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
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
|
; Milawa - A Reflective Theorem Prover
; Copyright (C) 2005-2009 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 "MILAWA")
(include-book "primitives-5")
(%interactive)
(%autoprove not-equal-when-less)
(%autoprove not-equal-when-less-two)
(%autoprove |a <= d, b+c <= e --> b+a+c <= d+e|)
(%autoprove |(< (+ a b) (+ c b d))|)
(%autoprove |(< (+ a b c)) (+ d c))|
(%disable default
|(< (+ b a c) (+ d a))|
|[OUTSIDE](< (+ b a c) (+ d a))|)
(%use (%instance (%thm |(< (+ b a c) (+ d a))|)
(a c) (b a) (c b) (d d))))
(%autoprove |a <= b, b <= c --> a < 1+c|)
(%autoprove |b <= c, a <= b --> a < 1+c|)
(%autoprove natp-of-max)
(%autoprove equal-of-max-and-zero)
(%autoprove max-of-zero-left)
(%autoprove max-of-zero-right)
(%autoprove natp-of-min)
(%autoprove equal-of-min-and-zero)
(%autoprove min-of-zero-left)
(%autoprove min-of-zero-right)
;; Special admission for ordp, ord<, and rank. We don't use %autoadmit, instead we add
;; their definitions as axioms, which we convert into theorems now.
(defsection ordp
(%prove (%rule ordp
:lhs (ordp x)
:rhs (if (not (consp x))
(natp x)
(and (consp (car x))
(ordp (car (car x)))
(not (equal (car (car x)) 0))
(< 0 (cdr (car x)))
(ordp (cdr x))
(if (consp (cdr x))
(ord< (car (car (cdr x))) (car (car x)))
t)))))
(%use (build.axiom (definition-of-ordp)))
(%cleanup)
(%qed)
;; Don't enable; definition
)
(defsection ord<
(%prove (%rule ord<
:lhs (ord< x y)
:rhs (cond ((not (consp x))
(if (consp y) t (< x y)))
((not (consp y)) nil)
((not (equal (car (car x)) (car (car y))))
(ord< (car (car x)) (car (car y))))
((not (equal (cdr (car x)) (cdr (car y))))
(< (cdr (car x)) (cdr (car y))))
(t (ord< (cdr x) (cdr y))))))
(%use (build.axiom (definition-of-ord<)))
(%cleanup)
(%qed)
;; Don't enable; definition
)
(defsection rank
(%prove (%rule rank
:lhs (rank x)
:rhs (if (consp x)
(+ 1 (+ (rank (car x))
(rank (cdr x))))
0)))
(%use (build.axiom (definition-of-rank)))
(%cleanup)
(%qed)
;; Don't enable; definition
)
;; NOTE: we had to move booleanp-of-ord< after the rank stuff.
;; NOTE: we had to move booleanp-of-ordp after the rank stuff.
(%autoprove ord<-when-naturals
(%restrict default ord< (equal x 'x)))
(%autoprove ordp-when-natp
(%restrict default ordp (equal x 'x)))
(%autoprove natp-of-rank
(%restrict default rank (equal x 'x)))
(%autoprove rank-when-not-consp
(%restrict default rank (equal x 'x)))
(%autoprove rank-of-cons
(%restrict default rank (equal x '(cons x y))))
(%autoprove |(< 0 (rank x))|
(%restrict default rank (equal x 'x)))
(%autoprove ordp-of-rank)
(%autoprove rank-of-car
(%restrict default rank (equal x 'x)))
(%autoprove rank-of-car-weak
(%restrict default rank (equal x 'x)))
(%autoprove rank-of-cdr
(%restrict default rank (equal x 'x)))
(%autoprove rank-of-cdr-weak
(%restrict default rank (equal x 'x)))
(%autoprove rank-of-second)
(%autoprove rank-of-second-weak
(%use (%instance (%thm transitivity-of-<-three)
(a (rank (car (cdr x))))
(b (rank (cdr x)))
(c (rank x)))))
(%autoprove rank-of-third)
(%autoprove rank-of-third-weak
(%use (%instance (%thm transitivity-of-<-three)
(a (rank (third x)))
(b (rank (cdr x)))
(c (rank x)))))
(%autoprove rank-of-fourth)
(%autoprove rank-of-fourth-weak
(%use (%instance (%thm transitivity-of-<-four)
(a (rank (fourth x)))
(b (rank (cdr x)))
(c (rank x)))))
(%autoprove booleanp-of-ord<
;; Note. This is a simpler induction scheme than ACL2 picks. Originally I
;; used ACL2's induction scheme, but with this simpler scheme the proof was
;; about 1/5 the size.
(%induct (rank x)
((not (consp x))
nil)
((consp x)
(((x (cdr x)) (y (cdr y)))
((x (car (car x))) (y (car (car y)))))))
(%split)
(%restrict default ord< (equal x 'x) (equal y 'y)))
(%autoprove booleanp-of-ordp
;; Again we use a simpler induction scheme than ACL2.
(%induct (rank x)
((not (consp x))
nil)
((consp x)
(((x (car (car x))))
((x (cdr x))))))
(%split)
(%restrict default ordp (equal x 'x)))
;; We don't need an equivalent of ord<-is-well-founded; that's just to instruct ACL2
;; about which well founded relation to use.
(%autoadmit two-nats-measure)
(%autoprove ordp-of-two-nats-measure
(%enable default two-nats-measure)
(%restrict default ordp (equal x '(CONS (CONS '1 (+ '1 A)) (NFIX B)))))
(%autoprove ord<-of-two-nats-measure
(%enable default two-nats-measure)
(%restrict default ord< (equal x '(CONS (CONS '1 (+ '1 A1)) (NFIX B1)))))
(%autoadmit three-nats-measure)
(%autoprove ordp-of-three-nats-measure
(%enable default three-nats-measure)
(%restrict default ordp (or (equal x '(CONS (CONS '2 (+ '1 A)) (CONS (CONS '1 (+ '1 B)) (NFIX C))))
(equal x '(CONS (CONS '1 (+ '1 B)) (NFIX C))))))
(%autoprove ord<-of-three-nats-measure
(%enable default three-nats-measure)
(%restrict default ord< (or (equal x '(CONS (CONS '2 (+ '1 A1)) (CONS (CONS '1 (+ '1 B1)) (NFIX C1))))
(equal x '(CONS (CONS '1 (+ '1 B1)) (NFIX C1))))))
|
