a)))
            (constraint-db-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-append

(defthm constraint-db-p-of-append
  (equal (constraint-db-p (append a b))
         (and (constraint-db-p (list-fix a))
              (constraint-db-p b)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-rev

(defthm constraint-db-p-of-rev
  (equal (constraint-db-p (rev x))
         (constraint-db-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-duplicated-members

(defthm constraint-db-p-of-duplicated-members
  (implies (constraint-db-p x)
           (constraint-db-p (acl2::duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-difference

(defthm constraint-db-p-of-difference
  (implies (constraint-db-p x)
           (constraint-db-p (set::difference x y)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-intersect-2

(defthm constraint-db-p-of-intersect-2
  (implies (constraint-db-p y)
           (constraint-db-p (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-intersect-1

(defthm constraint-db-p-of-intersect-1
  (implies (constraint-db-p x)
           (constraint-db-p (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-union

(defthm constraint-db-p-of-union
  (iff (constraint-db-p (set::union x y))
       (and (constraint-db-p (set::sfix x))
            (constraint-db-p (set::sfix y))))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-mergesort

(defthm constraint-db-p-of-mergesort
  (iff (constraint-db-p (set::mergesort x))
       (constraint-db-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-delete

(defthm constraint-db-p-of-delete
  (implies (constraint-db-p x)
           (constraint-db-p (set::delete k x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-insert

(defthm constraint-db-p-of-insert
  (iff (constraint-db-p (set::insert a x))
       (and (constraint-db-p (set::sfix x))
            (and (consp a)
                 (pseudo-fnsym-p (car a))
                 (constraint-tuplelist-p (cdr a)))))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-sfix

(defthm constraint-db-p-of-sfix
  (iff (constraint-db-p (set::sfix x))
       (or (constraint-db-p x)
           (not (set::setp x))))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-list-fix

(defthm constraint-db-p-of-list-fix
  (implies (constraint-db-p x)
           (constraint-db-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-constraint-db-p-compound-recognizer

(defthm true-listp-when-constraint-db-p-compound-recognizer
  (implies (constraint-db-p x)
           (true-listp x))
  :rule-classes :compound-recognizer)

Theorem: constraint-db-p-when-not-consp

(defthm constraint-db-p-when-not-consp
  (implies (not (consp x))
           (equal (constraint-db-p x) (not x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-cdr-when-constraint-db-p

(defthm constraint-db-p-of-cdr-when-constraint-db-p
  (implies (constraint-db-p (double-rewrite x))
           (constraint-db-p (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-cons

(defthm constraint-db-p-of-cons
  (equal (constraint-db-p (cons a x))
         (and (and (consp a)
                   (pseudo-fnsym-p (car a))
                   (constraint-tuplelist-p (cdr a)))
              (constraint-db-p x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-remove-assoc

(defthm constraint-db-p-of-remove-assoc
  (implies (constraint-db-p x)
           (constraint-db-p (remove-assoc-equal name x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-put-assoc

(defthm constraint-db-p-of-put-assoc
  (implies (and (constraint-db-p x))
           (iff (constraint-db-p (put-assoc-equal name acl2::val x))
                (and (pseudo-fnsym-p name)
                     (constraint-tuplelist-p acl2::val))))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-fast-alist-clean

(defthm constraint-db-p-of-fast-alist-clean
  (implies (constraint-db-p x)
           (constraint-db-p (fast-alist-clean x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-hons-shrink-alist

(defthm constraint-db-p-of-hons-shrink-alist
  (implies (and (constraint-db-p x)
                (constraint-db-p y))
           (constraint-db-p (hons-shrink-alist x y)))
  :rule-classes ((:rewrite)))

Theorem: constraint-db-p-of-hons-acons

(defthm constraint-db-p-of-hons-acons
  (equal (constraint-db-p (hons-acons a n x))
         (and (pseudo-fnsym-p a)
              (constraint-tuplelist-p n)
              (constraint-db-p x)))
  :rule-classes ((:rewrite)))

Theorem: constraint-tuplelist-p-of-cdr-of-hons-assoc-equal-when-constraint-db-p

(defthm
 constraint-tuplelist-p-of-cdr-of-hons-assoc-equal-when-constraint-db-p
 (implies (constraint-db-p x)
          (iff (constraint-tuplelist-p (cdr (hons-assoc-equal k x)))
               (or (hons-assoc-equal k x)
                   (constraint-tuplelist-p nil))))
 :rule-classes ((:rewrite)))

Theorem: alistp-when-constraint-db-p-rewrite

(defthm alistp-when-constraint-db-p-rewrite
  (implies (constraint-db-p x) (alistp x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-constraint-db-p

(defthm alistp-when-constraint-db-p
  (implies (constraint-db-p x) (alistp x))
  :rule-classes :tau-system)

Theorem: constraint-tuplelist-p-of-cdar-when-constraint-db-p

(defthm constraint-tuplelist-p-of-cdar-when-constraint-db-p
  (implies (constraint-db-p x)
           (iff (constraint-tuplelist-p (cdar x))
                (or (consp x)
                    (constraint-tuplelist-p nil))))
  :rule-classes ((:rewrite)))

Theorem: pseudo-fnsym-p-of-caar-when-constraint-db-p

(defthm pseudo-fnsym-p-of-caar-when-constraint-db-p
  (implies (constraint-db-p x)
           (iff (pseudo-fnsym-p (caar x))
                (or (consp x) (pseudo-fnsym-p nil))))
  :rule-classes ((:rewrite)))