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
|
open bossLib HolKernel boolLib listTheory rich_listTheory optionTheory combinTheory pairTheory;
open ottLib ottTheory caml_typedefTheory;
open utilTheory basicTheory environmentTheory validTheory weakenTheory shiftTheory type_substTheory;
open type_substsTheory;
val _ = new_theory "teq";
val EVERY_ZIP_SAME = Q.prove (
`!l f. EVERY (\(x, y). f x y) (ZIP (l, l)) = EVERY (\x. f x x) l`,
Induct THEN SRW_TAC [] []);
val EVERY_ZIP_SWAP = Q.prove (
`!l1 l2 f. (LENGTH l1 = LENGTH l2) ==>
(EVERY (\(x, y). f x y) (ZIP (l1, l2)) = EVERY (\(x, y). f y x) (ZIP (l2, l1)))`,
Induct THEN SRW_TAC [] [] THEN Cases_on `l2` THEN SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) []);
val EVERY_T = Q.prove (
`!l. EVERY (\x. T) l = T`,
Induct THEN SRW_TAC [] []);
local
val lem1 = Q.prove (
`!E. value_env E ==> ~MEM EB_tv E`,
Induct THEN SRW_TAC [] [value_env_def] THEN Cases_on `h` THEN FULL_SIMP_TAC (srw_ss ()) [value_env_def]);
in
val teq_thm = Q.store_thm ("teq_thm",
`(!Tsigma E e t. JTe Tsigma E e t ==> !t'. JTeq E t t' ==> JTe Tsigma E e t') /\
(!Tsigma E pm t1 t2. JTpat_matching Tsigma E pm t1 t2 ==>
!t'. JTeq E t2 t' ==> JTpat_matching Tsigma E pm t1 t') /\
(!Tsigma E e E'. JTlet_binding Tsigma E e E' ==> T) /\
(!Tsigma E e E'. JTletrec_binding Tsigma E e E' ==> T)`,
RULE_INDUCT_TAC JTe_sind [JTe_fun]
[([``"JTe_uprim"``, ``"JTe_bprim"``, ``"JTe_ident"``, ``"JTe_constant"``, ``"JTe_typed"``, ``"JTe_cons"``,
``"JTe_and"``, ``"JTe_or"``, ``"JTe_while"``, ``"JTe_for"``, ``"JTe_assert"``, ``"JTe_location"``,
``"JTe_record_proj"``, ``"JTe_function"``],
METIS_TAC [JTeq_rules]),
([``"JTe_match"``],
METIS_TAC []),
([``"JTe_tuple"``, ``"JTe_construct"``],
SRW_TAC [] [] THEN Q.EXISTS_TAC `e_t_list` THEN SRW_TAC [] [] THEN METIS_TAC [JTeq_rules]),
([``"JTe_record_constr"``],
SRW_TAC [] [] THEN MAP_EVERY Q.EXISTS_TAC [`field_name'_list`, `t'_list`, `field_name_e_t_list`] THEN
SRW_TAC [] [] THEN METIS_TAC [JTeq_rules]),
([``"JTe_record_with"``],
SRW_TAC [] [] THEN Q.EXISTS_TAC `field_name_e_t_list` THEN SRW_TAC [] [] THEN METIS_TAC [JTeq_rules]),
([``"JTe_apply"``],
METIS_TAC [JTeq_rules, ok_thm]),
([``"JTe_assertfalse"``],
SRW_TAC [] [JTconst_cases] THEN METIS_TAC [ok_ok_thm, JTeq_rules, teq_ok_thm]),
([``"JTe_let_mono"``],
SRW_TAC [] [] THEN DISJ1_TAC THEN Q.EXISTS_TAC `x_t_list` THEN SRW_TAC [] [] THEN1 METIS_TAC [] THEN
FULL_SIMP_TAC (srw_ss()) [REVERSE_EQ, EB_vn_list_thm] THEN SRW_TAC [] [] THEN
`~MEM EB_tv (REVERSE (MAP (\z. EB_vn (FST z) (TS_forall (shiftt 0 1 (SND z)))) x_t_list))` by
SRW_TAC [] [MEM_REVERSE, MEM_MAP] THEN
METIS_TAC [weak_teq_thm, ok_thm, APPEND, APPEND_ASSOC, ok_ok_thm]),
([``"JTe_let_poly"``],
SRW_TAC [] [] THEN DISJ2_TAC THEN Q.EXISTS_TAC `x_t_list` THEN SRW_TAC [] [] THEN1 METIS_TAC [] THEN
FULL_SIMP_TAC (srw_ss()) [REVERSE_EQ, EB_vn_list_thm] THEN SRW_TAC [] [] THEN
`~MEM EB_tv (REVERSE (MAP (\z. EB_vn (FST z) (TS_forall (SND z))) x_t_list))` by
SRW_TAC [] [MEM_REVERSE, MEM_MAP] THEN
METIS_TAC [weak_teq_thm, ok_thm, APPEND, APPEND_ASSOC, ok_ok_thm]),
([``"JTe_letrec"``],
SRW_TAC [] [] THEN Q.EXISTS_TAC `x_t_list` THEN SRW_TAC [] [] THEN
`~MEM EB_tv (REVERSE (MAP (\z. EB_vn (FST z) (TS_forall (SND z))) x_t_list))` by
SRW_TAC [] [MEM_REVERSE, MEM_MAP] THEN
METIS_TAC [weak_teq_thm, ok_thm, APPEND, APPEND_ASSOC, ok_ok_thm]),
([``"JTpat_matching_pm"``],
SRW_TAC [] [] THEN Q.EXISTS_TAC `pattern_e_E_list` THEN SRW_TAC [] [] THEN
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN SRW_TAC [] [] THEN
METIS_TAC [weak_teq_thm, ok_thm, APPEND, APPEND_ASSOC, ok_ok_thm, pat_env_lem, lem1])]);
end;
val num_abbrev_def = Define
`(num_abbrev [] = 0:num) /\
(num_abbrev (EB_ta x y z :: E) = 1 + num_abbrev E) /\
(num_abbrev (_ :: E) = num_abbrev E)`;
local
val lem1 = Q.prove (
`!l x. MEM x l ==> typexpr_size x <= typexpr1_size l`,
Induct THEN SRW_TAC [] [typexpr_size_def] THEN RES_TAC THEN DECIDE_TAC);
in
val apply_abbrev_def = tDefine "apply_abbrev"
`(apply_abbrev tpo tcn t (TE_var tv) = TE_var tv) /\
(apply_abbrev tpo tcn t (TE_idxvar i1 i2) = TE_idxvar i1 i2) /\
(apply_abbrev tpo tcn t TE_any = TE_any) /\
(apply_abbrev tpo tcn t (TE_arrow t1 t2) =
TE_arrow (apply_abbrev tpo tcn t t1) (apply_abbrev tpo tcn t t2)) /\
(apply_abbrev tpo tcn t (TE_tuple ts) = TE_tuple (MAP (apply_abbrev tpo tcn t) ts)) /\
(apply_abbrev tpo tcn t (TE_constr ts tc) =
if (tc = TC_name tcn) /\ (LENGTH ts = LENGTH tpo) then
substs_typevar_typexpr (ZIP (MAP tp_to_tv tpo, MAP (apply_abbrev tpo tcn t) ts)) t
else
TE_constr (MAP (apply_abbrev tpo tcn t) ts) tc)`
(WF_REL_TAC `measure (\(a, b, c, d). typexpr_size d)` THEN SRW_TAC [] [] THEN IMP_RES_TAC lem1 THEN
DECIDE_TAC);
end;
val remove_abbrev_def = Define
`(remove_abbrev [] = (NONE, [])) /\
(remove_abbrev (EB_ta tpo tcn t :: E) = (SOME (EB_ta tpo tcn t), E)) /\
(remove_abbrev (EB_td tcn k :: E) =
let (a, E') = remove_abbrev E in
(a, EB_td tcn k :: E')) /\
(remove_abbrev (EB_tr tcn k fns :: E) =
let (a, E') = remove_abbrev E in
(a, EB_tr tcn k fns :: E')) /\
(remove_abbrev (EB_tv :: E) =
let (a, E') = remove_abbrev E in
(OPTION_MAP (\EB. shiftEB 0 1 EB) a, EB_tv :: E')) /\
(remove_abbrev (EB_l k t :: E) = remove_abbrev E) /\
(remove_abbrev (EB_fn fn tpo tcn t :: E) = remove_abbrev E) /\
(remove_abbrev (EB_pc cn tpo tl tc :: E) = remove_abbrev E) /\
(remove_abbrev (EB_cc cn tc :: E) = remove_abbrev E) /\
(remove_abbrev (EB_vn vn ts :: E) = remove_abbrev E)`;
val remove_abbrev_result_thm = Q.prove (
`!E. ?E'. (remove_abbrev E = (NONE, E')) \/ ?tpo tcn t. remove_abbrev E = (SOME (EB_ta tpo tcn t), E')`,
Induct THEN SRW_TAC [] [remove_abbrev_def] THEN Cases_on `h` THEN
SRW_TAC [] [remove_abbrev_def, shiftEB_def]);
val remove_abbrev_num_abbrev_thm = Q.prove (
`!E E' a. (remove_abbrev E = (a, E')) ==> (num_abbrev E' = num_abbrev E - 1)`,
Induct THEN SRW_TAC [] [remove_abbrev_def, num_abbrev_def] THEN SRW_TAC [] [num_abbrev_def] THEN
Cases_on `h` THEN FULL_SIMP_TAC (srw_ss ()) [remove_abbrev_def, num_abbrev_def, LET_THM, LAMBDA_PROD2] THEN
SRW_TAC [] [num_abbrev_def] THEN Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) []);
val remove_abbrev_num_abbrev_thm2 = Q.prove (
`!E. (?E'. remove_abbrev E = (NONE, E')) = (num_abbrev E = 0)`,
Induct THEN SRW_TAC [] [remove_abbrev_def, num_abbrev_def] THEN SRW_TAC [] [num_abbrev_def] THEN
Cases_on `h` THEN FULL_SIMP_TAC (srw_ss ()) [remove_abbrev_def, num_abbrev_def, LET_THM, LAMBDA_PROD2] THEN
SRW_TAC [] [num_abbrev_def] THEN Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) []);
val Eok_lem1 = Q.prove (
`!EB E. Eok (EB::E) ==> Eok E`,
METIS_TAC [ok_ok_thm, APPEND]);
val remove_lookup_lem = Q.prove (
`!E tcn E' tpo t. Eok E /\ (remove_abbrev E = (SOME (EB_ta tpo tcn t), E')) ==>
(lookup E (name_tcn tcn) = SOME (EB_ta tpo tcn t))`,
Induct THEN SRW_TAC [] [lookup_def, remove_abbrev_def] THEN Cases_on `h` THEN
FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, lookup_def, domEB_def, LET_THM, LAMBDA_PROD2,
shiftEB_add_thm, Eok_def] THEN
IMP_RES_TAC Eok_lem1 THEN IMP_RES_TAC ok_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [] THEN
Cases_on `E'` THEN FULL_SIMP_TAC (srw_ss ()) [] THEN SRW_TAC [] [] THEN Cases_on `remove_abbrev E` THEN
FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THENL
[METIS_TAC [lookup_dom_thm],
METIS_TAC [lookup_dom_thm],
Cases_on `EB` THEN FULL_SIMP_TAC (srw_ss()) [shiftEB_def]]);
local
val lem1 = Q.prove (
`!EB E t1 t2. Eok (EB::E) /\ ~(EB = EB_tv) /\ JTeq E t1 t2 ==> JTeq (EB::E) t1 t2`,
METIS_TAC [weak_teq_thm, MEM, APPEND]);
val lem2 = Q.prove (
`!E. ?EBopt E'. remove_abbrev E = (EBopt, E')`,
SRW_TAC [] [] THEN Cases_on `remove_abbrev E` THEN METIS_TAC []);
val lem3 = Q.prove (
`!E1 EB E2 tpo tcn t E'.
(remove_abbrev (E1 ++ [EB] ++ E2) = (SOME (EB_ta (TPS_nary tpo) tcn t),E')) /\
(remove_abbrev (E1 ++ E2) = (SOME (EB_ta (TPS_nary tpo) tcn t),E')) ==>
~is_tc_EB EB`,
Induct THEN SRW_TAC [] [remove_abbrev_def, is_tc_EB_def] THENL
[Cases_on `EB` THEN FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, is_tc_EB_def, LET_THM] THEN
SRW_TAC [] [] THEN IMP_RES_TAC remove_abbrev_num_abbrev_thm THEN
`~(num_abbrev E' = 0)` by METIS_TAC [remove_abbrev_num_abbrev_thm2, NOT_SOME_NONE, PAIR_EQ] THEN
DECIDE_TAC,
Cases_on `h` THEN FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, is_tc_EB_def, LET_THM, LAMBDA_PROD2] THENL
[STRIP_ASSUME_TAC (Q.SPEC `E1++[EB]++E2` lem2) THEN
STRIP_ASSUME_TAC (Q.SPEC `E1++E2` lem2) THEN SRW_TAC [] [] THEN Cases_on `EB'` THEN
FULL_SIMP_TAC (srw_ss()) [shiftEB_def] THEN
Cases_on `EB''` THEN FULL_SIMP_TAC (srw_ss()) [shiftEB_def, shiftt_11] THEN
METIS_TAC [SND],
METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
STRIP_ASSUME_TAC (Q.SPEC `E1++[EB]++E2` lem2) THEN
STRIP_ASSUME_TAC (Q.SPEC `E1++E2` lem2) THEN SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [] THEN
SRW_TAC [] [] THEN METIS_TAC [SND],
STRIP_ASSUME_TAC (Q.SPEC `E1++[EB]++E2` lem2) THEN
STRIP_ASSUME_TAC (Q.SPEC `E1++E2` lem2) THEN SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [] THEN
SRW_TAC [] [] THEN METIS_TAC [SND],
SRW_TAC [] [] THEN Cases_on `E1` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
`LENGTH (t++[EB]) = LENGTH t` by METIS_TAC [] THEN
FULL_SIMP_TAC (srw_ss()) [],
METIS_TAC []]]);
val lem4 = Q.prove (
`!E1 EB E2'. Eok (E1++[EB]++E2) /\ ~is_tc_EB EB ==> Eok (E1++E2)`,
METIS_TAC [ok_str_thm, EXISTS_DEF]);
in
val remove_abbrev_split_lem = Q.prove (
`!E EB E'. (remove_abbrev E = (SOME EB, E')) ==>
?E1 EB' E2. ~(EB' = EB_tv) /\ (E = E1++[EB']++E2) /\
((remove_abbrev (E1++E2) = (SOME EB, E')) \/ (E' = E1++E2))`,
Induct THEN SRW_TAC [] [remove_abbrev_def] THEN Cases_on `h` THEN
FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, LET_THM] THEN
STRIP_ASSUME_TAC (Q.SPEC `E` lem2) THEN FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN SRW_TAC [] [] THENL
[MAP_EVERY Q.EXISTS_TAC [`EB_tv::E1`, `EB'''`, `E2`] THEN SRW_TAC [] [remove_abbrev_def],
METIS_TAC [APPEND, APPEND_ASSOC],
MAP_EVERY Q.EXISTS_TAC [`EB_vn v t::E1`, `EB''`, `E2`] THEN SRW_TAC [] [remove_abbrev_def],
MAP_EVERY Q.EXISTS_TAC [`[]`, `EB_vn v t`, `E1++[EB'']++E2`] THEN SRW_TAC [] [remove_abbrev_def],
MAP_EVERY Q.EXISTS_TAC [`EB_cc c t::E1`, `EB''`, `E2`] THEN SRW_TAC [] [remove_abbrev_def],
MAP_EVERY Q.EXISTS_TAC [`[]`, `EB_cc c t`, `E1++[EB'']++E2`] THEN SRW_TAC [] [remove_abbrev_def],
MAP_EVERY Q.EXISTS_TAC [`EB_pc c t t0 t1::E1`, `EB''`, `E2`] THEN SRW_TAC [] [remove_abbrev_def],
MAP_EVERY Q.EXISTS_TAC [`[]`, `EB_pc c t t0 t1`, `E1++[EB'']++E2`] THEN SRW_TAC [] [remove_abbrev_def],
MAP_EVERY Q.EXISTS_TAC [`EB_fn f t t0 t1::E1`, `EB''`, `E2`] THEN SRW_TAC [] [remove_abbrev_def],
MAP_EVERY Q.EXISTS_TAC [`[]`, `EB_fn f t t0 t1`, `E1++[EB'']++E2`] THEN SRW_TAC [] [remove_abbrev_def],
MAP_EVERY Q.EXISTS_TAC [`EB_td t n::E1`, `EB''`, `E2`] THEN SRW_TAC [] [remove_abbrev_def],
METIS_TAC [APPEND, APPEND_ASSOC],
MAP_EVERY Q.EXISTS_TAC [`EB_tr t n l::E1`, `EB''`, `E2`] THEN SRW_TAC [] [remove_abbrev_def],
METIS_TAC [APPEND, APPEND_ASSOC],
MAP_EVERY Q.EXISTS_TAC [`[]`, `EB_ta t t0 t1`, `E`] THEN SRW_TAC [] [],
MAP_EVERY Q.EXISTS_TAC [`EB_l n t::E1`, `EB''`, `E2`] THEN SRW_TAC [] [remove_abbrev_def],
MAP_EVERY Q.EXISTS_TAC [`[]`, `EB_l n t`, `E1++[EB'']++E2`] THEN SRW_TAC [] [remove_abbrev_def]]);
val remove_abbrev_teq_weak_thm = Q.prove (
`!E E' tpo tcn t t1 t2.
Eok E /\ (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t),E')) /\ JTeq E' t1 t2 ==>
JTeq E t1 t2`,
STRIP_TAC THEN Induct_on `LENGTH E` THEN SRW_TAC [] [] THEN Cases_on `E` THEN
FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def] THEN IMP_RES_TAC remove_abbrev_split_lem THENL
[`LENGTH (h::t') = LENGTH (E1++[EB']++E2)` by METIS_TAC [] THEN
FULL_SIMP_TAC (srw_ss()) [] THEN
`LENGTH t' = LENGTH (E1++E2)` by (SRW_TAC [] [] THEN DECIDE_TAC) THEN
IMP_RES_TAC lem3 THEN IMP_RES_TAC lem4 THEN
RES_TAC THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
IMP_RES_TAC weak_teq_thm THEN FULL_SIMP_TAC (srw_ss()) [],
FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
IMP_RES_TAC weak_teq_thm THEN FULL_SIMP_TAC (srw_ss()) []]);
end;
val abbrev_occurs_def = ottDefine "abbrev_occurs"
`(abbrev_occurs tcn (TE_var _) = F) /\
(abbrev_occurs tcn (TE_idxvar _ _) = F) /\
(abbrev_occurs tcn TE_any = F) /\
(abbrev_occurs tcn (TE_arrow t1 t2) = abbrev_occurs tcn t1 \/ abbrev_occurs tcn t2) /\
(abbrev_occurs tcn (TE_tuple ts) = EXISTS (abbrev_occurs tcn) ts) /\
(abbrev_occurs tcn (TE_constr ts tc) = (tc = TC_name tcn) \/ EXISTS (abbrev_occurs tcn) ts)`;
val apply_abbrev_no_occur_thm = Q.prove (
`(!t2 tpo tcn t1. ~abbrev_occurs tcn t2 ==> (apply_abbrev tpo tcn t1 t2 = t2)) /\
(!ts tpo tcn t1. ~EXISTS (abbrev_occurs tcn) ts ==> (MAP (\x. apply_abbrev tpo tcn t1 x) ts = ts))`,
Induct THEN SRW_TAC [] [apply_abbrev_def, abbrev_occurs_def] THEN FULL_SIMP_TAC (srw_ss()) [] THEN
METIS_TAC []);
val abbrev_occur_lem1 = Q.prove (
`(!t tcn E tp l.
~MEM (name_tcn tcn) (MAP domEB E) /\
tkind E (substs_typevar_typexpr (MAP (\tp. (tp_to_tv tp,TE_constr [] TC_unit)) l) t) ==>
~abbrev_occurs tcn t) /\
(!ts tcn E tp l.
~MEM (name_tcn tcn) (MAP domEB E) /\
EVERY (\t. tkind E (substs_typevar_typexpr (MAP (\tp. (tp_to_tv tp,TE_constr [] TC_unit)) l) t)) ts ==>
EVERY (\t. ~abbrev_occurs tcn t) ts)`,
Induct THEN SRW_TAC [] [substs_typevar_typexpr_def, Eok_def, abbrev_occurs_def, EVERY_MAP, o_DEF] THENL
[METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
CCONTR_TAC THEN Cases_on `t` THEN FULL_SIMP_TAC (srw_ss()) [Eok_def, COND_EXPAND_EQ] THEN
IMP_RES_TAC lookup_dom_thm THEN FULL_SIMP_TAC (srw_ss()) [],
METIS_TAC [],
METIS_TAC [],
METIS_TAC []]);
local
val lem2 = Q.prove (
`(!t x y tcn. abbrev_occurs tcn (shiftt x y t) = abbrev_occurs tcn t) /\
(!ts x y tcn. EXISTS (\t. abbrev_occurs tcn (shiftt x y t)) ts = EXISTS (\t. abbrev_occurs tcn t) ts)`,
Induct THEN SRW_TAC [] [abbrev_occurs_def, shiftt_def, EXISTS_MAP]);
in
val abbrev_no_occur_thm = Q.prove (
`!E tpo tcn t E'. Eok E /\ (remove_abbrev E = (SOME (EB_ta tpo tcn t),E')) ==> ~abbrev_occurs tcn t`,
recInduct (fetch "-" "remove_abbrev_ind") THEN
SRW_TAC [] [remove_abbrev_def, LAMBDA_PROD2, LET_THM, Eok_def] THEN
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THENL
[Cases_on `tpo` THEN FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN METIS_TAC [abbrev_occur_lem1],
Cases_on `EB` THEN FULL_SIMP_TAC (srw_ss()) [shiftEB_def] THEN SRW_TAC [] [] THEN METIS_TAC [lem2],
METIS_TAC [ok_ok_thm],
Cases_on `tpo` THEN FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN METIS_TAC [ok_ok_thm],
Cases_on `tpo` THEN Cases_on `tl` THEN Cases_on `tc` THEN Cases_on `l` THEN
FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN Cases_on `l'` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
METIS_TAC [ok_ok_thm],
Cases_on `tc` THEN FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN METIS_TAC [ok_ok_thm],
METIS_TAC [ok_ok_thm]]);
end;
val remove_abbrev_lem1 = Q.prove (
`!E tpo tcn t E'. (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t),E')) /\
MEM (name_tcn typeconstr_name) (MAP domEB E') ==>
MEM (name_tcn typeconstr_name) (MAP domEB E)`,
Induct THEN SRW_TAC [] [remove_abbrev_def] THEN Cases_on `h` THEN
FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, domEB_def, LET_THM, LAMBDA_PROD2] THEN
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
FULL_SIMP_TAC (srw_ss()) [domEB_def] THEN Cases_on `EB` THEN FULL_SIMP_TAC (srw_ss()) [shiftEB_def] THEN
SRW_TAC [] []);
val remove_abbrev_lem2 = Q.prove (
`!E tpo tcn t E' tcn2 EB. (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t),E')) /\ ~(tcn = tcn2) /\
(lookup E (name_tcn tcn2) = SOME EB) ==>
(lookup E' (name_tcn tcn2) = SOME EB)`,
Induct THEN SRW_TAC [] [remove_abbrev_def, lookup_def] THENL
[Cases_on `EB` THEN FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, domEB_def, LET_THM, LAMBDA_PROD2] THEN
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [lookup_def] THEN
FULL_SIMP_TAC (srw_ss()) [domEB_def],
Cases_on `h` THEN FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, domEB_def, LET_THM, LAMBDA_PROD2] THEN
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [lookup_def] THEN
FULL_SIMP_TAC (srw_ss()) [domEB_def] THEN Cases_on `EB` THEN FULL_SIMP_TAC (srw_ss()) [shiftEB_def] THEN
SRW_TAC [] [] THEN METIS_TAC [],
Cases_on `h` THEN
FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, domEB_def, LET_THM, LAMBDA_PROD2, shiftEB_add_thm] THEN
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [lookup_def] THEN
FULL_SIMP_TAC (srw_ss()) [domEB_def, shiftEB_add_thm]]);
local
val lem3 = Q.prove (
`!E tpo tcn t E' idx. (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t),E')) /\
idx_bound E idx ==> idx_bound E' idx`,
Induct THEN SRW_TAC [] [remove_abbrev_def, idx_bound_def] THEN Cases_on `h` THEN
FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, idx_bound_def, LET_THM, LAMBDA_PROD2] THEN
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [idx_bound_def] THEN
Cases_on `idx` THEN FULL_SIMP_TAC (srw_ss()) [idx_bound_def] THEN Cases_on `EB` THEN
FULL_SIMP_TAC (srw_ss()) [shiftEB_def] THEN SRW_TAC [] []);
val lem4 = Q.prove (
`!E tpo tcn t E'.
Eok E /\ (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t),E')) /\
(lookup E (name_tcn tcn) = SOME (EB_ta (TPS_nary tpo) tcn t)) ==>
tkind E' (substs_typevar_typexpr (MAP (\tp. (tp_to_tv tp,TE_constr [] TC_unit)) tpo) t)`,
Induct THEN SRW_TAC [] [lookup_def, remove_abbrev_def] THEN
FULL_SIMP_TAC (srw_ss()) [domEB_def, Eok_def, remove_abbrev_def] THEN Cases_on `h` THEN
FULL_SIMP_TAC (srw_ss()) [domEB_def, Eok_def, remove_abbrev_def, LET_THM, LAMBDA_PROD2, shiftEB_add_thm] THEN
SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [Eok_def] THENL
[Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN Cases_on `EB2` THEN
FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [shiftEB_def] THEN
SRW_TAC [] [] THEN Cases_on `EB` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
FULL_SIMP_TAC (srw_ss()) [shiftEB_def, shiftt_11] THEN SRW_TAC [] [] THEN
`tkind (EB_tv::r)
(shiftt 0 1 (substs_typevar_typexpr (MAP (\tp. (tp_to_tv tp,TE_constr [] TC_unit)) tpo) t1))` by
METIS_TAC [weak_one_tv_ok_thm, APPEND, num_tv_def, APPEND_ASSOC, shiftE_def] THEN
FULL_SIMP_TAC (srw_ss()) [subst_shiftt_com_lem, MAP_MAP, shiftt_def],
METIS_TAC [ok_ok_thm],
Cases_on `t'` THEN FULL_SIMP_TAC (srw_ss()) [Eok_def],
Cases_on `t'` THEN Cases_on `t0` THEN Cases_on `t1` THEN Cases_on `l` THEN
FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN Cases_on `l'` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
METIS_TAC [ok_ok_thm],
Cases_on `t'` THEN FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN METIS_TAC [ok_ok_thm],
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
`Eok (EB_td t' n::r)` by (SRW_TAC [] [Eok_def] THEN METIS_TAC [remove_abbrev_lem1, ok_ok_thm]) THEN
`~MEM EB_tv [EB_td t' n]` by SRW_TAC [] [] THEN METIS_TAC [weak_ok_thm, APPEND],
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
`Eok (EB_tr t' n l::r)` by (SRW_TAC [] [Eok_def] THEN METIS_TAC [remove_abbrev_lem1, ok_ok_thm]) THEN
`~MEM EB_tv [EB_tr t' n l]` by SRW_TAC [] [] THEN METIS_TAC [weak_ok_thm, APPEND],
METIS_TAC [ok_ok_thm]]);
in
val teq_remove_abbrev_ok_thm = Q.prove (
`(!E. Eok E ==> !tpo tcn t E'. (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t),E')) ==> Eok E') /\
(!E tc k. (typeconstr_kind E tc = SOME k) ==>
!tpo tcn t E'. (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t),E')) ==>
if (TC_name tcn = tc) /\ (LENGTH tpo = k) then
EBok E' (EB_ta (TPS_nary tpo) tcn t)
else
(typeconstr_kind E' tc = SOME k)) /\
(!E ts. tsok E ts ==> !tpo tcn t E'. (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t),E')) ==>
Eok E') /\
(!E tps t. ntsok E tps t ==> !tpo tcn t' E'. (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t'),E')) ==>
Eok E') /\
(!E t. tkind E t ==> !tpo tcn t' E'. (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t'),E')) ==>
tkind E' (apply_abbrev tpo tcn t' t))`,
HO_MATCH_MP_TAC Eok_ind THEN SRW_TAC [] [Eok_def] THEN
FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, LET_THM, LAMBDA_PROD2, apply_abbrev_def] THENL
[Cases_on `EB` THEN FULL_SIMP_TAC (srw_ss()) [shiftEB_def] THEN SRW_TAC [] [Eok_def] THEN
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [],
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN Cases_on `t_list` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
METIS_TAC [ok_ok_thm],
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN Cases_on `t_list` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
METIS_TAC [ok_ok_thm],
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN Cases_on `t_list` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
METIS_TAC [ok_ok_thm],
Cases_on `E'` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [Eok_def] THEN
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
METIS_TAC [remove_abbrev_lem1],
METIS_TAC [ok_ok_thm],
Cases_on `E'` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [Eok_def] THEN
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
METIS_TAC [remove_abbrev_lem1],
METIS_TAC [ok_ok_thm],
Cases_on `lookup E (name_tcn typeconstr_name)` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THENL
[IMP_RES_TAC lookup_name_thm THEN SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [] THEN
IMP_RES_TAC remove_lookup_lem THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
SRW_TAC [] [EBok_def, Eok_def] THEN FULL_SIMP_TAC (srw_ss()) [COND_EXPAND_EQ] THEN METIS_TAC [lem4],
IMP_RES_TAC lookup_name_thm THEN SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [] THEN
Cases_on `tcn = typeconstr_name` THEN SRW_TAC [] [] THEN IMP_RES_TAC remove_abbrev_lem2 THEN
SRW_TAC [] [] THEN
IMP_RES_TAC remove_lookup_lem THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
FULL_SIMP_TAC (srw_ss()) [COND_EXPAND_EQ]],
FULL_SIMP_TAC (srw_ss()) [shiftEB_def] THEN METIS_TAC [ok_ok_thm, APPEND],
METIS_TAC [ok_ok_thm],
SRW_TAC [] [Eok_def] THEN METIS_TAC [lem3],
SRW_TAC [] [Eok_def],
SRW_TAC [] [Eok_def, EVERY_MAP] THEN FULL_SIMP_TAC (srw_ss()) [EVERY_MEM],
SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [Eok_def, EVERY_MAP] THEN
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN MATCH_MP_TAC tvar_subst_ok_lem THEN
SRW_TAC [] [MAP_FST_ZIP, EVERY_REVERSE] THEN SRW_TAC [] [GSYM EVERY_MAP] THEN SRW_TAC [] [MAP_SND_ZIP] THEN
SRW_TAC [] [EVERY_MAP] THEN SRW_TAC [] [EVERY_MEM] THEN FULL_SIMP_TAC (srw_ss()) [EBok_def, Eok_def] THEN
Q.EXISTS_TAC `MAP (\x. (tp_to_tv x, TE_constr [] TC_unit)) tpo` THEN SRW_TAC [] [MAP_MAP, ETA_THM] THEN
METIS_TAC []]);
end;
val type_substs_com_thm = Q.prove (
`(!t subs1 subs2.
(!tv. MEM tv (ftv_typexpr t) ==> MEM tv (MAP FST subs1))
==>
(substs_typevar_typexpr subs2 (substs_typevar_typexpr subs1 t) =
substs_typevar_typexpr (MAP (\(tv,t). (tv,substs_typevar_typexpr subs2 t)) subs1) t)) /\
(!tlist subs1 subs2.
(!tv. MEM tv (FLAT (MAP ftv_typexpr tlist)) ==> MEM tv (MAP FST subs1))
==>
(MAP (\t. substs_typevar_typexpr subs2 (substs_typevar_typexpr subs1 t)) tlist =
MAP (\t. substs_typevar_typexpr (MAP (\(tv,t). (tv,substs_typevar_typexpr subs2 t)) subs1) t)
tlist))`,
Induct THEN SRW_TAC [] [substs_typevar_typexpr_def, ftv_typexpr_def, MAP_MAP] THENL
[Cases_on `list_assoc t subs1` THEN
SRW_TAC [] [substs_typevar_typexpr_def, list_assoc_map, LAMBDA_PROD2] THEN
IMP_RES_TAC mem_list_assoc THEN FULL_SIMP_TAC (srw_ss()) [],
METIS_TAC [],
METIS_TAC []]);
local
val lem1 = Q.prove (
`!tpo typexpr_list. (LENGTH tpo = LENGTH typexpr_list) ==>
(MAP (\p. tp_to_tv (FST p)) (ZIP (tpo,typexpr_list)) = MAP tp_to_tv tpo)`,
METIS_TAC [MAP_MAP, MAP_FST_ZIP, ETA_THM]);
in
val type_subst_apply_abbrev_thm = Q.prove (
`!substs t' tpo tcn t. (!tv. MEM tv (ftv_typexpr t) ==> MEM tv (MAP tp_to_tv tpo)) ==>
(apply_abbrev tpo tcn t (substs_typevar_typexpr substs t') =
substs_typevar_typexpr (MAP (\s. (FST s, apply_abbrev tpo tcn t (SND s))) substs)
(apply_abbrev tpo tcn t t'))`,
recInduct substs_typevar_typexpr_ind THEN SRW_TAC [] [] THEN
SRW_TAC [] [apply_abbrev_def, substs_typevar_typexpr_def, MAP_MAP, MAP_EQ, ZIP_MAP] THEN
FULL_SIMP_TAC (srw_ss()) [ftv_typexpr_def, MEM_FLAT, EXISTS_MAP] THEN
FULL_SIMP_TAC (srw_ss()) [EXISTS_MEM] THENL
[Cases_on `list_assoc typevar sub` THEN SRW_TAC [] [apply_abbrev_def, list_assoc_map],
MP_TAC ((SIMP_RULE (srw_ss()) [MAP_MAP] o
Q.SPECL [`t`,
`MAP (\p. (tp_to_tv (FST p),apply_abbrev tpo tcn t (SND p)))
(ZIP (tpo,typexpr_list))`,
`MAP (\s. (FST s,apply_abbrev tpo tcn t (SND s))) sub`])
(hd (CONJUNCTS type_substs_com_thm))) THEN
SRW_TAC [] [lem1] THEN
MATCH_MP_TAC (METIS_PROVE [] ``!x y f z. (x = y) ==> (f x z = f y z)``) THEN
SRW_TAC [] [MAP_EQ] THEN
`MEM (SND p) typexpr_list` by METIS_TAC [MEM_ZIP, SND, MEM_EL] THEN
METIS_TAC []]);
end;
local
val lem1 = Q.prove (
`!l1 l2.
tkind E (substs_typevar_typexpr (MAP (\x. (FST x,TE_constr [] TC_unit)) l1) (TE_var typevar)) /\
(MAP FST l1 = MAP FST l2) /\
EVERY (\(t,t'). JTeq E t t') (ZIP (MAP SND l1,MAP SND l2)) ==>
JTeq E (substs_typevar_typexpr l1 (TE_var typevar)) (substs_typevar_typexpr l2 (TE_var typevar))`,
Induct THEN SRW_TAC [] [substs_typevar_typexpr_def, list_assoc_def] THEN1 METIS_TAC [JTeq_rules] THEN
Cases_on `l2` THEN FULL_SIMP_TAC (srw_ss()) [substs_typevar_typexpr_def] THEN
Cases_on `h` THEN Cases_on `h'` THEN FULL_SIMP_TAC (srw_ss()) [list_assoc_def]);
in
val teq_substs_same_thm = Q.store_thm ("teq_substs_same_thm",
`!l1 t l2. tkind E (substs_typevar_typexpr (MAP (\x. (FST x, TE_constr [] TC_unit)) l1) t) /\
(MAP FST l1 = MAP FST l2) /\
EVERY (\(t, t'). JTeq E t t') (ZIP (MAP SND l1, MAP SND l2)) ==>
JTeq E (substs_typevar_typexpr l1 t) (substs_typevar_typexpr l2 t)`,
recInduct substs_typevar_typexpr_ind THEN SRW_TAC [] [lem1, substs_typevar_typexpr_def] THENL
[METIS_TAC [JTeq_rules],
FULL_SIMP_TAC (srw_ss()) [Eok_def],
FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN METIS_TAC [JTeq_rules],
FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN
ONCE_REWRITE_TAC [JTeq_cases] THEN NTAC 5 DISJ2_TAC THEN DISJ1_TAC THEN
Q.EXISTS_TAC `ZIP (MAP (\typexpr_. substs_typevar_typexpr sub typexpr_) typexpr_list,
MAP (\typexpr_. substs_typevar_typexpr l2 typexpr_) typexpr_list)` THEN
SRW_TAC [] [MAP_FST_ZIP, MAP_SND_ZIP, LENGTH_MAP, LENGTH_ZIP, ZIP_MAP, MAP_ZIP_SAME, EVERY_MAP] THEN
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM, MEM_MAP] THEN METIS_TAC [],
FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN
ONCE_REWRITE_TAC [JTeq_cases] THEN NTAC 6 DISJ2_TAC THEN
MAP_EVERY Q.EXISTS_TAC [`ZIP (MAP (\typexpr_. substs_typevar_typexpr sub typexpr_) typexpr_list,
MAP (\typexpr_. substs_typevar_typexpr l2 typexpr_) typexpr_list)`,
`typeconstr`] THEN
SRW_TAC [] [MAP_FST_ZIP, MAP_SND_ZIP, LENGTH_MAP, LENGTH_ZIP, ZIP_MAP, MAP_ZIP_SAME, EVERY_MAP] THEN
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM, MEM_MAP] THEN METIS_TAC []]);
end;
local
val teq_refl_lem = Q.prove (
`!E t. JTeq E t t = tkind E t`,
METIS_TAC [JTeq_rules, teq_ok_thm]);
val lem1 = Q.prove (
`(MAP f (MAP FST l) = MAP FST (MAP (\x. (f (FST x), f (SND x))) l)) /\
(MAP f (MAP SND l) = MAP SND (MAP (\x. (f (FST x), f (SND x))) l))`,
SRW_TAC [] [MAP_MAP]);
val lem2 = Q.prove (
`(MAP f (MAP FST l) = MAP FST (MAP (\x. (f (FST x), SND x)) l))`,
SRW_TAC [] [MAP_MAP]);
val lem3 = Q.prove (
`!t f l. (ftv_typexpr (substs_typevar_typexpr (MAP (\z. (f z, TE_constr [] TC_unit)) l) t) = []) ==>
(!tv. MEM tv (ftv_typexpr t) ==> MEM tv (MAP f l))`,
recInduct ftv_typexpr_ind THEN SRW_TAC [] [ftv_typexpr_def, substs_typevar_typexpr_def] THEN
FULL_SIMP_TAC (srw_ss()) [MEM_FLAT, MAP_MAP, EXISTS_MAP, FLAT_EQ_EMPTY, EVERY_MAP] THEN
FULL_SIMP_TAC (srw_ss()) [EXISTS_MEM, EVERY_MEM] THENL
[Cases_on `list_assoc typevar (MAP (\z. (f z,TE_constr [] TC_unit)) l)` THEN
FULL_SIMP_TAC (srw_ss()) [ftv_typexpr_def] THEN IMP_RES_TAC list_assoc_mem THEN
FULL_SIMP_TAC (srw_ss()) [MEM_MAP] THEN METIS_TAC [],
METIS_TAC [],
METIS_TAC []]);
val lem4 = Q.prove (
`!l. EVERY (\x. tkind E (FST x)) l ==>
EVERY (\s. ftv_typexpr (SND s) = []) (MAP (\z. (SND z,FST z)) l)`,
Induct THEN SRW_TAC [] [] THEN METIS_TAC [ftv_lem1]);
val lem5 = Q.prove (
`!l.
(!t E.
tkind E (substs_typevar_typexpr (MAP (\z. (SND z,TE_constr [] TC_unit)) l) t) ==>
!tv. MEM tv (ftv_typexpr t) ==> MEM tv (MAP SND l)) /\
(!ts E.
EVERY (\t. tkind E (substs_typevar_typexpr (MAP (\z. (SND z,TE_constr [] TC_unit)) l) t)) ts ==>
!tv. MEM tv (FLAT (MAP ftv_typexpr ts)) ==> MEM tv (MAP SND l))`,
STRIP_TAC THEN Induct THEN
SRW_TAC [] [Eok_def, ftv_typexpr_def, substs_typevar_typexpr_def, EVERY_MAP, ETA_THM] THENL
[Cases_on `list_assoc t (MAP (\z. (SND z,TE_constr [] TC_unit)) l)` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN IMP_RES_TAC list_assoc_mem THEN
FULL_SIMP_TAC (srw_ss()) [MEM_MAP] THEN SRW_TAC [] [] THEN METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
METIS_TAC []]);
val lem6 = Q.prove (
`!E tpo tcn t' E'. Eok E /\ (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t'),E')) ==>
(lookup E' (name_tcn tcn) = NONE)`,
Induct THEN SRW_TAC [] [remove_abbrev_def, lookup_def, domEB_def] THEN Cases_on `h` THEN
FULL_SIMP_TAC (srw_ss()) [remove_abbrev_def, lookup_def, domEB_def] THEN IMP_RES_TAC Eok_lem1 THEN
FULL_SIMP_TAC (srw_ss()) [LET_THM] THEN SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [Eok_def] THENL
[Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [lookup_def, domEB_def] THEN
Cases_on `EB` THEN FULL_SIMP_TAC (srw_ss()) [shiftEB_def] THEN SRW_TAC [] [],
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [lookup_def, domEB_def] THEN
METIS_TAC [remove_lookup_lem, lookup_dom_thm],
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [lookup_def, domEB_def] THEN
METIS_TAC [remove_lookup_lem, lookup_dom_thm],
METIS_TAC [lookup_dom_thm]]);
val lem7 = Q.prove (
`!tpo t_t'_list.
(LENGTH tpo = LENGTH t_t'_list) ==>
(MAP (\tp. (tp_to_tv tp,TE_constr [] TC_unit)) tpo =
MAP (\p. (tp_to_tv (FST p),TE_constr [] TC_unit)) (ZIP (tpo,t_t'_list)))`,
Induct THEN SRW_TAC [] [] THEN Cases_on `t_t'_list` THEN FULL_SIMP_TAC (srw_ss()) []);
val lem8 = Q.prove (
`!E tpo t t_t'_list.
(LENGTH tpo = LENGTH t_t'_list) /\ ntsok E tpo t /\ EVERY (\x. JTeq E (FST x) (SND x)) t_t'_list ==>
JTeq E (substs_typevar_typexpr (MAP (\p. (tp_to_tv (FST p), (FST (SND p)))) (ZIP (tpo,t_t'_list))) t)
(substs_typevar_typexpr (MAP (\p. (tp_to_tv (FST p), (SND (SND p)))) (ZIP (tpo,t_t'_list))) t)`,
SRW_TAC [] [Eok_def] THEN MATCH_MP_TAC teq_substs_same_thm THEN
SRW_TAC [] [MAP_FST_ZIP, MAP_SND_ZIP, MAP_MAP, MAP_ZIP_SAME, ZIP_MAP, ETA_THM, LAMBDA_PROD2] THEN
METIS_TAC [lem7]);
in
val apply_abbrev_str_thm = Q.prove (
`!E t1 t2. JTeq E t1 t2 ==> !tpo tcn t E'.
(remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t), E')) ==>
JTeq E' (apply_abbrev tpo tcn t t1) (apply_abbrev tpo tcn t t2)`,
RULE_INDUCT_TAC JTeq_ind [remove_abbrev_def, apply_abbrev_def, teq_refl_lem]
[([``"JTeq_sym"``, ``"JTeq_trans"``, ``"JTeq_arrow"``], METIS_TAC [JTeq_rules]),
([``"JTeq_refl"``], METIS_TAC [teq_remove_abbrev_ok_thm]),
([``"JTeq_tuple"``],
SRW_TAC [] [lem1] THEN FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN
FULL_SIMP_TAC (srw_ss()) [GSYM EVERY_MEM] THEN MATCH_MP_TAC (List.nth (CONJUNCTS JTeq_rules, 5)) THEN
SRW_TAC [] [EVERY_MAP]),
([``"JTeq_expand"``],
SRW_TAC [] [] THEN SRW_TAC [] [] THENL
[IMP_RES_TAC remove_lookup_lem THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
FULL_SIMP_TAC (srw_ss()) [LENGTH_MAP] THEN SRW_TAC [] [ZIP_MAP, MAP_ZIP_SAME] THEN
SRW_TAC [] [MAP_MAP, tp_to_tv_def] THEN IMP_RES_TAC lookup_ok_thm THEN
FULL_SIMP_TAC (srw_ss()) [EBok_def, Eok_def] THEN IMP_RES_TAC teq_remove_abbrev_ok_thm THEN
IMP_RES_TAC lem4 THEN FULL_SIMP_TAC (srw_ss()) [MAP_MAP] THEN
`EVERY (\s. ftv_typexpr (SND s) = [])
(MAP (\z. (tp_to_tv (TP_var (SND z)),TE_constr [] TC_unit)) t_typevar_list)`
by SRW_TAC [] [EVERY_MAP, ftv_typexpr_def] THEN
FULL_SIMP_TAC (srw_ss()) [tp_to_tv_def] THEN
`!tv. MEM tv (ftv_typexpr t) ==> MEM tv (MAP tp_to_tv (MAP (\z. TP_var (SND z)) t_typevar_list))`
by (SRW_TAC [] [MAP_MAP, tp_to_tv_def] THEN METIS_TAC [lem5]) THEN
FULL_SIMP_TAC (srw_ss()) [type_subst_apply_abbrev_thm] THEN
IMP_RES_TAC abbrev_no_occur_thm THEN
FULL_SIMP_TAC (srw_ss()) [apply_abbrev_no_occur_thm, tp_to_tv_def, MAP_MAP, apply_abbrev_def] THEN
MATCH_MP_TAC (List.nth (CONJUNCTS JTeq_rules, 0)) THEN
MATCH_MP_TAC tvar_subst_ok_lem THEN SRW_TAC [] [MAP_MAP, EVERY_REVERSE, EVERY_MAP] THEN
Q.EXISTS_TAC `MAP (\z. (SND z,TE_constr [] TC_unit)) t_typevar_list` THEN SRW_TAC [] [MAP_MAP] THEN
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN SRW_TAC [] [] THEN RES_TAC THEN
IMP_RES_TAC teq_remove_abbrev_ok_thm,
IMP_RES_TAC remove_lookup_lem THEN IMP_RES_TAC lookup_ok_thm THEN
FULL_SIMP_TAC (srw_ss()) [EBok_def, Eok_def] THEN SRW_TAC [] [] THEN IMP_RES_TAC ftv_lem1 THEN
FULL_SIMP_TAC (srw_ss()) [MAP_MAP, tp_to_tv_def] THEN IMP_RES_TAC lem3 THEN IMP_RES_TAC lem4 THEN
SRW_TAC [] [type_subst_apply_abbrev_thm, MAP_MAP] THEN
MATCH_MP_TAC ((SIMP_RULE (srw_ss()) [MAP_MAP] o
Q.SPEC `MAP (\x. (apply_abbrev tpo tcn t' (FST x), SND x)) t_typevar_list`)
(List.nth (CONJUNCTS JTeq_rules, 3))) THEN
SRW_TAC [] [EVERY_MAP] THEN IMP_RES_TAC teq_remove_abbrev_ok_thm THENL
[`lookup E' (name_tcn typeconstr_name) =
SOME (EB_ta (TPS_nary (MAP (\z. TP_var (SND z)) t_typevar_list)) typeconstr_name t)`
by METIS_TAC [remove_abbrev_lem2] THEN
IMP_RES_TAC lookup_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [EBok_def, Eok_def] THEN
METIS_TAC [lem6, abbrev_occur_lem1, lookup_dom_thm, apply_abbrev_no_occur_thm],
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN METIS_TAC [teq_remove_abbrev_ok_thm],
Cases_on `tcn = typeconstr_name` THENL
[SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [] THEN FULL_SIMP_TAC (srw_ss()) [] THEN
METIS_TAC [LENGTH_MAP],
`lookup E' (name_tcn typeconstr_name) =
SOME (EB_ta (TPS_nary (MAP (\z. TP_var (SND z)) t_typevar_list)) typeconstr_name t)`
by METIS_TAC [remove_abbrev_lem2] THEN
IMP_RES_TAC lookup_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [EBok_def, Eok_def] THEN
METIS_TAC [lem6, abbrev_occur_lem1, lookup_dom_thm, apply_abbrev_no_occur_thm]],
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN METIS_TAC [teq_remove_abbrev_ok_thm]]]),
([``"JTeq_constr"``],
SRW_TAC [] [lem1] THEN FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN
FULL_SIMP_TAC (srw_ss()) [GSYM EVERY_MEM] THEN SRW_TAC [] [] THENL
[IMP_RES_TAC teq_remove_abbrev_ok_thm THEN POP_ASSUM MP_TAC THEN
SRW_TAC [] [EBok_def] THEN
`EVERY (\x. JTeq E' (FST x) (SND x))
(MAP (\x. (apply_abbrev tpo tcn t (FST x), apply_abbrev tpo tcn t (SND x))) t_t'_list)`
by SRW_TAC [] [EVERY_MAP] THEN
IMP_RES_TAC lem8 THEN POP_ASSUM MP_TAC THEN SRW_TAC [] [MAP_MAP, ZIP_MAP],
MATCH_MP_TAC (List.nth (CONJUNCTS JTeq_rules, 6)) THEN SRW_TAC [] [EVERY_MAP] THEN
IMP_RES_TAC teq_remove_abbrev_ok_thm THEN METIS_TAC []])]);
end;
local
val lem1 = Q.prove (
`!l1 l2 P. (LENGTH l1 = LENGTH l2) ==> (EVERY (\x. P (FST x)) (ZIP (l1,l2)) = EVERY (\x. P x) l1)`,
Induct THEN SRW_TAC [] [] THEN Cases_on `l2` THEN FULL_SIMP_TAC (srw_ss()) []);
val lem2 = Q.prove (
`!tpo ts l tcn t'. (LENGTH ts = LENGTH tpo) ==>
(MAP (\p. (tp_to_tv (FST p),apply_abbrev l tcn t' (SND p))) (ZIP (tpo,ts)) =
MAP (\p. (tp_to_tv (SND p),apply_abbrev l tcn t' (FST p))) (ZIP (ts,tpo)))`,
Induct THEN SRW_TAC [] [] THEN Cases_on `ts` THEN FULL_SIMP_TAC (srw_ss()) []);
in
val abbrev_eq_thm = Q.prove (
`(!t tpo tcn t' E. tkind E t /\ (lookup E (name_tcn tcn) = SOME (EB_ta (TPS_nary tpo) tcn t')) ==>
JTeq E t (apply_abbrev tpo tcn t' t)) /\
(!ts tpo tcn t' E. EVERY (\x. tkind E x) ts /\
(lookup E (name_tcn tcn) = SOME (EB_ta (TPS_nary tpo) tcn t')) ==>
EVERY (\t. JTeq E t (apply_abbrev tpo tcn t' t)) ts)`,
Induct THEN SRW_TAC [] [apply_abbrev_def, Eok_def] THENL
[`tkind E (TE_idxvar n n0)` by SRW_TAC [] [Eok_def] THEN METIS_TAC [JTeq_rules],
METIS_TAC [JTeq_rules],
MATCH_MP_TAC ((SIMP_RULE (srw_ss()) [MAP_FST_ZIP, MAP_SND_ZIP, LENGTH_MAP] o
Q.SPEC `ZIP (ts, MAP (\a. apply_abbrev tpo tcn t' a) ts)`)
(List.nth (CONJUNCTS JTeq_rules, 5))) THEN
SRW_TAC [] [LENGTH_MAP, LENGTH_ZIP, ZIP_MAP, MAP_ZIP_SAME, EVERY_MAP],
FULL_SIMP_TAC (srw_ss()) [COND_EXPAND_EQ] THEN
IMP_RES_TAC lookup_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [COND_EXPAND_EQ, EBok_def, Eok_def] THEN
ONCE_REWRITE_TAC [JTeq_cases] THEN SRW_TAC [] [] THEN
NTAC 2 DISJ2_TAC THEN DISJ1_TAC THEN
Q.EXISTS_TAC `TE_constr (MAP (apply_abbrev tpo tcn t') ts) (TC_name tcn)` THEN
SRW_TAC [] [] THENL
[ONCE_REWRITE_TAC [JTeq_cases] THEN NTAC 6 DISJ2_TAC THEN SRW_TAC [] [] THEN
Q.EXISTS_TAC `ZIP (ts, MAP (apply_abbrev tpo tcn t') ts)` THEN
SRW_TAC [] [MAP_FST_ZIP, MAP_SND_ZIP, LENGTH_ZIP, ZIP_MAP, MAP_ZIP_SAME, EVERY_MAP, Eok_def],
ONCE_REWRITE_TAC [JTeq_cases] THEN NTAC 3 DISJ2_TAC THEN DISJ1_TAC THEN SRW_TAC [] [] THEN
Q.EXISTS_TAC `ZIP (MAP (apply_abbrev tpo tcn t') ts, MAP tp_to_tv tpo)` THEN
SRW_TAC [] [MAP_FST_ZIP, ZIP_MAP, LENGTH_ZIP, EVERY_MAP, lem1, MAP_MAP, tp_to_tv_thm, ETA_THM,
MAP_SND_ZIP] THENL
[METIS_TAC [lem2],
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM] THEN SRW_TAC [] [] THEN METIS_TAC [teq_ok_thm]]],
MATCH_MP_TAC ((SIMP_RULE (srw_ss()) [MAP_FST_ZIP, MAP_SND_ZIP, LENGTH_MAP] o
Q.SPEC `ZIP (ts, MAP (\a. apply_abbrev tpo tcn t' a) ts)`)
(List.nth (CONJUNCTS JTeq_rules, 6))) THEN
SRW_TAC [] [LENGTH_MAP, LENGTH_ZIP, ZIP_MAP, MAP_ZIP_SAME, EVERY_MAP]]);
end;
val not_abbrev_lem = Q.prove (
`!E ts1 tc2 tcn tpo t E'. Eok E /\ ~is_abbrev_tc E (TE_constr ts1 tc2) /\
(remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t), E')) ==>
~(tc2 = TC_name tcn)`,
Cases_on `tc2` THEN SRW_TAC [] [is_abbrev_tc_def] THEN Cases_on `lookup E (name_tcn t')` THEN
FULL_SIMP_TAC (srw_ss()) [] THEN1 METIS_TAC [remove_lookup_lem, NOT_SOME_NONE] THEN Cases_on `x` THEN
FULL_SIMP_TAC (srw_ss()) [] THEN IMP_RES_TAC remove_lookup_lem THEN CCONTR_TAC THEN
FULL_SIMP_TAC (srw_ss()) [] THEN METIS_TAC [environment_binding_distinct]);
val alg_eq_step_def = Define
`alg_eq_step E t1 t2 =
case t1 of
TE_var tv -> F
|| TE_idxvar i1 i2 -> (t2 = TE_idxvar i1 i2)
|| TE_any -> F
|| TE_arrow t1' t2' -> (?t1'' t2''. (t2 = TE_arrow t1'' t2'') /\ JTeq E t1' t1'' /\ JTeq E t2' t2'')
|| TE_tuple ts ->
(?ts'. (t2 = TE_tuple ts') /\ (LENGTH ts = LENGTH ts') /\ (LENGTH ts >= 2) /\
EVERY (\(t, t'). JTeq E t t') (ZIP (ts, ts')))
|| TE_constr ts tc ->
(?ts'. (t2 = TE_constr ts' tc) /\ (LENGTH ts = LENGTH ts') /\
(SOME (LENGTH ts) = typeconstr_kind E tc) /\
EVERY (\(t, t'). JTeq E t t') (ZIP (ts, ts')))`;
val apply_abbrev_weak_thm = Q.prove (
`!E E' tpo tcn t1 t2.
tkind E t1 /\ tkind E t2 /\
(remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t), E')) /\
JTeq E' (apply_abbrev tpo tcn t t1) (apply_abbrev tpo tcn t t2) ==>
JTeq E t1 t2`,
SRW_TAC [] [] THEN `Eok E` by METIS_TAC [ok_ok_thm] THEN
MAP_EVERY IMP_RES_TAC [remove_lookup_lem, abbrev_eq_thm, remove_abbrev_teq_weak_thm] THEN
METIS_TAC [JTeq_rules]);
local
val tac =
Q.PAT_ASSUM `EVERY A B` MP_TAC THEN SRW_TAC [] [LENGTH_MAP, ZIP_MAP, EVERY_MAP] THEN
FULL_SIMP_TAC (srw_ss()) [EVERY_MEM, LAMBDA_PROD2] THEN
IMP_RES_TAC teq_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [Eok_def, EVERY_MEM] THEN
`!a. MEM a (ZIP (l, l')) ==> tkind E (FST a) /\ tkind E (SND a)` by
(SRW_TAC [] [MEM_ZIP] THEN SRW_TAC [] [] THEN METIS_TAC [MEM_EL]) THEN
METIS_TAC [apply_abbrev_weak_thm];
in
val alg_eq_step_abbrev_lem = Q.prove (
`!E t1 t2 E' tpo tcn t. JTeq E t1 t2 /\ JTeq E' (apply_abbrev tpo tcn t t1) (apply_abbrev tpo tcn t t2) /\
(remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t), E')) /\
~is_abbrev_tc E t1 /\ ~is_abbrev_tc E t2 /\
alg_eq_step E' (apply_abbrev tpo tcn t t1) (apply_abbrev tpo tcn t t2) ==>
alg_eq_step E t1 t2`,
REWRITE_TAC [alg_eq_step_def] THEN Cases_on `t1` THEN SRW_TAC [] [apply_abbrev_def, COND_EXPAND_EQ] THEN
`Eok E` by METIS_TAC [teq_ok_thm, ok_ok_thm] THENL
[Cases_on `t2` THEN FULL_SIMP_TAC (srw_ss()) [apply_abbrev_def, COND_EXPAND_EQ] THEN
METIS_TAC [not_abbrev_lem],
Cases_on `t2` THEN FULL_SIMP_TAC (srw_ss()) [apply_abbrev_def, COND_EXPAND_EQ] THEN
IMP_RES_TAC teq_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN
METIS_TAC [apply_abbrev_weak_thm, not_abbrev_lem],
Cases_on `t2` THEN FULL_SIMP_TAC (srw_ss()) [apply_abbrev_def, COND_EXPAND_EQ] THEN
SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [LENGTH_MAP] THENL
[tac,
METIS_TAC [not_abbrev_lem]],
Cases_on `t2` THEN FULL_SIMP_TAC (srw_ss()) [apply_abbrev_def, COND_EXPAND_EQ] THEN
SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [LENGTH_MAP] THEN
METIS_TAC [not_abbrev_lem],
Cases_on `t2` THEN FULL_SIMP_TAC (srw_ss()) [apply_abbrev_def, COND_EXPAND_EQ] THEN
SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [LENGTH_MAP] THENL
[METIS_TAC [not_abbrev_lem],
IMP_RES_TAC teq_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [Eok_def],
tac,
IMP_RES_TAC teq_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [Eok_def],
tac,
METIS_TAC [not_abbrev_lem],
IMP_RES_TAC teq_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [Eok_def],
tac,
IMP_RES_TAC teq_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [Eok_def],
tac]]);
end;
local
val lem1 = Q.prove (
`!E tcn tpo tcn' t. (num_abbrev E = 0) ==> ~(lookup E (name_tcn tcn) = SOME (EB_ta tpo tcn' t))`,
Induct THEN SRW_TAC [] [lookup_def] THEN Cases_on `h` THEN
FULL_SIMP_TAC (srw_ss ()) [num_abbrev_def, lookup_def, domEB_def, shiftEB_add_thm] THEN1
(Cases_on `EB2` THEN FULL_SIMP_TAC (srw_ss ()) [num_abbrev_def, lookup_def, domEB_def, shiftEB_def])
THEN METIS_TAC []);
in
val no_abbrev_thm = Q.prove (
`!E t1 t2. JTeq E t1 t2 ==> (num_abbrev E = 0) ==> (t1 = t2)`,
RULE_INDUCT_TAC JTeq_ind []
[([``"JTeq_expand"``], METIS_TAC [lem1]),
([``"JTeq_tuple"``, ``"JTeq_constr"``], SRW_TAC [] [EVERY_MEM, MAP_EQ])]);
end;
local
val lem1 = Q.prove (
`!E tpo tcn t' E' n EB. Eok E /\ (remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t'),E')) /\
(lookup E' n = SOME EB) /\ (~?vn. (n = name_vn vn)) ==> (lookup E n = SOME EB)`,
recInduct (fetch "-" "remove_abbrev_ind") THEN
SRW_TAC [] [remove_abbrev_def, domEB_def, Eok_def, lookup_def] THEN
FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN IMP_RES_TAC Eok_lem1 THEN
FULL_SIMP_TAC (srw_ss()) [LET_THM, LAMBDA_PROD2] THEN
IMP_RES_TAC ok_ok_thm THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
FULL_SIMP_TAC (srw_ss()) [lookup_dom_thm, shiftEB_add_thm, lookup_def, domEB_def] THENL
[Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [],
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [],
Cases_on `remove_abbrev E` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
Cases_on `EB'` THEN FULL_SIMP_TAC (srw_ss()) [shiftEB_def] THEN SRW_TAC [] [] THEN
METIS_TAC [],
METIS_TAC [NOT_SOME_NONE, name_distinct],
Cases_on `tpo` THEN FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN METIS_TAC [lookup_dom_thm, name_distinct],
Cases_on `tpo` THEN Cases_on `tl` THEN Cases_on `tc` THEN Cases_on `l` THEN
FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN METIS_TAC [lookup_dom_thm, name_distinct],
Cases_on `tc` THEN FULL_SIMP_TAC (srw_ss()) [Eok_def] THEN METIS_TAC [lookup_dom_thm, name_distinct]]);
in
val remove_abbrev_is_abbrev_lem = Q.prove (
`!E tpo tcn t E' t'. Eok E /\
(remove_abbrev E = (SOME (EB_ta (TPS_nary tpo) tcn t), E')) /\ ~is_abbrev_tc E t' ==>
~is_abbrev_tc E' (apply_abbrev tpo tcn t t')`,
Cases_on `t'` THEN SRW_TAC [] [is_abbrev_tc_def, apply_abbrev_def] THENL
[Cases_on `lookup E (name_tcn tcn)` THEN FULL_SIMP_TAC (srw_ss()) [] THEN1
METIS_TAC [remove_lookup_lem, NOT_SOME_NONE] THEN Cases_on `x` THEN
FULL_SIMP_TAC (srw_ss()) [] THEN IMP_RES_TAC remove_lookup_lem THEN
FULL_SIMP_TAC (srw_ss()) [] THEN METIS_TAC [environment_binding_distinct],
Cases_on `t''` THEN FULL_SIMP_TAC (srw_ss()) [is_abbrev_tc_def] THEN
Cases_on `lookup E' (name_tcn t')` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
Cases_on `x` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
Cases_on `lookup E (name_tcn t'')` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
IMP_RES_TAC lem1 THEN FULL_SIMP_TAC (srw_ss ()) [] THEN
Cases_on `x` THEN FULL_SIMP_TAC (srw_ss()) []]);
end;
val alg_eq_step_complete_thm = Q.prove (
`!E t1 t2. ~is_abbrev_tc E t1 /\ ~is_abbrev_tc E t2 /\ JTeq E t1 t2 ==> alg_eq_step E t1 t2`,
STRIP_TAC THEN Induct_on `num_abbrev E` THENL
[SRW_TAC [] [] THEN `t1 = t2` by METIS_TAC [no_abbrev_thm] THEN SRW_TAC [] [alg_eq_step_def] THEN
Cases_on `t1` THEN SRW_TAC [] [] THEN
IMP_RES_TAC teq_ok_thm THEN FULL_SIMP_TAC (srw_ss ()) [Eok_def, EVERY_ZIP_SAME] THEN
FULL_SIMP_TAC (srw_ss ()) [EVERY_MEM] THEN METIS_TAC [JTeq_rules],
SRW_TAC [] [] THEN STRIP_ASSUME_TAC (Q.SPEC `E` remove_abbrev_result_thm) THEN1
(IMP_RES_TAC remove_abbrev_num_abbrev_thm2 THEN FULL_SIMP_TAC (srw_ss()) []) THEN
IMP_RES_TAC remove_abbrev_num_abbrev_thm THEN `v = num_abbrev E'` by DECIDE_TAC THEN
MATCH_MP_TAC alg_eq_step_abbrev_lem THEN Cases_on `tpo` THEN SRW_TAC [] [] THEN
IMP_RES_TAC remove_abbrev_is_abbrev_lem THEN METIS_TAC [apply_abbrev_str_thm, teq_ok_thm, ok_ok_thm]]);
val teq_fun1 = Q.prove (
`(!E t a. ~is_abbrev_tc E t ==> (JTeq E (TE_var a) t = F)) /\
(!E t i1 i2. ~is_abbrev_tc E t ==> (JTeq E (TE_idxvar i1 i2) t = (t = TE_idxvar i1 i2) /\ tkind E t)) /\
(!E t. ~is_abbrev_tc E t ==> (JTeq E TE_any t = F)) /\
(!E t t1 t2. ~is_abbrev_tc E t ==>
(JTeq E (TE_arrow t1 t2) t = ?t1' t2'. (t = TE_arrow t1' t2') /\ JTeq E t1 t1' /\ JTeq E t2 t2')) /\
(!E t ts. ~is_abbrev_tc E t ==>
(JTeq E (TE_tuple ts) t =
?ts'. (t = TE_tuple ts') /\ (LENGTH ts = LENGTH ts') /\ (LENGTH ts >= 2) /\
EVERY (\(t, t'). JTeq E t t') (ZIP (ts, ts')))) /\
(!E t ts tc. ~is_abbrev_tc E t /\ ~is_abbrev_tc E (TE_constr ts tc) ==>
(JTeq E (TE_constr ts tc) t =
?ts'. (t = TE_constr ts' tc) /\ (LENGTH ts = LENGTH ts') /\
(SOME (LENGTH ts) = typeconstr_kind E tc) /\
EVERY (\(t, t'). JTeq E t t') (ZIP (ts, ts'))))`,
SRW_TAC [] [] THENL
[CCONTR_TAC THEN FULL_SIMP_TAC (srw_ss()) [] THEN IMP_RES_TAC teq_ok_thm THEN
FULL_SIMP_TAC (srw_ss()) [Eok_def],
EQ_TAC THEN SRW_TAC [] [] THENL [ALL_TAC, METIS_TAC [teq_ok_thm], METIS_TAC [JTeq_rules]],
CCONTR_TAC THEN FULL_SIMP_TAC (srw_ss()) [] THEN IMP_RES_TAC teq_ok_thm THEN
FULL_SIMP_TAC (srw_ss()) [Eok_def],
EQ_TAC THEN SRW_TAC [] [] THENL [ALL_TAC, METIS_TAC [JTeq_rules]],
EQ_TAC THEN SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [LAMBDA_PROD2] THENL
[ALL_TAC,
IMP_RES_TAC JTeq_rules THEN METIS_TAC [MAP_FST_ZIP, MAP_SND_ZIP, LENGTH_ZIP, LENGTH_MAP]],
EQ_TAC THEN SRW_TAC [] [] THEN FULL_SIMP_TAC (srw_ss()) [LAMBDA_PROD2] THENL
[ALL_TAC,
IMP_RES_TAC JTeq_rules THEN METIS_TAC [MAP_FST_ZIP, MAP_SND_ZIP, LENGTH_ZIP, LENGTH_MAP]]] THEN
IMP_RES_TAC alg_eq_step_complete_thm THEN
FULL_SIMP_TAC (srw_ss()) [is_abbrev_tc_def, alg_eq_step_def, LAMBDA_PROD2]);
val _ = save_thm ("teq_fun1", teq_fun1);
val teq_fun2 = Q.store_thm ("teq_fun2",
`(!E t a. ~is_abbrev_tc E t ==> (JTeq E t (TE_var a) = F)) /\
(!E t i1 i2. ~is_abbrev_tc E t ==> (JTeq E t (TE_idxvar i1 i2) = (t = TE_idxvar i1 i2) /\ tkind E t)) /\
(!E t. ~is_abbrev_tc E t ==> ((JTeq E t TE_any) = F)) /\
(!E t t1 t2. ~is_abbrev_tc E t ==>
(JTeq E t (TE_arrow t1 t2) = ?t1' t2'. (t = TE_arrow t1' t2') /\ JTeq E t1 t1' /\ JTeq E t2 t2')) /\
(!E t ts. ~is_abbrev_tc E t ==>
(JTeq E t (TE_tuple ts) =
?ts'. (t = TE_tuple ts') /\ (LENGTH ts = LENGTH ts') /\ (LENGTH ts >= 2) /\
EVERY (\(t, t'). JTeq E t t') (ZIP (ts, ts')))) /\
(!E t ts tc. ~is_abbrev_tc E t /\ ~is_abbrev_tc E (TE_constr ts tc) ==>
(JTeq E t (TE_constr ts tc) =
?ts'. (t = TE_constr ts' tc) /\ (LENGTH ts = LENGTH ts') /\
(SOME (LENGTH ts) = typeconstr_kind E tc) /\
EVERY (\(t, t'). JTeq E t t') (ZIP (ts, ts'))))`,
METIS_TAC [teq_fun1, List.nth (CONJUNCTS JTeq_rules, 1)]);
val _ = export_theory ();
|
