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