Basic theorems about atj-check-no-aij-type-list, generated by std::deflist.
Theorem:
(defthm atj-check-no-aij-type-list-of-cons (equal (atj-check-no-aij-type-list (cons acl2::a acl2::x) fn) (and (atj-check-no-aij-type acl2::a fn) (atj-check-no-aij-type-list acl2::x fn))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-cdr-when-atj-check-no-aij-type-list (implies (atj-check-no-aij-type-list (double-rewrite acl2::x) fn) (atj-check-no-aij-type-list (cdr acl2::x) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-when-not-consp (implies (not (consp acl2::x)) (atj-check-no-aij-type-list acl2::x fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-of-car-when-atj-check-no-aij-type-list (implies (atj-check-no-aij-type-list acl2::x fn) (iff (atj-check-no-aij-type (car acl2::x) fn) (consp acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-append (equal (atj-check-no-aij-type-list (append acl2::a acl2::b) fn) (and (atj-check-no-aij-type-list acl2::a fn) (atj-check-no-aij-type-list acl2::b fn))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-list-fix (equal (atj-check-no-aij-type-list (list-fix acl2::x) fn) (atj-check-no-aij-type-list acl2::x fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-sfix (iff (atj-check-no-aij-type-list (sfix acl2::x) fn) (or (atj-check-no-aij-type-list acl2::x fn) (not (setp acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-insert (iff (atj-check-no-aij-type-list (insert acl2::a acl2::x) fn) (and (atj-check-no-aij-type-list (sfix acl2::x) fn) (atj-check-no-aij-type acl2::a fn))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-delete (implies (atj-check-no-aij-type-list acl2::x fn) (atj-check-no-aij-type-list (delete acl2::k acl2::x) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-mergesort (iff (atj-check-no-aij-type-list (mergesort acl2::x) fn) (atj-check-no-aij-type-list (list-fix acl2::x) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-union (iff (atj-check-no-aij-type-list (union acl2::x acl2::y) fn) (and (atj-check-no-aij-type-list (sfix acl2::x) fn) (atj-check-no-aij-type-list (sfix acl2::y) fn))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-intersect-1 (implies (atj-check-no-aij-type-list acl2::x fn) (atj-check-no-aij-type-list (intersect acl2::x acl2::y) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-intersect-2 (implies (atj-check-no-aij-type-list acl2::y fn) (atj-check-no-aij-type-list (intersect acl2::x acl2::y) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-difference (implies (atj-check-no-aij-type-list acl2::x fn) (atj-check-no-aij-type-list (difference acl2::x acl2::y) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-duplicated-members (implies (atj-check-no-aij-type-list acl2::x fn) (atj-check-no-aij-type-list (duplicated-members acl2::x) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-rev (equal (atj-check-no-aij-type-list (rev acl2::x) fn) (atj-check-no-aij-type-list (list-fix acl2::x) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-rcons (iff (atj-check-no-aij-type-list (rcons acl2::a acl2::x) fn) (and (atj-check-no-aij-type acl2::a fn) (atj-check-no-aij-type-list (list-fix acl2::x) fn))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-when-member-equal-of-atj-check-no-aij-type-list (and (implies (and (member-equal acl2::a acl2::x) (atj-check-no-aij-type-list acl2::x fn)) (atj-check-no-aij-type acl2::a fn)) (implies (and (atj-check-no-aij-type-list acl2::x fn) (member-equal acl2::a acl2::x)) (atj-check-no-aij-type acl2::a fn))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-when-subsetp-equal (and (implies (and (subsetp-equal acl2::x acl2::y) (atj-check-no-aij-type-list acl2::y fn)) (atj-check-no-aij-type-list acl2::x fn)) (implies (and (atj-check-no-aij-type-list acl2::y fn) (subsetp-equal acl2::x acl2::y)) (atj-check-no-aij-type-list acl2::x fn))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-set-equiv-congruence (implies (set-equiv acl2::x acl2::y) (equal (atj-check-no-aij-type-list acl2::x fn) (atj-check-no-aij-type-list acl2::y fn))) :rule-classes :congruence)
Theorem:
(defthm atj-check-no-aij-type-list-of-set-difference-equal (implies (atj-check-no-aij-type-list acl2::x fn) (atj-check-no-aij-type-list (set-difference-equal acl2::x acl2::y) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-intersection-equal-1 (implies (atj-check-no-aij-type-list (double-rewrite acl2::x) fn) (atj-check-no-aij-type-list (intersection-equal acl2::x acl2::y) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-intersection-equal-2 (implies (atj-check-no-aij-type-list (double-rewrite acl2::y) fn) (atj-check-no-aij-type-list (intersection-equal acl2::x acl2::y) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-union-equal (equal (atj-check-no-aij-type-list (union-equal acl2::x acl2::y) fn) (and (atj-check-no-aij-type-list (list-fix acl2::x) fn) (atj-check-no-aij-type-list (double-rewrite acl2::y) fn))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-take (implies (atj-check-no-aij-type-list (double-rewrite acl2::x) fn) (iff (atj-check-no-aij-type-list (take acl2::n acl2::x) fn) (or (atj-check-no-aij-type nil fn) (<= (nfix acl2::n) (len acl2::x))))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-repeat (iff (atj-check-no-aij-type-list (repeat acl2::n acl2::x) fn) (or (atj-check-no-aij-type acl2::x fn) (zp acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-of-nth-when-atj-check-no-aij-type-list (implies (atj-check-no-aij-type-list acl2::x fn) (iff (atj-check-no-aij-type (nth acl2::n acl2::x) fn) (< (nfix acl2::n) (len acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-update-nth (implies (atj-check-no-aij-type-list (double-rewrite acl2::x) fn) (iff (atj-check-no-aij-type-list (update-nth acl2::n acl2::y acl2::x) fn) (and (atj-check-no-aij-type acl2::y fn) (or (<= (nfix acl2::n) (len acl2::x)) (atj-check-no-aij-type nil fn))))) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-butlast (implies (atj-check-no-aij-type-list (double-rewrite acl2::x) fn) (atj-check-no-aij-type-list (butlast acl2::x acl2::n) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-nthcdr (implies (atj-check-no-aij-type-list (double-rewrite acl2::x) fn) (atj-check-no-aij-type-list (nthcdr acl2::n acl2::x) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-last (implies (atj-check-no-aij-type-list (double-rewrite acl2::x) fn) (atj-check-no-aij-type-list (last acl2::x) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-remove (implies (atj-check-no-aij-type-list acl2::x fn) (atj-check-no-aij-type-list (remove acl2::a acl2::x) fn)) :rule-classes ((:rewrite)))
Theorem:
(defthm atj-check-no-aij-type-list-of-revappend (equal (atj-check-no-aij-type-list (revappend acl2::x acl2::y) fn) (and (atj-check-no-aij-type-list (list-fix acl2::x) fn) (atj-check-no-aij-type-list acl2::y fn))) :rule-classes ((:rewrite)))