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(in-package "ACL2")
; member-intersectp.lisp Mihir Mehta
(include-book "not-intersectp-list")
(defund 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))))
:hints (("Goal" :in-theory (enable disjoint-list-listp))))
(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)))
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(defthm no-duplicates-listp-of-append
(equal (no-duplicates-listp (binary-append x y))
(and (no-duplicates-listp (true-list-fix x))
(no-duplicates-listp y))))
(defthm
not-intersectp-list-of-cons-1
(implies (case-split (consp y))
(equal (not-intersectp-list (cons x y) l)
(and (not-intersectp-list (list x) l)
(not-intersectp-list y l))))
:hints
(("goal" :do-not-induct t
:in-theory (disable not-intersectp-list-of-append-2)
:use (:instance not-intersectp-list-of-append-2
(x (list x))))))
(defun member-intersectp-equal (x y)
(declare (xargs :guard (and (true-list-listp x) (true-list-listp y))))
(and (consp x)
(or (not (not-intersectp-list (car x) y))
(member-intersectp-equal (cdr x) y))))
(defcong list-equiv
equal (member-intersectp-equal x y)
1
:hints (("goal" :in-theory (enable fast-list-equiv)
:induct (fast-list-equiv x x-equiv))))
(defthm when-append-is-disjoint-list-listp
(equal (disjoint-list-listp (binary-append x y))
(and (disjoint-list-listp (true-list-fix x))
(disjoint-list-listp y)
(not (member-intersectp-equal x y))))
:hints (("Goal" :in-theory (enable disjoint-list-listp))))
(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 (("Goal" :in-theory (enable not-intersectp-list))))
(local
(defthm member-intersectp-binary-append-lemma
(equal (member-intersectp-equal e (binary-append x y))
(or (member-intersectp-equal e x)
(member-intersectp-equal e y)))))
(local
(defthm member-intersectp-is-commutative-lemma-1
(implies (not (consp x))
(not (member-intersectp-equal y x)))
:hints (("Goal" :in-theory (enable not-intersectp-list)))))
(defthm member-intersectp-is-commutative-lemma-2
(implies (and (consp x)
(not (not-intersectp-list (car x) y)))
(member-intersectp-equal y x))
:hints (("Goal" :in-theory (enable not-intersectp-list))))
(defthmd 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 (("Goal" :in-theory (enable not-intersectp-list))))
(defthm member-intersectp-is-commutative
(equal (member-intersectp-equal x y)
(member-intersectp-equal y x))
:hints (("Goal" :in-theory (enable member-intersectp-is-commutative-lemma-3)) ))
;; This is useful, but might be useful at some point...
(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 member-intersectp-binary-append
(equal (member-intersectp-equal e (binary-append x y))
(or (member-intersectp-equal e x)
(member-intersectp-equal e y)))
:rule-classes
(:rewrite
(:rewrite
:corollary
(equal (member-intersectp-equal (binary-append x y) e)
(or (member-intersectp-equal x e)
(member-intersectp-equal y e))))))
(defthm not-intersectp-list-of-flatten
(equal (not-intersectp-list (flatten x) y)
(not (member-intersectp-equal x y))))
(defthm
not-intersectp-list-of-set-difference$-lemma-1
(implies (and (intersectp-equal x y)
(member-equal y l))
(not (not-intersectp-list x l)))
:hints
(("goal"
:in-theory (enable member-equal not-intersectp-list))))
(defthm
not-intersectp-list-of-set-difference$-lemma-2
(implies (and (subsetp-equal l1 (cons nil l2))
(not-intersectp-list x l2))
(not-intersectp-list x l1))
:hints (("goal" :in-theory (enable subsetp-equal not-intersectp-list)))
:rule-classes
(:rewrite
(:rewrite
:corollary
(implies (and
(not-intersectp-list x l2)
(subsetp-equal l1 (cons nil l2)))
(not-intersectp-list x l1)))))
(defthm not-intersectp-list-of-set-difference$-lemma-3
(implies (and (not (member-intersectp-equal x y1))
(subsetp-equal y2 (cons nil y1)))
(not (member-intersectp-equal x y2)))
:hints (("goal" :in-theory (enable member-intersectp-equal
subsetp-equal))))
(defthm
not-intersectp-list-of-set-difference$-lemma-4
(implies (and (disjoint-list-listp x2)
(member-equal x1 x2))
(not-intersectp-list x1 (remove-equal x1 x2)))
:hints
(("goal" :in-theory
(e/d (disjoint-list-listp subsetp-equal not-intersectp-list
set-difference$-redefinition)
(member-intersectp-is-commutative set-difference-equal)))))
(defthm
not-intersectp-list-of-set-difference$
(implies (and (disjoint-list-listp y)
(disjoint-list-listp x2)
(member-equal x1 y)
(subsetp-equal y x2))
(not-intersectp-list x1 (set-difference-equal x2 y)))
:hints
(("goal"
:in-theory
(e/d (disjoint-list-listp subsetp-equal not-intersectp-list)
(member-intersectp-is-commutative))
:induct (mv (subsetp-equal y x2)
(member-equal x1 y))
:expand (:with set-difference$-redefinition
(set-difference-equal x2 y)))))
(defthm
not-member-intersectp-of-set-difference$-1
(implies (and (disjoint-list-listp x2)
(subsetp-equal x1 y)
(subsetp-equal y x2)
(disjoint-list-listp y))
(not (member-intersectp-equal x1 (set-difference-equal x2 y))))
:hints
(("goal"
:in-theory (e/d (disjoint-list-listp subsetp-equal member-intersectp-equal
set-difference-equal)
(member-intersectp-is-commutative))
:expand
((:with member-intersectp-is-commutative
(:free (x)
(member-intersectp-equal x nil)))
(:with member-intersectp-is-commutative
(:free (x1 x2 y)
(member-intersectp-equal x1 (cons x2 y))))
(:with member-intersectp-is-commutative
(:free (x y1 y2)
(member-intersectp-equal (set-difference-equal x y1)
y2)))))))
(defthm
member-intersectp-of-set-difference$-lemma-1
(equal (member-intersectp-equal x1 (cons x2 y))
(or (not (not-intersectp-list x2 x1))
(member-intersectp-equal x1 y)))
:hints (("goal" :do-not-induct t
:in-theory (disable member-intersectp-is-commutative)
:expand ((:with member-intersectp-is-commutative
(member-intersectp-equal x1 (cons x2 y)))
(:with member-intersectp-is-commutative
(member-intersectp-equal y x1))))))
(defthmd member-intersectp-of-set-difference$-lemma-2
(implies (and (member-equal x y)
(case-split (consp x)))
(not (not-intersectp-list x y)))
:hints (("goal" :in-theory (enable not-intersectp-list))))
(defthm member-intersectp-of-set-difference$-1
(implies (not (member-intersectp-equal x y))
(equal (member-intersectp-equal (set-difference-equal x y)
z)
(member-intersectp-equal x z)))
:hints
(("Goal" :in-theory (enable member-intersectp-of-set-difference$-lemma-2)))
:rule-classes
(:rewrite
(:rewrite
:corollary
(implies (not (member-intersectp-equal x y))
(equal (member-intersectp-equal z (set-difference-equal x y))
(member-intersectp-equal z x))))))
(defthmd
not-intersectp-list-of-cons-2
(implies (and (member-equal i y)
(not-intersectp-list y l))
(not-intersectp-list (list i) l))
:hints (("goal" :in-theory (disable not-intersectp-list-when-subsetp-1)
:use (:instance not-intersectp-list-when-subsetp-1
(x (list i))))))
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