		Search-engine friendly clone of the ACL2 documentation.

Undocumented

Vcd-idxhash-p

(vcd-idxhash-p x len) recognizes association lists where every key satisfies vcd-index-p and each value satisfies not.

This is an ordinary defalist.

Function: vcd-idxhash-p

(defun vcd-idxhash-p (x len)
  (declare (xargs :guard (natp len)))
  (if (consp x)
      (and (consp (car x))
           (vcd-index-p (caar x) len)
           (not (cdar x))
           (vcd-idxhash-p (cdr x) len))
    t))

Definitions and Theorems

Function: vcd-idxhash-p

(defun vcd-idxhash-p (x len)
  (declare (xargs :guard (natp len)))
  (if (consp x)
      (and (consp (car x))
           (vcd-index-p (caar x) len)
           (not (cdar x))
           (vcd-idxhash-p (cdr x) len))
    t))

Theorem: vcd-idxhash-p-of-revappend

(defthm vcd-idxhash-p-of-revappend
  (equal (vcd-idxhash-p (revappend acl2::x acl2::y)
                        len)
         (and (vcd-idxhash-p (list-fix acl2::x) len)
              (vcd-idxhash-p acl2::y len)))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-remove

(defthm vcd-idxhash-p-of-remove
  (implies (vcd-idxhash-p acl2::x len)
           (vcd-idxhash-p (remove acl2::a acl2::x)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-last

(defthm vcd-idxhash-p-of-last
  (implies (vcd-idxhash-p (double-rewrite acl2::x)
                          len)
           (vcd-idxhash-p (last acl2::x) len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-nthcdr

(defthm vcd-idxhash-p-of-nthcdr
  (implies (vcd-idxhash-p (double-rewrite acl2::x)
                          len)
           (vcd-idxhash-p (nthcdr acl2::n acl2::x)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-butlast

(defthm vcd-idxhash-p-of-butlast
  (implies (vcd-idxhash-p (double-rewrite acl2::x)
                          len)
           (vcd-idxhash-p (butlast acl2::x acl2::n)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-update-nth

(defthm vcd-idxhash-p-of-update-nth
  (implies (vcd-idxhash-p (double-rewrite acl2::x)
                          len)
           (iff (vcd-idxhash-p (update-nth acl2::n acl2::y acl2::x)
                               len)
                (and (and (consp acl2::y)
                          (vcd-index-p (car acl2::y) len)
                          (not (cdr acl2::y)))
                     (or (<= (nfix acl2::n) (len acl2::x))
                         (and (consp nil)
                              (vcd-index-p (car nil) len)
                              (not (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-repeat

(defthm vcd-idxhash-p-of-repeat
  (iff (vcd-idxhash-p (repeat acl2::n acl2::x)
                      len)
       (or (and (consp acl2::x)
                (vcd-index-p (car acl2::x) len)
                (not (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-take

(defthm vcd-idxhash-p-of-take
  (implies (vcd-idxhash-p (double-rewrite acl2::x)
                          len)
           (iff (vcd-idxhash-p (take acl2::n acl2::x)
                               len)
                (or (and (consp nil)
                         (vcd-index-p (car nil) len)
                         (not (cdr nil)))
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-union-equal

(defthm vcd-idxhash-p-of-union-equal
  (equal (vcd-idxhash-p (union-equal acl2::x acl2::y)
                        len)
         (and (vcd-idxhash-p (list-fix acl2::x) len)
              (vcd-idxhash-p (double-rewrite acl2::y)
                             len)))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-intersection-equal-2

(defthm vcd-idxhash-p-of-intersection-equal-2
  (implies (vcd-idxhash-p (double-rewrite acl2::y)
                          len)
           (vcd-idxhash-p (intersection-equal acl2::x acl2::y)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-intersection-equal-1

(defthm vcd-idxhash-p-of-intersection-equal-1
  (implies (vcd-idxhash-p (double-rewrite acl2::x)
                          len)
           (vcd-idxhash-p (intersection-equal acl2::x acl2::y)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-set-difference-equal

(defthm vcd-idxhash-p-of-set-difference-equal
  (implies (vcd-idxhash-p acl2::x len)
           (vcd-idxhash-p (set-difference-equal acl2::x acl2::y)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-set-equiv-congruence

(defthm vcd-idxhash-p-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (vcd-idxhash-p acl2::x len)
                  (vcd-idxhash-p acl2::y len)))
  :rule-classes :congruence)

Theorem: vcd-idxhash-p-when-subsetp-equal

(defthm vcd-idxhash-p-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (vcd-idxhash-p acl2::y len))
                (vcd-idxhash-p acl2::x len))
       (implies (and (vcd-idxhash-p acl2::y len)
                     (subsetp-equal acl2::x acl2::y))
                (vcd-idxhash-p acl2::x len)))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-rcons

(defthm vcd-idxhash-p-of-rcons
  (iff (vcd-idxhash-p (acl2::rcons acl2::a acl2::x)
                      len)
       (and (and (consp acl2::a)
                 (vcd-index-p (car acl2::a) len)
                 (not (cdr acl2::a)))
            (vcd-idxhash-p (list-fix acl2::x) len)))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-rev

(defthm vcd-idxhash-p-of-rev
  (equal (vcd-idxhash-p (rev acl2::x) len)
         (vcd-idxhash-p (list-fix acl2::x) len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-duplicated-members

(defthm vcd-idxhash-p-of-duplicated-members
  (implies (vcd-idxhash-p acl2::x len)
           (vcd-idxhash-p (duplicated-members acl2::x)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-difference

(defthm vcd-idxhash-p-of-difference
  (implies (vcd-idxhash-p acl2::x len)
           (vcd-idxhash-p (difference acl2::x acl2::y)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-intersect-2

(defthm vcd-idxhash-p-of-intersect-2
  (implies (vcd-idxhash-p acl2::y len)
           (vcd-idxhash-p (intersect acl2::x acl2::y)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-intersect-1

(defthm vcd-idxhash-p-of-intersect-1
  (implies (vcd-idxhash-p acl2::x len)
           (vcd-idxhash-p (intersect acl2::x acl2::y)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-union

(defthm vcd-idxhash-p-of-union
  (iff (vcd-idxhash-p (union acl2::x acl2::y)
                      len)
       (and (vcd-idxhash-p (sfix acl2::x) len)
            (vcd-idxhash-p (sfix acl2::y) len)))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-mergesort

(defthm vcd-idxhash-p-of-mergesort
  (iff (vcd-idxhash-p (mergesort acl2::x) len)
       (vcd-idxhash-p (list-fix acl2::x) len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-delete

(defthm vcd-idxhash-p-of-delete
  (implies (vcd-idxhash-p acl2::x len)
           (vcd-idxhash-p (delete acl2::k acl2::x)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-insert

(defthm vcd-idxhash-p-of-insert
  (iff (vcd-idxhash-p (insert acl2::a acl2::x)
                      len)
       (and (vcd-idxhash-p (sfix acl2::x) len)
            (and (consp acl2::a)
                 (vcd-index-p (car acl2::a) len)
                 (not (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-sfix

(defthm vcd-idxhash-p-of-sfix
  (iff (vcd-idxhash-p (sfix acl2::x) len)
       (or (vcd-idxhash-p acl2::x len)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-list-fix

(defthm vcd-idxhash-p-of-list-fix
  (equal (vcd-idxhash-p (list-fix acl2::x) len)
         (vcd-idxhash-p acl2::x len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-append

(defthm vcd-idxhash-p-of-append
  (equal (vcd-idxhash-p (append acl2::a acl2::b)
                        len)
         (and (vcd-idxhash-p acl2::a len)
              (vcd-idxhash-p acl2::b len)))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-when-not-consp

(defthm vcd-idxhash-p-when-not-consp
  (implies (not (consp acl2::x))
           (vcd-idxhash-p acl2::x len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-cdr-when-vcd-idxhash-p

(defthm vcd-idxhash-p-of-cdr-when-vcd-idxhash-p
  (implies (vcd-idxhash-p (double-rewrite acl2::x)
                          len)
           (vcd-idxhash-p (cdr acl2::x) len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-cons

(defthm vcd-idxhash-p-of-cons
  (equal (vcd-idxhash-p (cons acl2::a acl2::x)
                        len)
         (and (and (consp acl2::a)
                   (vcd-index-p (car acl2::a) len)
                   (not (cdr acl2::a)))
              (vcd-idxhash-p acl2::x len)))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-make-fal

(defthm vcd-idxhash-p-of-make-fal
  (implies (and (vcd-idxhash-p acl2::x len)
                (vcd-idxhash-p acl2::y len))
           (vcd-idxhash-p (make-fal acl2::x acl2::y)
                          len))
  :rule-classes ((:rewrite)))

Theorem: not-of-cdr-when-member-equal-of-vcd-idxhash-p

(defthm not-of-cdr-when-member-equal-of-vcd-idxhash-p
  (and (implies (and (vcd-idxhash-p acl2::x len)
                     (member-equal acl2::a acl2::x))
                (not (cdr acl2::a)))
       (implies (and (member-equal acl2::a acl2::x)
                     (vcd-idxhash-p acl2::x len))
                (not (cdr acl2::a))))
  :rule-classes ((:rewrite)))

Theorem: vcd-index-p-of-car-when-member-equal-of-vcd-idxhash-p

(defthm vcd-index-p-of-car-when-member-equal-of-vcd-idxhash-p
  (and (implies (and (vcd-idxhash-p acl2::x len)
                     (member-equal acl2::a acl2::x))
                (vcd-index-p (car acl2::a) len))
       (implies (and (member-equal acl2::a acl2::x)
                     (vcd-idxhash-p acl2::x len))
                (vcd-index-p (car acl2::a) len)))
  :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-vcd-idxhash-p

(defthm consp-when-member-equal-of-vcd-idxhash-p
  (implies (and (vcd-idxhash-p acl2::x len)
                (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)
                                     (vcd-idxhash-p acl2::x len)
                                   'nil)
                                 (consp acl2::a)))))

Theorem: not-of-cdr-of-assoc-when-vcd-idxhash-p

(defthm not-of-cdr-of-assoc-when-vcd-idxhash-p
  (implies (vcd-idxhash-p acl2::x len)
           (not (cdr (assoc-equal acl2::k acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-fast-alist-clean

(defthm vcd-idxhash-p-of-fast-alist-clean
  (implies (vcd-idxhash-p acl2::x len)
           (vcd-idxhash-p (fast-alist-clean acl2::x)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-hons-shrink-alist

(defthm vcd-idxhash-p-of-hons-shrink-alist
  (implies (and (vcd-idxhash-p acl2::x len)
                (vcd-idxhash-p acl2::y len))
           (vcd-idxhash-p (hons-shrink-alist acl2::x acl2::y)
                          len))
  :rule-classes ((:rewrite)))

Theorem: vcd-idxhash-p-of-hons-acons

(defthm vcd-idxhash-p-of-hons-acons
  (equal (vcd-idxhash-p (hons-acons acl2::a acl2::n acl2::x)
                        len)
         (and (vcd-index-p acl2::a len)
              (not acl2::n)
              (vcd-idxhash-p acl2::x len)))
  :rule-classes ((:rewrite)))

Theorem: not-of-cdr-of-hons-assoc-equal-when-vcd-idxhash-p

(defthm not-of-cdr-of-hons-assoc-equal-when-vcd-idxhash-p
  (implies (vcd-idxhash-p acl2::x len)
           (not (cdr (hons-assoc-equal acl2::k acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vcd-index-p-of-caar-when-vcd-idxhash-p

(defthm vcd-index-p-of-caar-when-vcd-idxhash-p
  (implies (vcd-idxhash-p acl2::x len)
           (iff (vcd-index-p (caar acl2::x) len)
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: not-of-cdar-when-vcd-idxhash-p

(defthm not-of-cdar-when-vcd-idxhash-p
  (implies (vcd-idxhash-p acl2::x len)
           (not (cdar acl2::x)))
  :rule-classes ((:rewrite)))