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
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
|
;;
;; Copyright (C) 2018, Rockwell Collins
;; All rights reserved.
;;
;; This software may be modified and distributed under the terms
;; of the 3-clause BSD license. See the LICENSE file for details.
;;
;;
(in-package "ACL2")
(include-book "intersection")
(defun region-p (term)
(declare (type t term))
(case-match term
((`not x) (trapezoid-p x))
((x) (trapezoid-p x))
(& nil)))
(defun negated-region-p (term)
(declare (type t term))
(case-match term
((`not &) t)
(& nil)))
(def::un region-trapezoid (term)
(declare (xargs :signature ((region-p) trapezoid-p)))
(case-match term
((`not x) x)
((x) x)
(& nil)))
(def::un true-region (zoid)
(declare (xargs :signature ((trapezoid-p) region-p)))
(list zoid))
(defthm not-negated-region-p-true-region
(not (negated-region-p (true-region zoid)))
:rule-classes (:type-prescription
(:forward-chaining :trigger-terms ((true-region zoid)))))
(def::un negated-region (zoid)
(declare (xargs :signature ((trapezoid-p) region-p)))
`(not ,zoid))
(defthm negated-region-p-negated-region
(negated-region-p (negated-region zoid))
:rule-classes (:type-prescription
(:forward-chaining :trigger-terms ((negated-region zoid)))))
(defthm region-trapezoid-true-region
(equal (region-trapezoid (true-region zoid))
zoid))
(defthm region-trapezoid-negated-region
(equal (region-trapezoid (negated-region zoid))
zoid))
;; ---------------------------------------------------------------------------------
(defun eval-region (term env)
(declare (type t term))
(case-match term
((`not x) (not (eval-ineq-list x env)))
((x) (eval-ineq-list x env))
(& nil)))
(defthm alt-eval-region
(implies
(region-p term)
(iff (eval-region term env)
(if (negated-region-p term)
(not (eval-ineq-list (region-trapezoid term) env))
(eval-ineq-list (region-trapezoid term) env)))))
(defthm eval-region-false
(not (eval-region (negated-region nil) env)))
(defthm eval-region-true
(eval-region (true-region nil) env))
(in-theory (disable (true-region) (negated-region)))
(in-theory (disable eval-region))
(in-theory (disable region-p region-trapezoid true-region negated-region negated-region-p))
(defun wf-region-p (x env)
(declare (type t x env))
(and (region-p x)
(iff (eval-region x env)
(not (negated-region-p x)))))
(defthm wf-region-implies-region-p
(implies
(wf-region-p x env)
(region-p x))
:rule-classes (:forward-chaining))
;; ---------------------------------------------------------------------------------
(def::un not-region (term)
(declare (xargs :signature ((region-p) region-p)))
(if (negated-region-p term)
(true-region (region-trapezoid term))
(negated-region (region-trapezoid term))))
(def::signature not-region ((lambda (x) (wf-region-p x x2))) (lambda (x) (wf-region-p x x2)))
(defthm eval-region-not-region
(implies
(region-p term)
(iff (eval-region (not-region term) env)
(not (eval-region term env)))))
(in-theory (disable not-region))
;; ---------------------------------------------------------------------------------
(def::un and-regions (x y env)
(declare (xargs :signature ((region-p region-p env-p) region-p)))
(if (not (eval-region x env)) x
(if (not (eval-region y env)) y
(true-region (and-zoid (region-trapezoid x) (region-trapezoid y) env)))))
(def::signature and-regions ((lambda (x) (wf-region-p x x1)) (lambda (x) (wf-region-p x x1)) env-p) (lambda (x) (wf-region-p x x1)))
(defthm inv1-and-regions
(implies
(and
(wf-region-p x cex)
(wf-region-p y cex)
(env-p cex))
(iff (eval-region (and-regions x y cex) cex)
(and
(eval-region x cex)
(eval-region y cex)))))
(defthm inv2a-and-regions
(implies
(and
(and
(wf-region-p x cex)
(wf-region-p y cex)
(env-p cex))
(and
(eval-region x cex)
(eval-region y cex))
(not
(and
(eval-region x any)
(eval-region y any))))
(not (eval-region (and-regions x y cex) any))))
(defthm inv2b-and-regions
(implies
(and
(and
(wf-region-p x cex)
(wf-region-p y cex)
(env-p cex))
(not
(and
(eval-region x cex)
(eval-region y cex)))
(and
(eval-region x any)
(eval-region y any)))
(eval-region (and-regions x y cex) any)))
(defthm simplify-and-regions
(implies
(not (eval-region x env))
(equal (and-regions x y env) x)))
(defthm simplify-and-regions-2
(implies
(not (eval-region y env))
(equal (and-regions x y env)
(if (not (eval-region x env)) x y))))
(in-theory (disable and-regions))
;; ---------------------------------------------------------------------------------
(def::un normalize-equal-0 (poly env)
(declare (xargs :signature ((poly-p env-p) region-p)))
(let ((const (isConstant poly)))
(let ((value (eval-poly poly env)))
(if (equal value 0)
(if const (true-region nil)
(let ((xid (leading-variable poly)))
(let ((rst (solve xid poly)))
(let ((r (variableEquality xid rst)))
(true-region (list r))))))
(if const (negated-region nil)
(if (< value 0) (negated-region (list (solvePolyGreater0 :exclusive (mul -1 poly) env)))
(negated-region (list (solvePolyGreater0 :exclusive poly env)))))))))
(defthm inv1-normalize-equal-0
(implies
(and
(poly-p poly)
(env-p cex))
(iff (eval-region (normalize-equal-0 poly cex) cex)
(equal (eval-poly poly cex) 0)))
:hints (("Goal" :in-theory (enable eval-ineq-polyRelationp))))
(defthm inv2a-normalize-equal-0
(implies
(and
(poly-p poly)
(env-p cex)
(equal (eval-poly poly cex) 0)
(not (equal (eval-poly poly any) 0)))
(not (eval-region (normalize-equal-0 poly cex) any)))
:hints (("Goal" :in-theory (enable eval-ineq-polyRelationp))))
(defthm inv2b-normalize-equal-0
(implies
(and
(poly-p poly)
(env-p cex)
(not (equal (eval-poly poly cex) 0))
(equal (eval-poly poly any) 0))
(eval-region (normalize-equal-0 poly cex) any))
:hints (("Goal" :in-theory (enable eval-ineq-polyRelationp))))
(defthm wf-region-normalize-equal-0
(implies
(and
(poly-p poly)
(env-p env))
(wf-region-p (normalize-equal-0 poly env) env))
:hints (("Goal" :in-theory (enable eval-ineq-polyRelationp)))
:rule-classes (:rewrite
(:forward-chaining :trigger-terms ((normalize-equal-0 poly env)))))
(in-theory (disable normalize-equal-0))
;; ---------------------------------------------------------------------------------
(def::un normalize-gt-0 (relation poly env)
(declare (xargs :signature ((relation-p poly-p env-p) region-p)))
(let ((const (isConstant poly)))
(let ((value (eval-poly poly env)))
(if const (if (gt-relation-p relation value) (true-region nil) (negated-region nil))
(if (gt-relation-p relation value)
(true-region (list (solvePolyGreater0 relation poly env)))
(negated-region (list (solvePolyGreater0 (relation-not relation) (mul -1 poly) env))))))))
(defthm eval-region-normalize-gt-0
(implies
(and
(relation-p relation)
(poly-p poly)
(env-p env))
(iff (eval-region (normalize-gt-0 relation poly env) any)
(gt-relation-p relation (eval-poly poly any)))))
(defthm wf-region-normalize-gt-0
(implies
(and
(relation-p relation)
(poly-p poly)
(env-p env))
(wf-region-p (normalize-gt-0 relation poly env) env))
:rule-classes (:rewrite
(:forward-chaining :trigger-terms ((normalize-gt-0 relation poly env)))))
(in-theory (disable normalize-gt-0))
;; ---------------------------------------------------------------------------------
(def::un generalize-ineq (term env)
(declare (xargs :signature ((t env-p) region-p)))
(case-match term
(('and x y)
(let ((x (generalize-ineq x env))
(y (generalize-ineq y env)))
(and-regions x y env)))
(('or x y)
(let ((x (generalize-ineq x env))
(y (generalize-ineq y env)))
(not-region (and-regions (not-region x) (not-region y) env))))
(('not x)
(let ((x (generalize-ineq x env)))
(not-region x)))
(('= var poly)
(let ((x (bound-poly var))
(y (poly-fix! poly)))
(normalize-equal-0 (sub x y) env)))
(('!= var poly)
(let ((x (bound-poly var))
(y (poly-fix! poly)))
(not-region (normalize-equal-0 (sub x y) env))))
(('< x y)
(let ((x (bound-poly x))
(y (poly-fix! y)))
(normalize-gt-0 :exclusive (sub y x) env)))
(('<= x y)
(let ((x (bound-poly x))
(y (poly-fix! y)))
(normalize-gt-0 :inclusive (sub y x) env)))
(('> x y)
(let ((x (bound-poly x))
(y (poly-fix! y)))
(normalize-gt-0 :exclusive (sub x y) env)))
(('>= x y)
(let ((x (bound-poly x))
(y (poly-fix! y)))
(normalize-gt-0 :inclusive (sub x y) env)))
(& (negated-region nil))))
(include-book "arithmetic-5/top" :dir :system)
(defthm inv1-generalize-ineq
(implies
(env-p env)
(and (wf-region-p (generalize-ineq term env) env)
(iff (eval-region (generalize-ineq term env) env)
(eval-ineq term env))))
:hints (("Goal" :induct (generalize-ineq term env)
:do-not-induct t)
(and stable-under-simplificationp
'(:in-theory (enable eval-ineq)))))
(def::signature generalize-ineq (t env-p) (lambda (x) (wf-region-p x x1))
:hints (("Goal" :in-theory (disable wf-region-p))))
(in-theory (disable wf-region-p alt-eval-region))
(defthm inv2-generalize-ineq
(implies
(and
(env-p env)
(iff (eval-ineq term env)
(not (eval-ineq term any))))
(iff (eval-region (generalize-ineq term env) any)
(eval-ineq term any)))
:hints (("Goal" :induct (generalize-ineq term env)
:do-not-induct t)
(and stable-under-simplificationp
'(:in-theory (enable eval-ineq)))))
|
