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
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
|
;depends on drat.plf
(declare TestSuccess type)
; Test for clause_eq
(declare test_clause_eq
(! a clause
(! b clause
(! result bool
(! (^
(bool_and
(bool_eq (clause_eq a b) result)
(bool_and
(bool_eq (clause_eq b a) result)
(bool_and
(bool_eq (clause_eq a a) tt)
(bool_eq (clause_eq b b) tt))))
tt)
TestSuccess)))))
(check
(% a var
(% b var
(@ c1 (clc (pos a) (clc (neg b) cln))
(@ c2 (clc (pos a) (clc (neg b) cln))
(: TestSuccess
(test_clause_eq c1 c2 tt)))))))
(check
(% a var
(% b var
(@ c1 (clc (neg a) (clc (neg b) cln))
(@ c2 (clc (pos a) (clc (neg b) cln))
(: TestSuccess
(test_clause_eq c1 c2 ff)))))))
(check
(% a var
(% b var
(@ c1 (clc (pos a) (clc (pos b) cln))
(@ c2 (clc (pos a) (clc (neg b) cln))
(: TestSuccess
(test_clause_eq c1 c2 ff)))))))
(check
(% a var
(% b var
(@ c1 (clc (pos a) (clc (pos a) cln))
(@ c2 (clc (pos a) (clc (neg b) cln))
(: TestSuccess
(test_clause_eq c1 c2 ff)))))))
(check
(% a var
(% b var
(@ c1 (clc (neg b) (clc (pos a) (clc (neg a) cln)))
(@ c1 (clc (neg a) (clc (neg b) (clc (pos a) cln)))
(: TestSuccess
(test_clause_eq c1 c2 ff)))))))
(check
(% a var
(% b var
(@ c1 (clc (pos a) (clc (pos a) (clc (neg b) cln)))
(@ c2 (clc (pos a) (clc (neg b) cln))
(: TestSuccess
(test_clause_eq c1 c2 tt)))))))
(check
(% a var
(% b var
(@ c1 (clc (pos a) (clc (pos a) (clc (neg b) cln)))
(@ c2 (clc (pos a) (clc (neg b) (clc (neg b) cln)))
(: TestSuccess
(test_clause_eq c1 c2 tt)))))))
(check
(% a var
(% b var
(@ c1 (clc (pos a) (clc (pos a) (clc (neg a) cln)))
(@ c2 (clc (pos a) (clc (neg a) (clc (neg a) cln)))
(: TestSuccess
(test_clause_eq c1 c2 tt)))))))
(check
(% a var
(% b var
(@ c1 cln
(@ c2 (clc (pos a) (clc (neg a) (clc (neg a) cln)))
(: TestSuccess
(test_clause_eq c1 c2 ff)))))))
(declare check_rat
(! f cnf
(! c clause
(! b bool
(! sc (^ (is_rat f c) b)
bottom)))))
(declare trust_cnf (! f cnf (cnf_holds f)))
; RAT Test 1
; Formula: (-p, -a) ^ (-p, b) ^( b, c) ^ (-c, a)
; Candidate RAT: (p, a)
; Answer: true
(check
(% va var
(% vb var
(% vc var
(% vp var
(check_rat
(cnfc (clc (neg vp) (clc (neg va) cln))
(cnfc (clc (neg vp) (clc (pos vb) cln))
(cnfc (clc (pos vb) (clc (pos vc) cln))
(cnfc (clc (neg vc) (clc (pos va) cln)) cnfn))))
(clc (pos vp) (clc (pos va) cln))
tt))))))
; RAT Test 2
; Formula:
; p cnf 4 8
; 1 2 -3 0
; -1 -2 3 0
; 2 3 -4 0
; -2 -3 4 0
; -1 -3 -4 0
; 1 3 4 0
; -1 2 4 0
; 1 -2 -4 0
; Candidate RAT: -1
(check
(% v1 var
(% v2 var
(% v3 var
(% v4 var
(check_rat
(cnfc (clc (pos v1) (clc (pos v2) (clc (neg v3) cln)))
(cnfc (clc (neg v1) (clc (neg v2) (clc (pos v3) cln)))
(cnfc (clc (pos v2) (clc (pos v3) (clc (neg v4) cln)))
(cnfc (clc (neg v2) (clc (neg v3) (clc (pos v4) cln)))
(cnfc (clc (neg v1) (clc (neg v3) (clc (neg v4) cln)))
(cnfc (clc (pos v1) (clc (pos v3) (clc (pos v4) cln)))
(cnfc (clc (neg v1) (clc (pos v2) (clc (pos v4) cln)))
(cnfc (clc (pos v1) (clc (neg v2) (clc (neg v4) cln)))
cnfn))))))))
(clc (neg v1) cln)
tt))))))
; RAT Test 3
; Formula:
; p cnf 4 9
; 1 2 -3 0
; -1 -2 3 0
; 2 3 -4 0
; -2 -3 4 0
; -1 -3 -4 0
; 1 3 4 0
; -1 2 4 0
; 1 -2 -4 0
; -1 0
; Candidate RAT: 2
(check
(% v1 var
(% v2 var
(% v3 var
(% v4 var
(check_rat
(cnfc (clc (pos v1) (clc (pos v2) (clc (neg v3) cln)))
(cnfc (clc (neg v1) (clc (neg v2) (clc (pos v3) cln)))
(cnfc (clc (pos v2) (clc (pos v3) (clc (neg v4) cln)))
(cnfc (clc (neg v2) (clc (neg v3) (clc (pos v4) cln)))
(cnfc (clc (neg v1) (clc (neg v3) (clc (neg v4) cln)))
(cnfc (clc (pos v1) (clc (pos v3) (clc (pos v4) cln)))
(cnfc (clc (neg v1) (clc (pos v2) (clc (pos v4) cln)))
(cnfc (clc (pos v1) (clc (neg v2) (clc (neg v4) cln)))
(cnfc (clc (neg v1) cln)
cnfn)))))))))
(clc (pos v2) cln)
tt))))))
; RAT Test 4
; Formula:
; p cnf 4 2
; 2 -3 0
; 1 -4 0
; Candidate RAT: 3
; Answer: false
(check
(% v1 var
(% v2 var
(% v3 var
(% v4 var
(check_rat
(cnfc (clc (pos v2) (clc (neg v3) cln))
(cnfc (clc (pos v1) (clc (neg v4) cln)) cnfn))
(clc (pos v3) cln)
ff))))))
; DRAT Test 1 (from Example 1 @ https://www.cs.utexas.edu/~marijn/drat-trim/)
; without deletions
; Formula:
; p cnf 4 8
; 1 2 -3 0
; -1 -2 3 0
; 2 3 -4 0
; -2 -3 4 0
; -1 -3 -4 0
; 1 3 4 0
; -1 2 4 0
; 1 -2 -4
; Proof:
; -1 0
; 2 0
; 0
(check
(% v1 var
(% v2 var
(% v3 var
(% v4 var
(:
(holds cln)
(drat_proof_of_bottom _
(trust_cnf (cnfc (clc (pos v1) (clc (pos v2) (clc (neg v3) cln)))
(cnfc (clc (neg v1) (clc (neg v2) (clc (pos v3) cln)))
(cnfc (clc (pos v2) (clc (pos v3) (clc (neg v4) cln)))
(cnfc (clc (neg v2) (clc (neg v3) (clc (pos v4) cln)))
(cnfc (clc (neg v1) (clc (neg v3) (clc (neg v4) cln)))
(cnfc (clc (pos v1) (clc (pos v3) (clc (pos v4) cln)))
(cnfc (clc (neg v1) (clc (pos v2) (clc (pos v4) cln)))
(cnfc (clc (pos v1) (clc (neg v2) (clc (neg v4) cln)))
cnfn)))))))))
(DRATProofa (clc (neg v1) cln)
(DRATProofa (clc (pos v2) cln)
(DRATProofa cln
DRATProofn))))))))))
; DRAT Test 2 (from Example 1 @ https://www.cs.utexas.edu/~marijn/drat-trim/)
; with deletions
; Formula:
; p cnf 4 8
; 1 2 -3 0
; -1 -2 3 0
; 2 3 -4 0
; -2 -3 4 0
; -1 -3 -4 0
; 1 3 4 0
; -1 2 4 0
; 1 -2 -4
; Proof:
; -1 0
; d -1 -2 3 0
; d -1 -3 -4 0
; d -1 2 4 0
; 2 0
; d 1 2 -3 0
; d 2 3 -4 0
; 0
(check
(% v1 var (% v2 var (% v3 var (% v4 var
(: (holds cln)
(drat_proof_of_bottom _
(trust_cnf
(cnfc (clc (pos v1) (clc (pos v2) (clc (neg v3) cln)))
(cnfc (clc (neg v1) (clc (neg v2) (clc (pos v3) cln)))
(cnfc (clc (pos v2) (clc (pos v3) (clc (neg v4) cln)))
(cnfc (clc (neg v2) (clc (neg v3) (clc (pos v4) cln)))
(cnfc (clc (neg v1) (clc (neg v3) (clc (neg v4) cln)))
(cnfc (clc (pos v1) (clc (pos v3) (clc (pos v4) cln)))
(cnfc (clc (neg v1) (clc (pos v2) (clc (pos v4) cln)))
(cnfc (clc (pos v1) (clc (neg v2) (clc (neg v4) cln)))
cnfn)))))))))
(DRATProofa (clc (neg v1) cln)
(DRATProofd (clc (neg v1) (clc (neg v2) (clc (pos v3) cln)))
(DRATProofd (clc (neg v1) (clc (neg v3) (clc (neg v4) cln)))
(DRATProofd (clc (neg v1) (clc (pos v2) (clc (pos v4) cln)))
(DRATProofa (clc (pos v2) cln)
(DRATProofd (clc (pos v1) (clc (pos v2) (clc (neg v3) cln)))
(DRATProofd (clc (pos v2) (clc (pos v3) (clc (neg v4) cln)))
(DRATProofa cln
DRATProofn)))))))))))))))
; ===================================== ;
; Test Suite from "Two Flavors of DRAT" ;
; ===================================== ;
; The paper includes a number of proofs which explore specified and operational
; DRAT validity.
; Our test predicate for asserting the specified and operational validity of
; DRAT proofs
(declare spec_oper_test
(! f cnf
(! proof DRATProof
(! spec_validity bool
(! oper_validity bool
(! sc (^ (bool_and
(bool_eq (is_specified_drat_proof f proof) spec_validity)
(bool_eq (is_operational_drat_proof f proof) oper_validity)
) tt)
TestSuccess))))))
(declare x var)
(declare y var)
(declare z var)
(declare w var)
(define ex_1_formula
(cnfc (clc (pos x) (clc (pos y) (clc (pos z) cln)))
(cnfc (clc (neg x) (clc (pos y) (clc (pos z) cln)))
(cnfc (clc (pos x) (clc (neg y) (clc (pos z) cln)))
(cnfc (clc (neg x) (clc (neg y) (clc (pos z) cln)))
(cnfc (clc (pos x) (clc (pos y) (clc (neg z) cln)))
(cnfc (clc (neg x) (clc (pos y) (clc (neg z) cln)))
(cnfc (clc (pos x) (clc (neg y) (clc (neg z) cln)))
(cnfc (clc (neg x) (clc (neg y) (clc (neg z) cln)))
cnfn)))))))))
; Spec-valid, operationally-invalid
(define ex_1_pf_pi
(DRATProofa (clc (pos x) (clc (pos y) cln))
(DRATProofa (clc (pos x) cln)
(DRATProofa (clc (pos w) (clc (neg x) cln))
(DRATProofd (clc (pos w) (clc (neg x) cln))
(DRATProofa (clc (neg w) (clc (neg x) cln))
(DRATProofa (clc (pos w) (clc (pos x) cln))
(DRATProofa (clc (pos y) (clc (pos w) cln))
(DRATProofa cln
DRATProofn)))))))))
(check
(: TestSuccess
(spec_oper_test ex_1_formula ex_1_pf_pi tt ff)))
; Spec-invalid, operationally valid
(define ex_1_pf_sigma
(DRATProofa (clc (pos x) (clc (pos y) cln))
(DRATProofa (clc (pos x) cln)
(DRATProofd (clc (pos x) cln)
(DRATProofa (clc (pos w) (clc (neg y) cln))
(DRATProofa (clc (neg w) (clc (neg y) cln))
(DRATProofa (clc (pos w) cln)
(DRATProofa cln
DRATProofn))))))))
(check
(: TestSuccess
(spec_oper_test ex_1_formula ex_1_pf_sigma ff tt)))
(declare x1 var)
(declare x2 var)
(declare x3 var)
(declare x4 var)
(declare x5 var)
(declare x6 var)
(declare x7 var)
(declare x8 var)
(declare x9 var)
(declare x10 var)
(define ex_2_formula
(cnfc (clc (pos x1) cln)
(cnfc (clc (neg x1) (clc (pos x2) cln))
(cnfc (clc (neg x1) (clc (neg x2) (clc (pos x3) cln)))
(cnfc (clc (neg x1) (clc (neg x3) (clc (pos x4) cln)))
(cnfc (clc (pos x5) (clc (pos x6) cln))
(cnfc (clc (neg x2) (clc (neg x5) (clc (pos x7) cln)))
(cnfc (clc (neg x1) (clc (neg x5) (clc (pos x6) cln)))
(cnfc (clc (neg x5) (clc (neg x6) (clc (pos x4) cln)))
(cnfc (clc (neg x3) (clc (neg x6) (clc (pos x8) cln)))
(cnfc (clc (neg x6) (clc (neg x4) (clc (pos x3) cln)))
(cnfc (clc (neg x8) (clc (pos x5) cln))
(cnfc (clc (neg x3) (clc (pos x9) (clc (pos x10) cln)))
(cnfc (clc (neg x4) (clc (neg x9) (clc (pos x10) cln)))
(cnfc (clc (neg x10) (clc (pos x9) cln))
(cnfc (clc (neg x9) (clc (pos x7) cln))
(cnfc (clc (neg x7) (clc (neg x8) (clc (neg x9) (clc (neg x10) cln))))
cnfn)))))))))))))))))
; Spec-valid, operationally-valid
(define ex_2_pf_tau
(DRATProofa (clc (pos x5) cln)
(DRATProofa (clc (pos x4) cln)
(DRATProofa (clc (pos x9) cln)
(DRATProofa cln
DRATProofn)))))
(check
(: TestSuccess
(spec_oper_test ex_2_formula ex_2_pf_tau tt tt)))
; Spec-valid, operationally unspecified in the paper, but its operationally valid.
(define ex_3_pf_tau_prime
(DRATProofa (clc (pos x5) cln)
(DRATProofd (clc (neg x1) (clc (pos x2) cln))
(DRATProofa (clc (pos x9) cln)
(DRATProofa cln
DRATProofn)))))
(check
(: TestSuccess
(spec_oper_test ex_2_formula ex_3_pf_tau_prime tt tt)))
; Spec-invalid, operationally-invalid
(define ex_4_pf_pi_prime
(DRATProofa (clc (pos x) (clc (pos y) cln))
(DRATProofa (clc (pos x) cln)
(DRATProofa (clc (pos w) (clc (neg x) cln))
(DRATProofa (clc (neg w) (clc (neg x) cln))
(DRATProofa (clc (pos w) (clc (pos x) cln))
(DRATProofa (clc (pos y) (clc (pos w) cln))
(DRATProofa cln
DRATProofn))))))))
(check
(: TestSuccess
(spec_oper_test ex_1_formula ex_4_pf_pi_prime ff ff)))
; Spec-valid, operationally valid
(define ex_5_pf_sigma_prime
(DRATProofa (clc (pos x) (clc (pos y) cln))
(DRATProofa (clc (pos x) cln)
(DRATProofa (clc (pos w) (clc (neg y) cln))
(DRATProofa (clc (neg w) (clc (neg y) cln))
(DRATProofa (clc (pos w) cln)
(DRATProofa cln
DRATProofn)))))))
(check
(: TestSuccess
(spec_oper_test ex_1_formula ex_5_pf_sigma_prime tt tt)))
|
