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; Copyright (C) 2017, Regents of the University of Texas
; Written by Mihir Mehta
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
(in-package "ACL2")
(local (include-book "file-system-lemmas"))
(include-book "std/lists/flatten" :dir :system)
(defthm no-duplicatesp-of-member
(implies (and (member-equal x lst)
(no-duplicatesp (flatten lst)))
(no-duplicatesp x)))
(defun not-intersectp-list (x l)
(or (atom l)
(and (not (intersectp x (car l)))
(not-intersectp-list x (cdr l)))))
(defcong list-equiv equal (not-intersectp-list x l) 1)
(defthm not-intersectp-list-correctness-1
(equal (intersectp-equal x (flatten l))
(not (not-intersectp-list x l))))
(defthm not-intersectp-list-correctness-2
(implies (and (not-intersectp-list x l)
(member-equal y l))
(not (intersectp-equal x y))))
(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))))
(defthm not-intersectp-equal-if-subset
(implies (and (not-intersectp-list x l2)
(subsetp-equal l1 l2))
(not-intersectp-list x l1)))
(defthm flatten-subset-no-duplicatesp-lemma-1
(implies (and (consp z)
(no-duplicatesp (flatten z))
(member-equal y z)
(not (equal y (car z))))
(not (intersectp-equal (car z) y))))
(defthm
flatten-subset-no-duplicatesp-lemma-2
(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))))
(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)))
;; 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))))
(defun disjoint-list-listp (x)
(if (atom x)
(equal x nil)
(and (not-intersectp-list (car x) (cdr x))
(disjoint-list-listp (cdr x)))))
(defun no-duplicates-listp (x)
(if (atom x)
(equal x nil)
(and (no-duplicatesp (car x)) (no-duplicates-listp (cdr x)))))
(defthm flatten-disjoint-lists
(implies (true-listp l)
(equal (no-duplicatesp-equal (flatten l))
(and (disjoint-list-listp l) (no-duplicates-listp l)))))
;; This theorem won't go through because both
;; (disjoint-list-listp '((1 2) (3 4))) and
;; (subsetp-equal '((1 2) (1 2)) '((1 2) (3 4))) are t.
;; (verify (implies (and (subsetp-equal x y) (disjoint-list-listp y)) (disjoint-list-listp x)))
(defun member-intersectp-equal (x y)
(and (consp x)
(or (not (not-intersectp-list (car x) y))
(member-intersectp-equal (cdr x) y))))
(encapsulate ()
(local (include-book "std/basic/inductions" :dir :system))
(defcong list-equiv equal (member-intersectp-equal x y) 1
:hints
(("goal"
:induct (cdr-cdr-induct x x-equiv)))))
(defthm when-append-is-disjoint-list-listp
(implies (true-listp x)
(equal (disjoint-list-listp (binary-append x y))
(and (disjoint-list-listp x)
(disjoint-list-listp y) (not (member-intersectp-equal x y))))))
(defthm member-intersectp-with-subset
(implies (and (member-intersectp-equal z x)
(subsetp-equal x y))
(member-intersectp-equal z y)))
(defthm intersectp-member-when-not-member-intersectp
(implies (and (member-equal x lst2)
(not (member-intersectp-equal lst1 lst2)))
(not-intersectp-list x lst1))
:hints (("Subgoal *1/4''" :use (:instance intersectp-is-commutative (y (car lst1)))) ))
(defthm member-intersectp-binary-append
(equal (member-intersectp-equal e (binary-append x y))
(or (member-intersectp-equal e x)
(member-intersectp-equal e y))))
(defthm member-intersectp-is-commutative-lemma-1
(implies (not (consp x))
(not (member-intersectp-equal y x))))
(defthm
member-intersectp-is-commutative-lemma-2
(implies (and (consp x)
(not (not-intersectp-list (car x) y)))
(member-intersectp-equal y x))
:hints
(("Subgoal *1/2''" :use (:instance intersectp-is-commutative (x (car x))
(y (car y))))))
(defthm
member-intersectp-is-commutative-lemma-3
(implies (and (consp x)
(not-intersectp-list (car x) y))
(equal (member-intersectp-equal y (cdr x))
(member-intersectp-equal y x)))
:hints
(("Subgoal *1/1''" :use (:instance intersectp-is-commutative (x (car x))
(y (car y))))))
(defthm member-intersectp-is-commutative
(equal (member-intersectp-equal x y)
(member-intersectp-equal y x)))
(defthm
another-lemma-about-member-intersectp
(implies (or (member-intersectp-equal x z)
(member-intersectp-equal y z))
(member-intersectp-equal z (binary-append x y))))
(defthm not-intersectp-list-of-append-2
(equal (not-intersectp-list (binary-append x y) l)
(and (not-intersectp-list x l)
(not-intersectp-list y l))))
(defthm no-duplicates-listp-of-append
(implies (true-listp x)
(equal (no-duplicates-listp (binary-append x y))
(and (no-duplicates-listp x) (no-duplicates-listp y)))))
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