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