Basic theorems about abnf-tree-list-with-root-p, generated by std::deflist.
Theorem:
(defthm abnf-tree-list-with-root-p-of-cons (equal (abnf-tree-list-with-root-p (cons acl2::a acl2::x) rulename) (and (abnf-tree-with-root-p acl2::a rulename) (abnf-tree-list-with-root-p acl2::x rulename))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-cdr-when-abnf-tree-list-with-root-p (implies (abnf-tree-list-with-root-p (double-rewrite acl2::x) rulename) (abnf-tree-list-with-root-p (cdr acl2::x) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-when-not-consp (implies (not (consp acl2::x)) (equal (abnf-tree-list-with-root-p acl2::x rulename) (not acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-with-root-p-of-car-when-abnf-tree-list-with-root-p (implies (abnf-tree-list-with-root-p acl2::x rulename) (iff (abnf-tree-with-root-p (car acl2::x) rulename) (consp acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm true-listp-when-abnf-tree-list-with-root-p (implies (abnf-tree-list-with-root-p acl2::x rulename) (true-listp acl2::x)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-list-fix (implies (abnf-tree-list-with-root-p acl2::x rulename) (abnf-tree-list-with-root-p (list-fix acl2::x) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-sfix (iff (abnf-tree-list-with-root-p (sfix acl2::x) rulename) (or (abnf-tree-list-with-root-p acl2::x rulename) (not (setp acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-insert (iff (abnf-tree-list-with-root-p (insert acl2::a acl2::x) rulename) (and (abnf-tree-list-with-root-p (sfix acl2::x) rulename) (abnf-tree-with-root-p acl2::a rulename))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-delete (implies (abnf-tree-list-with-root-p acl2::x rulename) (abnf-tree-list-with-root-p (delete acl2::k acl2::x) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-mergesort (iff (abnf-tree-list-with-root-p (mergesort acl2::x) rulename) (abnf-tree-list-with-root-p (list-fix acl2::x) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-union (iff (abnf-tree-list-with-root-p (union acl2::x acl2::y) rulename) (and (abnf-tree-list-with-root-p (sfix acl2::x) rulename) (abnf-tree-list-with-root-p (sfix acl2::y) rulename))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-intersect-1 (implies (abnf-tree-list-with-root-p acl2::x rulename) (abnf-tree-list-with-root-p (intersect acl2::x acl2::y) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-intersect-2 (implies (abnf-tree-list-with-root-p acl2::y rulename) (abnf-tree-list-with-root-p (intersect acl2::x acl2::y) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-difference (implies (abnf-tree-list-with-root-p acl2::x rulename) (abnf-tree-list-with-root-p (difference acl2::x acl2::y) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-duplicated-members (implies (abnf-tree-list-with-root-p acl2::x rulename) (abnf-tree-list-with-root-p (duplicated-members acl2::x) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-rev (equal (abnf-tree-list-with-root-p (rev acl2::x) rulename) (abnf-tree-list-with-root-p (list-fix acl2::x) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-append (equal (abnf-tree-list-with-root-p (append acl2::a acl2::b) rulename) (and (abnf-tree-list-with-root-p (list-fix acl2::a) rulename) (abnf-tree-list-with-root-p acl2::b rulename))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-rcons (iff (abnf-tree-list-with-root-p (rcons acl2::a acl2::x) rulename) (and (abnf-tree-with-root-p acl2::a rulename) (abnf-tree-list-with-root-p (list-fix acl2::x) rulename))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-with-root-p-when-member-equal-of-abnf-tree-list-with-root-p (and (implies (and (member-equal acl2::a acl2::x) (abnf-tree-list-with-root-p acl2::x rulename)) (abnf-tree-with-root-p acl2::a rulename)) (implies (and (abnf-tree-list-with-root-p acl2::x rulename) (member-equal acl2::a acl2::x)) (abnf-tree-with-root-p acl2::a rulename))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-when-subsetp-equal (and (implies (and (subsetp-equal acl2::x acl2::y) (abnf-tree-list-with-root-p acl2::y rulename)) (equal (abnf-tree-list-with-root-p acl2::x rulename) (true-listp acl2::x))) (implies (and (abnf-tree-list-with-root-p acl2::y rulename) (subsetp-equal acl2::x acl2::y)) (equal (abnf-tree-list-with-root-p acl2::x rulename) (true-listp acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-set-difference-equal (implies (abnf-tree-list-with-root-p acl2::x rulename) (abnf-tree-list-with-root-p (set-difference-equal acl2::x acl2::y) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-intersection-equal-1 (implies (abnf-tree-list-with-root-p (double-rewrite acl2::x) rulename) (abnf-tree-list-with-root-p (intersection-equal acl2::x acl2::y) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-intersection-equal-2 (implies (abnf-tree-list-with-root-p (double-rewrite acl2::y) rulename) (abnf-tree-list-with-root-p (intersection-equal acl2::x acl2::y) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-union-equal (equal (abnf-tree-list-with-root-p (union-equal acl2::x acl2::y) rulename) (and (abnf-tree-list-with-root-p (list-fix acl2::x) rulename) (abnf-tree-list-with-root-p (double-rewrite acl2::y) rulename))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-take (implies (abnf-tree-list-with-root-p (double-rewrite acl2::x) rulename) (iff (abnf-tree-list-with-root-p (take acl2::n acl2::x) rulename) (or (abnf-tree-with-root-p nil rulename) (<= (nfix acl2::n) (len acl2::x))))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-repeat (iff (abnf-tree-list-with-root-p (repeat acl2::n acl2::x) rulename) (or (abnf-tree-with-root-p acl2::x rulename) (zp acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-with-root-p-of-nth-when-abnf-tree-list-with-root-p (implies (abnf-tree-list-with-root-p acl2::x rulename) (iff (abnf-tree-with-root-p (nth acl2::n acl2::x) rulename) (< (nfix acl2::n) (len acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-update-nth (implies (abnf-tree-list-with-root-p (double-rewrite acl2::x) rulename) (iff (abnf-tree-list-with-root-p (update-nth acl2::n acl2::y acl2::x) rulename) (and (abnf-tree-with-root-p acl2::y rulename) (or (<= (nfix acl2::n) (len acl2::x)) (abnf-tree-with-root-p nil rulename))))) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-butlast (implies (abnf-tree-list-with-root-p (double-rewrite acl2::x) rulename) (abnf-tree-list-with-root-p (butlast acl2::x acl2::n) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-nthcdr (implies (abnf-tree-list-with-root-p (double-rewrite acl2::x) rulename) (abnf-tree-list-with-root-p (nthcdr acl2::n acl2::x) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-last (implies (abnf-tree-list-with-root-p (double-rewrite acl2::x) rulename) (abnf-tree-list-with-root-p (last acl2::x) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-remove (implies (abnf-tree-list-with-root-p acl2::x rulename) (abnf-tree-list-with-root-p (remove acl2::a acl2::x) rulename)) :rule-classes ((:rewrite)))
Theorem:
(defthm abnf-tree-list-with-root-p-of-revappend (equal (abnf-tree-list-with-root-p (revappend acl2::x acl2::y) rulename) (and (abnf-tree-list-with-root-p (list-fix acl2::x) rulename) (abnf-tree-list-with-root-p acl2::y rulename))) :rule-classes ((:rewrite)))