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
|
; Last 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 "xdoc/top" :dir :system)
(include-book "list-defuns")
(include-book "abstract")
(local (include-book "list-fix"))
(local (include-book "append"))
(local (include-book "rev"))
(defsection std/lists/last
:parents (std/lists last)
:short "Lemmas about @(see last) available in the @(see std/lists) library."
(defthm last-when-atom
(implies (atom x)
(equal (last x)
x)))
(defthm last-when-atom-of-cdr
(implies (atom (cdr x))
(equal (last x)
x)))
(defthm last-of-cons
(equal (last (cons a x))
(if (consp x)
(last x)
(cons a x))))
(defthm consp-of-last
(equal (consp (last l))
(consp l)))
(defthm true-listp-of-last
(equal (true-listp (last l))
(true-listp l)))
(defthm last-of-list-fix
(equal (last (list-fix x))
(list-fix (last x))))
(defthm len-of-last
(equal (len (last l))
(if (consp l)
1
0)))
(defthm upper-bound-of-len-of-last
(< (len (last x)) 2)
:rule-classes :linear)
(defthm member-of-car-of-last
(iff (member (car (last x)) x)
(if (consp x)
t
nil)))
(defsection subsetp-of-last
(local (defthm l0
(implies (subsetp-equal a x)
(subsetp-equal a (cons b x)))))
(defthm subsetp-of-last
;; possibly good for forward chaining?
(subsetp (last x) x)))
(defthm last-of-append
(equal (last (append x y))
(if (consp y)
(last y)
(append (last x) y))))
(defthm last-of-rev
(equal (last (rev x))
(if (consp x)
(list (car x))
nil)))
(defthm last-of-revappend
(equal (last (revappend x y))
(cond ((consp y)
(last y))
((consp x)
(cons (car x) y))
(t
y))))
(def-listp-rule element-list-p-of-last
(implies (element-list-p (double-rewrite x))
(element-list-p (last x)))))
|
