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• Svstack-p

Svstack-p-basics

Basic theorems about svstack-p, generated by std::deflist.

Definitions and Theorems

Theorem: svstack-p-of-cons

(defthm svstack-p-of-cons
  (equal (svstack-p (cons acl2::a x))
         (and (svex-alist-p acl2::a)
              (svstack-p x)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-cdr-when-svstack-p

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

Theorem: svstack-p-when-not-consp

(defthm svstack-p-when-not-consp
  (implies (not (consp x)) (svstack-p x))
  :rule-classes ((:rewrite)))

Theorem: svex-alist-p-of-car-when-svstack-p

(defthm svex-alist-p-of-car-when-svstack-p
  (implies (svstack-p x)
           (iff (svex-alist-p (car x))
                (or (consp x) (svex-alist-p nil))))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-append

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

Theorem: svstack-p-of-list-fix

(defthm svstack-p-of-list-fix
  (equal (svstack-p (list-fix x))
         (svstack-p x))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-sfix

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

Theorem: svstack-p-of-insert

(defthm svstack-p-of-insert
  (iff (svstack-p (insert acl2::a x))
       (and (svstack-p (sfix x))
            (svex-alist-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-delete

(defthm svstack-p-of-delete
  (implies (svstack-p x)
           (svstack-p (delete acl2::k x)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-mergesort

(defthm svstack-p-of-mergesort
  (iff (svstack-p (mergesort x))
       (svstack-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-union

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

Theorem: svstack-p-of-intersect-1

(defthm svstack-p-of-intersect-1
  (implies (svstack-p x)
           (svstack-p (intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-intersect-2

(defthm svstack-p-of-intersect-2
  (implies (svstack-p y)
           (svstack-p (intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-difference

(defthm svstack-p-of-difference
  (implies (svstack-p x)
           (svstack-p (difference x y)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-duplicated-members

(defthm svstack-p-of-duplicated-members
  (implies (svstack-p x)
           (svstack-p (duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-rev

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

Theorem: svstack-p-of-rcons

(defthm svstack-p-of-rcons
  (iff (svstack-p (acl2::rcons acl2::a x))
       (and (svex-alist-p acl2::a)
            (svstack-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: svex-alist-p-when-member-equal-of-svstack-p

(defthm svex-alist-p-when-member-equal-of-svstack-p
  (and (implies (and (member-equal acl2::a x)
                     (svstack-p x))
                (svex-alist-p acl2::a))
       (implies (and (svstack-p x)
                     (member-equal acl2::a x))
                (svex-alist-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-when-subsetp-equal

(defthm svstack-p-when-subsetp-equal
  (and (implies (and (subsetp-equal x y) (svstack-p y))
                (svstack-p x))
       (implies (and (svstack-p y) (subsetp-equal x y))
                (svstack-p x)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-set-equiv-congruence

(defthm svstack-p-set-equiv-congruence
  (implies (set-equiv x y)
           (equal (svstack-p x) (svstack-p y)))
  :rule-classes :congruence)

Theorem: svstack-p-of-set-difference-equal

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

Theorem: svstack-p-of-intersection-equal-1

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

Theorem: svstack-p-of-intersection-equal-2

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

Theorem: svstack-p-of-union-equal

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

Theorem: svstack-p-of-take

(defthm svstack-p-of-take
  (implies (svstack-p (double-rewrite x))
           (iff (svstack-p (take acl2::n x))
                (or (svex-alist-p nil)
                    (<= (nfix acl2::n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-repeat

(defthm svstack-p-of-repeat
  (iff (svstack-p (repeat acl2::n x))
       (or (svex-alist-p x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: svex-alist-p-of-nth-when-svstack-p

(defthm svex-alist-p-of-nth-when-svstack-p
  (implies (and (svstack-p x)
                (< (nfix acl2::n) (len x)))
           (svex-alist-p (nth acl2::n x)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-update-nth

(defthm svstack-p-of-update-nth
  (implies (svstack-p (double-rewrite x))
           (iff (svstack-p (update-nth acl2::n y x))
                (and (svex-alist-p y)
                     (or (<= (nfix acl2::n) (len x))
                         (svex-alist-p nil)))))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-butlast

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

Theorem: svstack-p-of-nthcdr

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

Theorem: svstack-p-of-last

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

Theorem: svstack-p-of-remove

(defthm svstack-p-of-remove
  (implies (svstack-p x)
           (svstack-p (remove acl2::a x)))
  :rule-classes ((:rewrite)))

Theorem: svstack-p-of-revappend

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