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• Pos-listp

Pos-listp-basics

Basic theorems about pos-listp, generated by deflist.

Definitions and Theorems

Theorem: pos-listp-of-cons

(defthm pos-listp-of-cons
  (equal (pos-listp (cons acl2::a acl2::x))
         (and (posp acl2::a)
              (pos-listp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pos-listp-of-cdr-when-pos-listp

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

Theorem: pos-listp-when-not-consp

(defthm pos-listp-when-not-consp
  (implies (not (consp acl2::x))
           (pos-listp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: posp-of-car-when-pos-listp

(defthm posp-of-car-when-pos-listp
  (implies (pos-listp acl2::x)
           (iff (posp (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: pos-listp-of-append

(defthm pos-listp-of-append
  (equal (pos-listp (append acl2::a acl2::b))
         (and (pos-listp acl2::a)
              (pos-listp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: pos-listp-of-list-fix

(defthm pos-listp-of-list-fix
  (equal (pos-listp (list-fix acl2::x))
         (pos-listp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: pos-listp-of-sfix

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

Theorem: pos-listp-of-insert

(defthm pos-listp-of-insert
  (iff (pos-listp (insert acl2::a acl2::x))
       (and (pos-listp (sfix acl2::x))
            (posp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: pos-listp-of-delete

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

Theorem: pos-listp-of-mergesort

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

Theorem: pos-listp-of-union

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

Theorem: pos-listp-of-intersect-1

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

Theorem: pos-listp-of-intersect-2

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

Theorem: pos-listp-of-difference

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

Theorem: pos-listp-of-duplicated-members

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

Theorem: pos-listp-of-rev

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

Theorem: pos-listp-of-rcons

(defthm pos-listp-of-rcons
  (iff (pos-listp (acl2::rcons acl2::a acl2::x))
       (and (posp acl2::a)
            (pos-listp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: posp-when-member-equal-of-pos-listp

(defthm posp-when-member-equal-of-pos-listp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (pos-listp acl2::x))
                (posp acl2::a))
       (implies (and (pos-listp acl2::x)
                     (member-equal acl2::a acl2::x))
                (posp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: pos-listp-when-subsetp-equal

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

Theorem: pos-listp-set-equiv-congruence

(defthm pos-listp-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (pos-listp acl2::x)
                  (pos-listp acl2::y)))
  :rule-classes :congruence)

Theorem: pos-listp-of-set-difference-equal

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

Theorem: pos-listp-of-intersection-equal-1

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

Theorem: pos-listp-of-intersection-equal-2

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

Theorem: pos-listp-of-union-equal

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

Theorem: pos-listp-of-take

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

Theorem: pos-listp-of-repeat

(defthm pos-listp-of-repeat
  (iff (pos-listp (repeat acl2::n acl2::x))
       (or (posp acl2::x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: posp-of-nth-when-pos-listp

(defthm posp-of-nth-when-pos-listp
  (implies (pos-listp acl2::x)
           (iff (posp (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: pos-listp-of-update-nth

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

Theorem: pos-listp-of-butlast

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

Theorem: pos-listp-of-nthcdr

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

Theorem: pos-listp-of-last

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

Theorem: pos-listp-of-remove

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

Theorem: pos-listp-of-revappend

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