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
139
140
141
142
143
144
145
146
147
148
|
; Revappend lemmas
; Copyright (C) 2005-2013 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>
;
; revappend.lisp
; This file was originally part of the Unicode library.
(in-package "ACL2")
(include-book "list-fix")
(local (defthm len-when-consp
(implies (consp x)
(< 0 (len x)))
:rule-classes ((:linear) (:rewrite))))
(defsection std/lists/revappend
:parents (std/lists revappend)
:short "Lemmas about @(see revappend) available in the @(see std/lists)
library."
:long "<p>Note: we typically avoid reasoning about @('revappend') because
the following rule can always rewrite it away:</p>
@(thm revappend-removal)
<p>However, if for some reason you need to disable @('revappend-removal'),
these lemmas may be useful.</p>"
(defthm true-listp-of-revappend
(equal (true-listp (revappend x y))
(true-listp y)))
(defthm consp-of-revappend
(equal (consp (revappend x y))
(or (consp x)
(consp y))))
(defthm len-of-revappend
(equal (len (revappend x y))
(+ (len x)
(len y))))
;; (defthm nth-of-revappend
;; (equal (nth n (revappend x y))
;; (if (< (nfix n) (len x))
;; (nth (- (len x) (+ 1 (nfix n))) x)
;; (nth (- n (len x)) y))))
(defthm equal-when-revappend-same
(equal (equal (revappend x y1)
(revappend x y2))
(equal y1 y2)))
(defthm list-fix-of-revappend
(equal (list-fix (revappend x y))
(revappend x (list-fix y))))
(local (defthm revappend-lengths-x
(implies (not (equal (len x1) (len x2)))
(not (equal (revappend x1 y)
(revappend x2 y))))
:hints(("Goal" :use ((:instance len (x (revappend x1 y)))
(:instance len (x (revappend x2 y))))))))
(local (defthm revappend-nonempty-list
(implies (consp x)
(not (equal (revappend x y) y)))
:hints(("Goal" :use ((:instance len (x (revappend x y)))
(:instance len (x y)))))))
(local (defun revappend-induction-2-2 (x1 x2 a1 a2)
(if (and (consp x1)
(consp x2))
(revappend-induction-2-2 (cdr x1) (cdr x2) (cons (car x1) a1) (cons (car x2) a2))
(list x1 x2 a1 a2))))
(local (defthm revappend-unequal-accs
(implies (and (equal (len a1) (len a2))
(not (equal a1 a2)))
(not (equal (revappend x1 a1)
(Revappend x2 a2))))
:hints (("goal" :induct (revappend-induction-2-2 x1 x2 a1 a2))
(and stable-under-simplificationp
'(:cases ((consp x2))))
(and stable-under-simplificationp
'(:cases ((consp x1)))))))
(local (defun revappend-induction-2 (x y acc)
(if (and (consp x)
(consp y))
(revappend-induction-2 (cdr x) (cdr y) (cons (car x) acc))
(list x y acc))))
(defthm equal-of-revappends-when-true-listps
(implies (and (true-listp x1)
(true-listp x2))
(equal (equal (revappend x1 y)
(revappend x2 y))
(equal x1 x2)))
:hints(("Goal"
:induct (revappend-induction-2 x1 x2 y))
(and stable-under-simplificationp
'(:expand ((revappend x1 y)
(revappend x2 y))))))
(def-listp-rule element-list-p-of-revappend
(equal (element-list-p (revappend x y))
(and (element-list-p (list-fix x))
(element-list-p y))))
(def-listfix-rule element-list-fix-of-revappend
(equal (element-list-fix (revappend x y))
(revappend (element-list-fix x)
(element-list-fix y))))
(def-projection-rule elementlist-projection-of-revappend
(equal (elementlist-projection (revappend x y))
(revappend (elementlist-projection x)
(elementlist-projection y)))))
|
