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(in-package "ACL2")
; not-intersectp-list.lisp Mihir Mehta
(include-book "../file-system-lemmas")
(include-book "std/lists/flatten" :dir :system)
(defthm no-duplicatesp-of-member
(implies (and (not (no-duplicatesp x))
(no-duplicatesp (flatten lst)))
(not (member-equal x lst))))
(defthm len-of-flatten-of-update-nth
(equal (len (flatten (update-nth key val l)))
(- (+ (len (flatten l)) (len val))
(len (nth key l))))
:hints (("goal" :induct (update-nth key val l))))
(defund not-intersectp-list (x l)
(declare (xargs :guard (and (true-listp x) (true-list-listp l))))
(or (atom l)
(and (not (intersectp-equal x (car l)))
(not-intersectp-list x (cdr l)))))
(defcong list-equiv equal (not-intersectp-list x l) 1
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(defthm not-intersectp-list-correctness-1
(equal (intersectp-equal x (flatten l))
(not (not-intersectp-list x l)))
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(defthmd not-intersectp-list-correctness-2
(implies (and (not-intersectp-list x l)
(member-equal y l))
(not (intersectp-equal x y)))
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(defthm not-intersectp-list-of-append-1
(equal (not-intersectp-list x (binary-append l1 l2))
(and (not-intersectp-list x l1)
(not-intersectp-list x l2)))
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(defthm not-intersectp-list-when-atom
(implies (atom x) (not-intersectp-list x l))
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(defthm not-intersectp-list-when-subsetp-1
(implies (and (not-intersectp-list y l)
(subsetp-equal x y))
(not-intersectp-list x l))
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(defthm not-intersectp-list-when-subsetp-2
(implies (and (not-intersectp-list x l2)
(subsetp-equal l1 l2))
(not-intersectp-list x l1))
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(defthm not-intersectp-list-of-true-list-fix
(equal (not-intersectp-list x (true-list-fix l))
(not-intersectp-list x l))
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(defthm
flatten-subset-no-duplicatesp-lemma-1
(implies (and (no-duplicatesp (flatten z))
(consp z)
(member-equal x z)
(member-equal y z)
(not (equal y x)))
(not (intersectp-equal x y)))
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(local
(defthm flatten-subset-no-duplicatesp-lemma-3
(implies (and (member-equal z y)
(not (member-equal z x))
(subsetp-equal x y)
(no-duplicatesp-equal (flatten y)))
(not-intersectp-list z x))
:hints (("Goal" :in-theory (enable not-intersectp-list)))))
;; This is, sort of, the main lemma.
(defthm flatten-subset-no-duplicatesp
(implies (and (subsetp-equal x y)
(no-duplicatesp-equal (flatten y))
(no-duplicatesp-equal x))
(no-duplicatesp-equal (flatten x))))
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