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
|
; 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-1")
(%interactive)
;; BOZO reorganize these properly
(%autoprove natp-of-nfix
(%enable default nfix))
(%autoprove nfix-when-natp-cheap
(%enable default nfix))
(%autoprove nfix-when-not-natp-cheap
(%enable default nfix))
(%autoprove equal-of-nfix-of-self)
(defsection [outside]equal-of-nfix-of-self-alt
;; Can't rely on term-order for outside-in.
(%prove (%rule [outside]equal-of-nfix-of-self-alt
:type outside
:lhs (equal (nfix x) x)
:rhs (natp x)))
(%auto)
(%qed)
(%enable default [outside]equal-of-nfix-of-self-alt))
(%autoprove equal-of-zero-and-nfix
(%enable default nfix zp))
(defsection [outside]equal-of-zero-and-nfix-alt
;; Can't rely on term-order for outside-in.
(%prove (%rule [outside]equal-of-zero-and-nfix-alt
:type outside
:lhs (equal (nfix x) 0)
:rhs (zp x)))
(%auto)
(%qed)
(%enable default [outside]equal-of-zero-and-nfix-alt))
(%autoprove zp-when-natp-cheap
(%enable default zp))
(%autoprove zp-when-not-natp-cheap
(%enable default zp))
(%autoprove zp-of-nfix
(%enable default nfix))
(%autoprove nfix-of-nfix)
(%autoprove natp-when-not-zp-cheap)
(%autoprove natp-when-zp-cheap)
(%autoprove nfix-when-zp-cheap)
(%autoprove equal-of-nfix-with-positive-constant
(%enable default nfix))
;; Addition.
(%autoprove natp-of-plus
(%use (build.axiom (axiom-natp-of-plus))))
(%autoprove plus-under-iff
(%disable default natp-of-plus [outside]natp-of-plus)
(%use (%thm natp-of-plus)))
(%autoprove commutativity-of-+
(%use (build.axiom (axiom-commutativity-of-+))))
(%autoprove associativity-of-+
(%use (build.axiom (axiom-associativity-of-+))))
(%disable default [outside]associativity-of-+) ;; Interferes with constant gathering
(%autoprove commutativity-of-+-two
(%use (%instance (build.axiom (axiom-commutativity-of-+)) (b (+ b c)))))
(%autoprove gather-constants-from-plus-of-plus)
(%autoprove plus-completion-left
(%use (build.axiom (axiom-plus-when-not-natp-left)))
(%use (build.instantiation (build.axiom (axiom-plus-of-zero-when-natural))
(list (cons 'a 'b))))
(%use (build.axiom (axiom-plus-when-not-natp-left)))
(%use (build.instantiation (build.axiom (axiom-plus-when-not-natp-left))
(list (cons 'a 'b)
(cons 'b ''0)))))
(%autoprove plus-completion-right
(%disable default nfix plus-completion-left)
(%use (%instance (%thm plus-completion-left) (a b) (b a))))
(%autoprove plus-of-zero-right
(%enable default plus-completion-right)
(%use (build.axiom (axiom-plus-of-zero-when-natural))))
(%autoprove plus-of-zero-left
(%use (%instance (%thm commutativity-of-+) (a 0) (b a))))
(%autoprove plus-when-zp-left-cheap
(%use (%thm plus-completion-left)))
(%autoprove plus-when-zp-right-cheap
(%use (%thm plus-completion-right)))
(%autoprove plus-of-nfix-left
(%enable default nfix))
(%autoprove plus-of-nfix-right
(%enable default nfix))
;; Less-Than Relation.
(%autoprove booleanp-of-<
(%use (build.axiom (axiom-<-nil-or-t))))
(%autoprove irreflexivity-of-<
(%use (build.axiom (axiom-irreflexivity-of-<))))
(%autoprove less-of-zero-right
(%use (build.axiom (axiom-less-of-zero-right))))
(%autoprove less-completion-right
(%use (build.axiom (axiom-less-completion-right))))
(%autoprove less-when-zp-right-cheap
(%use (%thm less-completion-right)))
(%autoprove less-of-zero-left
(%use (build.axiom (axiom-less-of-zero-left-when-natp))))
(%autoprove less-completion-left
(%use (build.axiom (axiom-less-completion-left))))
(%autoprove less-when-zp-left-cheap
(%use (%thm less-completion-left)))
(%autoprove less-of-nfix-left
(%enable default nfix))
(%autoprove less-of-nfix-right
(%enable default nfix))
(%autoprove transitivity-of-<
(%use (build.axiom (axiom-transitivity-of-<))))
(%autoprove antisymmetry-of-<
(%disable default transitivity-of-<)
(%use (%instance (%thm transitivity-of-<) (a a) (b b) (c a))))
(%autoprove trichotomy-of-<
(%use (build.axiom (axiom-trichotomy-of-<-when-natp))))
(%autoprove one-plus-trick
(%use (build.axiom (axiom-one-plus-trick))))
(%autoprove less-of-one-right
(%use (build.axiom (axiom-natural-less-than-one-is-zero))))
(%autoprove less-of-one-left
(%enable default zp))
(%autoprove transitivity-of-<-two
(%enable default nfix)
(%disable default trichotomy-of-<)
(%use (%instance (%thm trichotomy-of-<) (a b) (b c))))
(%autoprove transitivity-of-<-three)
(%autoprove transitivity-of-<-four)
|
