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• Stackp

Stackp-basics

Basic theorems about stackp, generated by std::deflist.

Definitions and Theorems

Theorem: stackp-of-cons

(defthm stackp-of-cons
  (equal (stackp (cons acl2::a acl2::x))
         (and (framep acl2::a) (stackp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: stackp-of-cdr-when-stackp

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

Theorem: stackp-when-not-consp

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

Theorem: framep-of-car-when-stackp

(defthm framep-of-car-when-stackp
  (implies (stackp acl2::x)
           (iff (framep (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-stackp-compound-recognizer

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

Theorem: stackp-of-list-fix

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

Theorem: stackp-of-sfix

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

Theorem: stackp-of-insert

(defthm stackp-of-insert
  (iff (stackp (insert acl2::a acl2::x))
       (and (stackp (sfix acl2::x))
            (framep acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: stackp-of-delete

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

Theorem: stackp-of-mergesort

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

Theorem: stackp-of-union

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

Theorem: stackp-of-intersect-1

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

Theorem: stackp-of-intersect-2

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

Theorem: stackp-of-difference

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

Theorem: stackp-of-duplicated-members

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

Theorem: stackp-of-rev

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

Theorem: stackp-of-append

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

Theorem: stackp-of-rcons

(defthm stackp-of-rcons
  (iff (stackp (rcons acl2::a acl2::x))
       (and (framep acl2::a)
            (stackp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: framep-when-member-equal-of-stackp

(defthm framep-when-member-equal-of-stackp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (stackp acl2::x))
                (framep acl2::a))
       (implies (and (stackp acl2::x)
                     (member-equal acl2::a acl2::x))
                (framep acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: stackp-when-subsetp-equal

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

Theorem: stackp-of-set-difference-equal

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

Theorem: stackp-of-intersection-equal-1

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

Theorem: stackp-of-intersection-equal-2

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

Theorem: stackp-of-union-equal

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

Theorem: stackp-of-take

(defthm stackp-of-take
  (implies (stackp (double-rewrite acl2::x))
           (iff (stackp (take acl2::n acl2::x))
                (or (framep nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: stackp-of-repeat

(defthm stackp-of-repeat
  (iff (stackp (repeat acl2::n acl2::x))
       (or (framep acl2::x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: framep-of-nth-when-stackp

(defthm framep-of-nth-when-stackp
  (implies (stackp acl2::x)
           (iff (framep (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: stackp-of-update-nth

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

Theorem: stackp-of-butlast

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

Theorem: stackp-of-nthcdr

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

Theorem: stackp-of-last

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

Theorem: stackp-of-remove

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

Theorem: stackp-of-revappend

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