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
|
;;
;; Copyright (C) 2021, Collins Aerospace
;; 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 "../Base/flip")
(include-book "../Base/step")
; Matt K. mod: Avoid ACL2(p) error from computed hint that returns state.
(set-waterfall-parallelism nil)
(local (in-theory (disable rewrite-<-into-average
rewrite-<-into-average-alt-1
double-average-is-just-sum
reconstruct-average
rewrite-equality-into-just-average
reconstruct-average-difference
rewrite-equality-into-average
)))
;; I ran experiments with and without this:
;;
;; (local (in-average-theory))
;;
;; in average theory: 1210s
;; in standard theory: 547s
;;
(defun initially-synchronized-p (i ens)
(let* ((i (uav-id-fix i))
(uav (ith-uav i ens)))
;; L R
;; .....|.....|....
;; x
;;
(and
(implies
;;
;; If a UAV is inside of its segment moving left then it is
;; synchronized with its right neighbor until it reaches
;; its left segment boundary.
;;
(and
(< i (1- (N)))
(< (uav->direction uav) 0)
(< (uav-left-boundary uav) (uav->location uav))
(< (uav->location uav) (uav-right-boundary uav)))
(and
(< 0 (uav->direction (ith-uav (1+ i) ens)))
(equal (- (uav-right-boundary uav) (uav->location uav))
(- (uav->location (ith-uav (1+ i) ens)) (uav-right-boundary uav)))))
(implies
;;
;; If a UAV is inside of its segment moving right then it is
;; is synchronized with its left neighbor until it reaches
;; its right segment boundary
;;
(and
(< 0 i)
(< 0 (uav->direction uav))
(< (uav-left-boundary uav) (uav->location uav))
(< (uav->location uav) (uav-right-boundary uav)))
(and
(< (uav->direction (ith-uav (1- i) ens)) 0)
(equal (- (uav->location uav) (uav-left-boundary uav))
(- (uav-left-boundary uav) (uav->location (ith-uav (1- i) ens))))))
(implies
;;
;; If a UAV is approaching its segment from the left, it is
;; being escorted by its left neighbor. When departing its left
;; boundary and entering the segment, it is co-incident with its
;; left neighbor.
;;
(and
(< 0 i)
(< 0 (uav->direction uav))
(<= (uav->location uav) (uav-left-boundary uav)))
(and
(implies
(< (uav->location uav) (uav-left-boundary uav))
(< 0 (uav->direction (ith-uav (1- i) ens))))
(equal (uav->location uav)
(uav->location (ith-uav (1- i) ens)))))
(implies
;;
;; If a UAV is approaching its segment from the right, it is
;; being escorted by its right neighbor. When departing its right
;; boundary and entering the segment, it is co-incident with its
;; right neighbor.
;;
(and
(< i (1- (N)))
(< (uav->direction uav) 0)
(<= (uav-right-boundary uav) (uav->location uav)))
(and
(implies
(< (uav-right-boundary uav) (uav->location uav))
(< (uav->direction (ith-uav (1+ i) ens)) 0))
(equal (uav->location uav)
(uav->location (ith-uav (1+ i) ens)))))
)))
;;(in-theory (disable FUNCTIONAL-COMMUTATIVITY-OF-MINUS-*-LEFT))
(defun-sk initial-synchronization (ens)
(forall (i) (initially-synchronized-p i ens)))
(in-theory (disable initial-synchronization))
(defthm initial-synchronization-implies-initially-synchronized-p
(implies
(initial-synchronization ens)
(initially-synchronized-p i ens))
:hints (("Goal" :in-theory (disable initially-synchronized-p))))
(encapsulate
()
(local (in-theory (disable EQUAL-UAV-ID-FIX-1-TO-UAV-ID-EQUIV)))
(defthm degenerate-alpha-0
(iff (equal 1 (+ (* 1/2 (N)) (* -1/2 (UAV-ID-FIX I))))
(if (uav-id-p (+ -2 (N)))
(uav-id-equiv I (+ -2 (N)))
nil))
:hints (("Goal" :in-theory (enable natp
uav-id-equiv
uav-id-p
uav-id-fix))))
(defthm degenerate-alpha-1
(iff (UAV-ID-EQUIV 0 1)
(equal (N) 1))
:hints (("Goal" :in-theory (enable uav-id-equiv
uav-id-p
uav-id-fix))))
)
(encapsulate
()
(local (include-arithmetic))
(defthmd alt-right-perimeter-boundary
(equal (right-perimeter-boundary)
(* (N) (segment-length)))
:hints (("GOal" :in-theory (enable right-perimeter-boundary segment-length))))
)
(encapsulate
()
(defthm initially-synchronized-p-flip-on-events-helper
(implies
(and
(wf-ensemble ens)
(escort-condition ens)
(initially-synchronized-p i ens))
(initially-synchronized-p i (flip-on-events ens)))
:hints (("Goal" :do-not-induct t
:in-theory (enable flip-uav))
(pattern::hint*
ordering-rule
balanced-average
location-pinching-rule
escort-condition-implies
)
(average-hint)
))
(defthm initial-synchronization-flip-on-events
(implies
(and
(wf-ensemble ens)
(escort-condition ens)
(initial-synchronization ens))
(initial-synchronization (flip-on-events ens)))
:hints (("Goal" :do-not-induct t
:in-theory (disable initially-synchronized-p)
:expand (initial-synchronization (flip-on-events ens)))))
)
(in-theory (disable chasing-neighbor))
;; Painfully slow .. Subgoal 197.17.3.2
;; it would be nice to figure out what is going on (?)
;;
;;
(encapsulate
()
(local
(without-subsumption
(defthmd initially-synchronized-p-update-location-all-helper
(implies
(and
(wf-ensemble ens)
(escort-condition ens)
;; (implies
;; (< (uav-id-fix i) (+ -1 (N)))
;; (escort-condition-p i (+ 1 (uav-id-fix i)) ens))
;; (implies
;; (< 0 (uav-id-fix i))
;; (escort-condition-p (+ -1 (uav-id-fix i)) i ens))
(lte-min-time-to-impending-impact-event dt ens)
(implies
(and
(not (chasing-neighbor (+ -1 (uav-id-fix I)) ens))
(< 0 (uav-id-fix I)))
(<= (nnrat-fix dt) (MIN-TIME-TO-IMPACT-FOR-UAV (+ -1 (uav-id-fix I)) ENS)))
(implies
(and
(not (chasing-neighbor (+ 1 (uav-id-fix I)) ens))
(< (uav-id-fix I) (+ -1 (N))))
(<= (nnrat-fix dt) (MIN-TIME-TO-IMPACT-FOR-UAV (+ 1 (uav-id-fix I)) ENS)))
(implies
(not (chasing-neighbor i ens))
(<= (nnrat-fix dt) (MIN-TIME-TO-IMPACT-FOR-UAV I ENS)))
(initially-synchronized-p i ens))
(initially-synchronized-p i (update-location-all dt ens)))
:hints (("Goal" :do-not-induct t
:in-theory (disable CHASING-NEIGHBOR-IS-NOT-IMPENDING-IMPACT-EVENT-FOR-UAV))
;;(except "Subgoal 200.17.3.2" :print nil)
;;(except "Subgoal 197.17.3.2" :print nil)
;;(and stable-under-simplificationp
;; '(:do-not '(preprocess)))
(pattern::hint*
ordering-rule
expand-chasing-neighbor
expand-min-time-to-impact-for-uav
;;balanced-average
location-pinching-rule
escort-condition-implies
)
)))
)
(defthm initially-synchronized-p-update-location
(implies
(and
(wf-ensemble ens)
(escort-condition ens)
(lte-min-time-to-impending-impact-event dt ens)
(initially-synchronized-p i ens))
(initially-synchronized-p i (update-location-all dt ens)))
:hints (("Goal" :do-not-induct t
:use initially-synchronized-p-update-location-all-helper
:in-theory (e/d (CHASING-NEIGHBOR-IS-NOT-IMPENDING-IMPACT-EVENT-FOR-UAV)
(initially-synchronized-p)))
))
)
(defthm event-for-uav-ensures-initially-synchronized-p
(implies
(and
(wf-ensemble ens)
(escort-condition ens)
(event-for-uav i ens))
(initially-synchronized-p i (flip-on-events ens)))
:hints (("Goal" :do-not-induct t)
(and stable-under-simplificationp
'(:in-theory (enable flip-uav
ESCORT-EVENT-FOR-UAV
impact-event-for-uav)))
(pattern::hint*
escort-condition-implies
ordering-rule
)))
|
