-
Notifications
You must be signed in to change notification settings - Fork 96
Expand file tree
/
Copy pathcrep_inlineProofScript.sml
More file actions
4664 lines (4537 loc) · 179 KB
/
crep_inlineProofScript.sml
File metadata and controls
4664 lines (4537 loc) · 179 KB
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
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
(*
Correctness proof for function inlining pass
*)
Theory crep_inlineProof
Ancestors
crepLang crepSem crepProps crep_inline
pan_commonProps pan_common
prim_rec iterate
Libs
preamble
Definition state_rel_def:
state_rel s t ⇔
s.globals = t.globals ∧
s.code = t.code ∧
s.memory = t.memory ∧
s.memaddrs = t.memaddrs ∧
s.sh_memaddrs = t.sh_memaddrs ∧
s.clock = t.clock ∧
s.be = t.be ∧
s.ffi = t.ffi ∧
s.base_addr = t.base_addr ∧
s.top_addr = t.top_addr
End
Definition locals_rel_def:
locals_rel s t ⇔
s.locals SUBMAP t.locals
End
Definition locals_strong_rel_def:
locals_strong_rel s t ⇔
s.locals = t.locals
End
Theorem OPT_MMAP_SOME_ALL:
∀f l.
((∃x. OPT_MMAP f l = SOME x) ⇔ (∀x. MEM x l ⇒ ?y. f x = SOME y))
Proof
rpt strip_tac >>
Induct_on `l` >> simp[OPT_MMAP_def] >>
strip_tac >>
eq_tac >>
rpt strip_tac >> gvs[]
QED
Theorem OPT_MMAP_ALL_EQ:
∀f g l.
(∀x. MEM x l ==> f x = g x) ==> (OPT_MMAP f l = OPT_MMAP g l)
Proof
rpt strip_tac >>
Induct_on `l` >> gs[]
QED
Theorem eval_original_extend_locals:
∀s e wl l.
eval s e = SOME wl /\
s.locals SUBMAP l ⇒
eval (s with locals := l) e = SOME wl
Proof
recInduct eval_ind >>
rpt strip_tac >> gs[eval_def] >>
imp_res_tac SUBMAP_FLOOKUP_EQN >>
qpat_x_assum `_ = SOME wl` mp_tac
>>~- ([`OPT_MMAP`],
TOP_CASE_TAC >>
imp_res_tac $ iffLR OPT_MMAP_SOME_ALL >> gs[] >>
`!a. MEM a es ⇒ eval s a = eval (s with locals := l) a` by (
rpt strip_tac >> qpat_x_assum `!_. MEM _ _ ⇒ ?_. _` imp_res_tac >>
res_tac >> simp[]
) >>
drule OPT_MMAP_ALL_EQ >> gvs[] >>
disch_tac >>
`OPT_MMAP (\a. eval (s with locals := l) a) es = SOME x` by metis_tac[] >> gvs[]) >>
last_x_assum imp_res_tac >> fs[] >>
every_case_tac >> gs[mem_load_def]
QED
Theorem eval_original_extend_locals_rel:
∀s e wl t.
eval s e = SOME wl ==>
locals_rel s t ∧ state_rel s t ⇒
eval t e = SOME wl
Proof
simp[locals_rel_def, state_rel_def] >>
rpt strip_tac >>
drule eval_original_extend_locals >>
disch_then $ qspec_then `t.locals` assume_tac >> gs[] >>
`s with locals := t.locals = t` by simp[state_component_equality] >> gvs[]
QED
Theorem eval_state_locals_rel:
∀s e wl t.
eval s e = SOME wl ∧ state_rel s t ∧ locals_rel s t ⇒
eval t e = SOME wl
Proof
rpt strip_tac >>
irule eval_original_extend_locals_rel >>
qrefine `s` >> simp[]
QED
Theorem eval_optmmap_state_locals_rel:
∀s es ws t.
OPT_MMAP (eval s) es = SOME ws ∧ state_rel s t ∧ locals_rel s t ⇒
OPT_MMAP (eval t) es = SOME ws
Proof
gen_tac >> gen_tac >> qid_spec_tac `s` >>
Induct_on `es` >> gs[OPT_MMAP_def] >>
rpt strip_tac >>
`eval t h = SOME h'` by metis_tac[eval_state_locals_rel] >>
qrefine `h'` >> gs[] >>
last_x_assum $ qspecl_then [`s`, `t'`, `t`] assume_tac >> gs[]
QED
Theorem SUBMAP_IMP_FUPDATE_SUBMAP:
∀f g x y.
f SUBMAP g ⇒ f |+ (x, y) SUBMAP g |+ (x, y)
Proof
rpt strip_tac >>
gs[SUBMAP_DEF] >>
rpt strip_tac >>
gvs[FAPPLY_FUPDATE_THM]
QED
Theorem SUBMAP_IMP_DOMSUB_SUBMAP:
∀f g x.
f SUBMAP g ⇒ f \\ x SUBMAP g \\ x
Proof
rpt strip_tac >> gs[SUBMAP_DEF] >>
rpt strip_tac >> gvs[DOMSUB_FAPPLY_THM]
QED
Theorem SUBMAP_IMP_DOMSUB_FUPDATE:
∀f g x y.
f SUBMAP g ⇒ f \\ x SUBMAP g |+ (x, y)
Proof
rpt strip_tac >> gs[SUBMAP_DEF] >>
rpt strip_tac >> gvs[DOMSUB_FAPPLY_THM, FAPPLY_FUPDATE_THM]
QED
Theorem res_var_submap_res_var:
∀f g x y.
f SUBMAP g ⇒ res_var f (x,y) SUBMAP res_var g (x, y)
Proof
rpt strip_tac >>
Cases_on `y` >> gs[res_var_def] >> gs[SUBMAP_IMP_DOMSUB_SUBMAP, SUBMAP_IMP_FUPDATE_SUBMAP]
QED
Definition locals_ext_rel_def:
locals_ext_rel a b a' b' ⇔
FDIFF a'.locals (FDOM a.locals) = FDIFF b'.locals (FDOM b.locals)
End
Theorem locals_rel_dec_clock:
∀s t.
locals_rel s t ∧ state_rel s t ⇒
locals_rel (dec_clock s) (dec_clock t) ∧ state_rel (dec_clock s) (dec_clock t)
Proof
gvs[dec_clock_def, locals_rel_def, state_rel_def]
QED
Theorem evaluate_locals_same_fdom:
∀p s r s'.
evaluate (p, s) = (r, s') ∧
(case r of
| NONE => T
| SOME Continue => T
| SOME Break => T
| _ => F) = T ⇒
FDOM s.locals = FDOM s'.locals
Proof
recInduct evaluate_ind >> rpt conj_tac
>~ [`evaluate (While _ _, _) = _`]
>- (
rpt strip_tac >>
qpat_x_assum `evaluate _ = (r, s')` mp_tac >>
simp[Once evaluate_def] >>
rpt TOP_CASE_TAC
>- (
disch_tac >> fs[CaseEq "result"]
) >>
disch_tac >>
`(dec_clock s).locals = s.locals` by fs[dec_clock_def] >>
pairarg_tac >> gs[CaseEq "option", CaseEq "result"]
)
>~ [`evaluate (Dec _ _ _, _) = _`]
>- (
gs[evaluate_def, CaseEq "option", state_component_equality] >> rpt strip_tac >>
TRY (imp_res_tac EQ_FDOM_SUBMAP) >>
pairarg_tac >> gs[] >>
Cases_on `FLOOKUP s.locals v` >> gvs[res_var_def, flookup_thm] >>
fs[ABSORPTION_RWT] >>
qpat_x_assum `_ = FDOM st.locals` $ gs o single o GSYM >> fs[DELETE_INSERT, DELETE_NON_ELEMENT_RWT, ABSORPTION_RWT]
)
>~ [`evaluate (Seq _ _, _) = _`]
>- (
gs[evaluate_def] >> rpt strip_tac >>
pairarg_tac >> fs[] >>
Cases_on `res = NONE` >> fs[]
)
>~ [`evaluate (If _ _ _, _) = _`]
>- (
rpt strip_tac >>
fs[evaluate_def, CaseEq "option", CaseEq "word_lab"]
)
>~ [`evaluate (Call _ _ _, _) = _`]
>- (
rpt strip_tac >>
Cases_on `s.clock` >>
fs[evaluate_def, CaseEq "option", CaseEq "pair$prod", CaseEq "word_lab", CaseEq "result"] >> gvs[]
>>~- ([`FLOOKUP s.locals _ = SOME _`],
fs[flookup_thm] >>
qpat_x_assum `_ = FDOM s'.locals` $ rw o single o GSYM >> fs[ABSORPTION_RWT]) >>
Cases_on `eid = eid'` >> fs[]
) >>
rpt strip_tac >>
gvs[evaluate_def, CaseEq "option", CaseEq "word_lab", state_component_equality,
set_globals_def, CaseEq "ffi_result"
]
>- (
qpat_x_assum `_ = s'.locals` $ rw o single o GSYM >>
gvs[flookup_thm, ABSORPTION_RWT]
)
>>~- ([`sh_mem_op _ _ _ _`],
qpat_x_assum `_ = (_, s')` mp_tac >>
Cases_on `op` >>
fs[CaseEq "option", CaseEq "word_lab", sh_mem_op_def, sh_mem_load_def, sh_mem_store_def] >>
TRY (IF_CASES_TAC) >> fs[CaseEq "ffi_result", set_var_def] >>
TRY (TOP_CASE_TAC) >> fs[CaseEq "ffi_result"] >>
disch_tac >> gvs[flookup_thm, state_component_equality, CaseEq "result"] >>
qpat_x_assum `_ = s'.locals` $ rw o single o GSYM >> fs[ABSORPTION_RWT]
) >>
Cases_on `s.clock` >> gvs[dec_clock_def]
QED
Theorem evaluate_locals_same_fdom':
∀p s r s'.
evaluate (p, s) = (r, s') ∧
(r = NONE ∨ r = SOME Break ∨ r = SOME Continue) ⇒
FDOM s.locals = FDOM s'.locals
Proof
rpt strip_tac >>
drule evaluate_locals_same_fdom >> fs[]
QED
(* Need *)
Theorem evaluate_state_locals_rel_strong:
∀p s r s' t.
evaluate (p, s) = (r, s') ∧
r ≠ SOME Error ∧
locals_rel s t ∧ state_rel s t ⇒
∃r' t'.
evaluate (p, t) = (r, t') ∧ state_rel s' t' ∧
case r of
| NONE => r' = NONE ∧ locals_rel s' t' ∧ locals_ext_rel s s' t t'
| SOME Break => r' = SOME Break ∧ locals_rel s' t' ∧ locals_ext_rel s s' t t'
| SOME Continue => r' = SOME Continue ∧ locals_rel s' t' ∧ locals_ext_rel s s' t t'
| SOME (Return retv) => r' = SOME (Return retv)
| SOME (FinalFFI f) => r' = SOME (FinalFFI f)
| SOME TimeOut => r' = SOME TimeOut
| SOME (Exception e) => r' = SOME (Exception e)
| _ => F
Proof
recInduct evaluate_ind >>
rpt conj_tac
>~ [`evaluate (While _ _, _)`]
>- (
completeInduct_on `s.clock` >>
rpt strip_tac >>
qpat_x_assum `_ = (r, s')` mp_tac >>
PURE_ONCE_REWRITE_TAC[evaluate_def] >>
imp_res_tac eval_state_locals_rel >> fs[] >>
fs[CaseEq "option", CaseEq "word_lab"] >>
disch_tac >> fs[] >>
qpat_x_assum `!_ _. eval _ _ = _ ⇒ eval _ _ = _` imp_res_tac >> fs[] >>
Cases_on `w = 0w` >> fs[]
>- gs[locals_ext_rel_def] >>
`t.clock = s.clock` by fs[state_rel_def] >> fs[] >>
Cases_on `s.clock = 0` >> fs[]
>- gvs[empty_locals_def, state_rel_def] >>
pairarg_tac >> fs[] >>
pairarg_tac >> gvs[] >>
imp_res_tac locals_rel_dec_clock >>
qpat_x_assum `!_. _ < s.clock ⇒ _` $ qspec_then `s1'.clock` mp_tac >> impl_tac
>- (
drule evaluate_clock >> disch_tac >>
irule LET_TRANS >>
qrefine `(dec_clock s).clock` >> fs[dec_clock_def]
) >>
disch_then $ qspec_then `s1'` mp_tac >> fs[] >>
disch_tac >>
qpat_x_assum `!_. res' ≠ SOME Error ∧ _ ∧ _ ⇒ _` imp_res_tac >> fs[] >>
Cases_on `res' = SOME Error` >> fs[] >>
gs[CaseEq "option", CaseEq "result"] >>
TRY (
qpat_x_assum `!_. locals_rel s1' _ ∧ state_rel s1' _ ⇒ _` $ qspec_then `s1` mp_tac >> fs[] >>
disch_tac >> fs[] >>
qrefine `r''` >>
Cases_on `r` >> fs[]
) >>
TRY (
Cases_on `x` >> fs[]
) >>
`(dec_clock t).locals = t.locals` by fs[dec_clock_def] >>
`(dec_clock s).locals = s.locals` by fs[dec_clock_def] >>
fs[locals_ext_rel_def]
)
>~ [`evaluate (Dec _ _ _, _)`]
>- (
rpt strip_tac >> fs[evaluate_def] >>
imp_res_tac eval_state_locals_rel >>
gs[CaseEq "option", CaseEq "word_lab"] >>
first_x_assum imp_res_tac >> fs[] >>
pairarg_tac >> fs[] >>
pairarg_tac >> fs[] >>
qrefine `r` >> gvs[CaseEq "option" , CaseEq "result"] >>
last_x_assum $ qspec_then `t with locals := t.locals |+ (v, value)` mp_tac >> impl_tac
>- (
fs[state_rel_def, locals_rel_def] >>
imp_res_tac SUBMAP_IMP_FUPDATE_SUBMAP >>
pop_assum $ fs o single
) >>
disch_tac >> gvs[] >>
conj_tac
>- gs[state_rel_def] >>
Cases_on `r` >> TRY (Cases_on `x`) >> fs[] >>
fs[locals_rel_def, locals_ext_rel_def] >>
conj_tac
>>~- ([`res_var _ _ SUBMAP res_var _ _`],
Cases_on `FLOOKUP s.locals v` >> fs[res_var_def] >>
rev_drule $ iffLR SUBMAP_FLOOKUP_EQN >>
disch_tac >>
pop_assum imp_res_tac >> fs[res_var_def, SUBMAP_IMP_FUPDATE_SUBMAP] >>
Cases_on `FLOOKUP t.locals v` >> fs[res_var_def, SUBMAP_IMP_DOMSUB_FUPDATE, SUBMAP_IMP_DOMSUB_SUBMAP]) >>
Cases_on `FLOOKUP s.locals v` >> fs[res_var_def] >>
rev_drule $ iffLR SUBMAP_FLOOKUP_EQN >>
disch_then imp_res_tac >>
TRY (qpat_assum `FLOOKUP t.locals _ = SOME _` kall_tac >> gs[res_var_def]) >>
gs[flookup_thm] >>
imp_res_tac evaluate_locals_same_fdom' >> gs[] >>
qpat_x_assum `!_ _ _. _` kall_tac >>
`v ∈ FDOM st'.locals` by metis_tac[COMPONENT] >>
`v ∈ FDOM st.locals` by metis_tac[COMPONENT] >>
fs[ABSORPTION_RWT, GSYM DRESTRICT_DOMSUB] >>
qpat_x_assum `v INSERT FDOM s.locals = _` $ gs o single o GSYM >>
qpat_x_assum `v INSERT FDOM t.locals = _` $ gs o single o GSYM >>
fs[DELETE_INSERT, FDIFF_FUPDATE]
>>~- ([`t.locals ' v`], fs[ABSORPTION_RWT]) >>
Cases_on `FLOOKUP t.locals v` >> fs[res_var_def, FDIFF_FUPDATE, FDIFF_FDOMSUB_INSERT, DELETE_NON_ELEMENT_RWT]
>>~- ([`FLOOKUP t.locals v = NONE`],
fs[FDIFF_def, compl_insert, GSYM DRESTRICT_DOMSUB] >>
qpat_x_assum `_ \\ _ = _ \\ _` $ fs o single o GSYM >>
irule EQ_SYM >> fs[] >>
irule DOMSUB_NOT_IN_DOM >> fs[FDOM_DRESTRICT, flookup_thm]
) >>
fs[FDIFF_def, compl_insert, GSYM DRESTRICT_DOMSUB, flookup_thm, fmap_eq_flookup, DOMSUB_FLOOKUP_THM] >>
rpt strip_tac >>
Cases_on `v = x'` >>
qpat_x_assum `!_. _` $ qspec_then `x'` assume_tac >> gs[] >>
fs[FLOOKUP_SIMP] >> metis_tac[flookup_thm]
)
>~ [`evaluate (Seq _ _, _)`]
>- (
rpt strip_tac >> fs[evaluate_def] >>
pairarg_tac >> fs[] >>
pairarg_tac >> fs[] >>
Cases_on `res' = NONE` >> fs[] >>
qpat_x_assum `!_. locals_rel s _ ∧ state_rel s _ ⇒ _` $ qspec_then `t` assume_tac >> gs[] >>
TRY (last_x_assum $ qspec_then `s1` assume_tac >> gs[]) >>
Cases_on `r` >> fs[locals_rel_def, locals_ext_rel_def] >>
Cases_on `x` >> fs[]
)
>~ [`evaluate (If _ _ _, _)`]
>- (
rpt strip_tac >> fs[evaluate_def] >>
imp_res_tac eval_state_locals_rel >> fs[] >>
gs[CaseEq "option", CaseEq "word_lab"] >>
pop_assum imp_res_tac >> fs[]
)
>~ [`evaluate (Call _ _ _, _)`]
>- (
rpt strip_tac >> fs[evaluate_def] >>
imp_res_tac eval_optmmap_state_locals_rel >>
imp_res_tac eval_state_locals_rel >>
gs[CaseEq "option", CaseEq "word_lab", CaseEq "prod"] >>
first_assum imp_res_tac >>
`t.clock = s.clock` by fs[state_rel_def] >> fs[] >>
`t.code = s.code` by fs[state_rel_def] >> fs[] >>
qpat_assum `!_ _. OPT_MMAP _ _ = _ ⇒ _` imp_res_tac >> fs[] >>
Cases_on `s.clock = 0` >> fs[]
>- gvs[state_rel_def, empty_locals_def] >>
qpat_x_assum `_ = (r, s')` mp_tac >>
TOP_CASE_TAC >> fs[] >>
TOP_CASE_TAC >> fs[] >>
TOP_CASE_TAC >> fs[]
>>~- ([`_ = _ ∧ empty_locals _ = _ ⇒ _`],
disch_tac >> gvs[] >>
pop_assum $ qspec_then `dec_clock t with locals := newlocals` mp_tac >> impl_tac
>- fs[locals_rel_def, state_rel_def, dec_clock_def] >>
disch_tac >> fs[state_rel_def, empty_locals_def]) >>
gs[CaseEq "option", CaseEq "prod"] >> disch_tac >> gvs[] >>
qpat_x_assum `!_. locals_rel (dec_clock s with locals := newlocals) _ ∧ state_rel _ _ ⇒ _` $ qspec_then `dec_clock t with locals := newlocals` mp_tac >> impl_tac
>>~- ([`locals_rel (dec_clock _ with locals := _) (dec_clock _ with locals := _)`],
fs[locals_rel_def, state_rel_def, dec_clock_def]) >>
disch_tac >> fs[]
>>~- ([`state_rel (empty_locals _) (empty_locals _)`],
fs[state_rel_def, empty_locals_def])
>- (
qpat_x_assum `!_. locals_rel _ _ ∧ state_rel _ _ ⇒ _` $ qspec_then `t' with locals := t.locals` mp_tac >> impl_tac
>- fs[state_rel_def, locals_rel_def] >>
disch_tac >> fs[locals_rel_def, locals_ext_rel_def] >>
Cases_on `r` >> TRY (Cases_on `x`) >> fs[]
)
>- (
qpat_x_assum `!_. locals_rel _ _ ∧ state_rel _ _ ⇒ _` $ qspec_then `t' with locals := t.locals |+ (rt, w)` mp_tac >> impl_tac
>- fs[locals_rel_def, state_rel_def, SUBMAP_IMP_FUPDATE_SUBMAP] >>
fs[locals_rel_def] >>
drule $ iffLR SUBMAP_FLOOKUP_EQN >>
disch_then imp_res_tac >> fs[] >>
disch_tac >> fs[] >>
Cases_on `r` >> TRY (Cases_on `x`) >> fs[locals_ext_rel_def] >>
fs[FDIFF_def, compl_insert, GSYM DRESTRICT_DOMSUB] >>
pop_assum $ fs o single o GSYM >>
irule EQ_SYM >>
irule DOMSUB_NOT_IN_DOM >>
fs[FDOM_DRESTRICT, flookup_thm]
) >>
Cases_on `c = eid'` >> gs[]
>- (
qpat_x_assum `!_. locals_rel _ _ ∧ state_rel _ _ ⇒ _` $ qspec_then `t' with locals := t.locals` mp_tac >> impl_tac
>- fs[locals_rel_def, state_rel_def] >>
disch_tac >> fs[locals_ext_rel_def] >>
Cases_on `r` >> TRY (Cases_on `x`) >> fs[]
) >>
qrefine `r` >> conj_tac
>- gvs[state_rel_def, empty_locals_def] >>
Cases_on `r` >> TRY (Cases_on `x`) >> fs[]
) >>
fs[evaluate_def] >> rpt strip_tac
>- fs[locals_ext_rel_def] >>
imp_res_tac eval_state_locals_rel >> fs[] >>
gs[CaseEq "option", CaseEq "word_lab"] >>
pop_assum imp_res_tac >> fs[]
>>~- ([`locals_ext_rel a a b b`], fs[locals_ext_rel_def])
>>~- ([`_ with memory := _ = _`],
qrefine `t with memory := m` >>
gvs[state_rel_def, locals_rel_def, locals_ext_rel_def]
)
>- (
fs[locals_rel_def] >>
imp_res_tac SUBMAP_FLOOKUP_EQN >> fs[] >> conj_tac
>- gvs[state_rel_def] >>
gvs[SUBMAP_IMP_FUPDATE_SUBMAP] >>
gs[locals_ext_rel_def, FDIFF_def, compl_insert, flookup_thm, GSYM DRESTRICT_DOMSUB] >>
irule EQ_SYM >>
irule DOMSUB_NOT_IN_DOM >> fs[FDOM_DRESTRICT]
)
>- (
`t.globals = s.globals` by fs[state_rel_def] >>
gvs[set_globals_def, state_rel_def, locals_rel_def, locals_ext_rel_def]
)
>- (
fs[locals_rel_def] >>
Cases_on `is_load op` >> gs[CaseEq "option"] >>
drule $ iffLR SUBMAP_FLOOKUP_EQN >>
disch_then imp_res_tac >>
Cases_on `op` >>
fs[sh_mem_op_def, sh_mem_load_def, sh_mem_store_def] >>
`t.ffi = s.ffi ∧ t.sh_memaddrs = s.sh_memaddrs` by fs[state_rel_def] >> gs[CaseEq "word_lab"]
>>~- ([`addr ∈ _.sh_memaddrs `],
Cases_on `addr ∈ s.sh_memaddrs` >> gs[CaseEq "ffi_result"]
>- gvs[set_var_def, state_rel_def, locals_rel_def, locals_ext_rel_def, SUBMAP_IMP_FUPDATE_SUBMAP, flookup_thm, ABSORPTION, FDIFF_def] >>
gvs[state_rel_def, empty_locals_def, locals_rel_def, locals_ext_rel_def]
)
)
>- (
gvs[state_rel_def, empty_locals_def, locals_rel_def, locals_ext_rel_def]
)
>- (
fs[state_rel_def, empty_locals_def]
)
>- (
`t.clock = s.clock` by fs[state_rel_def] >> fs[] >>
Cases_on `s.clock = 0` >>
gvs[state_rel_def, dec_clock_def, locals_rel_def, locals_ext_rel_def, empty_locals_def]
) >>
fs[locals_rel_def] >>
drule $ iffLR SUBMAP_FLOOKUP_EQN >>
disch_then imp_res_tac >>
gvs[state_rel_def, CaseEq "ffi_result", locals_ext_rel_def]
QED
Theorem evaluate_state_locals_rel:
∀p s r s' t.
evaluate (p, s) = (r, s') ⇒
r ≠ SOME Error ==>
locals_rel s t ∧ state_rel s t ⇒
∃r' t'.
evaluate (p, t) = (r, t') ∧ state_rel s' t' ∧
case r of
| NONE => r' = NONE ∧ locals_rel s' t'
| SOME Break => r' = SOME Break ∧ locals_rel s' t'
| SOME Continue => r' = SOME Continue ∧ locals_rel s' t'
| SOME (Return retv) => r' = SOME (Return retv)
| SOME (FinalFFI f) => r' = SOME (FinalFFI f)
| SOME TimeOut => r' = SOME TimeOut
| SOME (Exception e) => r' = SOME (Exception e)
| _ => F
Proof
rpt strip_tac >>
drule_all evaluate_state_locals_rel_strong >>
disch_tac >>
Cases_on `r` >> TRY (Cases_on `x`) >> fs[]
QED
Theorem single_dec_evaluate:
∀p s r s' v e val .
eval s e = SOME val ∧
evaluate (p, s with locals := s.locals |+ (v, val)) = (r, s') ∧
r ≠ SOME Error ==>
∃t'. evaluate (Dec v e p, s) = (r, t') ∧ state_rel s' t'
Proof
rpt strip_tac >> gs[evaluate_def, state_rel_def]
QED
Theorem nested_decs_evaluate:
!vs es p s r s' vals.
OPT_MMAP (eval s) es = SOME vals ∧
LENGTH vs = LENGTH es ∧
ALL_DISTINCT vs /\
(!v. MEM v vs ⇒ !e. MEM e es ⇒ ¬MEM v (var_cexp e)) ∧
evaluate (p, s with locals := s.locals |++ ZIP (vs, vals)) = (r, s') ∧
r ≠ SOME Error ==>
∃t'.
evaluate (nested_decs vs es p, s) = (r, t') ∧ state_rel s' t'
Proof
Induct_on `vs` >> gs[nested_decs_def, evaluate_def]
>- (
rpt strip_tac >> gs[FUPDATE_LIST, FUPDATE_DEF] >>
`s with locals := s.locals = s` by simp[state_component_equality] >> gvs[state_rel_def]
) >>
rpt strip_tac >>
Cases_on `es` >> gs[nested_decs_def] >>
Cases_on `vals` >> gvs[FUPDATE_LIST_THM] >>
drule opt_mmap_length_eq >> disch_tac >> fs[] >>
`OPT_MMAP (eval (s with locals := s.locals |+ (h, h''))) t = SOME t'` by (
qpat_x_assum `OPT_MMAP (eval s) t = SOME t'` $ gvs o single o GSYM >>
irule OPT_MMAP_ALL_EQ >>
rpt strip_tac >>
irule update_locals_not_vars_eval_eq' >> gs[]
) >>
last_x_assum drule >> gs[] >>
disch_then $ qspecl_then [`p`, `r`, `s'`] assume_tac >> gs[] >>
rev_drule single_dec_evaluate >>
disch_then $ qspecl_then [`nested_decs vs t p`, `r`, `t''`, `h`] assume_tac >> gs[state_rel_def]
QED
Theorem genlist_less_than:
∀n a v. MEM v (GENLIST (λx. (a:num) + SUC x) n) ⇒ a < v
Proof
Induct >> gs[GENLIST] >>
rpt strip_tac >> gs[LESS_ADD_SUC]
QED
Theorem genlist_not_in:
∀n a v. v ≤ a ⇒ ¬MEM v (GENLIST (λx. (a:num) + SUC x) n)
Proof
spose_not_then assume_tac >> gs[] >>
drule genlist_less_than >> decide_tac
QED
Theorem genlist_all_distinct:
∀n a. ALL_DISTINCT (GENLIST (λx. a + SUC x) n)
Proof
Induct >> gs[GENLIST, ALL_DISTINCT_SNOC, MEM_GENLIST]
QED
Theorem eval_dec_clock_eq:
∀s e. eval (dec_clock s) e = eval s e
Proof
simp[eval_upd_clock_eq, dec_clock_def]
QED
Theorem opt_mmap_eval_dec_clock_eq:
∀s es. OPT_MMAP (eval (dec_clock s)) es = OPT_MMAP (eval s) es
Proof
rpt gen_tac >>
irule OPT_MMAP_CONG >> fs[] >>
rpt strip_tac >> gs[eval_dec_clock_eq]
QED
Theorem not_has_return_not_evaluate_return:
∀p s.
¬has_return p ⇒
∃r s'.
evaluate (p, s) = (r, s') ∧
case r of
| SOME (Return retv) => F
| _ => T
Proof
recInduct evaluate_ind >>
rw[has_return_def]
>~ [`While _ _`]
>- (
simp[Once evaluate_def] >> every_case_tac >> gs[] >> every_case_tac >> gs[]
) >>
gs[evaluate_def]
>~ [`sh_mem_op`]
>- (
rpt (TOP_CASE_TAC >> gs[]) >>
Cases_on `op` >> gs[sh_mem_op_def, sh_mem_load_def, sh_mem_store_def] >>
rpt (TOP_CASE_TAC >> gs[])
) >>
every_case_tac >> gs[]
QED
Theorem not_has_return_not_evaluate_return':
∀p s r s' retv.
¬(has_return p) ∧
evaluate (p, s) = (r, s') ⇒
r ≠ SOME (Return retv)
Proof
rpt strip_tac >>
dxrule not_has_return_not_evaluate_return >> gvs[] >>
qrefine `s` >> gvs[]
QED
Theorem evaluate_while_not_break_continue:
∀p e s r s'.
evaluate (While e p, s) = (r, s') ⇒
case r of
| SOME Break => F
| SOME Continue => F
| _ => T
Proof
completeInduct_on `s.clock` >>
rpt strip_tac >>
pop_assum mp_tac >>
fs[Once evaluate_def, CaseEq "option", CaseEq "word_lab"] >>
disch_tac >> fs[] >>
Cases_on `w ≠ 0w` >> fs[] >>
Cases_on `s.clock = 0` >> fs[] >>
pairarg_tac >> fs[] >>
`s1.clock < s.clock` by (
irule LET_TRANS >>
qrefine `(dec_clock s).clock` >>
dxrule evaluate_clock >> fs[dec_clock_def]
) >>
last_x_assum $ qspec_then `s1.clock` mp_tac >> fs[] >>
disch_then $ qspec_then `s1` mp_tac >> fs[] >>
disch_tac >>
qpat_x_assum `_ = (r, s')` mp_tac >>
PURE_ONCE_REWRITE_TAC[evaluate_def] >>
disch_tac >> gs[CaseEq "option", CaseEq "result", CaseEq "word_lab"] >>
res_tac >> gvs[]
QED
Theorem res_var_commutes_strong:
res_var (res_var lc (h,FLOOKUP lc' h)) (n,FLOOKUP lc' n) =
res_var (res_var lc (n,FLOOKUP lc' n)) (h,FLOOKUP lc' h)
Proof
Cases_on `n ≠ h` >> metis_tac[res_var_commutes]
QED
Theorem res_var_foldl_commutes_strong:
∀h vs lc1 lc2.
res_var (FOLDL res_var lc1 (ZIP (vs, MAP (FLOOKUP lc2) vs))) (h, FLOOKUP lc2 h) =
FOLDL res_var (res_var lc1 (h, FLOOKUP lc2 h)) (ZIP (vs, MAP (FLOOKUP lc2) vs))
Proof
Induct_on `vs` >> fs[] >>
rpt strip_tac >>
fs[res_var_commutes_strong]
QED
Theorem evaluate_nested_decs_locals_nested_res_var:
∀p s r s' vs es vals.
OPT_MMAP (eval s) es = SOME vals ∧
LENGTH vs = LENGTH es ∧
ALL_DISTINCT vs /\
(!v. MEM v vs ⇒ !e. MEM e es ⇒ ¬MEM v (var_cexp e)) ∧
evaluate (p, s with locals := s.locals |++ ZIP (vs, vals)) = (r, s') ==>
∃t'.
evaluate (nested_decs vs es p, s) = (r, t') ∧ state_rel s' t' ∧
t'.locals = FOLDL res_var s'.locals (ZIP (vs, MAP (FLOOKUP s.locals) vs))
Proof
Induct_on `vs` >> rw[]
>- (
Cases_on `vals` >> fs[FUPDATE_LIST, nested_decs_def] >>
qrefine `s'` >> fs[state_rel_def] >>
`s with locals := s.locals = s` by fs[state_component_equality] >> fs[]
) >>
Cases_on `es` >> Cases_on `vals` >> gs[nested_decs_def, FUPDATE_LIST_THM, evaluate_def] >>
pairarg_tac >> gs[] >>
last_x_assum $ qspecl_then [`p`, `s with locals := s.locals |+ (h, h'')`, `r`, `s'`, `t`, `t'`] mp_tac >> impl_tac
>- (
fs[] >>
qpat_x_assum `OPT_MMAP _ _ = SOME _` $ rw o single o GSYM >>
irule OPT_MMAP_ALL_EQ >>
rpt strip_tac >>
first_x_assum $ qspec_then `h` assume_tac >> rfs[] >>
pop_assum imp_res_tac >>
imp_res_tac update_locals_not_vars_eval_eq'' >> fs[state_component_equality] >>
`s with locals := s.locals = s` by fs[state_component_equality] >> simp[]
) >>
disch_tac >> fs[] >>
`MAP (FLOOKUP (s.locals |+ (h, h''))) vs = MAP (FLOOKUP s.locals) vs` by (
fs[MAP_EQ_f] >>
rpt strip_tac >>
qpat_x_assum `!_. _` imp_res_tac >>
fs[FLOOKUP_UPDATE] >>
Cases_on `h = e` >> fs[]
) >> fs[] >>
conj_tac
>- (
Cases_on `FLOOKUP s.locals h` >> gvs[res_var_def, state_rel_def]
) >>
gvs[res_var_foldl_commutes_strong]
QED
Theorem not_some_is_none:
∀a. (∀v. a ≠ SOME v) ⇔ a = NONE
Proof
Cases >> fs[]
QED
Theorem fdom_eq_flookup_thm:
∀f1 f2.
FDOM f1 = FDOM f2 ⇔
(∀x. (∃v. FLOOKUP f1 x = SOME v) ⇒ (∃v. FLOOKUP f2 x = SOME v)) ∧
(∀x. FLOOKUP f1 x = NONE ⇒ FLOOKUP f2 x = NONE)
Proof
fs[GSYM SUBSET_ANTISYM_EQ, SUBSET_DEF, FDOM_FLOOKUP] >>
rpt strip_tac >>
`∀x. ((∃v. FLOOKUP f2 x = SOME v) ⇒ (∃v. FLOOKUP f1 x = SOME v)) = (FLOOKUP f1 x = NONE ⇒ FLOOKUP f2 x = NONE)` by (
gen_tac >>
gs[Once MONO_NOT_EQ] >>
qspec_then `FLOOKUP f1 x` assume_tac not_some_is_none >>
qspec_then `FLOOKUP f2 x` assume_tac not_some_is_none >>
metis_tac[]
) >>
metis_tac[]
QED
Theorem flookup_res_var_is_mem_zip_eq:
∀xs x lc1 lc2.
MEM x xs ⇒
FLOOKUP (FOLDL res_var lc1 (ZIP (xs, MAP (FLOOKUP lc2) xs))) x = FLOOKUP lc2 x
Proof
Induct_on `xs` >>
gs[] >>
rpt strip_tac >>
gs[GSYM res_var_foldl_commutes_strong] >>
Cases_on `FLOOKUP lc2 h` >> gs[res_var_def, FLOOKUP_UPDATE]
QED
Theorem not_var_prog_flookup_eqn:
∀p s r s' x.
evaluate (p, s) = (r, s') ∧
¬MEM x (var_prog p) ∧
(case r of
| NONE => T
| SOME Break => T
| SOME Continue => T
| _ => F) = T ⇒
FLOOKUP s'.locals x = FLOOKUP s.locals x
Proof
recInduct evaluate_ind >>
rpt conj_tac >>
gs[var_prog_def]
>~ [`evaluate (While _ _, _)`]
>- (
rpt strip_tac >>
qpat_x_assum `_ = (r, s')` mp_tac >>
simp[Once evaluate_def] >>
gs[CaseEq "option", CaseEq "word_lab"] >>
disch_tac >> fs[] >>
Cases_on `w = 0w` >> Cases_on `s.clock = 0` >> fs[] >>
pairarg_tac >> gs[CaseEq "option", CaseEq "result", dec_clock_def]
)
>~ [`evaluate (Dec _ _ _, _)`]
>- (
gs[evaluate_def, CaseEq "option"] >>
rpt strip_tac >> fs[] >>
pairarg_tac >> fs[] >>
qpat_x_assum `_ = s'` $ fs o single o GSYM >>
fs[flookup_res_var_thm, FLOOKUP_UPDATE]
)
>~ [`evaluate (Seq _ _, _)`]
>- (
rpt strip_tac >>
gs[evaluate_def] >>
pairarg_tac >> fs[] >>
Cases_on `res = NONE` >> fs[]
)
>~ [`evaluate (If _ _ _, _)`]
>- (
rpt strip_tac >> gs[evaluate_def, CaseEq "option", CaseEq "word_lab"] >>
Cases_on `w ≠ 0w` >> fs[]
)
>~ [`evaluate (Call _ _ _, _)`]
>- (
rpt strip_tac >> gvs[evaluate_def, CaseEq "option", CaseEq "word_lab", CaseEq "prod"] >>
Cases_on `s.clock = 0` >> gvs[CaseEq "option", CaseEq "result", CaseEq "prod"] >>
TRY(Cases_on `v5` >> TRY (Cases_on `x'`) >> gs[MEM, MEM_APPEND, FLOOKUP_UPDATE]) >>
Cases_on `eid <> eid'` >> gvs[] >>
Cases_on `v1` >> TRY (Cases_on `x'`) >> gvs[MEM, MEM_APPEND]
)
>~ [`evaluate (ShMem _ _ _, _)`]
>- (
rpt strip_tac >> gvs[evaluate_def, CaseEq "option", CaseEq "word_lab"] >>
Cases_on `op` >> fs[sh_mem_op_def, sh_mem_load_def, sh_mem_store_def, set_var_def]
>>~ [`addr ∈ s.sh_memaddrs`] >>
Cases_on `addr ∈ s.sh_memaddrs` >>
gvs[CaseEq "option", CaseEq "ffi_result", CaseEq "result", FLOOKUP_UPDATE, CaseEq "word_lab"]
) >>
rpt strip_tac >>
gvs[evaluate_def, CaseEq "option", CaseEq "word_lab", CaseEq "ffi_result", set_globals_def, FLOOKUP_UPDATE] >>
Cases_on `s.clock = 0` >> gvs[dec_clock_def]
QED
Theorem not_var_prog_not_affect_evaluate:
∀p s r s' v val.
evaluate (p, s) = (r, s') ∧
¬MEM v (var_prog p) ∧
r ≠ SOME Error ⇒
∃t'.
evaluate (p, s with locals := s.locals |+ (v, val)) = (r, t') ∧
state_rel s' t' ∧
case r of
| NONE => t'.locals = s'.locals |+ (v, val)
| SOME Break => t'.locals = s'.locals |+ (v, val)
| SOME Continue => t'.locals = s'.locals |+ (v, val)
| _ => T
Proof
recInduct evaluate_ind >>
rpt conj_tac
>~ [`evaluate (While _ _, _)`]
>- (
rpt strip_tac >>
qpat_x_assum `_ = (r, s')` mp_tac >>
once_rewrite_tac[evaluate_def] >>
gs[CaseEq "option", CaseEq "word_lab", var_prog_def] >>
disch_tac >> fs[] >>
drule_all update_locals_not_vars_eval_eq >>
disch_tac >> fs[] >>
Cases_on `w = 0w` >> fs[]
>- gvs[state_rel_def] >>
Cases_on `s.clock = 0` >> fs[]
>- gvs[state_rel_def, empty_locals_def] >>
pairarg_tac >> fs[] >>
pairarg_tac >> gvs[] >>
`∀s lc. dec_clock (s with locals := lc) = dec_clock s with locals := lc` by fs[dec_clock_def, state_component_equality] >> fs[] >>
qpat_x_assum `!_ _. ~MEM _ _ ∧ _ ≠ SOME Error ⇒ _` drule >> fs[] >>
Cases_on `res' = SOME Error` >> gs[] >>
disch_then $ qspec_then `val` mp_tac >> fs[] >>
disch_tac >> fs[] >>
Cases_on `res'` >> TRY (Cases_on `x`) >> `(dec_clock s).locals = s.locals` by fs[dec_clock_def] >>
gvs[] >>
last_x_assum drule >> fs[] >>
disch_then $ qspec_then `val` assume_tac >> fs[] >>
`s1 = s1' with locals := s1'.locals |+ (v, val)` by fs[state_component_equality, state_rel_def] >>
gvs[]
)
>~ [`evaluate (Dec _ _ _, _)`]
>- (
rpt strip_tac >>
gs[evaluate_def, var_prog_def, CaseEq "option"] >>
drule_all update_locals_not_vars_eval_eq >> fs[] >>
disch_tac >> fs[] >>
pairarg_tac >> fs[] >>
pairarg_tac >> fs[] >>
last_x_assum drule >>
rev_drule $ INST_TYPE [``:'a`` |-> ``:num``, ``:'b`` |-> ``:'a word_lab``] FUPDATE_COMMUTES >>
disch_then $ qspec_then `s.locals` mp_tac >>
disch_then $ fs o single >>
disch_then $ qspec_then `val` assume_tac >> gvs[] >>
conj_tac
>- (
Cases_on `FLOOKUP s.locals v` >> Cases_on `FLOOKUP (s.locals |+ (v', val)) v` >>
gvs[state_rel_def, res_var_def]
) >>
Cases_on `r` >> TRY (Cases_on `x`) >> fs[FLOOKUP_UPDATE] >>
Cases_on `FLOOKUP s.locals v` >> fs[res_var_def] >>
drule $ INST_TYPE [``:'a`` |-> ``:num``, ``:'b`` |-> ``:'a word_lab``] DOMSUB_FUPDATE_NEQ >>
disch_tac >>
drule $ INST_TYPE [``:'a`` |-> ``:num``, ``:'b`` |-> ``:'a word_lab``] FUPDATE_COMMUTES >>
disch_tac >>
gvs[]
)
>~ [`evaluate (Seq _ _, _)`]
>- (
rpt strip_tac >> gs[evaluate_def, var_prog_def] >>
pairarg_tac >> fs[] >>
pairarg_tac >> fs[] >>
qpat_x_assum `!_ _. ¬MEM _ _ ∧ _ ⇒ _` drule >>
Cases_on `res' = SOME Error` >> fs[] >>
disch_then $ qspec_then `val` assume_tac >> fs[] >>
Cases_on `res' = NONE` >> fs[]
>- (
last_x_assum drule >>
disch_then $ qspec_then `val` assume_tac >> gvs[] >>
`s1 = s1' with locals := s1'.locals |+ (v, val)` by gvs[state_rel_def, state_component_equality] >> fs[]
) >>
gvs[]
)
>~ [`evaluate (If _ _ _, _)`]
>- (
rpt strip_tac >> gs[evaluate_def, var_prog_def, CaseEq "option", CaseEq "word_lab"] >>
drule_all update_locals_not_vars_eval_eq >>
disch_tac >> fs[] >>
Cases_on `w = 0w` >> fs[]
)
>~ [`evaluate (Call _ _ _, _)`]
>- (
rpt strip_tac >> gs[evaluate_def, var_prog_def, CaseEq "option", CaseEq "word_lab"] >>
`OPT_MMAP (eval (s with locals := s.locals |+ (v, val))) argexps = SOME args` by (
qpat_x_assum `_ = SOME args` $ fs o single o GSYM >>
irule OPT_MMAP_CONG >> fs[] >>
rpt strip_tac >>
fs[MEM_FLAT, MEM_MAP] >>
qpat_x_assum `!_. (!_. _) ∨ ¬_` $ qspec_then `var_cexp x` assume_tac >> fs[] >>
TRY (pop_assum $ qspec_then `x` assume_tac >> fs[]) >>
drule update_locals_not_vars_eval_eq' >> fs[]
) >>
gs[CaseEq "prod"] >>
Cases_on `s.clock = 0` >> fs[]
>- gvs[state_rel_def, empty_locals_def] >>
`dec_clock (s with locals := s.locals |+ (v, val)) with locals := newlocals = dec_clock s with locals := newlocals` by gs[dec_clock_def, state_component_equality] >> fs[] >>
gs[CaseEq "option", CaseEq "result", CaseEq "prod"]
>- fs[state_rel_def] >> gvs[]
>>~- ([`state_rel (empty_locals _) _`], gvs[state_rel_def, empty_locals_def]) >>
`~MEM v (var_prog p)` by (
TRY (Cases_on `v5` >> TRY (Cases_on `x`) >> fs[]) >>
TRY (Cases_on `v1` >> TRY (Cases_on `x`) >> fs[])
)
>- (
last_x_assum drule >>
disch_then $ qspec_then `val` assume_tac >> fs[]
)
>- (
last_x_assum drule >>
disch_then $ qspec_then `val` assume_tac >> fs[] >>
`v ≠ rt` by (
Cases_on `v5` >> TRY (Cases_on `x`) >> fs[]
) >> fs[FLOOKUP_UPDATE] >>
drule $ INST_TYPE [``:'a`` |-> ``:num``, ``:'b`` |-> ``:'a word_lab``] FUPDATE_COMMUTES >> fs[]
) >>
Cases_on `eid = eid'` >> gvs[state_rel_def]
)
>~ [`evaluate (ShMem _ _ _, _)`]
>- (
rpt strip_tac >> gs[evaluate_def, CaseEq "option", CaseEq "word_lab", var_prog_def] >>
drule_all update_locals_not_vars_eval_eq >>
disch_tac >> fs[FLOOKUP_UPDATE] >>
Cases_on `op` >> fs[sh_mem_op_def, sh_mem_store_def, sh_mem_load_def, set_var_def]
>>~ [`addr ∈ s.sh_memaddrs`] >>
drule $ INST_TYPE [``:'a`` |-> ``:num``, ``:'b`` |-> ``:'a word_lab``] FUPDATE_COMMUTES >>
disch_then assume_tac >> fs[] >>
Cases_on `addr ∈ s.sh_memaddrs` >> gvs[AllCaseEqs(), state_rel_def, empty_locals_def, FLOOKUP_UPDATE] >>
qabbrev_tac `z = Word (word_of_bytes F 0w new_bytes)` >> fs[]
) >>
gs[evaluate_def, var_prog_def, CaseEq "option", CaseEq "word_lab", FLOOKUP_UPDATE] >>
rpt strip_tac >>
TRY (imp_res_tac update_locals_not_vars_eval_eq >> gvs[state_rel_def, set_globals_def, empty_locals_def, CaseEq "ffi_result"]) >>
res_tac
>- (
drule $ INST_TYPE [``:'a`` |-> ``:num``, ``:'b`` |-> ``:'a word_lab``] FUPDATE_COMMUTES >> fs[]
) >>
Cases_on `s.clock = 0` >> gvs[dec_clock_def]
QED
Theorem SUBMAP_DIFF_LIST:
∀l vs vals.
LENGTH vs = LENGTH vals ∧
ALL_DISTINCT vs ∧
(∀v. MEM v vs ⇒ v ∉ FDOM l) ⇒
l SUBMAP l |++ ZIP (vs, vals)
Proof
Induct_on `vs` >>
rpt strip_tac >>
Cases_on `vals` >> fs[]
>- fs[FUPDATE_LIST] >>
fs[FUPDATE_LIST_THM] >>
`~MEM h (MAP FST (ZIP (vs, t)))` by fs[MAP_ZIP] >>
drule FUPDATE_FUPDATE_LIST_COMMUTES >>
disch_then $ qspecl_then [`h'`, `l`] assume_tac >> fs[] >>
last_x_assum $ qspecl_then [`l`, `t`] mp_tac >> fs[] >>
disch_tac >>
drule SUBMAP_TRANS >> disch_then irule >>
fs[SUBMAP_FUPDATE_FLOOKUP] >>
disj1_tac >>
drule_all flookup_fupdate_zip_not_mem >>
disch_then $ qspec_then `l` assume_tac >> fs[] >>
fs[flookup_thm]
QED