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• Tree-list-match-element-p

Tree-list-match-element-p-basics

Basic theorems about tree-list-match-element-p, generated by std::deflist.

Definitions and Theorems

Theorem: tree-list-match-element-p-of-cons

(defthm tree-list-match-element-p-of-cons
  (equal (tree-list-match-element-p (cons acl2::a acl2::x)
                                    element rules)
         (and (tree-match-element-p acl2::a element rules)
              (tree-list-match-element-p acl2::x element rules)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-cdr-when-tree-list-match-element-p

(defthm
    tree-list-match-element-p-of-cdr-when-tree-list-match-element-p
  (implies (tree-list-match-element-p (double-rewrite acl2::x)
                                      element rules)
           (tree-list-match-element-p (cdr acl2::x)
                                      element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-when-not-consp

(defthm tree-list-match-element-p-when-not-consp
  (implies (not (consp acl2::x))
           (tree-list-match-element-p acl2::x element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-match-element-p-of-car-when-tree-list-match-element-p

(defthm tree-match-element-p-of-car-when-tree-list-match-element-p
  (implies (tree-list-match-element-p acl2::x element rules)
           (iff (tree-match-element-p (car acl2::x)
                                      element rules)
                (or (consp acl2::x)
                    (tree-match-element-p nil element rules))))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-append

(defthm tree-list-match-element-p-of-append
  (equal (tree-list-match-element-p (append acl2::a acl2::b)
                                    element rules)
         (and (tree-list-match-element-p acl2::a element rules)
              (tree-list-match-element-p acl2::b element rules)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-list-fix

(defthm tree-list-match-element-p-of-list-fix
  (equal (tree-list-match-element-p (list-fix acl2::x)
                                    element rules)
         (tree-list-match-element-p acl2::x element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-sfix

(defthm tree-list-match-element-p-of-sfix
  (iff (tree-list-match-element-p (sfix acl2::x)
                                  element rules)
       (or (tree-list-match-element-p acl2::x element rules)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-insert

(defthm tree-list-match-element-p-of-insert
  (iff (tree-list-match-element-p (insert acl2::a acl2::x)
                                  element rules)
       (and (tree-list-match-element-p (sfix acl2::x)
                                       element rules)
            (tree-match-element-p acl2::a element rules)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-delete

(defthm tree-list-match-element-p-of-delete
  (implies (tree-list-match-element-p acl2::x element rules)
           (tree-list-match-element-p (delete acl2::k acl2::x)
                                      element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-mergesort

(defthm tree-list-match-element-p-of-mergesort
  (iff (tree-list-match-element-p (mergesort acl2::x)
                                  element rules)
       (tree-list-match-element-p (list-fix acl2::x)
                                  element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-union

(defthm tree-list-match-element-p-of-union
  (iff (tree-list-match-element-p (union acl2::x acl2::y)
                                  element rules)
       (and (tree-list-match-element-p (sfix acl2::x)
                                       element rules)
            (tree-list-match-element-p (sfix acl2::y)
                                       element rules)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-intersect-1

(defthm tree-list-match-element-p-of-intersect-1
  (implies (tree-list-match-element-p acl2::x element rules)
           (tree-list-match-element-p (intersect acl2::x acl2::y)
                                      element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-intersect-2

(defthm tree-list-match-element-p-of-intersect-2
  (implies (tree-list-match-element-p acl2::y element rules)
           (tree-list-match-element-p (intersect acl2::x acl2::y)
                                      element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-difference

(defthm tree-list-match-element-p-of-difference
  (implies (tree-list-match-element-p acl2::x element rules)
           (tree-list-match-element-p (difference acl2::x acl2::y)
                                      element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-duplicated-members

(defthm tree-list-match-element-p-of-duplicated-members
  (implies (tree-list-match-element-p acl2::x element rules)
           (tree-list-match-element-p (duplicated-members acl2::x)
                                      element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-rev

(defthm tree-list-match-element-p-of-rev
  (equal (tree-list-match-element-p (rev acl2::x)
                                    element rules)
         (tree-list-match-element-p (list-fix acl2::x)
                                    element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-rcons

(defthm tree-list-match-element-p-of-rcons
  (iff (tree-list-match-element-p (rcons acl2::a acl2::x)
                                  element rules)
       (and (tree-match-element-p acl2::a element rules)
            (tree-list-match-element-p (list-fix acl2::x)
                                       element rules)))
  :rule-classes ((:rewrite)))

Theorem: tree-match-element-p-when-member-equal-of-tree-list-match-element-p

(defthm
 tree-match-element-p-when-member-equal-of-tree-list-match-element-p
 (and
    (implies (and (member-equal acl2::a acl2::x)
                  (tree-list-match-element-p acl2::x element rules))
             (tree-match-element-p acl2::a element rules))
    (implies (and (tree-list-match-element-p acl2::x element rules)
                  (member-equal acl2::a acl2::x))
             (tree-match-element-p acl2::a element rules)))
 :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-when-subsetp-equal

(defthm tree-list-match-element-p-when-subsetp-equal
 (and
    (implies (and (subsetp-equal acl2::x acl2::y)
                  (tree-list-match-element-p acl2::y element rules))
             (tree-list-match-element-p acl2::x element rules))
    (implies (and (tree-list-match-element-p acl2::y element rules)
                  (subsetp-equal acl2::x acl2::y))
             (tree-list-match-element-p acl2::x element rules)))
 :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-set-equiv-congruence

(defthm tree-list-match-element-p-set-equiv-congruence
 (implies (set-equiv acl2::x acl2::y)
          (equal (tree-list-match-element-p acl2::x element rules)
                 (tree-list-match-element-p acl2::y element rules)))
 :rule-classes :congruence)

Theorem: tree-list-match-element-p-of-set-difference-equal

(defthm tree-list-match-element-p-of-set-difference-equal
 (implies
   (tree-list-match-element-p acl2::x element rules)
   (tree-list-match-element-p (set-difference-equal acl2::x acl2::y)
                              element rules))
 :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-intersection-equal-1

(defthm tree-list-match-element-p-of-intersection-equal-1
 (implies
     (tree-list-match-element-p (double-rewrite acl2::x)
                                element rules)
     (tree-list-match-element-p (intersection-equal acl2::x acl2::y)
                                element rules))
 :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-intersection-equal-2

(defthm tree-list-match-element-p-of-intersection-equal-2
 (implies
     (tree-list-match-element-p (double-rewrite acl2::y)
                                element rules)
     (tree-list-match-element-p (intersection-equal acl2::x acl2::y)
                                element rules))
 :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-union-equal

(defthm tree-list-match-element-p-of-union-equal
  (equal (tree-list-match-element-p (union-equal acl2::x acl2::y)
                                    element rules)
         (and (tree-list-match-element-p (list-fix acl2::x)
                                         element rules)
              (tree-list-match-element-p (double-rewrite acl2::y)
                                         element rules)))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-take

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

Theorem: tree-list-match-element-p-of-repeat

(defthm tree-list-match-element-p-of-repeat
  (iff (tree-list-match-element-p (repeat acl2::n acl2::x)
                                  element rules)
       (or (tree-match-element-p acl2::x element rules)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: tree-match-element-p-of-nth-when-tree-list-match-element-p

(defthm tree-match-element-p-of-nth-when-tree-list-match-element-p
  (implies (and (tree-list-match-element-p acl2::x element rules)
                (< (nfix acl2::n) (len acl2::x)))
           (tree-match-element-p (nth acl2::n acl2::x)
                                 element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-update-nth

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

Theorem: tree-list-match-element-p-of-butlast

(defthm tree-list-match-element-p-of-butlast
  (implies (tree-list-match-element-p (double-rewrite acl2::x)
                                      element rules)
           (tree-list-match-element-p (butlast acl2::x acl2::n)
                                      element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-nthcdr

(defthm tree-list-match-element-p-of-nthcdr
  (implies (tree-list-match-element-p (double-rewrite acl2::x)
                                      element rules)
           (tree-list-match-element-p (nthcdr acl2::n acl2::x)
                                      element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-last

(defthm tree-list-match-element-p-of-last
  (implies (tree-list-match-element-p (double-rewrite acl2::x)
                                      element rules)
           (tree-list-match-element-p (last acl2::x)
                                      element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-remove

(defthm tree-list-match-element-p-of-remove
  (implies (tree-list-match-element-p acl2::x element rules)
           (tree-list-match-element-p (remove acl2::a acl2::x)
                                      element rules))
  :rule-classes ((:rewrite)))

Theorem: tree-list-match-element-p-of-revappend

(defthm tree-list-match-element-p-of-revappend
  (equal (tree-list-match-element-p (revappend acl2::x acl2::y)
                                    element rules)
         (and (tree-list-match-element-p (list-fix acl2::x)
                                         element rules)
              (tree-list-match-element-p acl2::y element rules)))
  :rule-classes ((:rewrite)))