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(in-package "ACL2")
(include-book "member-intersectp")
(include-book "std/lists/sets" :dir :system)
(include-book "centaur/fty/top" :dir :system)
; flatten-equiv.lisp Mihir Mehta
(defcong
set-equiv
equal (not-intersectp-list x l)
2
:hints
(("goal"
:in-theory
(e/d nil
((:rewrite not-intersectp-list-when-subsetp-2)))
:use
((:instance (:rewrite not-intersectp-list-when-subsetp-2)
(l1 l)
(x x)
(l2 l-equiv))
(:instance (:rewrite not-intersectp-list-when-subsetp-2)
(l2 l)
(x x)
(l1 l-equiv))))))
(defund flatten-equiv (x y)
(set-equiv (remove-equal nil (true-list-list-fix x))
(remove-equal nil (true-list-list-fix y))))
(defequiv flatten-equiv
:hints (("goal" :in-theory (enable flatten-equiv))))
(defthmd
flatten-equiv-implies-equal-not-intersectp-list-2-lemma-1
(equal (not-intersectp-list x
(remove-equal nil (true-list-list-fix l)))
(not-intersectp-list x l))
:hints (("goal" :in-theory (enable not-intersectp-list
true-list-list-fix intersectp-equal))))
(defcong
flatten-equiv
equal (not-intersectp-list x l)
2
:hints
(("goal"
:in-theory (enable flatten-equiv)
:use
((:instance
flatten-equiv-implies-equal-not-intersectp-list-2-lemma-1
(l l-equiv))
flatten-equiv-implies-equal-not-intersectp-list-2-lemma-1))))
(defthmd
flatten-equiv-implies-equal-member-intersectp-equal-1-lemma-1
(equal (member-intersectp-equal (remove-equal nil (true-list-list-fix x))
y)
(member-intersectp-equal x y))
:hints (("goal" :in-theory (enable not-intersectp-list
true-list-list-fix intersectp-equal))))
(defcong
set-equiv
equal (member-intersectp-equal x y)
1
:hints
(("goal" :do-not-induct t
:in-theory (e/d (set-equiv)
(member-intersectp-with-subset))
:use ((:instance member-intersectp-with-subset (z y)
(x x)
(y x-equiv))
(:instance member-intersectp-with-subset (z y)
(x x-equiv)
(y x))))))
(defcong
set-equiv
equal (member-intersectp-equal y x)
2
:hints
(("goal" :do-not-induct t
:in-theory (e/d (set-equiv)
(member-intersectp-with-subset))
:use ((:instance member-intersectp-with-subset (z y)
(x x)
(y x-equiv))
(:instance member-intersectp-with-subset (z y)
(x x-equiv)
(y x))))))
(defcong
flatten-equiv
equal (member-intersectp-equal x y)
1
:hints
(("goal" :do-not-induct t
:in-theory (enable member-intersectp-equal flatten-equiv)
:use ((:instance
flatten-equiv-implies-equal-member-intersectp-equal-1-lemma-1
(x x-equiv))
flatten-equiv-implies-equal-member-intersectp-equal-1-lemma-1))))
(defcong
flatten-equiv
equal (member-intersectp-equal y x)
2
:hints
(("goal" :do-not-induct t
:in-theory (enable member-intersectp-equal flatten-equiv)
:use ((:instance
flatten-equiv-implies-equal-member-intersectp-equal-1-lemma-1
(x x-equiv))
flatten-equiv-implies-equal-member-intersectp-equal-1-lemma-1))))
(defthmd set-equiv-implies-equal-true-list-listp-of-list-fix-1-lemma-1
(implies (and (not (true-list-listp x))
(true-list-listp y)
(true-listp x))
(not (subsetp-equal x y)))
:hints (("goal" :in-theory (enable subsetp-equal true-list-listp))))
(defthm
set-equiv-implies-equal-true-list-listp-of-list-fix-1
(implies (set-equiv x y)
(equal (true-list-listp (true-list-fix x))
(true-list-listp (true-list-fix y))))
:rule-classes :congruence
:hints
(("goal"
:in-theory (enable set-equiv)
:use
((:instance
set-equiv-implies-equal-true-list-listp-of-list-fix-1-lemma-1
(x (true-list-fix x))
(y (true-list-fix y)))
(:instance
set-equiv-implies-equal-true-list-listp-of-list-fix-1-lemma-1
(y (true-list-fix x))
(x (true-list-fix y)))))))
(defthm commutativity-2-of-cons-under-flatten-equiv-lemma-1
(set-equiv (list* x y z) (list* y x z))
:hints (("goal" :in-theory (enable set-equiv))))
(defthm commutativity-2-of-cons-under-flatten-equiv
(flatten-equiv (list* x y z)
(list* y x z))
:hints (("goal" :in-theory (enable flatten-equiv))))
(defcong flatten-equiv flatten-equiv (append x y) 2
:hints (("goal" :in-theory (e/d (flatten-equiv)
()))))
(defthm flatten-equiv-of-remove-of-nil
(flatten-equiv (remove-equal nil x) x)
:hints (("goal" :in-theory (e/d (flatten-equiv)
()))))
(defthm flatten-equiv-of-cons-when-atom
(implies (atom x)
(flatten-equiv (cons x y) y))
:hints (("goal" :in-theory (e/d (flatten-equiv)
()))))
(defcong flatten-equiv flatten-equiv (cons x y) 2
:hints (("goal" :in-theory (e/d (flatten-equiv)
()))))
(defthmd cons-equal-under-set-equiv-1
(iff (set-equiv (cons x y1) (cons x y2))
(set-equiv (remove-equal x y1)
(remove-equal x y2)))
:instructions ((:= (cons x y2)
(cons x (remove-equal x y2))
:equiv set-equiv)
(:= (cons x y1)
(cons x (remove-equal x y1))
:equiv set-equiv)
(:dive 1)
:x (:dive 1)
(:= (subsetp-equal (remove-equal x y1)
(remove-equal x y2)))
:top
(:= (subsetp-equal (remove-equal x y2)
(cons x (remove-equal x y1)))
(subsetp-equal (remove-equal x y2)
(remove-equal x y1)))
(:dive 2)
:x
:top :bash))
(defthm flatten-equiv-of-append-of-cons-1
(flatten-equiv (append x (cons y z))
(cons y (append x z)))
:hints (("goal" :in-theory (e/d (flatten-equiv)
()))))
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