1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
|
; Intersection$ Lemmas
; Copyright (C) 2013 Kookamara LLC
;
; Contact:
;
; Kookamara LLC
; 11410 Windermere Meadows
; Austin, TX 78759, USA
; http://www.kookamara.com/
;
; License: (An MIT/X11-style license)
;
; Permission is hereby granted, free of charge, to any person obtaining a
; copy of this software and associated documentation files (the "Software"),
; to deal in the Software without restriction, including without limitation
; the rights to use, copy, modify, merge, publish, distribute, sublicense,
; and/or sell copies of the Software, and to permit persons to whom the
; Software is furnished to do so, subject to the following conditions:
;
; The above copyright notice and this permission notice shall be included in
; all copies or substantial portions of the Software.
;
; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
; IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
; FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
; AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
; LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
; DEALINGS IN THE SOFTWARE.
;
; Original author: Jared Davis <jared@kookamara.com>
(in-package "ACL2")
(include-book "list-defuns")
(include-book "xdoc/top" :dir :system)
(local (include-book "sets"))
(local (in-theory (enable intersection$ intersectp)))
(defsection std/lists/intersection$
:parents (std/lists intersection$)
:short "Lemmas about @(see intersection$) available in the @(see std/lists)
library."
"<p>We'll take @(see intersectp) as the desired normal form for asking
whether intersections are empty.</p>"
(defthm intersection$-under-iff
(iff (intersection$ x y)
(intersectp x y)))
(defthm consp-of-intersection$
(equal (consp (intersection$ x y))
(intersectp x y)))
"<p>Basic atom/cons rules.</p>"
(defthm intersection$-when-atom-left
(implies (atom x)
(equal (intersection$ x y)
nil)))
(defthm intersection$-of-cons-left
(equal (intersection$ (cons a x) y)
(if (member a y)
(cons a (intersection$ x y))
(intersection$ x y))))
(defthm intersection$-when-atom-right
(implies (atom y)
(equal (intersection$ x y)
nil)))
"<p>We don't have a very nice rule for @(see cons) on the right if we're
trying to maintain @('equal'), because we don't know where in @('x') the
element occurs. However, if we're only maintaining @(see set-equiv), then we
can just put the element on the front and we get a perfectly nice rule:</p>"
(defthm intersection$-of-cons-right-under-set-equiv
(set-equiv (intersection$ x (cons a y))
(if (member a x)
(cons a (intersection$ x y))
(intersection$ x y)))
:hints(("Goal" :in-theory (enable set-equiv))))
"<h5>Basic set reasoning</h5>"
(defthm member-of-intersection$
(iff (member a (intersection$ x y))
(and (member a x)
(member a y)))
:rule-classes
(:rewrite
(:type-prescription
:corollary
(implies (not (member a x))
(not (member a (intersection$ x y)))))
(:type-prescription
:corollary
(implies (not (member a y))
(not (member a (intersection$ x y)))))))
(defthm subsetp-equal-of-intersection$-1
(subsetp-equal (intersection$ x y) x))
(defthm subsetp-equal-of-intersection$-2
(subsetp-equal (intersection$ x y) y))
(defthm subsetp-intersection-equal
(iff (subsetp a (intersection-equal b c))
(and (subsetp a b)
(subsetp a c))))
(defthm intersection$-commutes-under-set-equiv
(set-equiv (intersection$ x y)
(intersection$ y x))
:hints(("Goal" :in-theory (enable set-equiv))))
;; These are redundant with sets.lisp
(defcong set-equiv equal (intersection-equal x y) 2)
(defcong set-equiv set-equiv (intersection-equal x y) 1)
"<h5>Length bound</h5>
<p>Here is a nice bounding theorem. Note that there is no analogous rule for
@('-right'), because, e.g., X could have multiple copies of some member in Y,
and if so we end up reproducing them. Consider for instance:</p>
@({ (intersection$ '(a a a) '(a)) ==> '(a a a) })"
(defthm len-of-intersection$-upper-bound
(<= (len (intersection$ x y))
(len x))
:rule-classes ((:rewrite) (:linear))))
|
