forked from CakeML/cakeml
-
Notifications
You must be signed in to change notification settings - Fork 0
/
candle_kernel_funsScript.sml
2204 lines (2176 loc) · 83.6 KB
/
candle_kernel_funsScript.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
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
(*
Prove that kernel functions maintain Candle prover's invariants
*)
open preamble helperLib;
open semanticPrimitivesTheory semanticPrimitivesPropsTheory
evaluateTheory namespacePropsTheory evaluatePropsTheory
sptreeTheory holKernelProofTheory ml_hol_kernel_funsProgTheory
candle_kernel_permsTheory candle_kernelProgTheory computeProofTheory;
open permsTheory candle_kernel_valsTheory candle_prover_invTheory ast_extrasTheory;
local open ml_progLib in end
val _ = new_theory "candle_kernel_funs";
val _ = set_grammar_ancestry [
"candle_kernel_vals", "candle_prover_inv", "ast_extras", "evaluate",
"namespaceProps", "perms", "semanticPrimitivesProps", "misc"];
Theorem Arrow1:
(A --> B) f fv ∧
do_opapp [fv; av] = SOME (env, exp) ∧
evaluate (s:'ffi semanticPrimitives$state) env [exp] = (s', res) ∧
A a av ∧
perms_ok ∅ av ∧
perms_ok ∅ fv ⇒
s.ffi = s'.ffi ∧
((res = Rerr (Rabort Rtimeout_error)) ∨
(res = Rerr (Rabort Rtype_error)) ∨
s.refs = s'.refs ∧
s.eval_state = s'.eval_state ∧
s.next_type_stamp = s'.next_type_stamp ∧
∃rv. res = Rval [rv] ∧
B (f a) rv)
Proof
strip_tac
\\ qabbrev_tac ‘env' = write "a" av (write "f" fv ARB)’
\\ ‘Eval env' (Var (Short "a")) (A a)’
by simp [Abbr ‘env'’, ml_translatorTheory.Eval_Var_SIMP]
\\ ‘Eval env' (Var (Short "f")) ((A --> B) f)’
by simp [Abbr ‘env'’, ml_translatorTheory.Eval_Var_SIMP]
\\ qpat_x_assum ‘(_ --> _) _ _’ kall_tac
\\ qpat_x_assum ‘A _ _’ kall_tac
\\ dxrule_all ml_translatorTheory.Eval_Arrow
\\ simp [ml_translatorTheory.Eval_def]
\\ disch_then (qspec_then ‘s.refs’ strip_assume_tac)
\\ dxrule ml_translatorTheory.evaluate_empty_state_IMP
\\ simp [ml_progTheory.eval_rel_def, evaluate_def, Abbr ‘env'’,
ml_progTheory.nsLookup_write]
\\ strip_tac
\\ gvs [AllCaseEqs(),evaluateTheory.dec_clock_def]
\\ drule_then (qspec_then ‘s.clock’ mp_tac) evaluate_set_init_clock
\\ simp [with_same_clock]
\\ strip_tac \\ gvs []
THEN1
(drule evaluate_empty_perms
\\ reverse impl_tac THEN1 fs []
\\ fs [perms_ok_state_def]
\\ drule_all perms_ok_do_opapp \\ fs [])
\\ mp_tac (CONJUNCT1 is_clock_io_mono_evaluate)
\\ qmatch_asmsub_abbrev_tac ‘evaluate s env1 [e]’
\\ disch_then (qspecl_then [`s`,`env1`,`[e]`] mp_tac)
\\ rw [is_clock_io_mono_def]
\\ gs [io_events_mono_antisym]
QED
Theorem Arrow2:
(A --> B --> C) f fv ∧
do_partial_app fv av = SOME gv ∧
do_opapp [gv; bv] = SOME (env, exp) ∧
evaluate (s:'ffi semanticPrimitives$state) env [exp] = (s', res) ∧
A a av ∧ B b bv ∧
perms_ok ∅ av ∧
perms_ok ∅ bv ∧
perms_ok ∅ fv ⇒
s.ffi = s'.ffi ∧
((res = Rerr (Rabort Rtimeout_error)) ∨
(res = Rerr (Rabort Rtype_error)) ∨
s.refs = s'.refs ∧
s.eval_state = s'.eval_state ∧
s.next_type_stamp = s'.next_type_stamp ∧
∃rv. res = Rval [rv] ∧
C (f a b) rv)
Proof
strip_tac
\\ ‘LENGTH s'.refs = LENGTH s.refs’
by (gvs [do_partial_app_def, AllCaseEqs(), do_opapp_cases,
perms_ok_def]
\\ drule evaluate_empty_perms
\\ impl_tac \\ simp []
\\ gs [perms_ok_state_def, perms_ok_env_def]
\\ Cases \\ simp [ml_progTheory.nsLookup_nsBind_compute]
\\ rw [] \\ gs []
\\ first_x_assum irule
\\ first_assum (irule_at (Pos last)) \\ gs [])
\\ qabbrev_tac ‘env' = write "a" av (write "b" bv (write "f" fv ARB))’
\\ ‘Eval env' (Var (Short "a")) (A a)’
by simp [Abbr ‘env'’, ml_translatorTheory.Eval_Var_SIMP]
\\ ‘Eval env' (Var (Short "b")) (B b)’
by simp [Abbr ‘env'’, ml_translatorTheory.Eval_Var_SIMP]
\\ ‘Eval env' (Var (Short "f")) ((A --> B --> C) f)’
by simp [Abbr ‘env'’, ml_translatorTheory.Eval_Var_SIMP]
\\ qpat_x_assum ‘(_ --> _) _ _’ kall_tac
\\ qpat_x_assum ‘A _ _’ kall_tac
\\ qpat_x_assum ‘B _ _’ kall_tac
\\ dxrule_all ml_translatorTheory.Eval_Arrow \\ strip_tac
\\ dxrule_all ml_translatorTheory.Eval_Arrow
\\ simp [ml_translatorTheory.Eval_def]
\\ disch_then (qspec_then ‘s.refs’ strip_assume_tac)
\\ dxrule ml_translatorTheory.evaluate_empty_state_IMP
\\ simp [ml_progTheory.eval_rel_def, evaluate_def, Abbr ‘env'’,
ml_progTheory.nsLookup_write]
\\ qpat_x_assum ‘do_partial_app _ _ = _’ mp_tac
\\ simp [do_partial_app_def, Once do_opapp_def, AllCaseEqs (), PULL_EXISTS]
\\ rpt gen_tac \\ strip_tac
\\ rpt gen_tac \\ strip_tac \\ gvs []
\\ qpat_x_assum ‘do_opapp _ = SOME (env,exp)’ mp_tac
\\ simp [do_opapp_cases]
\\ strip_tac \\ gvs [evaluate_def, do_opapp_cases,
evaluateTheory.dec_clock_def]
\\ dxrule_then (qspec_then ‘s.clock’ mp_tac) evaluate_set_init_clock
\\ simp [with_same_clock]
\\ strip_tac \\ gvs []
\\ mp_tac (CONJUNCT1 is_clock_io_mono_evaluate)
\\ qmatch_asmsub_abbrev_tac ‘evaluate s env1 [e]’
\\ disch_then (qspecl_then [`s`,`env1`,`[e]`] mp_tac)
\\ rw [is_clock_io_mono_def]
\\ gs [io_events_mono_antisym]
QED
Triviality GC_T:
GC s = T
Proof
fs [cfHeapsBaseTheory.GC_def,set_sepTheory.SEP_EXISTS_THM]
\\ qexists_tac ‘λx. T’ \\ fs []
QED
Theorem MEM_st2heap: (* TODO: move *)
Mem l x ∈ st2heap p s ⇔
l < LENGTH s.refs ∧ x = EL l s.refs
Proof
PairCases_on ‘p’
\\ fs [cfStoreTheory.st2heap_def,cfStoreTheory.ffi2heap_def,
cfAppTheory.store2heap_MAPi,MEM_MAPi]
\\ IF_CASES_TAC \\ fs []
QED
Theorem HOL_STORE_not_kernel_locs:
(∀loc. loc ∈ kernel_locs ⇒ kernel_loc_ok refs loc s.refs) ∧
l < LENGTH s.refs ∧ l ∉ kernel_locs ⇒
(HOL_STORE refs * one (Mem l (EL l s.refs)) * GC) (st2heap p s)
Proof
fs [ml_monad_translatorBaseTheory.REFS_PRED_def,HOL_STORE_def]
\\ fs [ml_monad_translatorBaseTheory.REF_REL_def,set_sepTheory.SEP_EXISTS_THM,
set_sepTheory.SEP_CLAUSES,cfHeapsBaseTheory.REF_def]
\\ full_simp_tac (std_ss ++ helperLib.sep_cond_ss) [set_sepTheory.cond_STAR]
\\ fs [cfHeapsBaseTheory.cell_def,set_sepTheory.one_STAR,GSYM set_sepTheory.STAR_ASSOC]
\\ fs [GC_T,MEM_st2heap]
\\ fs [kernel_locs_def, SF DNF_ss, MEM_st2heap]
\\ fs [the_type_constants_def,the_term_constants_def,the_axioms_def,
the_context_def,kernel_loc_ok_def,listTheory.oEL_EQ_EL]
\\ metis_tac []
QED
Theorem REFS_PRED_HOL_STORE:
(∀loc. loc ∈ kernel_locs ⇒ kernel_loc_ok refs loc s.refs) ⇒
REFS_PRED (HOL_STORE,p) refs s
Proof
fs [ml_monad_translatorBaseTheory.REFS_PRED_def,HOL_STORE_def]
\\ fs [ml_monad_translatorBaseTheory.REF_REL_def,set_sepTheory.SEP_EXISTS_THM,
set_sepTheory.SEP_CLAUSES,cfHeapsBaseTheory.REF_def]
\\ full_simp_tac (std_ss ++ helperLib.sep_cond_ss) [set_sepTheory.cond_STAR]
\\ fs [cfHeapsBaseTheory.cell_def,set_sepTheory.one_STAR,GSYM set_sepTheory.STAR_ASSOC]
\\ fs [GC_T,MEM_st2heap]
\\ fs [kernel_locs_def, SF DNF_ss]
\\ fs [the_type_constants_def,the_term_constants_def,the_axioms_def,
the_context_def,kernel_loc_ok_def,listTheory.oEL_EQ_EL]
\\ metis_tac []
QED
Theorem HOL_STORE_kernel_loc_ok:
(HOL_STORE r * frame) (st2heap p s) ∧ loc ∈ kernel_locs ⇒
kernel_loc_ok r loc s.refs
Proof
fs [HOL_STORE_def]
\\ simp [ml_monad_translatorBaseTheory.REF_REL_def,set_sepTheory.SEP_EXISTS_THM,
set_sepTheory.SEP_CLAUSES,cfHeapsBaseTheory.REF_def]
\\ full_simp_tac (std_ss ++ helperLib.sep_cond_ss) [set_sepTheory.cond_STAR]
\\ fs [cfHeapsBaseTheory.cell_def,set_sepTheory.one_STAR,GSYM set_sepTheory.STAR_ASSOC]
\\ fs [GC_T,MEM_st2heap]
\\ strip_tac
\\ fs [kernel_loc_ok_def]
\\ fs [kernel_locs_def]
\\ pop_assum (fn th => fs [GSYM th] THEN assume_tac th)
\\ gvs [listTheory.oEL_EQ_EL]
\\ first_x_assum $ irule_at $ Pos hd \\ fs []
\\ fs [the_type_constants_def,
the_term_constants_def,
the_axioms_def,
the_context_def]
QED
Triviality IMP_perms_ok_lemma:
(LIST_TYPE (PAIR_TYPE STRING_TYPE NUM) x1 v ⇒ perms_ok ps v) ∧
(LIST_TYPE (PAIR_TYPE STRING_TYPE TYPE_TYPE) x2 v ⇒ perms_ok ps v) ∧
(LIST_TYPE THM_TYPE x3 v ⇒ perms_ok ps v) ∧
(LIST_TYPE UPDATE_TYPE x4 v ⇒ perms_ok ps v)
Proof
rw []
\\ drule_at (Pos last) LIST_TYPE_perms_ok \\ disch_then irule
\\ rw []
\\ TRY (drule_at (Pos last) PAIR_TYPE_perms_ok \\ disch_then irule)
\\ rw []
\\ imp_res_tac TYPE_TYPE_perms_ok \\ fs []
\\ imp_res_tac TERM_TYPE_perms_ok \\ fs []
\\ imp_res_tac THM_TYPE_perms_ok \\ fs []
\\ imp_res_tac STRING_TYPE_perms_ok \\ fs []
\\ imp_res_tac NUM_perms_ok \\ fs []
\\ imp_res_tac UPDATE_TYPE_perms_ok \\ fs []
QED
val p = “p:('a -> (string |-> ffi)) # (string list #
(string -> word8 list -> word8 list -> ffi -> ffi ffi_result option)) list”
Theorem ArrowM1:
ArrowP F (HOL_STORE,^p) (PURE A) (MONAD B D) f fv ∧
do_opapp [fv; av] = SOME (env, exp) ∧
evaluate (s:'a semanticPrimitives$state) env [exp] = (s', res) ∧
A a av ∧ STATE ctxt refs ∧
(∀loc. loc ∈ kernel_locs ⇒ kernel_loc_ok refs loc s.refs) ∧
(∀loc r. loc ∉ kernel_locs ∧ LLOOKUP s.refs loc = SOME r ⇒ ref_ok ctxt r) ∧
perms_ok kernel_perms av ∧
perms_ok kernel_perms fv ⇒
s.ffi = s'.ffi ∧
((res = Rerr (Rabort Rtimeout_error)) ∨
(res = Rerr (Rabort Rtype_error)) ∨
s.next_type_stamp = s'.next_type_stamp ∧
s.eval_state = s'.eval_state ∧
∃r refs2.
f a refs = (r,refs2) ∧
(∀loc. loc ∈ kernel_locs ⇒ kernel_loc_ok refs2 loc s'.refs) ∧
(∀loc r. loc ∉ kernel_locs ∧ LLOOKUP s'.refs loc = SOME r ⇒ ref_ok ctxt r) ∧
(∀x. r = M_success x ⇒ ∃rv. res = Rval [rv] ∧ B x rv) ∧
(∀x. r = M_failure x ⇒ ∃rv. res = Rerr (Rraise rv) ∧ D x rv))
Proof
fs [ml_monad_translatorTheory.ArrowP_def,ml_monad_translatorTheory.PURE_def,PULL_EXISTS]
\\ strip_tac \\ last_x_assum drule \\ fs []
\\ fs [state_ok_def]
\\ rename [‘STATE ctxt refs’]
\\ disch_then (qspecl_then [‘refs’,‘s’] mp_tac)
\\ impl_tac >- fs [REFS_PRED_HOL_STORE]
\\ disch_then (qspec_then ‘[]’ strip_assume_tac)
\\ fs [ml_monad_translatorBaseTheory.REFS_PRED_FRAME_def]
\\ drule_all perms_ok_do_opapp \\ strip_tac
\\ drule evaluate_kernel_perms
\\ disch_then drule
\\ impl_tac
>- (fs [kernel_perms_def,perms_ok_state_def]
\\ rw [] \\ first_x_assum drule
\\ fs [kernel_loc_ok_def,LLOOKUP_THM]
\\ strip_tac \\ gvs [perms_ok_ref_def]
\\ fs [kernel_locs_def]
\\ gvs [IN_kernel_locs]
\\ imp_res_tac IMP_perms_ok_lemma \\ fs [])
\\ simp [kernel_perms_def]
\\ csimp [GSYM kernel_perms_def]
\\ strip_tac
\\ drule evaluate_set_init_clock
\\ disch_then (qspec_then ‘s.clock’ mp_tac)
\\ impl_tac
>- (strip_tac \\ fs [ml_monad_translatorTheory.MONAD_def]
\\ every_case_tac \\ fs [])
\\ fs []
\\ ‘(s with refs := s.refs) = s’ by fs [state_component_equality]
\\ fs [] \\ reverse strip_tac \\ gvs []
THEN1
(imp_res_tac evaluatePropsTheory.evaluate_io_events_mono_imp
\\ fs [evaluatePropsTheory.io_events_mono_def]
\\ imp_res_tac rich_listTheory.IS_PREFIX_ANTISYM \\ fs [])
\\ gvs [] \\ pop_assum kall_tac
\\ Cases_on ‘res = Rerr (Rabort Rtimeout_error)’ \\ fs []
\\ Cases_on ‘res = Rerr (Rabort Rtype_error)’ \\ fs []
\\ fs [ml_monad_translatorTheory.MONAD_def]
\\ Cases_on ‘f a refs’ \\ fs []
\\ rewrite_tac [CONJ_ASSOC]
\\ reverse conj_tac THEN1 (every_case_tac \\ fs [])
\\ reverse conj_tac THEN1 (every_case_tac \\ fs [])
\\ ‘r = st3’ by (every_case_tac \\ fs [])
\\ rw []
THEN1
(‘REFS_PRED (HOL_STORE,p) refs s’ by fs [REFS_PRED_HOL_STORE]
\\ first_x_assum drule
\\ fs [ml_monad_translatorBaseTheory.REFS_PRED_def]
\\ first_x_assum drule \\ strip_tac
\\ fs [GSYM set_sepTheory.STAR_ASSOC]
\\ drule_all HOL_STORE_kernel_loc_ok \\ fs [])
\\ fs [oEL_THM] \\ rw [] \\ fs []
\\ ‘(HOL_STORE refs * one (Mem loc (EL loc s.refs)) * GC) (st2heap p s)’
by fs [HOL_STORE_not_kernel_locs]
\\ fs [GSYM set_sepTheory.STAR_ASSOC]
\\ first_x_assum drule
\\ fs [GSYM set_sepTheory.STAR_ASSOC]
\\ once_rewrite_tac [set_sepTheory.STAR_COMM]
\\ fs [GSYM set_sepTheory.STAR_ASSOC,set_sepTheory.one_STAR,MEM_st2heap]
\\ strip_tac
\\ qsuff_tac ‘ref_ok ctxt (EL loc s.refs)’ THEN1 asm_rewrite_tac []
\\ fs []
QED
Theorem ArrowM2:
ArrowP F (HOL_STORE,^p) (PURE A1) (ArrowM F (HOL_STORE,p) (PURE A2) (MONAD B D)) f fv ∧
do_partial_app fv a1v = SOME gv ∧
do_opapp [gv; a2v] = SOME (env, exp) ∧
evaluate (s:'a semanticPrimitives$state) env [exp] = (s', res) ∧
A1 a1 a1v ∧ A2 a2 a2v ∧ STATE ctxt refs ∧
(∀loc. loc ∈ kernel_locs ⇒ kernel_loc_ok refs loc s.refs) ∧
(∀loc r. loc ∉ kernel_locs ∧ LLOOKUP s.refs loc = SOME r ⇒ ref_ok ctxt r) ∧
perms_ok kernel_perms a1v ∧
perms_ok kernel_perms a2v ∧
perms_ok kernel_perms fv ⇒
s.ffi = s'.ffi ∧
((res = Rerr (Rabort Rtimeout_error)) ∨
(res = Rerr (Rabort Rtype_error)) ∨
s.next_type_stamp = s'.next_type_stamp ∧
s.eval_state = s'.eval_state ∧
∃r refs2.
f a1 a2 refs = (r,refs2) ∧
(∀loc. loc ∈ kernel_locs ⇒ kernel_loc_ok refs2 loc s'.refs) ∧
(∀loc r. loc ∉ kernel_locs ∧ LLOOKUP s'.refs loc = SOME r ⇒ ref_ok ctxt r) ∧
(∀x. r = M_success x ⇒ ∃rv. res = Rval [rv] ∧ B x rv) ∧
(∀x. r = M_failure x ⇒ ∃rv. res = Rerr (Rraise rv) ∧ D x rv))
Proof
strip_tac \\ irule ArrowM1 \\ fs [SF SFY_ss]
\\ first_assum $ irule_at $ Pos hd
\\ first_assum $ irule_at $ Pos hd
\\ first_assum $ irule_at $ Pos hd \\ fs []
\\ gvs [do_partial_app_def,AllCaseEqs()]
\\ qexists_tac ‘p’
\\ conj_tac THEN1
(simp [Once perms_ok_cases]
\\ pop_assum mp_tac
\\ simp [Once perms_ok_cases]
\\ simp [perms_ok_env_def]
\\ fs [ml_progTheory.nsLookup_nsBind_compute]
\\ strip_tac \\ reverse Cases
\\ fs [ml_progTheory.nsLookup_nsBind_compute]
\\ rw [] \\ fs []
\\ metis_tac [namespaceTheory.id_distinct,namespaceTheory.id_11])
\\ fs [ml_monad_translatorTheory.ArrowP_def]
\\ fs [ml_monad_translatorTheory.ArrowM_def]
\\ fs [ml_monad_translatorTheory.PURE_def,PULL_EXISTS]
\\ rpt strip_tac
\\ last_x_assum drule_all
\\ fs [do_opapp_def]
\\ simp [Once evaluate_def]
\\ fs [ml_monad_translatorTheory.ArrowP_def]
\\ fs [ml_monad_translatorTheory.PURE_def,PULL_EXISTS]
\\ fs [PULL_FORALL] \\ gen_tac
\\ disch_then (qspec_then ‘junk’ strip_assume_tac)
\\ fs [GSYM PULL_FORALL]
\\ first_x_assum drule
\\ disch_then drule
\\ fs [do_opapp_def]
QED
Theorem state_ok_with_clock[simp]:
state_ok ctxt (s with clock := k) ⇔ state_ok ctxt s
Proof
fs [state_ok_def]
QED
Theorem EqualityType_11:
∀abs x v v'. EqualityType abs ∧ abs x v ∧ abs x v' ⇒ v = v'
Proof
fs [ml_translatorTheory.EqualityType_def] \\ metis_tac []
QED
Theorem IMP_EVERY_v_ok:
(∀loc. if loc ∈ kernel_locs then kernel_loc_ok refs loc s'.refs
else LLOOKUP s'.refs loc = LLOOKUP s.refs loc) ∧
(∀loc. loc ∈ kernel_locs ⇒ kernel_loc_ok refs loc s.refs) ∧
EVERY (ref_ok ctxt) s.refs ∧ STATE ctxt refs ⇒
EVERY (ref_ok ctxt) s'.refs
Proof
rw [EVERY_EL]
\\ last_x_assum $ qspec_then ‘n’ mp_tac
\\ reverse (rw [])
>- (gvs [LLOOKUP_THM] \\ res_tac \\ fs [] \\ gvs [] \\ metis_tac [])
\\ last_x_assum drule
\\ strip_tac
\\ fs [kernel_loc_ok_def]
\\ gvs [IN_kernel_locs]
\\ gvs [LLOOKUP_THM,ref_ok_def]
\\ last_x_assum drule
\\ fs [ref_ok_def]
\\ ‘EqualityType (LIST_TYPE (PAIR_TYPE STRING_TYPE NUM)) ∧
EqualityType (LIST_TYPE (PAIR_TYPE STRING_TYPE TYPE_TYPE)) ∧
EqualityType (LIST_TYPE THM_TYPE) ∧
EqualityType (LIST_TYPE UPDATE_TYPE)’ by
(rw []
\\ irule cfLetAutoTheory.EqualityType_LIST_TYPE
\\ fs [EqualityType_THM_TYPE]
\\ fs [EqualityType_UPDATE_TYPE]
\\ TRY (irule cfLetAutoTheory.EqualityType_PAIR_TYPE)
\\ fs [EqualityType_TYPE_TYPE]
\\ fs [ml_translatorTheory.EqualityType_NUM_BOOL])
\\ imp_res_tac EqualityType_11 \\ fs []
QED
Theorem HOL_EXN_TYPE_Fail_v_ok:
HOL_EXN_TYPE (Failure m) v ⇒ v_ok ctxt v
Proof
Cases_on ‘m’
\\ rw [HOL_EXN_TYPE_def,ml_translatorTheory.STRING_TYPE_def]
\\ irule v_ok_Conv \\ fs []
\\ fs [Once v_ok_cases]
QED
Theorem inferred_APPEND:
!c v. inferred c v ==>
!x s. STATE (x ++ c) s /\ (!th. THM c th ==> THM (x ++ c) th) ==>
inferred (x ++ c) v
Proof
ho_match_mp_tac inferred_strongind
\\ rw[]
\\ rw[Once inferred_cases]
>- metis_tac[TYPE_APPEND_EXTEND]
>- metis_tac[TERM_APPEND_EXTEND]
>- metis_tac[]
QED
Theorem v_ok_APPEND:
(!c v. kernel_vals c v ⇒
!x s. STATE (x ++ c) s /\ (!th. THM c th ==> THM (x ++ c) th)
==> kernel_vals (x ++ c) v) ∧
(!c v. v_ok c v ⇒ ∀x s.
!x s. STATE (x ++ c) s /\ (!th. THM c th ==> THM (x ++ c) th)
==> v_ok (x ++ c) v) ∧
(∀c e. env_ok c e ⇒ ∀x s.
!x s. STATE (x ++ c) s /\ (!th. THM c th ==> THM (x ++ c) th)
==> env_ok (x ++ c) e)
Proof
ho_match_mp_tac v_ok_strongind
\\ rw[]
\\ rw[Once v_ok_cases]
>- metis_tac[inferred_APPEND]
\\ fs[EVERY_MEM] \\ metis_tac[]
QED
Theorem ref_ok_APPEND:
!c v. ref_ok c v ⇒ ∀x s.
!x s. STATE (x ++ c) s /\ (!th. THM c th ==> THM (x ++ c) th)
==> ref_ok (x ++ c) v
Proof
gen_tac \\ Cases \\ rw[ref_ok_def]
\\ fs[EVERY_MEM] \\ metis_tac[v_ok_APPEND]
QED
Theorem inferred_ok:
inferred ctxt f ∧
state_ok ctxt s ∧
v_ok ctxt v ∧
do_opapp [f; v] = SOME (env, exp) ∧
evaluate (s:'ffi state) env [exp] = (s', res) ⇒
(∃abort. s'.ffi = s.ffi ∧ res = Rerr (Rabort abort)) ∨
∃ctxt'.
state_ok ctxt' s' ∧
(∀v. v_ok ctxt v ⇒ v_ok ctxt' v) ∧
(∀v. kernel_vals ctxt v ⇒ kernel_vals ctxt' v) ∧
(∀vs. res = Rval vs ⇒ EVERY (v_ok ctxt') vs) ∧
(∀v. res = Rerr (Rraise v) ⇒ v_ok ctxt' v)
Proof
rw [Once inferred_cases]
>~ [‘TYPE ctxt ty’] >- (
Cases_on ‘ty’ \\ gs [TYPE_TYPE_def, do_opapp_cases])
>~ [‘TERM ctxt tm’] >- (
Cases_on ‘tm’ \\ gs [TERM_TYPE_def, do_opapp_cases])
>~ [‘THM ctxt th’] >- (
Cases_on ‘th’ \\ gs [THM_TYPE_def, do_opapp_cases])
\\ rename [‘f ∈ kernel_funs’]
\\ Cases_on ‘f ∈ { call_type_subst_v; call_freesin_v; call_vfree_in_v;
call_variant_v; vsubst_v; inst_v; trans_v; abs_v; eq_mp_v;
deduct_antisym_rule_v; inst_type_v; inst_1_v; trans_v;
compute_add_v; compute_v }’ THEN1
(gvs []
\\ qpat_x_assum ‘do_opapp _ = _’ mp_tac
\\ last_x_assum mp_tac
\\ rewrite_tac [kernel_funs_v_def]
\\ rename [‘Closure env1 a1 (Fun a2 ee)’]
\\ fs [do_opapp_def] \\ rw [] \\ gvs [evaluate_def]
\\ qexists_tac ‘ctxt’ \\ fs []
\\ irule_at (Pos last) v_ok_KernelVals
\\ irule_at Any v_ok_PartialApp
\\ first_assum $ irule_at $ Pos last
\\ irule_at Any v_ok_Inferred
\\ irule_at Any inferred_KernelFuns
\\ first_assum $ irule_at Any
\\ fs [do_partial_app_def])
\\ gvs [kernel_funs_def]
\\ rpt (qpat_x_assum ‘_ ≠ (_:v)’ kall_tac)
>~ [‘do_opapp [concl_v; v]’] >-
(drule_all concl_v_head \\ strip_tac \\ gvs []
\\ TRY (first_x_assum $ irule_at Any \\ gvs [] \\ NO_TAC)
\\ rename [‘THM_TYPE_HEAD _’]
\\ assume_tac concl_v_thm
\\ drule_all v_ok_THM_TYPE_HEAD \\ strip_tac
\\ (drule_then $ drule_then $ drule_then $ drule) Arrow1
\\ impl_tac
THEN1 (imp_res_tac THM_TYPE_perms_ok \\ fs [])
\\ strip_tac \\ gvs []
\\ qexists_tac ‘ctxt’ \\ fs []
\\ fs [state_ok_def]
\\ irule_at (Pos last) v_ok_KernelVals
\\ irule_at Any v_ok_Inferred
\\ irule_at Any inferred_TERM
\\ first_assum $ irule_at Any
\\ irule_at Any (holKernelProofTheory.concl_thm |> GEN_ALL |> SIMP_RULE std_ss [])
\\ imp_res_tac v_ok_THM \\ fs []
\\ first_assum $ irule_at Any
\\ first_assum $ irule_at Any
\\ fs [SF SFY_ss])
>~ [‘do_opapp [beta_v; v]’] >-
(drule_all beta_v_head \\ strip_tac \\ gvs []
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ drule_all v_ok_TERM_TYPE_HEAD \\ strip_tac
\\ imp_res_tac v_ok_TERM \\ fs []
\\ assume_tac beta_v_thm
\\ fs [state_ok_def]
\\ ‘perms_ok kernel_perms v ∧ perms_ok kernel_perms beta_v’ by
fs [TERM_TYPE_perms_ok, SF SFY_ss]
\\ drule_all ArrowM1 \\ strip_tac \\ fs []
\\ disj2_tac
\\ drule_all holKernelProofTheory.BETA_thm \\ strip_tac \\ gvs []
\\ qexists_tac ‘ctxt’ \\ fs []
\\ fs [PULL_EXISTS]
\\ first_assum $ irule_at $ Pos $ hd
\\ fs [SF SFY_ss]
\\ Cases_on ‘r’ \\ fs []
\\ imp_res_tac THM_IMP_v_ok \\ gvs []
\\ rename [‘M_failure ff’] \\ Cases_on ‘ff’ \\ fs []
\\ fs [HOL_EXN_TYPE_Fail_v_ok, SF SFY_ss])
>~ [‘do_opapp [types_v; v]’] >- (
drule_all types_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac types_v_thm
\\ fs[state_ok_def]
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms types_v`
by simp[UNIT_TYPE_perms_ok]
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ fs[holKernelTheory.types_def]
\\ fs[holKernelTheory.get_the_type_constants_def]
\\ gvs[]
\\ conj_tac >- metis_tac[]
\\ drule_then irule LIST_TYPE_v_ok
\\ rw[EVERY_MEM]
\\ drule_then irule PAIR_TYPE_v_ok
\\ conj_tac >- MATCH_ACCEPT_TAC NUM_v_ok
\\ MATCH_ACCEPT_TAC STRING_TYPE_v_ok
)
>~ [‘do_opapp [get_type_arity_v; v]’] >- (
drule_all get_type_arity_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac get_type_arity_v_thm
\\ fs[state_ok_def, STRING_TYPE_HEAD_def]
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms get_type_arity_v`
by simp[STRING_TYPE_perms_ok, SF SFY_ss]
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ fs [SF SFY_ss]
\\ fs[holKernelTheory.get_type_arity_def, ml_monadBaseTheory.st_ex_bind_def]
\\ fs[holKernelTheory.get_the_type_constants_def]
\\ drule assoc_thm
\\ strip_tac
\\ fs[SF SFY_ss]
\\ Cases_on`r` \\ fs[NUM_v_ok, SF SFY_ss]
\\ rename [‘M_failure ff’] \\ Cases_on ‘ff’ \\ fs []
\\ fs [HOL_EXN_TYPE_Fail_v_ok, SF SFY_ss])
>~ [‘do_opapp [call_new_type_v; v]’] >- (
drule_all call_new_type_v_head \\ strip_tac \\ gvs[]
>- (first_assum $ irule_at Any \\ simp[])
\\ assume_tac call_new_type_v_thm
\\ fs[state_ok_def]
\\ `∃pa. PAIR_TYPE STRING_TYPE INT pa v`
by (
irule v_ok_PAIR_TYPE_HEAD
\\ first_assum $ irule_at Any
\\ first_assum $ irule_at Any
\\ simp[STRING_TYPE_HEAD_def, INT_HEAD_def])
\\ PairCases_on`pa`
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms call_new_type_v`
by (
simp[]
\\ drule_at_then Any irule PAIR_TYPE_perms_ok
\\ simp[STRING_TYPE_perms_ok, INT_perms_ok, SF SFY_ss])
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ reverse(Cases_on`0 <= pa1` \\ fs[holKernelTheory.raise_Failure_def])
>- (
gvs[]
\\ first_assum $ irule_at Any
\\ simp[SF SFY_ss]
\\ drule_then irule HOL_EXN_TYPE_Fail_v_ok )
\\ drule new_type_thm
\\ disch_then(qspecl_then[`pa0`,`Num(ABS pa1)`]mp_tac)
\\ simp[]
\\ reverse TOP_CASE_TAC \\ simp[]
>- (
strip_tac \\ gvs[]
\\ first_assum $ irule_at Any
\\ simp[SF SFY_ss]
\\ rename1 `M_failure ff`
\\ Cases_on`ff` \\ gvs[]
\\ drule_then irule HOL_EXN_TYPE_Fail_v_ok )
\\ strip_tac \\ gvs[ml_translatorTheory.UNIT_TYPE_def,v_ok_Conv_NONE]
\\ first_assum $ irule_at Any
\\ simp[]
\\ reverse conj_tac >- metis_tac[v_ok_APPEND, CONS_APPEND]
\\ metis_tac[ref_ok_APPEND, CONS_APPEND])
>~ [‘do_opapp [mk_type_v; v]’] >- (
drule_all mk_type_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac mk_type_v_thm
\\ fs[state_ok_def]
\\ `∃pa. PAIR_TYPE STRING_TYPE (LIST_TYPE TYPE_TYPE) pa v`
by (
irule v_ok_PAIR_TYPE_HEAD
\\ first_assum $ irule_at Any
\\ first_assum $ irule_at Any
\\ simp[STRING_TYPE_HEAD_def]
\\ rpt strip_tac \\ irule v_ok_LIST_TYPE_HEAD
\\ first_assum $ irule_at Any
\\ first_assum $ irule_at Any
\\ MATCH_ACCEPT_TAC v_ok_TYPE_TYPE_HEAD )
\\ PairCases_on`pa`
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms mk_type_v`
by (
simp[]
\\ drule_at_then Any irule PAIR_TYPE_perms_ok
\\ simp[STRING_TYPE_perms_ok, LIST_TYPE_TYPE_perms_ok, SF SFY_ss])
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ fs [SF SFY_ss]
\\ fs[ml_translatorTheory.PAIR_TYPE_def]
\\ rveq \\ fs[v_ok_Conv_NONE]
\\ drule_then drule v_ok_LIST_TYPE
\\ strip_tac
\\ drule_all mk_type_thm \\ strip_tac
\\ Cases_on`r` \\ fs[]
>- metis_tac[TYPE_IMP_v_ok]
\\ rename [‘M_failure ff’] \\ Cases_on ‘ff’ \\ fs []
\\ fs [HOL_EXN_TYPE_Fail_v_ok, SF SFY_ss])
>~ [‘do_opapp [mk_vartype_v; v]’] >- (
drule_all mk_vartype_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac mk_vartype_v_thm
\\ fs[state_ok_def]
\\ `perms_ok ∅ v ∧ perms_ok ∅ mk_vartype_v`
by simp[STRING_TYPE_perms_ok, SF SFY_ss]
\\ drule_all Arrow1 \\ strip_tac \\ fs[]
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss]
\\ irule TYPE_IMP_v_ok
\\ first_assum $ irule_at $ Any
\\ EVAL_TAC)
>~ [‘do_opapp [dest_type_v; v]’] >- (
drule_all dest_type_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac dest_type_v_thm
\\ fs[state_ok_def]
\\ drule_all v_ok_TYPE_TYPE_HEAD \\ strip_tac
\\ imp_res_tac v_ok_TYPE \\ fs []
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms dest_type_v`
by simp[TYPE_TYPE_perms_ok, SF SFY_ss]
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ fs [SF SFY_ss]
\\ drule_all dest_type_thm \\ strip_tac
\\ Cases_on ‘r’ \\ fs []
>- (
drule_then irule PAIR_TYPE_v_ok
\\ rw[STRING_TYPE_v_ok, SF SFY_ss]
\\ drule_then irule LIST_TYPE_v_ok
\\ fs[EVERY_MEM]
\\ metis_tac[TYPE_IMP_v_ok])
\\ rename [‘M_failure ff’] \\ Cases_on ‘ff’ \\ fs []
\\ fs [HOL_EXN_TYPE_Fail_v_ok, SF SFY_ss])
>~ [‘do_opapp [dest_vartype_v; v]’] >- (
drule_all dest_vartype_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac dest_vartype_v_thm
\\ fs[state_ok_def]
\\ drule_all v_ok_TYPE_TYPE_HEAD \\ strip_tac
\\ imp_res_tac v_ok_TYPE \\ fs []
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms dest_vartype_v`
by simp[TYPE_TYPE_perms_ok, SF SFY_ss]
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ fs [SF SFY_ss]
\\ drule_all dest_vartype_thm \\ strip_tac
\\ Cases_on ‘r’ \\ fs []
>- rw[STRING_TYPE_v_ok, SF SFY_ss]
\\ rename [‘M_failure ff’] \\ Cases_on ‘ff’ \\ fs []
\\ fs [HOL_EXN_TYPE_Fail_v_ok, SF SFY_ss]
\\ fs[holKernelTheory.dest_vartype_def]
\\ Cases_on`ty` \\ fs[ml_monadBaseTheory.st_ex_return_def]
\\ fs[holKernelTheory.raise_Failure_def])
>~ [‘do_opapp [is_type_v; v]’] >- (
drule_all is_type_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac is_type_v_thm
\\ fs[state_ok_def]
\\ drule_all v_ok_TYPE_TYPE_HEAD \\ strip_tac
\\ `perms_ok ∅ v ∧ perms_ok ∅ is_type_v`
by simp[TYPE_TYPE_perms_ok, SF SFY_ss]
\\ drule_all Arrow1 \\ strip_tac \\ fs[]
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss, BOOL_v_ok])
>~ [‘do_opapp [is_vartype_v; v]’] >- (
drule_all is_vartype_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac is_vartype_v_thm
\\ fs[state_ok_def]
\\ drule_all v_ok_TYPE_TYPE_HEAD \\ strip_tac
\\ `perms_ok ∅ v ∧ perms_ok ∅ is_vartype_v`
by simp[TYPE_TYPE_perms_ok, SF SFY_ss]
\\ drule_all Arrow1 \\ strip_tac \\ fs[]
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss, BOOL_v_ok])
>~ [‘do_opapp [call_tyvars_v; v]’] >- (
drule_all call_tyvars_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac call_tyvars_v_thm
\\ fs[state_ok_def]
\\ drule_all v_ok_TYPE_TYPE_HEAD \\ strip_tac
\\ `perms_ok ∅ v ∧ perms_ok ∅ call_tyvars_v`
by simp[TYPE_TYPE_perms_ok, SF SFY_ss]
\\ drule_all Arrow1
\\ drule_all Arrow1 \\ strip_tac \\ fs[]
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss]
\\ drule_then irule LIST_TYPE_v_ok
\\ simp[STRING_TYPE_v_ok, SF SFY_ss])
>~ [‘do_opapp [constants_v; v]’] >- (
drule_all constants_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac constants_v_thm
\\ fs[state_ok_def]
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms constants_v`
by simp[UNIT_TYPE_perms_ok]
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ fs[holKernelTheory.constants_def]
\\ fs[holKernelTheory.get_the_term_constants_def]
\\ gvs[SF SFY_ss]
\\ drule_then irule LIST_TYPE_v_ok
\\ rw[EVERY_MEM]
\\ drule_then irule PAIR_TYPE_v_ok
\\ conj_tac >- MATCH_ACCEPT_TAC STRING_TYPE_v_ok
\\ rw[]
\\ irule TYPE_IMP_v_ok
\\ first_assum $ irule_at $ Any
\\ drule the_term_constants_TYPE
\\ simp[EVERY_MEM, FORALL_PROD]
\\ Cases_on`x` \\ fs[SF SFY_ss])
>~ [‘do_opapp [get_const_type_v; v]’] >- (
drule_all get_const_type_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac get_const_type_v_thm
\\ fs[state_ok_def]
\\ fs[STRING_TYPE_HEAD_def]
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms get_const_type_v`
by simp[STRING_TYPE_perms_ok, SF SFY_ss]
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ fs[holKernelTheory.get_const_type_def, ml_monadBaseTheory.st_ex_bind_def]
\\ fs[holKernelTheory.get_the_term_constants_def]
\\ drule assoc_thm
\\ strip_tac
\\ fs[SF SFY_ss]
\\ drule the_term_constants_TYPE
\\ simp[EVERY_MEM, FORALL_PROD] \\ strip_tac
\\ Cases_on`r` \\ fs[]
>- (drule_at_then Any irule TYPE_IMP_v_ok \\ simp[SF SFY_ss])
\\ rename [‘M_failure ff’] \\ Cases_on ‘ff’ \\ fs []
\\ fs [HOL_EXN_TYPE_Fail_v_ok, SF SFY_ss])
>~ [‘do_opapp [new_constant_v; v]’] >- (
drule_all new_constant_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac new_constant_v_thm
\\ fs[state_ok_def]
\\ `∃pa. PAIR_TYPE STRING_TYPE TYPE_TYPE pa v`
by (
irule v_ok_PAIR_TYPE_HEAD
\\ first_assum $ irule_at Any
\\ first_assum $ irule_at Any
\\ simp[STRING_TYPE_HEAD_def]
\\ MATCH_ACCEPT_TAC v_ok_TYPE_TYPE_HEAD )
\\ PairCases_on`pa`
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms new_constant_v`
by (
simp[]
\\ drule_at_then Any irule PAIR_TYPE_perms_ok
\\ simp[STRING_TYPE_perms_ok, TYPE_TYPE_perms_ok, SF SFY_ss])
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ fs[ml_translatorTheory.PAIR_TYPE_def]
\\ fs[v_ok_Conv_NONE]
\\ drule_then drule v_ok_TYPE \\ strip_tac
\\ drule_then drule new_constant_thm
\\ disch_then(qspec_then`pa0`mp_tac) \\ simp[]
\\ reverse TOP_CASE_TAC \\ simp[] \\ strip_tac \\ fs[ml_translatorTheory.UNIT_TYPE_def, v_ok_Conv_NONE]
>- (
first_assum $ irule_at Any
\\ fs[SF SFY_ss]
\\ rename1 `M_failure ff`
\\ Cases_on`ff` \\ gvs[]
\\ drule_then irule HOL_EXN_TYPE_Fail_v_ok )
\\ first_assum $ irule_at Any
\\ simp[]
\\ reverse conj_tac >- metis_tac[v_ok_APPEND, CONS_APPEND]
\\ metis_tac[ref_ok_APPEND, CONS_APPEND])
>~ [‘do_opapp [call_type_of_v; v]’] >- (
drule_all call_type_of_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac call_type_of_v_thm
\\ fs[state_ok_def]
\\ drule_all v_ok_TERM_TYPE_HEAD \\ strip_tac
\\ imp_res_tac v_ok_TERM \\ fs []
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms call_type_of_v`
by simp[TERM_TYPE_perms_ok, SF SFY_ss]
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ first_assum $ irule_at $ Pos $ hd
\\ fs [SF SFY_ss]
\\ drule_then drule type_of_thm
\\ strip_tac \\ fs[] \\ gvs[]
\\ drule_at_then Any irule TYPE_IMP_v_ok
\\ drule_at Any term_type
\\ fs[STATE_def] \\ rw[] \\ gvs[])
>~ [‘do_opapp [is_var_v; v]’] >- (
drule_all is_var_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac is_var_v_thm
\\ fs[state_ok_def]
\\ drule_all v_ok_TERM_TYPE_HEAD \\ strip_tac
\\ `perms_ok ∅ v ∧ perms_ok ∅ is_var_v`
by simp[TERM_TYPE_perms_ok, SF SFY_ss]
\\ drule_all Arrow1 \\ strip_tac \\ fs[]
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss, BOOL_v_ok])
>~ [‘do_opapp [is_const_v; v]’] >- (
drule_all is_const_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac is_const_v_thm
\\ fs[state_ok_def]
\\ drule_all v_ok_TERM_TYPE_HEAD \\ strip_tac
\\ `perms_ok ∅ v ∧ perms_ok ∅ is_const_v`
by simp[TERM_TYPE_perms_ok, SF SFY_ss]
\\ drule_all Arrow1 \\ strip_tac \\ fs[]
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss, BOOL_v_ok])
>~ [‘do_opapp [is_abs_v; v]’] >- (
drule_all is_abs_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac is_abs_v_thm
\\ fs[state_ok_def]
\\ drule_all v_ok_TERM_TYPE_HEAD \\ strip_tac
\\ `perms_ok ∅ v ∧ perms_ok ∅ is_abs_v`
by simp[TERM_TYPE_perms_ok, SF SFY_ss]
\\ drule_all Arrow1 \\ strip_tac \\ fs[]
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss, BOOL_v_ok])
>~ [‘do_opapp [is_comb_v; v]’] >- (
drule_all is_comb_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac is_comb_v_thm
\\ fs[state_ok_def]
\\ drule_all v_ok_TERM_TYPE_HEAD \\ strip_tac
\\ `perms_ok ∅ v ∧ perms_ok ∅ is_comb_v`
by simp[TERM_TYPE_perms_ok, SF SFY_ss]
\\ drule_all Arrow1 \\ strip_tac \\ fs[]
\\ qexists_tac`ctxt` \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss, BOOL_v_ok])
>~ [‘do_opapp [mk_var_v; v]’] >- (
drule_all mk_var_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac mk_var_v_thm
\\ fs[state_ok_def]
\\ `∃pa. PAIR_TYPE STRING_TYPE TYPE_TYPE pa v`
by (
irule v_ok_PAIR_TYPE_HEAD
\\ first_assum $ irule_at Any
\\ first_assum $ irule_at Any
\\ simp[STRING_TYPE_HEAD_def]
\\ MATCH_ACCEPT_TAC v_ok_TYPE_TYPE_HEAD )
\\ PairCases_on`pa`
\\ `perms_ok ∅ v ∧ perms_ok ∅ mk_var_v` by (
simp[]
\\ drule_at_then Any irule PAIR_TYPE_perms_ok
\\ simp[STRING_TYPE_perms_ok, TYPE_TYPE_perms_ok, SF SFY_ss])
\\ drule_all Arrow1 \\ strip_tac \\ fs[]
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss]
\\ irule TERM_IMP_v_ok
\\ first_assum $ irule_at $ Any
\\ irule mk_var_thm
\\ simp[SF SFY_ss]
\\ fs[ml_translatorTheory.PAIR_TYPE_def]
\\ rveq \\ fs[v_ok_Conv_NONE]
\\ drule_then (drule_then irule) v_ok_TYPE)
>~ [‘do_opapp [mk_const_v; v]’] >- (
drule_all mk_const_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac mk_const_v_thm
\\ fs[state_ok_def]
\\ `∃pa. PAIR_TYPE STRING_TYPE (LIST_TYPE (PAIR_TYPE TYPE_TYPE TYPE_TYPE)) pa v`
by (
irule v_ok_PAIR_TYPE_HEAD
\\ first_assum $ irule_at Any
\\ first_assum $ irule_at Any
\\ simp[STRING_TYPE_HEAD_def]
\\ rpt strip_tac
\\ irule v_ok_LIST_PAIR_TYPE_HEAD
\\ first_assum $ irule_at Any
\\ first_assum $ irule_at Any
\\ MATCH_ACCEPT_TAC v_ok_TYPE_TYPE_HEAD )
\\ PairCases_on`pa`
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms mk_const_v` by (
simp[]
\\ drule_at_then Any irule PAIR_TYPE_perms_ok
\\ simp[STRING_TYPE_perms_ok, SF SFY_ss]
\\ rpt strip_tac \\ drule_at_then Any irule LIST_TYPE_perms_ok
\\ rpt strip_tac \\ drule_at_then Any irule PAIR_TYPE_perms_ok
\\ MATCH_ACCEPT_TAC TYPE_TYPE_perms_ok)
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss]
\\ drule_then drule mk_const_thm
\\ fs[ml_translatorTheory.PAIR_TYPE_def]
\\ rveq \\ fs[v_ok_Conv_NONE]
\\ impl_tac
>- (
`EVERY ((λctxt p. TYPE ctxt (FST p) ∧ TYPE ctxt (SND p)) ctxt) pa1`
suffices_by rw[EVERY_MEM, FORALL_PROD]
\\ drule_at_then Any irule v_ok_LIST
\\ simp[ml_translatorTheory.PAIR_TYPE_def, FORALL_PROD, PULL_EXISTS]
\\ rw[v_ok_Conv_NONE]
\\ metis_tac[v_ok_TYPE])
\\ strip_tac
\\ Cases_on`r` \\ gvs[]
>- simp[SF SFY_ss, TERM_IMP_v_ok]
\\ rename [‘M_failure ff’] \\ Cases_on ‘ff’ \\ fs []
\\ fs [HOL_EXN_TYPE_Fail_v_ok, SF SFY_ss])
>~ [‘do_opapp [mk_abs_v; v]’] >- (
drule_all mk_abs_v_head \\ strip_tac \\ gvs[]
>- (qexists_tac ‘ctxt’ \\ fs [])
\\ assume_tac mk_abs_v_thm
\\ fs[state_ok_def]
\\ `∃pa. PAIR_TYPE TERM_TYPE TERM_TYPE pa v`
by (
irule v_ok_PAIR_TYPE_HEAD
\\ first_assum $ irule_at Any
\\ first_assum $ irule_at Any
\\ MATCH_ACCEPT_TAC v_ok_TERM_TYPE_HEAD )
\\ PairCases_on`pa`
\\ `perms_ok kernel_perms v ∧ perms_ok kernel_perms mk_abs_v` by (
simp[]
\\ drule_at_then Any irule PAIR_TYPE_perms_ok
\\ MATCH_ACCEPT_TAC TERM_TYPE_perms_ok)
\\ drule_all ArrowM1 \\ strip_tac \\ fs[]
\\ disj2_tac
\\ first_assum $ irule_at $ Pos $ hd
\\ simp[SF SFY_ss]
\\ drule mk_abs_thm
\\ fs[ml_translatorTheory.PAIR_TYPE_def]
\\ rveq \\ fs[v_ok_Conv_NONE]
\\ imp_res_tac v_ok_TERM