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