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• Tree-list-list-terminatedp

Tree-list-list-terminatedp-basics

Basic theorems about tree-list-list-terminatedp, generated by std::deflist.

Definitions and Theorems

Theorem: tree-list-list-terminatedp-of-cons

(defthm tree-list-list-terminatedp-of-cons
  (equal (tree-list-list-terminatedp (cons acl2::a acl2::x))
         (and (tree-list-terminatedp acl2::a)
              (tree-list-list-terminatedp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-cdr-when-tree-list-list-terminatedp

(defthm
  tree-list-list-terminatedp-of-cdr-when-tree-list-list-terminatedp
  (implies (tree-list-list-terminatedp (double-rewrite acl2::x))
           (tree-list-list-terminatedp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-when-not-consp

(defthm tree-list-list-terminatedp-when-not-consp
  (implies (not (consp acl2::x))
           (tree-list-list-terminatedp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: tree-list-terminatedp-of-car-when-tree-list-list-terminatedp

(defthm tree-list-terminatedp-of-car-when-tree-list-list-terminatedp
  (implies (tree-list-list-terminatedp acl2::x)
           (tree-list-terminatedp (car acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-append

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

Theorem: tree-list-list-terminatedp-of-list-fix

(defthm tree-list-list-terminatedp-of-list-fix
  (equal (tree-list-list-terminatedp (list-fix acl2::x))
         (tree-list-list-terminatedp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-sfix

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

Theorem: tree-list-list-terminatedp-of-insert

(defthm tree-list-list-terminatedp-of-insert
  (iff (tree-list-list-terminatedp (insert acl2::a acl2::x))
       (and (tree-list-list-terminatedp (sfix acl2::x))
            (tree-list-terminatedp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-delete

(defthm tree-list-list-terminatedp-of-delete
  (implies (tree-list-list-terminatedp acl2::x)
           (tree-list-list-terminatedp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-mergesort

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

Theorem: tree-list-list-terminatedp-of-union

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

Theorem: tree-list-list-terminatedp-of-intersect-1

(defthm tree-list-list-terminatedp-of-intersect-1
  (implies (tree-list-list-terminatedp acl2::x)
           (tree-list-list-terminatedp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-intersect-2

(defthm tree-list-list-terminatedp-of-intersect-2
  (implies (tree-list-list-terminatedp acl2::y)
           (tree-list-list-terminatedp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-difference

(defthm tree-list-list-terminatedp-of-difference
 (implies (tree-list-list-terminatedp acl2::x)
          (tree-list-list-terminatedp (difference acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-duplicated-members

(defthm tree-list-list-terminatedp-of-duplicated-members
 (implies (tree-list-list-terminatedp acl2::x)
          (tree-list-list-terminatedp (duplicated-members acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-rev

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

Theorem: tree-list-list-terminatedp-of-rcons

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

Theorem: tree-list-terminatedp-when-member-equal-of-tree-list-list-terminatedp

(defthm
 tree-list-terminatedp-when-member-equal-of-tree-list-list-terminatedp
 (and (implies (and (member-equal acl2::a acl2::x)
                    (tree-list-list-terminatedp acl2::x))
               (tree-list-terminatedp acl2::a))
      (implies (and (tree-list-list-terminatedp acl2::x)
                    (member-equal acl2::a acl2::x))
               (tree-list-terminatedp acl2::a)))
 :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-when-subsetp-equal

(defthm tree-list-list-terminatedp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (tree-list-list-terminatedp acl2::y))
                (tree-list-list-terminatedp acl2::x))
       (implies (and (tree-list-list-terminatedp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (tree-list-list-terminatedp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-set-equiv-congruence

(defthm tree-list-list-terminatedp-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (tree-list-list-terminatedp acl2::x)
                  (tree-list-list-terminatedp acl2::y)))
  :rule-classes :congruence)

Theorem: tree-list-list-terminatedp-of-set-difference-equal

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

Theorem: tree-list-list-terminatedp-of-intersection-equal-1

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

Theorem: tree-list-list-terminatedp-of-intersection-equal-2

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

Theorem: tree-list-list-terminatedp-of-union-equal

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

Theorem: tree-list-list-terminatedp-of-take

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

Theorem: tree-list-list-terminatedp-of-repeat

(defthm tree-list-list-terminatedp-of-repeat
  (iff (tree-list-list-terminatedp (repeat acl2::n acl2::x))
       (or (tree-list-terminatedp acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-terminatedp-of-nth-when-tree-list-list-terminatedp

(defthm tree-list-terminatedp-of-nth-when-tree-list-list-terminatedp
  (implies (tree-list-list-terminatedp acl2::x)
           (tree-list-terminatedp (nth acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-update-nth

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

Theorem: tree-list-list-terminatedp-of-butlast

(defthm tree-list-list-terminatedp-of-butlast
  (implies (tree-list-list-terminatedp (double-rewrite acl2::x))
           (tree-list-list-terminatedp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-nthcdr

(defthm tree-list-list-terminatedp-of-nthcdr
  (implies (tree-list-list-terminatedp (double-rewrite acl2::x))
           (tree-list-list-terminatedp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-last

(defthm tree-list-list-terminatedp-of-last
  (implies (tree-list-list-terminatedp (double-rewrite acl2::x))
           (tree-list-list-terminatedp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-remove

(defthm tree-list-list-terminatedp-of-remove
  (implies (tree-list-list-terminatedp acl2::x)
           (tree-list-list-terminatedp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-list-terminatedp-of-revappend

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