-
Notifications
You must be signed in to change notification settings - Fork 0
/
milMaxExeTraceExampleSpectreOOOScript.sml
791 lines (757 loc) · 44.4 KB
/
milMaxExeTraceExampleSpectreOOOScript.sml
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
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
open HolKernel boolLib Parse bossLib wordsLib wordsTheory finite_mapTheory pred_setTheory relationTheory listTheory rich_listTheory pairTheory milUtilityTheory milTheory milSemanticsUtilityTheory milMetaTheory milTracesTheory milMetaIOTheory milExecutableUtilityTheory milExecutableTransitionTheory milExecutableIOTheory milExecutableIOTraceTheory milExecutableIOCompletenessTheory milExecutableExamplesTheory milNoninterferenceTheory milMaxExeTraceUtilityTheory;
(* ================================================== *)
(* Maximal execution and trace of example spectreOoO *)
(* ================================================== *)
val _ = new_theory "milMaxExeTraceExampleSpectreOOO";
(* example Spectre OoO after initialization *)
Definition example_spectre_OoO_exe_init:
example_spectre_OoO_exe_init a b c d r1 r2 z b1 b2 t =
[i_assign (t+1) (e_val 1w) (o_internal (e_val b1));
i_assign (t+2) (e_val 1w) (o_internal (e_val a));
i_assign (t+3) (e_val 1w) (o_store res_MEM (t+1) (t+2));
i_assign (t+4) (e_val 1w) (o_internal (e_val z));
i_assign (t+5) (e_val 1w) (o_internal (e_val b));
i_assign (t+6) (e_val 1w) (o_store res_REG (t+4) (t+5));
i_assign (t+7) (e_val 1w) (o_internal (e_val r2));
i_assign (t+8) (e_val 1w) (o_internal (e_val c));
i_assign (t+9) (e_val 1w) (o_store res_REG (t+7) (t+8));
i_assign (t+10) (e_val 1w) (o_internal (e_val 0w));
i_assign (t+11) (e_val 1w) (o_internal (e_val d));
i_assign (t+12) (e_val 1w) (o_store res_PC (t+10) (t+11));
i_assign (t+13) (e_val 1w) (o_internal (e_val 0w));
i_assign (t+14) (e_val 1w) (o_internal (e_val r1));
i_assign (t+15) (e_val 1w) (o_internal (e_val r2));
i_assign (t+16) (e_val 1w) (o_internal (e_val z));
i_assign (t+17) (e_val 1w) (o_internal (e_val b1));
i_assign (t+18) (e_val 1w) (o_internal (e_val b2));
i_assign (t+19) (e_val 1w) (o_load res_MEM (t+17));
i_assign (t+20) (e_val 1w) (o_store res_REG (t+14) (t+19));
i_assign (t+21) (e_val 1w) (o_load res_REG (t+16));
i_assign (t+22) (e_eq (e_name (t+21)) (e_val 1w)) (o_load res_REG (t+14));
i_assign (t+23) (e_eq (e_name (t+21)) (e_val 1w)) (o_store res_REG (t+15) (t+22));
i_assign (t+24) (e_val 1w) (o_load res_REG (t+15));
i_assign (t+25) (e_val 1w) (o_store res_MEM (t+18) (t+24));
i_assign (t+26) (e_val 1w) (o_load res_PC (t+13));
i_assign (t+27) (e_val 1w) (o_internal (e_add (e_name (t+26)) (e_val 4w)));
i_assign (t+28) (e_val 1w) (o_store res_PC (t+13) (t+27))]
End
Definition example_spectre_OoO_list:
example_spectre_OoO_list a b c d r1 r2 z b1 b2 = example_spectre_OoO_exe_init a b c d r1 r2 z b1 b2 0
End
Definition example_spectre_OoO_store:
example_spectre_OoO_store a b c d r2 z b1 : num |-> v =
(FEMPTY |+ (1,b1) |+ (2,a) |+ (3,a) |+ (4,z) |+ (5,b) |+ (6,b) |+
(7,r2) |+ (8,c) |+ (9,c) |+ (10,0w) |+ (11,d) |+ (12,d))
End
Definition example_spectre_OoO_state_st_list:
example_spectre_OoO_state_st_list a b c d r1 r2 z b1 b2 =
State_st_list (example_spectre_OoO_list a b c d r1 r2 z b1 b2)
(example_spectre_OoO_store a b c d r2 z b1) [3] [12]
End
Definition example_spectre_OoO_state_st_list_final:
example_spectre_OoO_state_st_list_final a b c d b1 b2 =
State_st_list
[i_assign 1 (e_val 1w) (o_internal (e_val b1));
i_assign 2 (e_val 1w) (o_internal (e_val a));
i_assign 3 (e_val 1w) (o_store res_MEM 1 2);
i_assign 4 (e_val 1w) (o_internal (e_val 3w));
i_assign 5 (e_val 1w) (o_internal (e_val b));
i_assign 6 (e_val 1w) (o_store res_REG 4 5);
i_assign 7 (e_val 1w) (o_internal (e_val 2w));
i_assign 8 (e_val 1w) (o_internal (e_val c));
i_assign 9 (e_val 1w) (o_store res_REG 7 8);
i_assign 10 (e_val 1w) (o_internal (e_val 0w));
i_assign 11 (e_val 1w) (o_internal (e_val d));
i_assign 12 (e_val 1w) (o_store res_PC 10 11);
i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]
(FEMPTY |+ (1,b1) |+ (2,a) |+ (3,a) |+ (4,3w) |+ (5,b) |+
(6,b) |+ (7,2w) |+ (8,c) |+ (9,c) |+ (10,0w) |+ (11,d) |+
(12,d) |+ (13,0w) |+ (14,1w) |+ (15,2w) |+ (16,3w) |+ (17,b1) |+
(18,b2) |+ (19,a) |+ (20,a) |+ (21,b) |+ (24,c) |+ (25,c) |+
(26,d) |+ (27,d + 4w) |+ (28,d + 4w)) [25; 3] [28; 12]
End
(* Use 1 - 5 for r1, r2, z, a, b to execute
* a, b, c and d are initial values for b1, z, r2, PC respectively.
*)
Definition example_spectre_OoO_maximal_execution:
example_spectre_OoO_maximal_execution a b c d b1 b2 =
IO_bounded_execution (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 13 16
End
Definition example_spectre_OoO_maximal_trace_aux:
example_spectre_OoO_maximal_trace_aux a b c d b1 b2 =
IO_bounded_trace_aux (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 13 16
End
Definition example_spectre_OoO_maximal_trace:
example_spectre_OoO_maximal_trace a b c d b1 b2 =
IO_bounded_trace (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 13 16
End
Theorem IO_bounded_trace_last_example_spectre_OoO[local]:
translate_val_list = (\v t. []) ==>
sem_expr = sem_expr_exe ==>
!f_tran stl' tr a b c d b1 b2.
IO_bounded_trace_acc (\v t. []) sem_expr_exe (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [] = SOME (stl',tr) ==>
SND (SND (LAST (THE (IO_bounded_execution_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [])))) = stl'
Proof
rw [example_spectre_OoO_state_st_list] >>
`DROP 12 (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2) =
[i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]`
by (fs [example_spectre_OoO_list,example_spectre_OoO_exe_init] >> EVAL_TAC) >>
Q.ABBREV_TAC `i = i_assign 13 (e_val 1w) (o_internal (e_val 0w))` >>
Q.ABBREV_TAC `il = [i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]` >>
`~(Completed_list sem_expr (State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) i)`
by (fs [example_spectre_OoO_list,example_spectre_OoO_exe_init,example_spectre_OoO_store,Abbr `i`] >> EVAL_TAC) >>
`?x. IO_bounded_execution_acc (\v t. []) sem_expr_exe
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) 12 16 [] = SOME x`
by METIS_TAC [IO_bounded_trace_execution_acc_SOME] >> fs [] >>
`IO_bounded_execution_acc translate_val_list sem_expr
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) 12 (SUC 15) [] = SOME x` by fs [] >>
`IO_bounded_trace_acc translate_val_list sem_expr
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) 12 (SUC 15) [] = SOME (stl',tr)` by fs [] >>
Q.ABBREV_TAC `l = example_spectre_OoO_list a b c d 1w 2w 3w b1 b2` >>
Q.ABBREV_TAC `s = example_spectre_OoO_store a b c d 2w 3w b1` >>
PROVE_TAC [IO_bounded_trace_acc_empty_LAST]
QED
Theorem example_spectre_OoO_IO_bounded_trace_acc_not_true[local]:
!a b c d b1 b2.
b <> 1w ==>
IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_dl b1; obs_ds b2; obs_il (d + 4w)])
Proof
rw [example_spectre_OoO_state_st_list,Once IO_bounded_trace_acc,example_spectre_OoO_list,
example_spectre_OoO_exe_init,example_spectre_OoO_store,
FLOOKUP_DEF,OoO_Ftc_list_stored_incomplete] >>
NTAC 6 (fs [Once IO_bounded_trace_acc,FLOOKUP_DEF,sem_expr_exe,val_false,sem_instr_exe,sem_expr_exe,
OoO_Exe_list_instr_not_stored_guard_true_sem_instr,obs_of_ll,obs_visible]) >>
fs [Once IO_bounded_trace_acc] >>
fs [Once FLOOKUP_DEF,sem_expr_exe,val_false,sem_instr_exe,sem_expr_exe] >>
qmatch_goalsub_abbrev_tac `State_st_list instr_list store _ _` >>
`str_act_list sem_expr_exe (State_st_list instr_list store [3] [12]) 19 =
[i_assign 3 (e_val 1w) (o_store res_MEM 1 2)]` by (fs [Abbr `instr_list`,Abbr `store`] >> EVAL_TAC) >>
fs [bound_names_program_list,bound_name_instr,FLOOKUP_DEF,
OoO_Exe_list_instr_not_stored_guard_true_sem_instr,obs_of_ll,obs_visible,Abbr `store`] >>
qmatch_goalsub_abbrev_tac `store ' 3` >>
`store ' 3 = a` by (fs [Abbr `store`] >> EVAL_TAC) >>
`store ' 17 = b1` by (fs [Abbr `store`] >> EVAL_TAC) >> fs [Abbr `instr_list`,Abbr `store`] >>
fs [Once IO_bounded_trace_acc] >>
fs [FLOOKUP_DEF,sem_expr_exe,val_false,sem_instr_exe,sem_expr_exe,
OoO_Exe_list_instr_not_stored_guard_true_sem_instr,obs_of_ll,obs_visible] >>
fs [Once IO_bounded_trace_acc] >>
fs [Once FLOOKUP_DEF,sem_expr_exe,val_false,sem_instr_exe,sem_expr_exe] >>
qmatch_goalsub_abbrev_tac `State_st_list instr_list store _ _` >>
`str_act_list sem_expr_exe (State_st_list instr_list store [3] [12]) 21 =
[i_assign 6 (e_val 1w) (o_store res_REG 4 5)]` by (fs [Abbr `instr_list`,Abbr `store`] >> EVAL_TAC) >>
fs [bound_names_program_list,bound_name_instr,Once FLOOKUP_DEF,Abbr `store`] >>
fs [Once FLOOKUP_DEF,OoO_Exe_list_instr_not_stored_guard_true_sem_instr,
obs_of_ll,obs_visible,Abbr `instr_list`] >>
fs [Once IO_bounded_trace_acc] >>
fs [FLOOKUP_DEF,sem_expr_exe,v_eq,val_false] >>
qmatch_goalsub_abbrev_tac `store ' 6` >>
`store ' 6 = b` by (fs [Abbr `store`] >> EVAL_TAC) >> fs [Abbr `store`] >>
NTAC 3 (rw [Once IO_bounded_trace_acc] >> fs [FLOOKUP_DEF,sem_expr_exe,v_eq,val_false]) >>
rw [Once sem_instr_exe] >>
fs [str_act_list,addr_of_list,FIND_instr,bound_name_instr,str_may_list] >>
qmatch_goalsub_abbrev_tac `State_st_list instr_list store _ _` >>
`str_may_list_find sem_expr_exe instr_list store 24 res_REG 15 = [i_assign 9 (e_val 1w) (o_store res_REG 7 8)]`
by (fs [Abbr `instr_list`,Abbr `store`] >>EVAL_TAC >> fs []) >>
fs [str_act_list_find,str_act_list_cond,sem_expr_exe,Abbr `instr_list`,Abbr `store`] >>
NTAC 3 (rw [Once FLOOKUP_DEF]) >>
fs [bound_names_program_list] >>
rw [Once FLOOKUP_DEF] >>
fs [bound_name_instr,FLOOKUP_DEF,OoO_Exe_list_instr_not_stored_guard_true_sem_instr,
obs_visible,obs_of_ll] >>
EVAL_TAC >>
Cases_on `b1 = b2` >> fs [] >> EVAL_TAC
QED
Theorem example_spectre_OoO_maximal_execution_last_state_all_completed[local]:
translate_val_list = (\v t. []) ==>
sem_expr = sem_expr_exe ==>
!a b c d b1 b2.
all_instructions_completed_list sem_expr_exe
(SND (SND (LAST (THE (example_spectre_OoO_maximal_execution a b c d b1 b2)))))
Proof
rw [] >>
Cases_on `b = 1w` >-
(rw [] >> EVAL_TAC >> Cases_on `b1 <> b2` >> fs [] >> EVAL_TAC >> rw [] >> EVAL_TAC) >>
rw [example_spectre_OoO_maximal_execution,IO_bounded_execution] >>
`IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_dl b1; obs_ds b2; obs_il (d + 4w)])`
by fs [example_spectre_OoO_IO_bounded_trace_acc_not_true] >>
`SND (SND (LAST (THE (IO_bounded_execution_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [])))) =
example_spectre_OoO_state_st_list_final a b c d b1 b2`
by METIS_TAC [IO_bounded_trace_last_example_spectre_OoO] >>
rw [example_spectre_OoO_state_st_list_final] >> EVAL_TAC >> rw [] >> EVAL_TAC >> fs []
QED
Theorem State_st_list_well_formed_ok_example_spectre_OoO_state_st_list:
sem_expr_exe = sem_expr ==>
!a b c d b1 b2.
State_st_list_well_formed_ok (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)
Proof
rw [State_st_list_well_formed_ok,example_spectre_OoO_state_st_list,
example_spectre_OoO_list,example_spectre_OoO_exe_init,example_spectre_OoO_store] >-
EVAL_TAC >>
rw [state_list_to_state] >>
SIMP_TAC std_ss [well_formed_state] >>
REPEAT STRIP_TAC >|[
EVAL_TAC,
EVAL_TAC,
rw [bound_names_program] >> METIS_TAC [bound_name_instr],
fs [bound_name_instr] >> rw [] >>
fs [free_names_instr,names_e,names_o],
fs [] >> rw [] >> fs [bound_name_instr],
fs [] >> rw [] >>
fs [free_names_instr,names_e,names_o] >> METIS_TAC [bound_name_instr],
fs [],
fs [],
fs [map_down,FLOOKUP_DEF],
fs [map_down,FLOOKUP_DEF] >> rw [] >> fs [] >> EVAL_TAC,
fs [map_down,FLOOKUP_DEF] >> rw [] >> fs [sem_expr_correct,val_false],
fs [val_true,val_zero],
fs [val_true] >> rw [] >> fs [names_e],
fs [] >> rw [] >> fs [names_o,names_e,sem_expr_correct,val_false],
fs [sem_instr,sem_expr_correct,FLOOKUP_DEF] >> rw [] >> fs [] >> EVAL_TAC,
fs [val_true,val_zero],
(* sem_expr = sem_expr_exe needed here *)
`NO_DUPLICATES (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)`
by (fs [example_spectre_OoO_state_st_list] >> EVAL_TAC) >>
`State_st_list_ok (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)`
by fs [example_spectre_OoO_state_st_list,State_st_list_ok] >>
`str_may_list sem_expr_exe (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 3 = []` by EVAL_TAC >>
`str_may (state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)) 3 =
LIST_TO_SET (str_may_list sem_expr (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 3)`
by METIS_TAC [str_may_list_correct] >>
`str_may (state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)) 3 = {}`
by METIS_TAC [LIST_TO_SET] >>
fs [state_list_to_state,example_spectre_OoO_state_st_list,
example_spectre_OoO_list,example_spectre_OoO_exe_init,
example_spectre_OoO_store,bound_names_program],
`NO_DUPLICATES (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)`
by (fs [example_spectre_OoO_state_st_list] >> EVAL_TAC) >>
`State_st_list_ok (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)`
by fs [example_spectre_OoO_state_st_list,State_st_list_ok] >>
`str_may_list sem_expr_exe (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 = []` by EVAL_TAC >>
`str_may (state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)) 12 =
LIST_TO_SET (str_may_list sem_expr (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12)`
by METIS_TAC [str_may_list_correct] >>
`str_may (state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)) 12 = {}`
by METIS_TAC [LIST_TO_SET] >>
fs [state_list_to_state,example_spectre_OoO_state_st_list,
example_spectre_OoO_list,example_spectre_OoO_exe_init,
example_spectre_OoO_store,bound_names_program]
]
QED
Theorem HD_MAP_step_list_to_step_and_state_list_to_state[local]:
!pi stl.
pi <> [] ==>
FST (HD pi) = stl ==>
FST (HD (MAP step_list_to_step pi)) = state_list_to_state stl
Proof
Cases_on `pi` >> rw [] >>
Cases_on `h` >> Cases_on `r` >> rw [step_list_to_step]
QED
Theorem LAST_MAP_step_list_to_step_and_state_list_to_state[local]:
!pi.
pi <> [] ==>
SND (SND (LAST (MAP step_list_to_step pi))) = state_list_to_state (SND (SND (LAST pi)))
Proof
Cases_on `pi` >> rw [LAST_MAP] >>
Cases_on `LAST (h::t)` >> Cases_on `r` >> rw [step_list_to_step]
QED
Theorem example_spectre_OoO_maximal_execution_sound[local]:
sem_expr = sem_expr_exe ==>
translate_val_list = (\v t. []) ==>
!a b c d b1 b2 pi.
example_spectre_OoO_maximal_execution a b c d b1 b2 = SOME pi ==>
State_st_list_well_formed_ok (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) ==>
step_execution in_order_step (MAP step_list_to_step pi) /\
FST (HD pi) = example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2
Proof
rw [] >>
`translate_val_list_SORTED` by fs [translate_val_list_SORTED] >>
fs [example_spectre_OoO_maximal_execution,example_spectre_OoO_state_st_list] >>
`SORTED bound_name_instr_le (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)` by (fs [] >> EVAL_TAC) >>
`Completed_list_up_to sem_expr (State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) (PRE 13)`
by (fs [example_spectre_OoO_list,example_spectre_OoO_exe_init,example_spectre_OoO_store] >> EVAL_TAC >> rw [] >> EVAL_TAC) >>
`DROP (PRE 13) (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2) =
[i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]`
by (fs [example_spectre_OoO_list,example_spectre_OoO_exe_init] >> EVAL_TAC) >>
Q.ABBREV_TAC `i = i_assign 13 (e_val 1w) (o_internal (e_val 0w))` >>
Q.ABBREV_TAC `il = [i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]` >>
`~(Completed_list sem_expr (State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) i)`
by (fs [example_spectre_OoO_list,example_spectre_OoO_exe_init,example_spectre_OoO_store,Abbr `i`] >> EVAL_TAC) >>
`IO_bounded_execution translate_val_list sem_expr
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) 13 (SUC 15) = SOME pi` by fs [] >>
PROVE_TAC [IO_bounded_execution_in_order_step_sound]
QED
Theorem example_spectre_OoO_maximal_execution_correct[local]:
sem_expr = sem_expr_exe ==>
translate_val_list = (\v t. []) ==>
!a b c d b1 b2 pi.
example_spectre_OoO_maximal_execution a b c d b1 b2 = SOME pi ==>
pi <> [] ==>
maximal_execution_form in_order_step (state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2))
(MAP step_list_to_step pi)
Proof
rw [maximal_execution_form] >>
`State_st_list_well_formed_ok (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)`
by fs [State_st_list_well_formed_ok_example_spectre_OoO_state_st_list] >-
METIS_TAC [example_spectre_OoO_maximal_execution_sound] >-
(`FST (HD pi) = example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2`
by METIS_TAC [example_spectre_OoO_maximal_execution_sound] >>
fs [HD_MAP_step_list_to_step_and_state_list_to_state]) >>
`SND (SND (LAST (MAP step_list_to_step (THE (example_spectre_OoO_maximal_execution a b c d b1 b2))))) =
state_list_to_state (SND (SND (LAST (THE (example_spectre_OoO_maximal_execution a b c d b1 b2)))))`
by fs [LAST_MAP_step_list_to_step_and_state_list_to_state] >>
`all_instructions_completed_list sem_expr_exe
(SND (SND (LAST (THE (example_spectre_OoO_maximal_execution a b c d b1 b2)))))`
by fs [example_spectre_OoO_maximal_execution_last_state_all_completed] >>
`THE (example_spectre_OoO_maximal_execution a b c d b1 b2) = pi` by fs [] >> fs [] >>
`all_instructions_completed_list sem_expr (SND (SND (LAST pi)))` by fs [] >>
`?stl'.(SND (SND (LAST pi))) = stl'` by fs [] >>
`well_formed_state (state_list_to_state stl')`
by (fs [example_spectre_OoO_maximal_execution,example_spectre_OoO_state_st_list] >>
`DROP (PRE 13) (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2) =
[i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]`
by (fs [example_spectre_OoO_list] >> EVAL_TAC) >>
Q.ABBREV_TAC `i = i_assign 13 (e_val 1w) (o_internal (e_val 0w))` >>
Q.ABBREV_TAC `il = [i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]` >>
`~(Completed_list sem_expr (State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) i)`
by (fs [example_spectre_OoO_list,example_spectre_OoO_store,Abbr `i`] >> EVAL_TAC) >>
`IO_bounded_execution translate_val_list sem_expr
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) 13 (SUC 15) = SOME pi` by fs [] >>
PROVE_TAC [State_st_list_well_formed_ok_IO_bounded_execution_last_state]) >>
`SND (SND (step_list_to_step (LAST pi))) = SND (SND (LAST (MAP step_list_to_step pi)))` by METIS_TAC [LAST_MAP] >>
fs [all_instructions_completed_list_implies_termination_IO]
QED
Theorem example_spectre_OoO_maximal_trace_result[local]:
!a b c d b1 b2.
example_spectre_OoO_maximal_trace a b c d b1 b2 =
SOME [obs_dl b1; obs_ds b2; obs_il (d + 4w)]
Proof
rw [] >> Cases_on `b = 1w` >-
(rw [] >> EVAL_TAC >> Cases_on `b1 = b2` >> fs [] >> EVAL_TAC) >>
rw [example_spectre_OoO_maximal_trace,IO_bounded_trace,IO_bounded_trace_aux] >>
`IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,[obs_dl b1; obs_ds b2; obs_il (d + 4w)])`
by fs [example_spectre_OoO_IO_bounded_trace_acc_not_true] >>
fs [IO_bounded_trace_aux]
QED
Theorem example_spectre_OoO_maximal_execution_exists_result[local]:
!a b c d b1 b2.
?pi. example_spectre_OoO_maximal_execution a b c d b1 b2 = SOME pi /\ pi <> []
Proof
rw [] >>
`example_spectre_OoO_maximal_trace a b c d b1 b2 = SOME [obs_dl b1; obs_ds b2; obs_il (d + 4w)]`
by rw [example_spectre_OoO_maximal_trace_result] >>
fs [example_spectre_OoO_maximal_trace,example_spectre_OoO_maximal_execution] >>
`?pi. IO_bounded_execution (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 13 16 = SOME pi /\
[obs_dl b1; obs_ds b2; obs_il (d + 4w)] = trace obs_of_ll obs_visible pi`
by fs [IO_bounded_execution_for_trace] >>
Q.EXISTS_TAC `pi` >> fs [trace] >>
Cases_on `pi` >> fs []
QED
Theorem example_spectre_OoO_same_obs[local]:
!a a' b b' c c' d d' b1 b1' b2 b2'.
example_spectre_OoO_maximal_trace a b c d b1 b2 =
example_spectre_OoO_maximal_trace a' b' c' d' b1' b2' ==>
d = d' /\ b1 = b1' /\ b2 = b2'
Proof
rw [] >> fs [example_spectre_OoO_maximal_trace_result]
QED
Theorem noninterference_example_spectre_OoO_trace:
sem_expr = sem_expr_exe ==>
translate_val_list = (\v t. []) ==>
!a a' b b' c c' d d' b1 b1' b2 b2' s1 s2.
state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) = s1 ==>
state_list_to_state (example_spectre_OoO_state_st_list a' b' c' d' 1w 2w 3w b1' b2') = s2 ==>
trace_indist_IO s1 s2 ==>
d = d' /\ b1 = b1' /\ b2 = b2'
Proof
REWRITE_TAC [trace_indist_IO,trace_indist] >>
NTAC 2 STRIP_TAC >>
REPEAT GEN_TAC >> NTAC 3 STRIP_TAC >> fs [] >>
`?p1 p2. example_spectre_OoO_maximal_execution a b c d b1 b2 = SOME p1 /\
example_spectre_OoO_maximal_execution a' b' c' d' b1' b2' = SOME p2 /\
p1 <> [] /\ p2 <> []` by METIS_TAC [example_spectre_OoO_maximal_execution_exists_result] >>
`maximal_execution_form in_order_step (state_list_to_state
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)) (MAP step_list_to_step p1) /\
maximal_execution_form in_order_step (state_list_to_state
(example_spectre_OoO_state_st_list a' b' c' d' 1w 2w 3w b1' b2')) (MAP step_list_to_step p2)`
by fs [example_spectre_OoO_maximal_execution_correct] >>
Q.ABBREV_TAC `stl1 = example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2` >>
Q.ABBREV_TAC `stl2 = example_spectre_OoO_state_st_list a' b' c' d' 1w 2w 3w b1' b2'` >>
`State_st_list_well_formed_ok stl1 /\ State_st_list_well_formed_ok stl2`
by fs [Abbr `stl1`,Abbr `stl2`,State_st_list_well_formed_ok_example_spectre_OoO_state_st_list] >>
`well_formed_state (state_list_to_state stl1) /\ well_formed_state (state_list_to_state stl2)`
by (Cases_on `stl1` >> Cases_on `stl2` >> fs [State_st_list_well_formed_ok]) >>
`well_formed_state s1 /\ well_formed_state s2` by METIS_TAC [] >>
Q.ABBREV_TAC `max_org_pi1 = MAP step_list_to_step p1` >>
Q.ABBREV_TAC `max_org_pi2 = MAP step_list_to_step p2` >>
`maximal_execution_form in_order_step s1 max_org_pi1 /\
maximal_execution_form in_order_step s2 max_org_pi2` by METIS_TAC [] >>
`maximal_execution_form in_order_step' s1 (MAP update_lbl max_org_pi1) /\
maximal_execution_form in_order_step' s2 (MAP update_lbl max_org_pi2)`
by fs [max_execution_format_step_and_step'] >>
Q.ABBREV_TAC `max_pi1 = MAP update_lbl max_org_pi1` >>
Q.ABBREV_TAC `max_pi2 = MAP update_lbl max_org_pi2` >>
`trace obs_of_l obs_visible max_pi1 = trace obs_of_l obs_visible max_pi2`
by REWRITE_TAC [UNDISCH_ALL (SPEC_ALL noninterference_implies_maximal_execution_form_in_order_step')] >>
`trace obs_of_l obs_visible max_org_pi1 = trace obs_of_l obs_visible max_org_pi2` by METIS_TAC [MAP_update_lbl_same_obs,Abbr `max_pi1`,Abbr `max_pi2`] >>
`example_spectre_OoO_maximal_trace a b c d b1 b2 <> NONE /\
example_spectre_OoO_maximal_trace a' b' c' d' b1' b2' <> NONE`
by fs [example_spectre_OoO_maximal_trace_result] >>
`?tr1 tr2. example_spectre_OoO_maximal_trace a b c d b1 b2 = SOME tr1 /\
example_spectre_OoO_maximal_trace a' b' c' d' b1' b2' = SOME tr2`
by (Cases_on `example_spectre_OoO_maximal_trace a b c d b1 b2` >>
Cases_on `example_spectre_OoO_maximal_trace a' b' c' d' b1' b2'` >> fs []) >>
`(trace obs_of_l obs_visible max_org_pi1 = tr1) /\ (trace obs_of_l obs_visible max_org_pi2 = tr2)`
by METIS_TAC [example_spectre_OoO_maximal_execution,example_spectre_OoO_maximal_trace,
IO_bounded_execution_and_trace,Abbr `max_org_pi1`,Abbr `max_org_pi2`] >>
`example_spectre_OoO_maximal_trace a b c d b1 b2 = example_spectre_OoO_maximal_trace a' b' c' d' b1' b2'` by fs [] >>
METIS_TAC [example_spectre_OoO_same_obs]
QED
(* Unused definitions and theorems of example SpectreOoO
Definition example_spectre_OoO_state_st_list_mid:
example_spectre_OoO_state_st_list_mid a b c d b1 b2 =
State_st_list
[i_assign 1 (e_val 1w) (o_internal (e_val b1));
i_assign 2 (e_val 1w) (o_internal (e_val a));
i_assign 3 (e_val 1w) (o_store res_MEM 1 2);
i_assign 4 (e_val 1w) (o_internal (e_val 3w));
i_assign 5 (e_val 1w) (o_internal (e_val b));
i_assign 6 (e_val 1w) (o_store res_REG 4 5);
i_assign 7 (e_val 1w) (o_internal (e_val 2w));
i_assign 8 (e_val 1w) (o_internal (e_val c));
i_assign 9 (e_val 1w) (o_store res_REG 7 8);
i_assign 10 (e_val 1w) (o_internal (e_val 0w));
i_assign 11 (e_val 1w) (o_internal (e_val d));
i_assign 12 (e_val 1w) (o_store res_PC 10 11);
i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]
(FEMPTY |+ (1,b1) |+ (2,a) |+ (3,a) |+ (4,3w) |+ (5,b) |+ (6,b) |+
(7,2w) |+ (8,c) |+ (9,c) |+ (10,0w) |+ (11,d) |+ (12,d) |+ (13,0w) |+
(14,1w) |+ (15,2w) |+ (16,3w) |+ (17,b1) |+ (18,b2) |+ (19,a) |+ (20,a) |+ (21,b)) [3] [12]
End
Theorem IO_bounded_execution_acc_complete_steps[local]:
translate_val_list_SORTED ==>
!l s cs fs l' s' cs' fs' pi n pos.
State_st_list_well_formed_initialized_ok (State_st_list l s cs fs) ==>
SORTED bound_name_instr_le l ==>
Completed_list_up_to sem_expr (State_st_list l s cs fs) pos ==>
(!t. MEM t (bound_names_program_list l) /\ t > pos ==> t NOTIN FDOM s) ==>
FST (HD pi) = State_st_list l s cs fs ==>
step_execution in_order_step (MAP step_list_to_step pi) ==>
SND (SND (LAST pi)) = State_st_list l' s' cs' fs' ==>
Completed_list_up_to sem_expr (State_st_list l' s' cs' fs') (pos + n) ==>
(!t. MEM t (bound_names_program_list l') /\ t > (pos+n) ==> t NOTIN FDOM s') ==>
IO_bounded_execution_acc translate_val_list sem_expr (State_st_list l s cs fs) pos n [] = SOME pi
Proof
cheat
QED
Theorem APPEND_HD_FST[local]:
!l1 l2 l.
l = l1 ++ l2 ==> l1 <> [] ==> HD l = HD l1
Proof
rw [] >> Cases_on `l1` >> fs []
QED
Theorem IO_bounded_execution_acc_subresults[local]:
translate_val_list_SORTED ==>
!l l' l'' s s' s'' cs cs' cs'' fs fs' fs'' stl'' pos m n pi pi' i il.
State_st_list_well_formed_initialized_ok (State_st_list l s cs fs) ==>
State_st_list_well_formed_ok (State_st_list l' s' cs' fs') ==>
(!t. MEM t (bound_names_program_list l) /\ t > pos ==> t NOTIN FDOM s) ==>
(!t. MEM t (bound_names_program_list l'') /\ t > (pos+m+n) ==> t NOTIN FDOM s'') ==>
SORTED bound_name_instr_le l ==>
SORTED bound_name_instr_le l' ==>
Completed_list_up_to sem_expr (State_st_list l s cs fs) pos ==>
DROP pos l = i::il ==>
~Completed_list sem_expr (State_st_list l s cs fs) i ==>
pi <> [] ==>
IO_bounded_execution_acc translate_val_list sem_expr (State_st_list l s cs fs) pos m [] = SOME pi ==>
SND (SND (LAST pi)) = State_st_list l' s' cs' fs' ==>
SND (SND (LAST pi')) = State_st_list l'' s'' cs'' fs'' ==>
IO_bounded_execution_acc translate_val_list sem_expr (State_st_list l' s' cs' fs') (pos+m) n pi = SOME pi' ==>
IO_bounded_execution_acc translate_val_list sem_expr (State_st_list l s cs fs) pos (m+n) [] = SOME pi'
Proof
rw [] >>
Cases_on `m` >-
fs [Once IO_bounded_execution_acc] >>
`State_st_list_well_formed_ok (State_st_list l s cs fs)` by fs [State_st_list_well_formed_ok,State_st_list_well_formed_initialized_ok] >>
`step_execution_list_IO_HD (State_st_list l s cs fs) pi pos (SUC n')` by fs [IO_bounded_execution_acc_IO_empty_sound] >>
fs [step_execution_list_IO_HD] >>
`Completed_list_up_to sem_expr (State_st_list l' s' cs' fs') (pos + SUC n')` by METIS_TAC [] >>
`?pi''. pi' = pi ++ pi'' /\ step_execution in_order_step (MAP step_list_to_step pi') /\
Completed_list_up_to sem_expr (SND (SND (LAST pi'))) (pos + SUC n' + n) /\
LENGTH pi'' <= 2 * n` by fs [IO_bounded_execution_acc_IO_execution_sound] >>
`HD pi' = HD pi` by fs [APPEND_HD_FST] >>
`FST (HD pi') = State_st_list l s cs fs` by fs [] >>
`pos + SUC n' + n = pos + (n + SUC n') /\ n + (pos + SUC n') = pos + (n + SUC n')` by rw [] >>
`Completed_list_up_to sem_expr (SND (SND (LAST pi'))) (pos + (n + SUC n'))` by METIS_TAC [] >>
`(!t. MEM t (bound_names_program_list l'') /\ t > (pos+(n + SUC n')) ==> t NOTIN FDOM s'')` by METIS_TAC [] >>
METIS_TAC [IO_bounded_execution_acc_complete_steps]
QED
Theorem IO_bounded_trace_acc_subresults[local]:
!f_tran f_sem l s s' cs cs' fs fs' stl'' pos m n tr tr'.
IO_bounded_trace_acc f_tran f_sem (State_st_list l s cs fs) pos m [] = SOME (State_st_list l s' cs' fs',tr) ==>
IO_bounded_trace_acc f_tran f_sem (State_st_list l s' cs' fs') (pos+m) n tr = SOME (stl'',tr') ==>
IO_bounded_trace_acc f_tran f_sem (State_st_list l s cs fs) pos (m+n) [] = SOME (stl'',tr')
Proof
cheat
QED
Theorem example_spectre_OoO_IO_bounded_trace_acc_not_true_part1[local]:
!a b c d b1 b2.
b <> 1w ==>
IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 11 10 [] =
SOME (example_spectre_OoO_state_st_list_mid a b c d b1 b2, [obs_il d; obs_dl b1])
Proof
rw [example_spectre_OoO_state_st_list_mid] >> EVAL_TAC
QED
Theorem example_spectre_OoO_IO_bounded_trace_acc_not_true_part2[local]:
!a b c d b1 b2.
b <> 1w ==>
IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list_mid a b c d b1 b2) 21 7 [obs_il d; obs_dl b1] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_il d; obs_dl b1; obs_ds b2; obs_il (d + 4w)])
Proof
rw [example_spectre_OoO_state_st_list_mid,example_spectre_OoO_state_st_list_final] >>
NTAC 3 (rw [Once IO_bounded_trace_acc] >> fs [FLOOKUP_DEF,sem_expr_exe,v_eq,val_false]) >>
rw [Once sem_instr_exe] >>
fs [str_act_list,addr_of_list,FIND_instr,bound_name_instr,str_may_list] >>
Q.ABBREV_TAC `l1 = [i_assign 1 (e_val 1w) (o_internal (e_val b1));
i_assign 2 (e_val 1w) (o_internal (e_val a));
i_assign 3 (e_val 1w) (o_store res_MEM 1 2);
i_assign 4 (e_val 1w) (o_internal (e_val 3w));
i_assign 5 (e_val 1w) (o_internal (e_val b));
i_assign 6 (e_val 1w) (o_store res_REG 4 5);
i_assign 7 (e_val 1w) (o_internal (e_val 2w));
i_assign 8 (e_val 1w) (o_internal (e_val c));
i_assign 9 (e_val 1w) (o_store res_REG 7 8);
i_assign 10 (e_val 1w) (o_internal (e_val 0w));
i_assign 11 (e_val 1w) (o_internal (e_val d));
i_assign 12 (e_val 1w) (o_store res_PC 10 11);
i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]` >>
Q.ABBREV_TAC `(s1:num |-> v) = (FEMPTY |+ (1,b1) |+ (2,a) |+ (3,a) |+ (4,3w) |+
(5,b) |+ (6,b) |+ (7,2w) |+ (8,c) |+ (9,c) |+
(10,0w) |+ (11,d) |+ (12,d) |+ (13,0w) |+ (14,1w) |+
(15,2w) |+ (16,3w) |+ (17,b1) |+ (18,b2) |+ (19,a) |+
(20,a) |+ (21,b))` >>
`str_may_list_find sem_expr_exe l1 s1 24 res_REG 15 = [i_assign 9 (e_val 1w) (o_store res_REG 7 8)]`
by (fs [Abbr `l1`,Abbr `s1`] >> EVAL_TAC >> fs []) >>
fs [str_act_list_find,str_act_list_cond,sem_expr_exe,Abbr `s1`] >>
NTAC 3 (rw [Once FLOOKUP_DEF]) >>
fs [bound_names_program_list] >>
rw [Once FLOOKUP_DEF] >>
fs [bound_name_instr,FLOOKUP_DEF,OoO_Exe_list_instr_not_stored_guard_true_sem_instr,
obs_visible,obs_of_ll] >>
fs [Abbr `l1`] >> EVAL_TAC >>
Cases_on `b1 = b2` >> fs [] >> EVAL_TAC
QED
Theorem example_spectre_OoO_IO_bounded_trace_acc_not_true[local]:
!a b c d b1 b2.
b <> 1w ==>
IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 11 17 [] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_il d; obs_dl b1; obs_ds b2; obs_il (d + 4w)])
Proof
rw [] >>
`IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 11 10 [] =
SOME (example_spectre_OoO_state_st_list_mid a b c d b1 b2, [obs_il d; obs_dl b1])`
by fs [example_spectre_OoO_IO_bounded_trace_acc_not_true_part1] >>
`IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list_mid a b c d b1 b2) (11+10) 7 [obs_il d; obs_dl b1] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_il d; obs_dl b1; obs_ds b2; obs_il (d + 4w)])`
by fs [example_spectre_OoO_IO_bounded_trace_acc_not_true_part2] >>
fs [example_spectre_OoO_state_st_list,example_spectre_OoO_state_st_list_mid] >>
`example_spectre_OoO_list a b c d 1w 2w 3w b1 b2 =
[i_assign 1 (e_val 1w) (o_internal (e_val b1));
i_assign 2 (e_val 1w) (o_internal (e_val a));
i_assign 3 (e_val 1w) (o_store res_MEM 1 2);
i_assign 4 (e_val 1w) (o_internal (e_val 3w));
i_assign 5 (e_val 1w) (o_internal (e_val b));
i_assign 6 (e_val 1w) (o_store res_REG 4 5);
i_assign 7 (e_val 1w) (o_internal (e_val 2w));
i_assign 8 (e_val 1w) (o_internal (e_val c));
i_assign 9 (e_val 1w) (o_store res_REG 7 8);
i_assign 10 (e_val 1w) (o_internal (e_val 0w));
i_assign 11 (e_val 1w) (o_internal (e_val d));
i_assign 12 (e_val 1w) (o_store res_PC 10 11);
i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]` by fs [example_spectre_OoO_list] >>
`IO_bounded_trace_acc (\v t. []) sem_expr_exe
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] []) 11 (10+7) [] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_il d; obs_dl b1; obs_ds b2; obs_il (d + 4w)])` by fs [IO_bounded_trace_acc_subresults] >>
FULL_SIMP_TAC arith_ss []
QED
*)
val _ = export_theory ();