/
outline.html
575 lines (522 loc) · 25.1 KB
/
outline.html
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
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<base href="html/">
<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<style>
#correspondence th, td {
padding: 10px;
border-bottom:1px solid black;
}
#correspondence table {
border-collapse: collapse;
}
</style>
<title>Overview of Coq Development</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<div id="toc">
<h2> Correspondence Table </h2>
<div id="correspondence">
<table>
<tr>
<th>Definition/Lemma</th>
<th>Location in Paper</th>
<th>File(s)</th>
<th>Name in Formalization</th>
<th>Remarks</th>
</tr>
<tr>
<td> Indexed Valuation </td>
<td> Section 2.2 </td>
<td> <a href="discprob.idxval.ival.html">proba/theories/idxval/ival.v</a>,<br />
<a href="discprob.idxval.ival_dist.html">proba/theories/idxval/ival_dist.v</a>
</td>
<td> <a href="discprob.idxval.ival.html#ival">ival</a>,
<a href="discprob.idxval.ival_dist.html#ivdist">ivdist</a>
</td>
<td> We first define indexed valuations as a record type without the requirement that values sum to one, and then later define an "extended" version of the record in which there is an additional proof that values sum to 1. </td>
</tr>
<tr>
<td> Equivalence of Indexed Valuations </td>
<td> Section 2.2 </td>
<td> <a href="discprob.idxval.ival.html">proba/theories/idxval/ival.v</a><br /> </td>
<td> <a href="discprob.idxval.ival.html#eq_ival">eq_ival</a>,
<a href="discprob.idxval.ival_dist.html#eq_ivd">eq_ivd</a> </td>
<td></td>
</tr>
<tr>
<td> Probabilistice Choice between Indexed Valuations </td>
<td> Section 2.2 </td>
<td> <a href="discprob.idxval.ival_dist.html">proba/theories/idxval/ival_dist.v</a> </td>
<td> <a href="discprob.idxval.ival_dist.html#ivdplus">ivdplus</a> </td>
<td>In the formalization, this is defined in terms of simpler operations (<a href="discprob.idxval.ival.html#iplus">iplus</a> and <a href="discprob.idxval.ival.html#iscale">iscale</a>) as in the work of Varacca and Winskel, whereas the paper presents the "beta reduced" definition.</td>
</tr>
<tr>
<td> Non-Empty Sets of Indexed Valuations </td>
<td> Section 2.2 </td>
<td> <a href="discprob.idxval.pival.html">proba/theories/idxval/pival.v</a><br />
<a href="discprob.idxval.pival_dist.html">proba/theories/idxval/pival_dist.v</a><br /> </td>
<td> <a href="discprob.idxval.pival.html#pival">pival</a>,
<a href="discprob.idxval.pival_dist.html#pidist">pidist</a> </td>
<td>As with indexed valuations, we first define a version where every element of the set is merely an <a href="discprob.idxval.ival.html#ival">ival</a>, and another where their values must also sum to 1</td>
</tr>
<tr>
<td> Equivalence/Subset Ordering for Sets of Indexed Valuations </td>
<td> Section 2.2 </td>
<td> <a href="discprob.idxval.pival.html">proba/theories/idxval/pival.v</a><br />
<a href="discprob.idxval.pival_dist.html">proba/theories/idxval/pival_dist.v</a><br /> </td>
<td> <a href="discprob.idxval.pival.html#eq_pival">eq_pival</a>,
<a href="discprob.idxval.pival.html#le_pival">le_pival</a>,
<a href="discprob.idxval.pival_dist.html#eq_pidist">eq_pidist</a>,
<a href="discprob.idxval.pival_dist.html#le_pidist">le_pidist</a> </td>
<td></td>
</tr>
<tr>
<td> Equational Laws for Combined Monad </td>
<td> Section 2.2, Fig. 2 </td>
<td> <a href="discprob.idxval.pival_dist.html">proba/theories/idxval/pival_dist.v</a></td>
<td>
<a href="discprob.idxval.pival_dist.html#pidist_plus_comm">pidist_plus_comm</a>,
<a href="discprob.idxval.pival_dist.html#pidist_plus_choice1">pidist_plus_choice1</a>,
<a href="discprob.idxval.pival_dist.html#pidist_union_idemp">pidist_union_idemp</a>,
<a href="discprob.idxval.pival_dist.html#pidist_union_comm">pidist_union_comm</a>,
<a href="discprob.idxval.pival_dist.html#pidist_union_assoc">pidist_union_assoc</a>,
<a href="discprob.idxval.pival_dist.html#pidist_union_bind">pidist_union_bind</a>,
<a href="discprob.idxval.pival_dist.html#pidist_plus_bind">pidist_plus_bind</a>
</td>
<td> Various other laws are also described in this file, including the monad laws
(<a href="discprob.idxval.pival_dist.html#pidist_assoc">pidist_assoc</a>,
<a href="discprob.idxval.pival_dist.html#pidist_left_id">pidist_left_id</a>,
<a href="discprob.idxval.pival_dist.html#pidist_right_id">pidist_right_id</a>)
</td>
</tr>
<tr>
<td> Monadic encoding of approximate counters </td>
<td> Section 2.3, Fig. 3 </td>
<td> <a href="iris.tests.approx_counter.faa_approx_counter.html">theories/tests/approx_counter/faa_approx_counter.v</a></td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html#approx_incr">approx_incr</a>
<a href="iris.tests.approx_counter.faa_approx_counter.html#approx_n">approx_n</a>
</td>
<td> </td>
</tr>
<tr>
<td> Expected value of indexed valuation </td>
<td> Section 2.4 </td>
<td> <a href="discprob.idxval.extrema.html">proba/theories/idxval/extrema.v</a></td>
<td>
<a href="discprob.idxval.extrema.html#Ex_ival">Ex_ival</a>,
<a href="discprob.idxval.extrema.html#ex_Ex_ival">ex_Ex_ival</a>,
<a href="discprob.idxval.extrema.html#is_Ex_ival">is_Ex_ival</a>,
<a href="discprob.idxval.extrema.html#Ex_ivd">Ex_ivd</a>,
<a href="discprob.idxval.extrema.html#ex_Ex_ivd">ex_Ex_ivd</a>,
<a href="discprob.idxval.extrema.html#is_Ex_ivd">is_Ex_ivd</a>,
</td>
<td> Recall that expected values may not exist (because they are infinite series). We follow what Boldo et al. do in their Coquelicot analysis library for infinite series: Ex_ival f I is a real number, which is equal to the expected value, if the expected value exists. ex_Ex_ival f I is a proposition stating stating this existence property, and is_Ex_ival f I v says that the expected value exists and is equal to v. <br /><br /> We first define these for ival that do not have probabilities summing to 1, then the _ivd variants are the versions with the latter restriction.
</td>
</tr>
<tr>
<td> Minimal and Maximal Expected Values </td>
<td> Section 2.4 </td>
<td> <a href="discprob.idxval.extrema.html">proba/theories/idxval/extrema.v</a></td>
<td>
<a href="discprob.idxval.extrema.html#Ex_min">Ex_min</a>,
<a href="discprob.idxval.extrema.html#Ex_max">Ex_max</a>,
<a href="discprob.idxval.extrema.html#ex_Ex_ival">ex_Ex_extrema</a>,
</td>
<td> </td>
</tr>
<tr>
<td> Rules for calculating extrema</td>
<td> Section 2.4, Fig. 4 </td>
<td> <a href="discprob.idxval.extrema.html">proba/theories/idxval/extrema.v</a></td>
<td>
<a href="discprob.idxval.extrema.html#Ex_min_mret">Ex_min_mret</a>,
<a href="discprob.idxval.extrema.html#Ex_min_plus_const_r">Ex_min_plus_const_r</a>,
<a href="discprob.idxval.extrema.html#Ex_min_scale_const">Ex_min_scale_const</a>,
<a href="discprob.idxval.extrema.html#Ex_min_pidist_plus">Ex_min_pidist_plus</a>,
<a href="discprob.idxval.extrema.html#Ex_min_comp">Ex_min_comp</a>,
<a href="discprob.idxval.extrema.html#Ex_min_bind_const_le">Ex_min_bind_const_le</a>,
<a href="discprob.idxval.extrema.html#Ex_min_bind_const_ge">Ex_min_bind_const_ge</a>
</td>
<td>As stated in the paper, the rules are given there without the side conditions about the existence/finiteness of extrema.</td>
</tr>
<tr>
<td> Expected value of monadic specification of approximate counter</td>
<td> Section 2.4 </td>
<td> <a href="iris.tests.approx_counter.faa_approx_counter.html">theories/tests/approx_counter/faa_approx_counter.v</a></td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html#Ex_min_approx_incr">Ex_min_approx_incr</a>
<a href="iris.tests.approx_counter.faa_approx_counter.html#Ex_min_approx_n">Ex_min_approx_n</a>
<a href="iris.tests.approx_counter.faa_approx_counter.html#Ex_max_approx_incr">Ex_max_approx_incr</a>
<a href="iris.tests.approx_counter.faa_approx_counter.html#Ex_max_approx_n">Ex_max_approx_n</a>
</td>
<td> </td>
</tr>
<tr>
<td>Coarser equivalence relation on indexed valuations</td>
<td> Section 2.4, Fig. 5 </td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a><br />
<a href="discprob.idxval.irrel_coupling_alt.html">proba/theories/idxval/irrel_coupling_alt.v</a>
</td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_ivd">irrel_ivd</a>,
<a href="discprob.idxval.irrel_coupling_alt.html#irrel_ivd_alt">irrel_ivd_alt</a>,
</td>
<td>For historical reasons, much of the development uses irrel_ivd, which is
specified inductively using the rules in Fig. 5. However,
<a href="discprob.idxval.irrel_coupling_alt.html#irrel_ivd_alt">irrel_ivd_alt</a> show that this is equivalent to the definition used in the (revised) paper in terms of expected values of bounded functions.</td>
</tr>
<tr>
<td>Coarser subset ordering on sets of indexed valuations</td>
<td>Section 2.4</td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a><br />
</td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist">irrel_pidist</a>,
</td>
<td> It is stated in Coq in terms of Ex_min instead of Ex_max, but see just
below for the Ex_max version.</td>
</tr>
<tr>
<td>Rules for coarser subset ordering on sets of indexed valuations</td>
<td>Section 2.4, Fig. 6.</td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a><br /></td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_refl">irrel_pidist_refl</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_trans">irrel_pidist_trans</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_proper">irrel_pidist_proper</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_bind">irrel_pidist_bind</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_choice">irrel_pidist_choice</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_irrel">irrel_pidist_irrel</a>,
</td>
<td></td>
</tr>
<tr>
<td>Lemma 2.1 </td>
<td>Section 2.4</td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a><br /></td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_bounded_supp_fun">
irrel_bounded_supp_fun</a>
<a href="discprob.idxval.irrel_equiv.html#irrel_pidist_bounded_supp_Ex_max">
irrel_pidist_bounded_supp_Ex_max</a>
</td>
<td></td>
</tr>
<tr>
<td>Non-deterministic Couplings Definition </td>
<td>Section 2.5</td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a><br />
<a href="discprob.idxval.irrel_coupling_alt.html">proba/theories/idxval/irrel_coupling_alt.v</a><br /></td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_couplingP">
irrel_couplingP</a>
<a href="discprob.idxval.irrel_coupling_alt.html#idist_pidist_couplingP'">
idist_pidist_couplingP'</a>
</td>
<td>Again, for historical reasons, the development primarily uses
<a href="discprob.idxval.irrel_equiv.html#irrel_couplingP">
irrel_couplingP</a>, which is what is called a "non-deterministic
coupling up to irrelevance" in the original submitted
appendix. However, the definition currently given in the paper
corresponds
to <a href="discprob.idxval.irrel_equiv.html#idist_pidist_couplingP'">
idist_pidist_couplingP'</a> which seems more natural to us in
hindsight (because one does not need to first introduce a
different notion and then quotient to get the "up to irrelevance"
version).
<a href="discprob.idxval.irrel_coupling_alt.html#irrel_couplingP_to_alt">irrel_couplingP_to_alt</a>
and
<a href="discprob.idxval.irrel_coupling_alt.html#alt_to_irrel_couplingP">alt_to_irrel_couplingP</a> show that these definitions are equivalent. Note that these are stated as records, and for technical reasons it's sometimes easier to use a version that is a proposition, <a href="discprob.idxval.irrel_equiv.html#irrel_coupling_propP">irrel_coupling_propP</a>
</td>
</tr>
<tr>
<td>Rules for Non-deterministic Couplings</td>
<td>Section 2.5, Fig. 7</td>
<td>
<a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a></td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_mret">
irrel_coupling_mret</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_proper">
irrel_coupling_proper</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_conseq">
irrel_coupling_conseq</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_plus">
irrel_coupling_plus</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_trivial">
irrel_coupling_trivial</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Theorem 2.2</td>
<td>Section 2.5</td>
<td> <a href="discprob.idxval.irrel_equiv.html">proba/theories/idxval/irrel_equiv.v</a></td>
<td>
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_eq_ex_Ex_supp">
irrel_coupling_eq_ex_Ex_supp</a>,
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_eq_Ex_min_supp">
irrel_coupling_eq_Ex_min_supp</a>
<a href="discprob.idxval.irrel_equiv.html#irrel_coupling_eq_Ex_max_supp">
irrel_coupling_eq_Ex_max_supp</a>,
</td>
<td>
</td>
</tr>
<tr>
<td>Language Per-Thread Head Step Reduction</td>
<td>Section 3.1, Fig. 8</td>
<td> <a href="iris.heap_lang.lang.html">theories/heap_lang/lang.v</a></td>
<td>
<a href="iris.heap_lang.lang.html#heap_lang.head_step">head_step</a>,
<a href="iris.heap_lang.lang.html#heap_lang.head_step_prob">head_step_prob</a>,
<a href="iris.heap_lang.lang.html#heap_lang.head_step_prob_rel">head_step_prob_rel</a>
</td>
<td>
The paper presents the semantics as a relation indexed by a
probability. For backwards compatibility reasons with Iris, we found it
easier to instead work with a plain reduction relation
(<a href="iris.heap_lang.lang.html#heap_lang.head_step">head_step</a>) and
a function which assigns probabilities to a reduction
(<a href="iris.heap_lang.lang.html#heap_lang.head_step_prob">head_step_prob</a>). However,
we also prove that this is equivalent to a version where the relation is
directly annotated with the probability
(<a href="iris.heap_lang.lang.html#heap_lang.head_step_prob_rel">head_step_prob_rel</a>).
(see the equivalence proofs <a href="iris.heap_lang.lang.html#heap_lang.head_step_prob_rel_to_fun">head_step_prob_rel_to_fun</a> and <a href="iris.heap_lang.lang.html#heap_lang.head_step_prob_to_rel">head_step_prob_to_rel</a>) <br /> <br />
The other main difference is that the paper just describes a single
primitive called "flip" for simulating Bernoulli random variables. To make
it easier to add other primitives to the language, we allow various probabilistic operations (<a href="iris.heap_lang.lang.html#heap_lang.prob_bin_op">prob_bin_op</a>) to be added (which are supposed to represent different distributions, possibly taking some parameters), and then the semantics of these operations are specified by giving an indexed valuation that they represent (<a href="iris.heap_lang.lang.html#heap_lang.prob_bin_op_eval">prob_bin_op_eval</a>).
</td>
</tr>
<tr>
<td>Concurrent Thread Reduction</td>
<td>Section 3.1, Fig. 8</td>
<td> <a href="iris.program_logic.language.html">theories/program_logic/language.v</a><br />
<a href="iris.program_logic.prob_language.html">theories/program_logic/prob_language.v</a></td>
<td>
<a href="iris.program_logic.language.html#istep">istep</a>,
<a href="iris.program_logic.prob_language.html#trace_step">trace_step</a>,
</td>
<td>
These are for a "generic" language satisfying certain axioms, of which the example language we give is one of them (as is done in regular Iris). Again, the paper presents them annotated with a probability, but it is just easier in Coq to give the non-annotated version, and then we subsequently define the indexed valuation representing the probabilistic transition system (see next entry).
</td>
</tr>
<tr>
<td>Probabilistic Thread Reduction as an Indexed Valuation</td>
<td>Section 3.1</td>
<td>
<a href="iris.program_logic.prob_language.html">theories/program_logic/prob_language.v</a></td>
<td>
<a href="iris.program_logic.prob_language.html#ivdist_trace_stepN_aux">ivdist_trace_stepN_aux</a>,
<a href="iris.program_logic.prob_language.html#ivdist_tpool_stepN">ivdist_tpool_stepN</a>,
</td>
<td> Note that it returns the whole threadpool and state rather than just the first thread. The version written on paper would have to be partial unless we used dependent types to ensure the number of threads never goes to 0. It seemed useful to be able to possibly make claims about probabilistic aspects of state rather than just the first thread in the future.
</td>
</tr>
<tr>
<td>Definition of bounded termination under a scheduler</td>
<td>Section 3.1</td>
<td>
<a href="iris.program_logic.prob_language.html">theories/program_logic/prob_language.v</a>
</td>
<td>
<a href="iris.program_logic.prob_language.html#terminates">terminates</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Background on Iris</td>
<td>Section 3.2</td>
<td> (various) <br />
<a href="iris.heap_lang.lifting.html">theories/heap_lang/lifting.v</a> <br />
<a href="iris.program_logic.weakestpre.html">theories/program_logic/weakestpre.v</a> <br />
<a href="iris.algebra.frac_auth.html">theories/algebra/frac_auth.v</a></td>
<td>
(various)
</td>
<td>
Iris actually uses weakest preconditions rather than Hoare triples. The various rules shown here are simplified or derived forms of other rules from Iris. It would take a very long time to document the Iris development in this outline. Since the information here is about aspects of Iris that are preserved unchanged in our extensions, it seems less important to do so as part of this artifact guide.
</td>
</tr>
<tr>
<td>New Probabilistic Rules</td>
<td>Section 3.3, Fig. 12</td>
<td>
<a href="iris.heap_lang.lifting.html">theories/heap_lang/lifting.v</a><br />
<a href="iris.program_logic.prob_lifting.html">theories/program_logic/prob_lifting.v</a>
</td>
<td>
<a href="iris.heap_lang.lifting.html#wp_flip">wp_flip</a>,
<a href="iris.heap_lang.lifting.html#wp_irrel_flip">wp_irrel_flip</a>,
<a href="iris.program_logic.prob_lifting.html#ownProb_anti_irrel'">ownProb_anti_irrel'</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Theorem 3.1</td>
<td>Section 3.4</td>
<td>
<a href="iris.heap_lang.adequacy.html">theories/heap_lang/adequacy.v</a><br />
</td>
<td>
<a href="iris.heap_lang.adequacy.html#heap_prob_adequacy_Ex_max">heap_prob_adequacy_Ex_max</a>
<a href="iris.heap_lang.adequacy.html#heap_prob_adequacy_Ex_min">heap_prob_adequacy_Ex_min</a>
</td>
<td>
These are stated in terms of weakest preconditions instead of the Hoare triple formulation in the paper, in keeping with the normal approach in Iris. Also note that the function f has type Val -> R. By assumption, the program terminates in at most n steps, and the normal adequacy statement for Iris ensures that the first thread in the pool will therefore be a value, so this makes sense. However, to convince Coq of that, we would have to define a variant of ivdist_tpool_stepN that would ensure that the first thread in the pool is indeed a value under these additional assumptions. <br /><br />
Instead, we define a
function <a href="iris.heap_lang.adequacy.html#coerce_cfg">coerce_cfg</a>
which lifts a function from val -> R to a function from arbitrary program
configurations to R by applying an arbitrary default value in the case
that the the first thread in the pool is not a value. The adequacy
statements quantify over this d, showing that the bounds do not depend on
what we pick for this arbitrary value.
</td>
</tr>
<tr>
<td>Approximate counter algorithm</td>
<td>Section 1.1, Fig 1(c)</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html">theories/tests/approx_counter/faa_approx_counter.v</a>
</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html#newcounter">newcounter</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#incr">incr</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#read">read</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Approximate counter rules</td>
<td>Section 4.1, Fig. 13</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html">theories/tests/approx_counter/faa_approx_counter.v</a>
</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html#newcounter_spec'">newcounter_spec'</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#Acounter_sep">Acounter_sep</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#Acounter_join">Acounter_join</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#incr_spec'">incr_spec'</a>,
<a href="iris.tests.approx_counter.faa_approx_counter.html#read_spec'">read_spec'</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Example client for Approximate Counter</td>
<td>Section 4.1, Fig. 14</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html">theories/tests/approx_counter/faa_approx_counter.v</a>
</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter.html#list_client">list_client</a>
<a href="iris.tests.approx_counter.faa_approx_counter.html#list_client_Ex_ival">list_client_Ex_ival</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Variations of Approximate Counter Example</td>
<td>Section 4.3</td>
<td>
<a href="iris.tests.approx_counter.faa_approx_counter_nomax.html">theories/tests/approx_counter/faa_approx_counter_nomax.v</a>
<a href="iris.tests.approx_counter.faa_approx_counter_nomax.html">theories/tests/approx_counter/faa_approx_counter_arb.v</a>
</td>
<td>
(various)
</td>
<td>
</td>
</tr>
<tr>
<td>Skiplist Code</td>
<td>Section 5.1</td>
<td>
<a href="iris.tests.skiplist.code.html">theories/tests/skiplist/code.v</a>
</td>
<td>
(entire file)
</td>
<td>
</td>
</tr>
<tr>
<td>Skiplist Monadic Model</td>
<td>Section 5.2</td>
<td>
<a href="iris.tests.skiplist.idxval.html">theories/tests/skiplist/idxval.v</a>
</td>
<td>
<a href="iris.tests.skiplist.idxval.html#skiplist">skiplist</a>
</td>
<td>
Note that it is more complicated looking than the version in the paper because what is written in the paper is not structurally recursive, so we have to include additional information to convince Coq that it is terminating.
</td>
</tr>
<tr>
<td>Expected value for Skiplist Monadic Model</td>
<td>Section 5.2</td>
<td>
<a href="iris.tests.skiplist.idxval_expected.html">theories/tests/skiplist/idxval_expected.v</a>
</td>
<td>
<a href="iris.tests.skiplist.idxval_expected.html#Skiplist_Expected.skiplist_Ex_max">skiplist_Ex_max</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Rules for Skiplist</td>
<td>Section 5.3, Fig. 17</td>
<td>
<a href="iris.tests.skiplist.spec_bundled.html">theories/tests/skiplist/spec_bundled.v</a>
</td>
<td>
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.new_spec">new_spec</a>,
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.add_spec">add_spec</a>,
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.mem_full_spec">mem_full_spec</a>,
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.skip_prop_sep">skip_prop_sep</a>,
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.skip_prop_join">skip_prop_join</a>
<a href="iris.tests.skiplist.spec_bundled.html#Skiplist_Spec.skip_prob_spec'">skip_prob_spec'</a>
</td>
<td>
</td>
</tr>
<tr>
<td>Example client for Skiplist</td>
<td>Section 5.3, mentioned but not shown</td>
<td>
<a href="iris.tests.skiplist.client.html">theories/tests/skiplist/client.v</a>
</td>
<td>
<a href="iris.tests.skiplist.client.html#skip_list_client">skip_list_client</a>,
<a href="iris.tests.skiplist.client.html#list_client_Ex_ival">list_client_Ex_ival</a>,
</td>
<td>
</td>
</tr>
</table>
</div>
</div>
</div>
</div>
</body>
</html>