-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathGeneralReflection.html
1207 lines (1035 loc) · 262 KB
/
GeneralReflection.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
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
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<title>GeneralReflection</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Library GeneralReflection</h1>
<div class="code">
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="var">MetaCoq.Checker.Checker</span>.<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">MetaCoq.Template</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="var">utils</span> <span class="id" title="keyword">All</span> <span class="id" title="var">Pretty</span>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="var">List</span> <span class="id" title="var">String</span>.<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="var">MonadNotation</span>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="var">FOL</span>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="var">Deduction</span>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="var">Tarski</span>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="var">VectorTech</span>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="var">Lia</span>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab1"></a><h1 class="section">Reification</h1>
Please read the PDF file giving a detailed description.
<div class="paragraph"> </div>
We are given a Coq term and try to find a reification of it in the syntax of a specific kind of first-order theory.
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <span class="id" title="var">FailureMonad</span>.<br/>
</div>
<div class="doc">
<a id="lab2"></a><h2 class="section">FailureMonad</h2>
Since our function is partial, and we want to give helpful error messages to the User, we use this monad, which represents either success or failure, along with an error message
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">A</span>:<span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Type</span> := <span class="id" title="var">ret</span> : <span class="id" title="var">A</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">A</span> | <span class="id" title="tactic">fail</span> : <span class="id" title="var">string</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">A</span>.<br/>
<span class="id" title="var">Arguments</span> <span class="id" title="var">ret</span> {<span class="id" title="var">_</span>} <span class="id" title="var">_</span>.<br/>
<span class="id" title="var">Arguments</span> <span class="id" title="tactic">fail</span> {<span class="id" title="var">_</span>} <span class="id" title="var">_</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">bind</span> {<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>} (<span class="id" title="var">k</span>:<span class="id" title="var">FailureMonad</span> <span class="id" title="var">A</span>) (<span class="id" title="var">f</span>:<span class="id" title="var">A</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">B</span>) := <span class="id" title="keyword">match</span> <span class="id" title="var">k</span> <span class="id" title="keyword">return</span> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">B</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">fail</span> <span class="id" title="var">x</span> => <span class="id" title="tactic">fail</span> <span class="id" title="var">x</span> | <span class="id" title="var">ret</span> <span class="id" title="var">k</span> => <span class="id" title="var">f</span> <span class="id" title="var">k</span> <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Notation</span> "x <- c1 ;; c2" := (<span class="id" title="var">bind</span> <span class="id" title="var">c1</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> => <span class="id" title="var">c2</span>)).<br/>
</div>
<div class="doc">
"Converts" from our monad to the TemplateMonad of MetaCoq. This is used to pass error messages back to the user
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">f2t</span> {<span class="id" title="var">T</span>:<span class="id" title="keyword">Type</span>} (<span class="id" title="var">a</span>:<span class="id" title="var">FailureMonad</span> <span class="id" title="var">T</span>) : <span class="id" title="var">TemplateMonad</span> <span class="id" title="var">T</span> := <span class="id" title="keyword">match</span> <span class="id" title="var">a</span> <span class="id" title="keyword">with</span> <span class="id" title="var">ret</span> <span class="id" title="var">k</span> => <span class="id" title="var">monad_utils.ret</span> <span class="id" title="var">k</span> | <span class="id" title="tactic">fail</span> <span class="id" title="var">s</span> => <span class="id" title="var">tmFail</span> <span class="id" title="var">s</span> <span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
Structurally recursive definition of a monadic map
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">flatten_monad</span> {<span class="id" title="var">A</span>:<span class="id" title="keyword">Type</span>} (<span class="id" title="var">l</span>:<span class="id" title="var">list</span> (<span class="id" title="var">FailureMonad</span> <span class="id" title="var">A</span>)) : <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">A</span>) := <span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nil</span> => <span class="id" title="var">ret</span> <span class="id" title="var">nil</span> | <span class="id" title="var">x</span>::<span class="id" title="var">xr</span> => <span class="id" title="var">xm</span> <- <span class="id" title="var">x</span>;; <span class="id" title="var">xrm</span> <- <span class="id" title="var">flatten_monad</span> <span class="id" title="var">xr</span>;; <span class="id" title="var">ret</span> (<span class="id" title="var">xm</span>::<span class="id" title="var">xrm</span>) <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">map_monad</span> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>:<span class="id" title="keyword">Type</span>} (<span class="id" title="var">f</span>:<span class="id" title="var">A</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">B</span>) (<span class="id" title="var">l</span>:<span class="id" title="var">list</span> <span class="id" title="var">A</span>) : <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">B</span>) := <span class="id" title="var">flatten_monad</span> (<span class="id" title="var">map</span> <span class="id" title="var">f</span> <span class="id" title="var">l</span>).<br/>
</div>
<div class="doc">
If the first monad fails, use the second. Used to cascade failure handling
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">orelse</span> {<span class="id" title="var">A</span>:<span class="id" title="keyword">Type</span>} (<span class="id" title="var">F1</span> <span class="id" title="var">F2</span> :<span class="id" title="var">FailureMonad</span> <span class="id" title="var">A</span>) := <span class="id" title="keyword">match</span> <span class="id" title="var">F1</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">fail</span> <span class="id" title="var">_</span> => <span class="id" title="var">F2</span> | <span class="id" title="var">_</span> => <span class="id" title="var">F1</span> <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">FailureMonad</span>.<br/>
<br/>
<span class="id" title="var">Arguments</span> <span class="id" title="var">ret</span> {<span class="id" title="var">_</span>} <span class="id" title="var">_</span>.<br/>
<span class="id" title="var">Arguments</span> <span class="id" title="tactic">fail</span> {<span class="id" title="var">_</span>} <span class="id" title="var">_</span>.<br/>
<span class="id" title="var">Arguments</span> <span class="id" title="var">orelse</span> {<span class="id" title="var">_</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
<span class="id" title="keyword">Notation</span> "x <- c1 ;; c2" := (<span class="id" title="var">bind</span> <span class="id" title="var">c1</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> => <span class="id" title="var">c2</span>)).<br/>
<span class="id" title="var">Arguments</span> <span class="id" title="var">Vector.cons</span> {<span class="id" title="var">_</span>} <span class="id" title="var">_</span> {<span class="id" title="var">_</span>} <span class="id" title="var">_</span>, <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">MetaCoqUtils</span>.<br/>
</div>
<div class="doc">
<a id="lab3"></a><h2 class="section">MetaCoqUtils</h2>
Here, we define various useful functions for working with MetaCoq terms
</div>
<div class="code">
<span class="id" title="keyword">Notation</span> <span class="id" title="var">vectorCons</span> <span class="id" title="var">x</span> <span class="id" title="var">T</span> <span class="id" title="var">n</span> <span class="id" title="var">xr</span> := <br/>
(<span class="id" title="var">tApp</span><br/>
(<span class="id" title="var">tConstruct</span><br/>
{|<br/>
<span class="id" title="var">inductive_mind</span> := (<span class="id" title="var">MPfile</span> (["VectorDef"; "Vectors"; "Coq"]), "t");<br/>
<span class="id" title="var">inductive_ind</span> := 0 |} 1 <span class="id" title="var">nil</span>)<br/>
([<span class="id" title="var">T</span>; <span class="id" title="var">x</span>; <span class="id" title="var">n</span>; <span class="id" title="var">xr</span>])).<br/>
<span class="id" title="keyword">Notation</span> <span class="id" title="var">vectorNil</span> <span class="id" title="var">T</span> :=<br/>
(<span class="id" title="var">tApp</span><br/>
(<span class="id" title="var">tConstruct</span><br/>
{|<br/>
<span class="id" title="var">inductive_mind</span> := (<span class="id" title="var">MPfile</span> (["VectorDef"; "Vectors"; "Coq"]), "t");<br/>
<span class="id" title="var">inductive_ind</span> := 0 |} 0 <span class="id" title="var">nil</span>) ([<span class="id" title="var">T</span>])).<br/>
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">recoverVector</span> (<span class="id" title="var">f</span> : <span class="id" title="var">Ast.term</span>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">f</span>}: <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) := <span class="id" title="keyword">let</span> <span class="id" title="var">ffail</span> := <span class="id" title="tactic">fail</span> "not a vector application" <span class="id" title="tactic">in</span> <span class="id" title="keyword">match</span> <span class="id" title="var">f</span> <span class="id" title="keyword">with</span><br/>
<span class="id" title="var">vectorNil</span> <span class="id" title="var">_</span> => <span class="id" title="var">ret</span> <span class="id" title="var">nil</span><br/>
| <span class="id" title="var">vectorCons</span> <span class="id" title="var">x</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">xr</span> => <span class="id" title="var">xrl</span> <- <span class="id" title="var">recoverVector</span> <span class="id" title="var">xr</span>;;<span class="id" title="var">ret</span> (<span class="id" title="var">x</span>::<span class="id" title="var">xrl</span>)<br/>
| <span class="id" title="var">_</span> => <span class="id" title="var">ffail</span><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab4"></a><h3 class="section">General utils for quoted terms</h3>
Remove the n first elements from a list
</div>
<div class="code">
<span class="id" title="var">Existing</span> <span class="id" title="keyword">Instance</span> <span class="id" title="var">config.default_checker_flags</span>.<br/>
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">popNElements</span> (<span class="id" title="var">l</span> : <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) : <span class="id" title="var">option</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) := <span class="id" title="keyword">match</span> (<span class="id" title="var">l</span>,<span class="id" title="var">n</span>) <span class="id" title="keyword">with</span><br/>
(<span class="id" title="var">a</span>,0) => <span class="id" title="var">Some</span> <span class="id" title="var">a</span><br/>
| (<span class="id" title="var">x</span>::<span class="id" title="var">xr</span>, <span class="id" title="var">S</span> <span class="id" title="var">n</span>) => <span class="id" title="var">popNElements</span> <span class="id" title="var">xr</span> <span class="id" title="var">n</span><br/>
| <span class="id" title="var">_</span> => <span class="id" title="var">None</span> <span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
If ls is a valid prefix of l, return the corresponding "suffix" (the remaining elements of l). Otherwise, return None
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">popListStart</span> (<span class="id" title="var">l</span> : <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) (<span class="id" title="var">ls</span> : <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) : <span class="id" title="var">option</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) := <span class="id" title="keyword">match</span> (<span class="id" title="var">l</span>,<span class="id" title="var">ls</span>) <span class="id" title="keyword">with</span><br/>
(<span class="id" title="var">a</span>,<span class="id" title="var">nil</span>)=> <span class="id" title="var">Some</span> <span class="id" title="var">a</span><br/>
| (<span class="id" title="var">lx</span>::<span class="id" title="var">lxr</span>, <span class="id" title="var">lsx</span>::<span class="id" title="var">lsxr</span>) => <span class="id" title="keyword">if</span> <span class="id" title="var">Checker.eq_term</span> <span class="id" title="var">init_graph</span> <span class="id" title="var">lx</span> <span class="id" title="var">lsx</span> <span class="id" title="keyword">then</span> <span class="id" title="var">popListStart</span> <span class="id" title="var">lxr</span> <span class="id" title="var">lsxr</span> <span class="id" title="keyword">else</span> <span class="id" title="var">None</span><br/>
| <span class="id" title="var">_</span> => <span class="id" title="var">None</span> <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qNatZero</span> := 0.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qNatSucc</span> := <span class="id" title="var">S</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qeq_refl</span> := (@<span class="id" title="var">eq_refl</span>).<br/>
</div>
<div class="doc">
Given n, yield a MetaCoq Ast.term representation of n
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">quoteNumber</span> (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) : <span class="id" title="var">Ast.term</span>:= <span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span> 0 => <span class="id" title="var">qNatZero</span> | <span class="id" title="var">S</span> <span class="id" title="var">n</span> => <span class="id" title="var">tApp</span> <span class="id" title="var">qNatSucc</span> ([<span class="id" title="var">quoteNumber</span> <span class="id" title="var">n</span>]) <span class="id" title="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Increases the indices of all tRel in t by amnt, assuming they are already larger than minn. If minn = 0, this increases all "free" tRel reference indices
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">addRelIndex</span> (<span class="id" title="var">minn</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">amnt</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">t</span>:<span class="id" title="var">Ast.term</span>) : <span class="id" title="var">Ast.term</span> := <span class="id" title="keyword">match</span> <span class="id" title="var">t</span> <span class="id" title="keyword">with</span><br/>
<span class="id" title="var">tRel</span> <span class="id" title="var">n</span> => <span class="id" title="keyword">if</span> <span class="id" title="var">Compare_dec.le_gt_dec</span> <span class="id" title="var">minn</span> <span class="id" title="var">n</span> <span class="id" title="keyword">then</span> <span class="id" title="var">tRel</span> (<span class="id" title="var">amnt</span>+<span class="id" title="var">n</span>) <span class="id" title="keyword">else</span> <span class="id" title="var">tRel</span> <span class="id" title="var">n</span><br/>
| <span class="id" title="var">tVar</span> <span class="id" title="var">k</span> => <span class="id" title="var">tVar</span> <span class="id" title="var">k</span><br/>
| <span class="id" title="var">tEvar</span> <span class="id" title="var">n</span> <span class="id" title="var">ls</span> => <span class="id" title="var">tEvar</span> <span class="id" title="var">n</span> (<span class="id" title="var">map</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span>) <span class="id" title="var">ls</span>)<br/>
| <span class="id" title="var">tSort</span> <span class="id" title="var">u</span> => <span class="id" title="var">tSort</span> <span class="id" title="var">u</span><br/>
| <span class="id" title="var">tCast</span> <span class="id" title="var">t</span> <span class="id" title="var">ck</span> <span class="id" title="var">t2</span> => <span class="id" title="var">tCast</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">t</span>) <span class="id" title="var">ck</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">t2</span>)<br/>
| <span class="id" title="var">tProd</span> <span class="id" title="var">n</span> <span class="id" title="var">t</span> <span class="id" title="var">v</span> => <span class="id" title="var">tProd</span> <span class="id" title="var">n</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">t</span>) (<span class="id" title="var">addRelIndex</span> (<span class="id" title="var">S</span> <span class="id" title="var">minn</span>) <span class="id" title="var">amnt</span> <span class="id" title="var">v</span>)<br/>
| <span class="id" title="var">tLambda</span> <span class="id" title="var">n</span> <span class="id" title="var">t</span> <span class="id" title="var">v</span> => <span class="id" title="var">tLambda</span> <span class="id" title="var">n</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">t</span>) (<span class="id" title="var">addRelIndex</span> (<span class="id" title="var">S</span> <span class="id" title="var">minn</span>) <span class="id" title="var">amnt</span> <span class="id" title="var">v</span>)<br/>
| <span class="id" title="var">tLetIn</span> <span class="id" title="var">n</span> <span class="id" title="var">t</span> <span class="id" title="var">v</span> <span class="id" title="var">e</span> => <span class="id" title="var">tLetIn</span> <span class="id" title="var">n</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">t</span>) (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">v</span>) (<span class="id" title="var">addRelIndex</span> (<span class="id" title="var">S</span> <span class="id" title="var">minn</span>) <span class="id" title="var">amnt</span> <span class="id" title="var">e</span>)<br/>
| <span class="id" title="var">tApp</span> <span class="id" title="var">f</span> <span class="id" title="var">al</span> => <span class="id" title="var">tApp</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">f</span>) (<span class="id" title="var">map</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span>) <span class="id" title="var">al</span>)<br/>
| <span class="id" title="var">tConst</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> => <span class="id" title="var">tConst</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span><br/>
| <span class="id" title="var">tInd</span> <span class="id" title="var">i</span> <span class="id" title="var">t</span> => <span class="id" title="var">tInd</span> <span class="id" title="var">i</span> <span class="id" title="var">t</span><br/>
| <span class="id" title="var">tConstruct</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> => <span class="id" title="var">tConstruct</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span><br/>
| <span class="id" title="var">tCase</span> <span class="id" title="var">k</span> <span class="id" title="var">i</span> <span class="id" title="var">r</span> <span class="id" title="var">m</span> => <span class="id" title="var">tCase</span> <span class="id" title="var">k</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">i</span>) (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">r</span>) (<span class="id" title="var">map</span> (<span class="id" title="keyword">fun</span> '(<span class="id" title="var">a</span>,<span class="id" title="var">b</span>) => (<span class="id" title="var">a</span>,<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">b</span>)) <span class="id" title="var">m</span>)<br/>
| <span class="id" title="var">tProj</span> <span class="id" title="var">p</span> <span class="id" title="var">t</span> => <span class="id" title="var">tProj</span> <span class="id" title="var">p</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span> <span class="id" title="var">t</span>)<br/>
| <span class="id" title="var">tFix</span> <span class="id" title="var">mft</span> <span class="id" title="var">n</span> => <span class="id" title="var">tFix</span> (<span class="id" title="var">map</span> (<span class="id" title="var">map_def</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span>) (<span class="id" title="var">addRelIndex</span> (<span class="id" title="var">S</span> <span class="id" title="var">minn</span>) <span class="id" title="var">amnt</span>)) <span class="id" title="var">mft</span>) <span class="id" title="var">n</span><br/>
| <span class="id" title="var">tCoFix</span> <span class="id" title="var">mft</span> <span class="id" title="var">n</span> => <span class="id" title="var">tCoFix</span> (<span class="id" title="var">map</span> (<span class="id" title="var">map_def</span> (<span class="id" title="var">addRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">amnt</span>) (<span class="id" title="var">addRelIndex</span> (<span class="id" title="var">S</span> <span class="id" title="var">minn</span>) <span class="id" title="var">amnt</span>)) <span class="id" title="var">mft</span>) <span class="id" title="var">n</span><br/>
<span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">map_def_monad</span> {<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>} (<span class="id" title="var">tyf</span> <span class="id" title="var">bodyf</span> : <span class="id" title="var">A</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">B</span>) (<span class="id" title="var">d</span>:<span class="id" title="var">def</span> <span class="id" title="var">A</span>) : <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">def</span> <span class="id" title="var">B</span>) := <span class="id" title="var">dtr</span> <- <span class="id" title="var">tyf</span> (<span class="id" title="var">dbody</span> <span class="id" title="var">d</span>);;<span class="id" title="var">dbr</span> <- <span class="id" title="var">bodyf</span> (<span class="id" title="var">dbody</span> <span class="id" title="var">d</span>);; <br/>
<span class="id" title="var">ret</span> {| <span class="id" title="var">dname</span> := <span class="id" title="var">dname</span> <span class="id" title="var">d</span>; <span class="id" title="var">dtype</span> := <span class="id" title="var">dtr</span>; <span class="id" title="var">dbody</span> := <span class="id" title="var">dbr</span>; <span class="id" title="var">rarg</span> := <span class="id" title="var">rarg</span> <span class="id" title="var">d</span> |}.<br/>
</div>
<div class="doc">
Inverse operation to addRelIndex. Can fail, if the term for instance contains a tRel 0, which would need to be lowered to tRel (-1), which does not exist
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">lowerRelIndex</span> (<span class="id" title="var">minn</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">tv</span>:<span class="id" title="var">FailureMonad</span> <span class="id" title="var">Ast.term</span>) (<span class="id" title="var">t</span>:<span class="id" title="var">Ast.term</span>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">t</span>}: <span class="id" title="var">FailureMonad</span> <span class="id" title="var">Ast.term</span> := <span class="id" title="keyword">match</span> <span class="id" title="var">t</span> <span class="id" title="keyword">with</span><br/>
<span class="id" title="var">tRel</span> <span class="id" title="var">n</span> => <span class="id" title="keyword">if</span> <span class="id" title="var">Compare_dec.le_gt_dec</span> <span class="id" title="var">minn</span> <span class="id" title="var">n</span> <span class="id" title="keyword">then</span> <span class="id" title="keyword">if</span> <span class="id" title="var">Compare_dec.le_gt_dec</span> <span class="id" title="var">minn</span> (<span class="id" title="var">S</span> <span class="id" title="var">n</span>) <span class="id" title="keyword">then</span> <span class="id" title="var">ret</span> (<span class="id" title="var">tRel</span> (<span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span> 0 => 0 | <span class="id" title="var">S</span> <span class="id" title="var">n</span> => <span class="id" title="var">n</span> <span class="id" title="keyword">end</span>)) <span class="id" title="keyword">else</span> <span class="id" title="var">tv</span> <span class="id" title="keyword">else</span> <span class="id" title="var">ret</span> (<span class="id" title="var">tRel</span> <span class="id" title="var">n</span>)<br/>
| <span class="id" title="var">tVar</span> <span class="id" title="var">k</span> => <span class="id" title="var">ret</span> (<span class="id" title="var">tVar</span> <span class="id" title="var">k</span>)<br/>
| <span class="id" title="var">tEvar</span> <span class="id" title="var">n</span> <span class="id" title="var">ls</span> => <span class="id" title="var">m</span> <- <span class="id" title="var">map_monad</span> (<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span>) <span class="id" title="var">ls</span>;;<span class="id" title="var">ret</span> (<span class="id" title="var">tEvar</span> <span class="id" title="var">n</span> <span class="id" title="var">m</span>)<br/>
| <span class="id" title="var">tSort</span> <span class="id" title="var">u</span> => <span class="id" title="var">ret</span> (<span class="id" title="var">tSort</span> <span class="id" title="var">u</span>)<br/>
| <span class="id" title="var">tCast</span> <span class="id" title="var">t</span> <span class="id" title="var">ck</span> <span class="id" title="var">t2</span> => <span class="id" title="var">mt</span> <- (<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">t</span>);;<span class="id" title="var">mt2</span><-(<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">t2</span>);;<span class="id" title="var">ret</span> (<span class="id" title="var">tCast</span> <span class="id" title="var">mt</span> <span class="id" title="var">ck</span> <span class="id" title="var">mt2</span>)<br/>
| <span class="id" title="var">tProd</span> <span class="id" title="var">n</span> <span class="id" title="var">t</span> <span class="id" title="var">v</span> => <span class="id" title="var">mt</span><-(<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">t</span>);;<span class="id" title="var">mv</span><-(<span class="id" title="var">lowerRelIndex</span> (<span class="id" title="var">S</span> <span class="id" title="var">minn</span>) <span class="id" title="var">tv</span> <span class="id" title="var">v</span>);;<span class="id" title="var">ret</span> (<span class="id" title="var">tProd</span> <span class="id" title="var">n</span> <span class="id" title="var">mt</span> <span class="id" title="var">mv</span>)<br/>
| <span class="id" title="var">tLambda</span> <span class="id" title="var">n</span> <span class="id" title="var">t</span> <span class="id" title="var">v</span> => <span class="id" title="var">mt</span><-(<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">t</span>);;<span class="id" title="var">mv</span><-(<span class="id" title="var">lowerRelIndex</span> (<span class="id" title="var">S</span> <span class="id" title="var">minn</span>) <span class="id" title="var">tv</span> <span class="id" title="var">v</span>);;<span class="id" title="var">ret</span> (<span class="id" title="var">tLambda</span> <span class="id" title="var">n</span> <span class="id" title="var">mt</span> <span class="id" title="var">mv</span>)<br/>
| <span class="id" title="var">tLetIn</span> <span class="id" title="var">n</span> <span class="id" title="var">t</span> <span class="id" title="var">v</span> <span class="id" title="var">e</span> => <span class="id" title="var">mt</span><-(<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">t</span>);;<span class="id" title="var">mv</span><-(<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">v</span>);;<span class="id" title="var">me</span><-(<span class="id" title="var">lowerRelIndex</span> (<span class="id" title="var">S</span> <span class="id" title="var">minn</span>) <span class="id" title="var">tv</span> <span class="id" title="var">e</span>);;<span class="id" title="var">ret</span> (<span class="id" title="var">tLetIn</span> <span class="id" title="var">n</span> <span class="id" title="var">mt</span> <span class="id" title="var">mv</span> <span class="id" title="var">me</span>)<br/>
| <span class="id" title="var">tApp</span> <span class="id" title="var">f</span> <span class="id" title="var">al</span> => <span class="id" title="var">mal</span><-(<span class="id" title="var">map_monad</span> (<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span>) <span class="id" title="var">al</span>);;<span class="id" title="var">ff</span> <- <span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">f</span>;;<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">ff</span> <span class="id" title="var">mal</span>)<br/>
| <span class="id" title="var">tConst</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> => <span class="id" title="var">ret</span> (<span class="id" title="var">tConst</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>)<br/>
| <span class="id" title="var">tInd</span> <span class="id" title="var">i</span> <span class="id" title="var">t</span> => <span class="id" title="var">ret</span> (<span class="id" title="var">tInd</span> <span class="id" title="var">i</span> <span class="id" title="var">t</span>)<br/>
| <span class="id" title="var">tConstruct</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> => <span class="id" title="var">ret</span> (<span class="id" title="var">tConstruct</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span>)<br/>
| <span class="id" title="var">tCase</span> <span class="id" title="var">k</span> <span class="id" title="var">i</span> <span class="id" title="var">r</span> <span class="id" title="var">m</span> => <span class="id" title="var">mi</span><-(<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">i</span>);;<span class="id" title="var">mr</span><-(<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">r</span>);;<span class="id" title="var">mm</span><-(<span class="id" title="var">map_monad</span> (<span class="id" title="keyword">fun</span> '(<span class="id" title="var">a</span>,<span class="id" title="var">b</span>) => <span class="id" title="var">mb</span> <- <span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">b</span>;; <span class="id" title="var">ret</span> (<span class="id" title="var">a</span>,<span class="id" title="var">mb</span>)) <span class="id" title="var">m</span>);;<span class="id" title="var">ret</span> (<span class="id" title="var">tCase</span> <span class="id" title="var">k</span> <span class="id" title="var">mi</span> <span class="id" title="var">mr</span> <span class="id" title="var">mm</span>)<br/>
| <span class="id" title="var">tProj</span> <span class="id" title="var">p</span> <span class="id" title="var">t</span> => <span class="id" title="var">mt</span><-(<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span> <span class="id" title="var">t</span>);;<span class="id" title="var">ret</span> (<span class="id" title="var">tProj</span> <span class="id" title="var">p</span> <span class="id" title="var">mt</span>)<br/>
| <span class="id" title="var">tFix</span> <span class="id" title="var">mft</span> <span class="id" title="var">n</span> => <span class="id" title="var">mmft</span><-(<span class="id" title="var">map_monad</span> (<span class="id" title="var">map_def_monad</span> (<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span>) (<span class="id" title="var">lowerRelIndex</span> (<span class="id" title="var">S</span> <span class="id" title="var">minn</span>) <span class="id" title="var">tv</span>)) <span class="id" title="var">mft</span>);;<span class="id" title="var">ret</span> (<span class="id" title="var">tFix</span> <span class="id" title="var">mmft</span> <span class="id" title="var">n</span>)<br/>
| <span class="id" title="var">tCoFix</span> <span class="id" title="var">mft</span> <span class="id" title="var">n</span> => <span class="id" title="var">mmft</span><-(<span class="id" title="var">map_monad</span> (<span class="id" title="var">map_def_monad</span> (<span class="id" title="var">lowerRelIndex</span> <span class="id" title="var">minn</span> <span class="id" title="var">tv</span>) (<span class="id" title="var">lowerRelIndex</span> (<span class="id" title="var">S</span> <span class="id" title="var">minn</span>) <span class="id" title="var">tv</span>)) <span class="id" title="var">mft</span>);;<span class="id" title="var">ret</span> (<span class="id" title="var">tCoFix</span> <span class="id" title="var">mmft</span> <span class="id" title="var">n</span>)<br/>
<span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">MetaCoqUtils</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">AbstractReflectionDefinitions</span>.<br/>
</div>
<div class="doc">
<a id="lab5"></a><h2 class="section">AbstractReflectionDefinitions</h2>
Here, we define basic terms and types used when reifying Type of term reification helper, which is given to extension points / base connective reifier.
The arguments are:
<ul class="doclist">
<li> tct:Ast.term -- the current instance of the tarski_reflector, quoted
</li>
<li> t:Ast.term -- the actual term to be reified
</li>
<li> envTerm:Ast.term -- the env we are reifying into, quoted
</li>
<li> env: -- a helper, which gives the env index for Coq terms
</li>
<li> returning a pair of the reified term and a proof that it is correct, both quoted
</li>
</ul>
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">helperTermReifierType</span> := (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">Ast.term</span> -> (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">prod</span> <span class="id" title="var">Ast.term</span> <span class="id" title="var">Ast.term</span>)).<br/>
</div>
<div class="doc">
The same, but used during the env constructing phase. The only arg is the term itself
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">helperTermVarsType</span> := (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>)).<br/>
</div>
<div class="doc">
Similar, but this helper is used to reify terms, not forms. Args:
<ul class="doclist">
<li> tct -- as above
</li>
<li> frees:nat -- when reifying a nary predicate, this is n.
</li>
<li> t:Ast.term -- the actual term/form to be reified
</li>
<li> envTerm -- as above
</li>
<li> env -- as above
</li>
<li> returning again a apair of reified term and proof
</li>
</ul>
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">helperFormReifierType</span> := (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">nat</span> -> <span class="id" title="var">Ast.term</span> -> (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">prod</span> <span class="id" title="var">Ast.term</span> <span class="id" title="var">Ast.term</span>)).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">helperFormVarsType</span> := (<span class="id" title="var">Ast.term</span>-><span class="id" title="var">nat</span>-><span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>)).<br/>
</div>
<div class="doc">
A base connective is an inductive in Coq.Init.Logic, like and. They are common and have their own subsystem. Arguments:
<ul class="doclist">
<li> tct -- as above
</li>
<li> lst:list Ast.term -- for e.g. (and X Y), this list contains <span class="inlinecode"><span class="id" title="var">X</span>;<span class="id" title="var">Y</span></span>.
</li>
<li> fuel:nat -- fuel (when 0, we stop)
</li>
<li> envTerm
</li>
<li> env
</li>
<li> a helper to recursively reify forms
</li>
<li> a helper to recursively reify terms
</li>
<li> return again a a pair of term and proof
</li>
</ul>
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">baseConnectiveReifier</span> := <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">nat</span> -> <span class="id" title="var">Ast.term</span> -> (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) -> <span class="id" title="var">helperFormReifierType</span> -> <span class="id" title="var">helperTermReifierType</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">prod</span> <span class="id" title="var">Ast.term</span> <span class="id" title="var">Ast.term</span>).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">baseConnectiveVars</span> := <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">nat</span> -> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">helperFormVarsType</span> -> <span class="id" title="var">helperTermVarsType</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>).<br/>
</div>
<div class="doc">
Extension point for term reification. Args are:
<ul class="doclist">
<li> tct:Ast.term
</li>
<li> fuel:nat (also for var finder)
</li>
<li> t:term (also for var finder) -- the actual term
</li>
<li> envTerm
</li>
<li> env
</li>
<li> recursion helper (also for var finder)
</li>
<li> returning the usual
</li>
</ul>
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">termFinderVars</span> := <span class="id" title="var">nat</span> -> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">helperTermVarsType</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">termFinderReifier</span> := <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">nat</span> -> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">Ast.term</span> -> (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) <br/>
-> <span class="id" title="var">helperTermReifierType</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">prod</span> <span class="id" title="var">Ast.term</span> <span class="id" title="var">Ast.term</span>).<br/>
</div>
<div class="doc">
Extension point for form reification. Args are:
<ul class="doclist">
<li> tct:Ast.term
</li>
<li> fuel:nat (also for var finder)
</li>
<li> t:term (also for var finder) -- the actual term
</li>
<li> frees
</li>
<li> envTerm
</li>
<li> env
</li>
<li> recursion helper for forms (also for var finder)
</li>
<li> recursion helper for terms (also for var finder)
</li>
<li> returning the usual
</li>
</ul>
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">propFinderVars</span> := <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">nat</span> -> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">nat</span> -> <span class="id" title="var">helperFormVarsType</span> -> <span class="id" title="var">helperTermVarsType</span> -> (<span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>)).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">propFinderReifier</span> := <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">nat</span> -> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">nat</span> -> <span class="id" title="var">Ast.term</span> -> (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) -> <span class="id" title="var">helperFormReifierType</span> -> <span class="id" title="var">helperTermReifierType</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">prod</span> <span class="id" title="var">Ast.term</span> <span class="id" title="var">Ast.term</span>).<br/>
<br/>
</div>
<div class="doc">
When running in no proof mode, extension points still expect proofs. We give this dummy term. If the extension point only uses this to build further proofs, the whole term is discarded in the end
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">noProofDummy</span> := <span class="id" title="var">tVar</span> "NoProofGivenBecauseWeAreInNoProofMode".<br/>
</div>
<div class="doc">
An instance of this class is required for the reifier to work. From it the basic reification building blocks are derived
</div>
<div class="code">
<span class="id" title="keyword">Class</span> <span class="id" title="var">tarski_reflector</span> := {<br/>
<span class="id" title="var">fs</span> : <span class="id" title="var">funcs_signature</span>; <br/>
<span class="id" title="var">ps</span> : <span class="id" title="var">preds_signature</span>; <br/>
<span class="id" title="var">D</span> : <span class="id" title="keyword">Type</span>;<br/>
<span class="id" title="var">I</span> : @<span class="id" title="var">interp</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span>; <br/>
<span class="id" title="var">emptyEnv</span> : <span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>; <br/>
<span class="id" title="var">isD</span> : <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">bool</span>; <br/>
}.<br/>
</div>
<div class="doc">
Extension point to extend the libary to more syntactic constructs
</div>
<div class="code">
<span class="id" title="keyword">Class</span> <span class="id" title="var">tarski_reflector_extensions</span> (<span class="id" title="var">t</span>:<span class="id" title="var">tarski_reflector</span>) := {<br/>
</div>
<div class="doc">
for reifying instances of inductive types Coq.Init.Logic.* applied to arguments. Mostly used for reifying eq
</div>
<div class="code">
<span class="id" title="var">baseLogicConnHelper</span> : <span class="id" title="var">option</span> (<span class="id" title="var">string</span> -> <span class="id" title="var">baseConnectiveReifier</span>); <br/>
<span class="id" title="var">baseLogicVarHelper</span> : <span class="id" title="var">option</span> (<span class="id" title="var">string</span> -> <span class="id" title="var">baseConnectiveVars</span>);<br/>
</div>
<div class="doc">
for reifying all terms that we fail to reify
</div>
<div class="code">
<span class="id" title="var">termReifierVarHelper</span> : <span class="id" title="var">option</span> <span class="id" title="var">termFinderVars</span>;<br/>
<span class="id" title="var">termReifierReifyHelper</span> : <span class="id" title="var">option</span> <span class="id" title="var">termFinderReifier</span>;<br/>
</div>
<div class="doc">
for reifying all forms that we fail to reify
</div>
<div class="code">
<span class="id" title="var">formReifierVarHelper</span> : <span class="id" title="var">option</span> <span class="id" title="var">propFinderVars</span>;<br/>
<span class="id" title="var">formReifierReifyHelper</span> : <span class="id" title="var">option</span> <span class="id" title="var">propFinderReifier</span><br/>
}.<br/>
</div>
<div class="doc">
Used if no instance of the above extension class can be found
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">defaultExtensions</span> <span class="id" title="var">tr</span> : <span class="id" title="var">tarski_reflector_extensions</span> <span class="id" title="var">tr</span> := <span class="id" title="var">Build_tarski_reflector_extensions</span> <span class="id" title="var">tr</span> <span class="id" title="var">None</span> <span class="id" title="var">None</span> <span class="id" title="var">None</span> <span class="id" title="var">None</span> <span class="id" title="var">None</span> <span class="id" title="var">None</span>.<br/>
</div>
<div class="doc">
Useful builder to build the main tarski_reflector class
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">buildDefaultTarski</span> {<span class="id" title="var">fs</span>:<span class="id" title="var">funcs_signature</span>} {<span class="id" title="var">ps</span>:<span class="id" title="var">preds_signature</span>} {<span class="id" title="var">D</span>:<span class="id" title="keyword">Type</span>} {<span class="id" title="var">I</span>:@<span class="id" title="var">interp</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span>} (<span class="id" title="var">point</span>:<span class="id" title="var">D</span>) (<span class="id" title="var">isD</span>:<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">bool</span>) := <span class="id" title="var">Build_tarski_reflector</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span> <span class="id" title="var">I</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">n</span>:<span class="id" title="var">nat</span> => <span class="id" title="var">point</span>) <span class="id" title="var">isD</span>.<br/>
<span class="id" title="keyword">Context</span> {<span class="id" title="var">tr</span> : <span class="id" title="var">tarski_reflector</span>}.<br/>
</div>
<div class="doc">
These types are used to define representability. A n-ary formula is representable if there is a syntactic formula and an environment that you can plug the relevant values into that is equivalent
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">naryProp</span> (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) : <span class="id" title="keyword">Type</span> := <span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span> 0 => <span class="id" title="keyword">Prop</span> | <span class="id" title="var">S</span> <span class="id" title="var">nn</span> => <span class="id" title="var">D</span> -> <span class="id" title="var">naryProp</span> <span class="id" title="var">nn</span> <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">representsP</span> {<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>} <span class="id" title="var">phi</span> <span class="id" title="var">rho</span> : (<span class="id" title="keyword">forall</span> (<span class="id" title="var">P</span>:<span class="id" title="var">naryProp</span> <span class="id" title="var">n</span>), <span class="id" title="keyword">Prop</span>) := <span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">return</span> (<span class="id" title="keyword">forall</span> (<span class="id" title="var">P</span>:<span class="id" title="var">naryProp</span> <span class="id" title="var">n</span>), <span class="id" title="keyword">Prop</span>) <span class="id" title="keyword">with</span><br/>
0 => (<span class="id" title="keyword">fun</span> (<span class="id" title="var">P</span>:<span class="id" title="keyword">Prop</span>) => <span class="id" title="var">P</span> <-> @<span class="id" title="var">sat</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span> <span class="id" title="var">I</span> <span class="id" title="var">rho</span> <span class="id" title="var">phi</span>)<br/>
| <span class="id" title="var">S</span> <span class="id" title="var">n</span> => (<span class="id" title="keyword">fun</span> (<span class="id" title="var">P</span>:<span class="id" title="var">D</span> -> <span class="id" title="var">naryProp</span> <span class="id" title="var">n</span>) => <span class="id" title="keyword">forall</span> <span class="id" title="var">d</span>, @<span class="id" title="var">representsP</span> <span class="id" title="var">n</span> <span class="id" title="var">phi</span> (<span class="id" title="var">d</span>.:<span class="id" title="var">rho</span>) (<span class="id" title="var">P</span> <span class="id" title="var">d</span>)) <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">representableP</span> (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">P</span> : <span class="id" title="var">naryProp</span> <span class="id" title="var">n</span>) := <span class="id" title="tactic">exists</span> <span class="id" title="var">phi</span> <span class="id" title="var">rho</span>, <span class="id" title="var">representsP</span> <span class="id" title="var">phi</span> <span class="id" title="var">rho</span> <span class="id" title="var">P</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">representsF</span> (<span class="id" title="var">d</span>:<span class="id" title="var">D</span>) <span class="id" title="var">trm</span> <span class="id" title="var">rho</span> := @<span class="id" title="tactic">eval</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span> <span class="id" title="var">I</span> <span class="id" title="var">rho</span> <span class="id" title="var">trm</span> = <span class="id" title="var">d</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">representableF</span> (<span class="id" title="var">d</span>:<span class="id" title="var">D</span>) := <span class="id" title="tactic">exists</span> <span class="id" title="var">trm</span> <span class="id" title="var">rho</span>, <span class="id" title="var">representsF</span> <span class="id" title="var">d</span> <span class="id" title="var">trm</span> <span class="id" title="var">rho</span>.<br/>
<br/>
</div>
<div class="doc">
Functions that allow us to construct syntactic connective applications without building vectors in MetaCoq
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">naryGFunc</span> (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">A</span> <span class="id" title="var">R</span> : <span class="id" title="keyword">Type</span>) := <span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span> 0 => <span class="id" title="var">R</span> | <span class="id" title="var">S</span> <span class="id" title="var">n</span> => <span class="id" title="var">A</span> -> @<span class="id" title="var">naryGFunc</span> <span class="id" title="var">n</span> <span class="id" title="var">A</span> <span class="id" title="var">R</span> <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">takeMultiple</span> {<span class="id" title="var">n</span> : <span class="id" title="var">nat</span>} (<span class="id" title="var">X</span> <span class="id" title="var">Y</span>:<span class="id" title="keyword">Type</span>) : (<span class="id" title="var">Vector.t</span> <span class="id" title="var">X</span> <span class="id" title="var">n</span> -> <span class="id" title="var">Y</span>) -> @<span class="id" title="var">naryGFunc</span> <span class="id" title="var">n</span> <span class="id" title="var">X</span> <span class="id" title="var">Y</span> := <br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">as</span> <span class="id" title="var">nn</span> <span class="id" title="keyword">return</span> (<span class="id" title="var">Vector.t</span> <span class="id" title="var">X</span> <span class="id" title="var">nn</span> -> <span class="id" title="var">Y</span>) -> @<span class="id" title="var">naryGFunc</span> <span class="id" title="var">nn</span> <span class="id" title="var">X</span> <span class="id" title="var">Y</span><br/>
<span class="id" title="keyword">with</span> 0 => <span class="id" title="keyword">fun</span> <span class="id" title="var">R</span> => <span class="id" title="var">R</span> (<span class="id" title="var">Vector.nil</span> <span class="id" title="var">X</span>) <br/>
| <span class="id" title="var">S</span> <span class="id" title="var">n</span> => <span class="id" title="keyword">fun</span> <span class="id" title="var">R</span> <span class="id" title="var">v</span> => @<span class="id" title="var">takeMultiple</span> <span class="id" title="var">n</span> <span class="id" title="var">X</span> <span class="id" title="var">Y</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">vr</span> => <span class="id" title="var">R</span> (<span class="id" title="var">Vector.cons</span> <span class="id" title="var">X</span> <span class="id" title="var">v</span> <span class="id" title="var">n</span> <span class="id" title="var">vr</span>)) <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">constructTerm</span> (<span class="id" title="var">k</span>:@<span class="id" title="var">syms</span> (@<span class="id" title="var">fs</span> <span class="id" title="var">tr</span>)) := @<span class="id" title="var">takeMultiple</span> (<span class="id" title="var">ar_syms</span> <span class="id" title="var">k</span>) <span class="id" title="var">term</span> <span class="id" title="var">term</span> (<span class="id" title="var">func</span> <span class="id" title="var">k</span>).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">constructForm</span> (<span class="id" title="var">k</span>:(@<span class="id" title="var">preds</span> (@<span class="id" title="var">ps</span> <span class="id" title="var">tr</span>))) := @<span class="id" title="var">takeMultiple</span> (@<span class="id" title="var">ar_preds</span> (@<span class="id" title="var">ps</span> <span class="id" title="var">tr</span>) <span class="id" title="var">k</span>) (@<span class="id" title="var">term</span> (@<span class="id" title="var">fs</span> <span class="id" title="var">tr</span>)) <span class="id" title="var">form</span> (<span class="id" title="var">atom</span> <span class="id" title="var">k</span>).<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">nary3GFunc</span> (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">A'</span> <span class="id" title="var">B'</span>:<span class="id" title="keyword">Type</span>) (<span class="id" title="var">C</span>:<span class="id" title="var">A</span> -> <span class="id" title="var">B</span> -> <span class="id" title="keyword">Type</span>) (<span class="id" title="var">C'</span>:<span class="id" title="var">A'</span>-><span class="id" title="var">B'</span>-><span class="id" title="keyword">Type</span>) : (<span class="id" title="var">Vector.t</span> <span class="id" title="var">A</span> <span class="id" title="var">n</span> -> <span class="id" title="var">A'</span>) -> (<span class="id" title="var">Vector.t</span> <span class="id" title="var">B</span> <span class="id" title="var">n</span> -> <span class="id" title="var">B'</span>) -> <span class="id" title="keyword">Type</span><br/>
:= <span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span> 0 => <span class="id" title="keyword">fun</span> <span class="id" title="var">mA</span> <span class="id" title="var">mB</span> => <span class="id" title="var">C'</span> (<span class="id" title="var">mA</span> (<span class="id" title="var">Vector.nil</span> <span class="id" title="var">A</span>)) (<span class="id" title="var">mB</span> (<span class="id" title="var">Vector.nil</span> <span class="id" title="var">B</span>))<br/>
| <span class="id" title="var">S</span> <span class="id" title="var">n</span> => <span class="id" title="keyword">fun</span> <span class="id" title="var">mA</span> <span class="id" title="var">mB</span> => <span class="id" title="keyword">forall</span> (<span class="id" title="var">a</span>:<span class="id" title="var">A</span>) (<span class="id" title="var">b</span>:<span class="id" title="var">B</span>), <span class="id" title="var">C</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> -> @<span class="id" title="var">nary3GFunc</span> <span class="id" title="var">n</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">A'</span> <span class="id" title="var">B'</span> <span class="id" title="var">C</span> <span class="id" title="var">C'</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">v</span> => <span class="id" title="var">mA</span> (<span class="id" title="var">Vector.cons</span> <span class="id" title="var">A</span> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">v</span>)) (<span class="id" title="keyword">fun</span> <span class="id" title="var">v</span> => <span class="id" title="var">mB</span> (<span class="id" title="var">Vector.cons</span> <span class="id" title="var">B</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">v</span>)) <span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeTermBase</span> (<span class="id" title="var">c</span>:@<span class="id" title="var">syms</span> (@<span class="id" title="var">fs</span> <span class="id" title="var">tr</span>)) : <span class="id" title="keyword">Type</span> := <br/>
<span class="id" title="keyword">forall</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>), @<span class="id" title="var">nary3GFunc</span> (<span class="id" title="var">ar_syms</span> <span class="id" title="var">c</span>) <span class="id" title="var">term</span> <span class="id" title="var">D</span> <span class="id" title="var">term</span> <span class="id" title="var">D</span> <br/>
(<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> <span class="id" title="var">d</span> => <span class="id" title="var">representsF</span> <span class="id" title="var">d</span> <span class="id" title="var">t</span> <span class="id" title="var">rho</span>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> <span class="id" title="var">d</span> => <span class="id" title="var">representsF</span> <span class="id" title="var">d</span> <span class="id" title="var">t</span> <span class="id" title="var">rho</span>) <br/>
(<span class="id" title="var">func</span> <span class="id" title="var">c</span>) (@<span class="id" title="var">i_f</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span> <span class="id" title="var">I</span> <span class="id" title="var">c</span>).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeTermProtoType</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">fZ</span>:<span class="id" title="var">vec</span> <span class="id" title="var">term</span> <span class="id" title="var">n</span> -> <span class="id" title="var">term</span>) (<span class="id" title="var">ifZ</span> : <span class="id" title="var">vec</span> <span class="id" title="var">D</span> <span class="id" title="var">n</span> -> <span class="id" title="var">D</span>) := <br/>
(<span class="id" title="keyword">forall</span> <span class="id" title="var">v</span> : <span class="id" title="var">vec</span> <span class="id" title="var">term</span> <span class="id" title="var">n</span>, @<span class="id" title="tactic">eval</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span> <span class="id" title="var">I</span> <span class="id" title="var">rho</span> (<span class="id" title="var">fZ</span> <span class="id" title="var">v</span>) = <span class="id" title="var">ifZ</span> (<span class="id" title="var">Vector.map</span> (@<span class="id" title="tactic">eval</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span> <span class="id" title="var">I</span> <span class="id" title="var">rho</span>) <span class="id" title="var">v</span>)) <br/>
-> @<span class="id" title="var">nary3GFunc</span> <span class="id" title="var">n</span> <span class="id" title="var">term</span> <span class="id" title="var">D</span> <span class="id" title="var">term</span> <span class="id" title="var">D</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> <span class="id" title="var">d</span> => <span class="id" title="var">representsF</span> <span class="id" title="var">d</span> <span class="id" title="var">t</span> <span class="id" title="var">rho</span>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> <span class="id" title="var">d</span> => <span class="id" title="var">representsF</span> <span class="id" title="var">d</span> <span class="id" title="var">t</span> <span class="id" title="var">rho</span>) <span class="id" title="var">fZ</span> <span class="id" title="var">ifZ</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeTermProto</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">fZ</span>:<span class="id" title="var">vec</span> <span class="id" title="var">term</span> <span class="id" title="var">n</span> -> <span class="id" title="var">term</span>) (<span class="id" title="var">ifZ</span> : <span class="id" title="var">vec</span> <span class="id" title="var">D</span> <span class="id" title="var">n</span> -> <span class="id" title="var">D</span>) : <span class="id" title="var">mergeTermProtoType</span> <span class="id" title="var">rho</span> <span class="id" title="var">n</span> <span class="id" title="var">fZ</span> <span class="id" title="var">ifZ</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">induction</span> <span class="id" title="var">n</span> <span class="id" title="keyword">as</span> [|<span class="id" title="var">n</span> <span class="id" title="var">IH</span>].<br/>
* <span class="id" title="var">cbn</span>. <span class="id" title="tactic">unfold</span> <span class="id" title="var">representsF</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>. <span class="id" title="var">cbn</span>. <span class="id" title="var">easy</span>.<br/>
* <span class="id" title="var">cbn</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">t</span> <span class="id" title="var">d</span> <span class="id" title="var">r</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>. <span class="id" title="var">cbn</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">v</span>. <span class="id" title="tactic">specialize</span> (<span class="id" title="var">H</span> (<span class="id" title="var">Vector.cons</span> <span class="id" title="var">t</span> <span class="id" title="var">v</span>)). <span class="id" title="tactic">unfold</span> <span class="id" title="var">representsF</span> <span class="id" title="tactic">in</span> <span class="id" title="var">r</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>. <span class="id" title="var">cbn</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">r</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeTerm</span> (<span class="id" title="var">c</span>:<span class="id" title="var">syms</span>) : <span class="id" title="var">mergeTermBase</span> <span class="id" title="var">c</span>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">rho</span>. <span class="id" title="tactic">eapply</span> <span class="id" title="var">mergeTermProto</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">intros</span> <span class="id" title="var">v</span>. <span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeFormBase</span> (<span class="id" title="var">c</span>:@<span class="id" title="var">preds</span> (@<span class="id" title="var">ps</span> <span class="id" title="var">tr</span>)) : <span class="id" title="keyword">Type</span> := <br/>
<span class="id" title="keyword">forall</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>), @<span class="id" title="var">nary3GFunc</span> (<span class="id" title="var">ar_preds</span> <span class="id" title="var">c</span>) <span class="id" title="var">term</span> <span class="id" title="var">D</span> <span class="id" title="var">form</span> (<span class="id" title="var">naryProp</span> 0) <br/>
(<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> <span class="id" title="var">d</span> => <span class="id" title="var">representsF</span> <span class="id" title="var">d</span> <span class="id" title="var">t</span> <span class="id" title="var">rho</span>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> <span class="id" title="var">P</span> => <span class="id" title="var">representsP</span> <span class="id" title="var">t</span> <span class="id" title="var">rho</span> <span class="id" title="var">P</span>) <br/>
(<span class="id" title="var">atom</span> <span class="id" title="var">c</span>) (@<span class="id" title="var">i_P</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span> <span class="id" title="var">I</span> <span class="id" title="var">c</span>).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeFormProtoType</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">fZ</span>:<span class="id" title="var">vec</span> <span class="id" title="var">term</span> <span class="id" title="var">n</span> -> <span class="id" title="var">form</span>) (<span class="id" title="var">ifZ</span> : <span class="id" title="var">vec</span> <span class="id" title="var">D</span> <span class="id" title="var">n</span> -> <span class="id" title="var">naryProp</span> 0) := <br/>
(<span class="id" title="keyword">forall</span> <span class="id" title="var">v</span> : <span class="id" title="var">vec</span> <span class="id" title="var">term</span> <span class="id" title="var">n</span>, @<span class="id" title="var">sat</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span> <span class="id" title="var">I</span> <span class="id" title="var">rho</span> (<span class="id" title="var">fZ</span> <span class="id" title="var">v</span>) = <span class="id" title="var">ifZ</span> (<span class="id" title="var">Vector.map</span> (@<span class="id" title="tactic">eval</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">D</span> <span class="id" title="var">I</span> <span class="id" title="var">rho</span>) <span class="id" title="var">v</span>)) <br/>
-> @<span class="id" title="var">nary3GFunc</span> <span class="id" title="var">n</span> <span class="id" title="var">term</span> <span class="id" title="var">D</span> <span class="id" title="var">form</span> (<span class="id" title="var">naryProp</span> 0) (<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> <span class="id" title="var">d</span> => <span class="id" title="var">representsF</span> <span class="id" title="var">d</span> <span class="id" title="var">t</span> <span class="id" title="var">rho</span>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> <span class="id" title="var">P</span> => <span class="id" title="var">representsP</span> <span class="id" title="var">t</span> <span class="id" title="var">rho</span> <span class="id" title="var">P</span>) <span class="id" title="var">fZ</span> <span class="id" title="var">ifZ</span>.<br/>
</div>
<div class="doc">
Proof that the functions built using the above helper actually represent the syntactic construct they are supposed to represent
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeFormProto</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">fZ</span>:<span class="id" title="var">vec</span> <span class="id" title="var">term</span> <span class="id" title="var">n</span> -> <span class="id" title="var">form</span>) (<span class="id" title="var">ifZ</span> : <span class="id" title="var">vec</span> <span class="id" title="var">D</span> <span class="id" title="var">n</span> -> <span class="id" title="var">naryProp</span> 0) : <span class="id" title="var">mergeFormProtoType</span> <span class="id" title="var">rho</span> <span class="id" title="var">n</span> <span class="id" title="var">fZ</span> <span class="id" title="var">ifZ</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">induction</span> <span class="id" title="var">n</span> <span class="id" title="keyword">as</span> [|<span class="id" title="var">n</span> <span class="id" title="var">IH</span>].<br/>
* <span class="id" title="var">cbn</span>. <span class="id" title="tactic">unfold</span> <span class="id" title="var">representsP</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>. <span class="id" title="var">cbn</span>. <span class="id" title="var">easy</span>.<br/>
* <span class="id" title="var">cbn</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">t</span> <span class="id" title="var">d</span> <span class="id" title="var">r</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>. <span class="id" title="var">cbn</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">v</span>. <span class="id" title="tactic">specialize</span> (<span class="id" title="var">H</span> (<span class="id" title="var">Vector.cons</span> <span class="id" title="var">t</span> <span class="id" title="var">v</span>)). <span class="id" title="tactic">unfold</span> <span class="id" title="var">representsF</span> <span class="id" title="tactic">in</span> <span class="id" title="var">r</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>. <span class="id" title="var">cbn</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">r</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeForm</span> (<span class="id" title="var">c</span>:<span class="id" title="var">preds</span>) : <span class="id" title="var">mergeFormBase</span> <span class="id" title="var">c</span>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">rho</span>. <span class="id" title="tactic">eapply</span> <span class="id" title="var">mergeFormProto</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">intros</span> <span class="id" title="var">v</span>. <span class="id" title="keyword">Defined</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qConstructTerm</span> := <span class="id" title="var">constructTerm</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeTerm</span> := <span class="id" title="var">mergeTerm</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qConstructForm</span> := <span class="id" title="var">constructForm</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeForm</span> := <span class="id" title="var">mergeForm</span>.<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">AbstractReflectionDefinitions</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">TarskiMerging</span>.<br/>
</div>
<div class="doc">
**TarskiMerging
Here we have reifiers for basic logical connectives which are part of the core Tarski semantics and thus every model.
These are and, or, falsity, implication and both quantifiers
</div>
<div class="code">
<span class="id" title="keyword">Context</span> {<span class="id" title="var">tr</span> : <span class="id" title="var">tarski_reflector</span>}.<br/>
<span class="id" title="keyword">Context</span> {<span class="id" title="var">te</span> : <span class="id" title="var">tarski_reflector_extensions</span> <span class="id" title="var">tr</span>}.<br/>
</div>
<div class="doc">
For each connective, we have a "merger" that takes subproof that a term is represented by its reification, and proof that the larger term also represents its subterm.
For falsity, there are not subterms, so we just prove that fal represents False, which is trivial
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeFalse</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) : @<span class="id" title="var">representsP</span> <span class="id" title="var">tr</span> 0 <span class="id" title="var">fal</span> <span class="id" title="var">rho</span> <span class="id" title="var">False</span>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="var">easy</span>. <span class="id" title="keyword">Defined</span>.<br/>
</div>
<div class="doc">
We then define a quoted version of the above proof which is later used to build subterms
</div>
<div class="code">
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeFalse</span> := @<span class="id" title="var">mergeFalse</span>.<br/>
</div>
<div class="doc">
We also define a form merger, which is a function that takes the reified subterms (not the proofs) and builds the reified term. For False, this is just a term that represents false (here given without syntact sugar)
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mFalse</span> := (@<span class="id" title="var">fal</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">full_operators</span>).<br/>
</div>
<div class="doc">
Finally we define a quoted form merger which will then later be applied to the subforms
</div>
<div class="code">
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeFormFalse</span> := @<span class="id" title="var">mFalse</span>.<br/>
<br/>
</div>
<div class="doc">
The same development for And. Note that this time we have subproofs. Our function arguments follow this pattern:
<ul class="doclist">
<li> tr_quoted - quoted tarski_reflector (not visible in the arguments, but it's a context variable. Note that since the functions don't acually use te, it does not become part of the arguments outside the section
</li>
<li> environment
</li>
<li> coq subterms that are to be reified
</li>
<li> reified subterms
</li>
<li> proof that the reified subterms are actually the reifications
</li>
</ul>
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeAnd</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) (<span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <span class="id" title="var">naryProp</span> 0) (<span class="id" title="var">fP</span> <span class="id" title="var">fQ</span> : <span class="id" title="var">form</span>) : <span class="id" title="var">representsP</span> <span class="id" title="var">fP</span> <span class="id" title="var">rho</span> <span class="id" title="var">P</span> -> <span class="id" title="var">representsP</span> <span class="id" title="var">fQ</span> <span class="id" title="var">rho</span> <span class="id" title="var">Q</span> -> @<span class="id" title="var">representsP</span> <span class="id" title="var">tr</span> 0 (<span class="id" title="var">fP</span>∧<span class="id" title="var">fQ</span>) <span class="id" title="var">rho</span> (<span class="id" title="var">P</span> /\ <span class="id" title="var">Q</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> [<span class="id" title="var">pPl</span> <span class="id" title="var">pPr</span>] [<span class="id" title="var">pQl</span> <span class="id" title="var">pQr</span>]. <span class="id" title="tactic">split</span>.<br/>
* <span class="id" title="tactic">intros</span> [<span class="id" title="var">pP</span> <span class="id" title="var">pQ</span>]. <span class="id" title="tactic">split</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">pPl</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">pQl</span>.<br/>
* <span class="id" title="tactic">intros</span> [<span class="id" title="var">pP</span> <span class="id" title="var">pQ</span>]. <span class="id" title="tactic">split</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">pPr</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">pQr</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeAnd</span> := @<span class="id" title="var">mergeAnd</span>.<br/>
</div>
<div class="doc">
This is the form merger for and. It is unfolded syntactic sugar, once one adds the arguments x and y, this will read x ∧ y.
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mAnd</span> := (@<span class="id" title="var">bin</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">full_operators</span> <span class="id" title="var">Conj</span>).<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeFormAnd</span> := @<span class="id" title="var">mAnd</span>.<br/>
<br/>
</div>
<div class="doc">
The same development for or
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeOr</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) (<span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <span class="id" title="var">naryProp</span> 0) (<span class="id" title="var">fP</span> <span class="id" title="var">fQ</span> : <span class="id" title="var">form</span>) : <span class="id" title="var">representsP</span> <span class="id" title="var">fP</span> <span class="id" title="var">rho</span> <span class="id" title="var">P</span> -> <span class="id" title="var">representsP</span> <span class="id" title="var">fQ</span> <span class="id" title="var">rho</span> <span class="id" title="var">Q</span> -> @<span class="id" title="var">representsP</span> <span class="id" title="var">tr</span> 0 (<span class="id" title="var">fP</span>∨<span class="id" title="var">fQ</span>) <span class="id" title="var">rho</span> (<span class="id" title="var">P</span> \/ <span class="id" title="var">Q</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> [<span class="id" title="var">pPl</span> <span class="id" title="var">pPr</span>] [<span class="id" title="var">pQl</span> <span class="id" title="var">pQr</span>]. <span class="id" title="tactic">split</span>.<br/>
* <span class="id" title="tactic">intros</span> [<span class="id" title="var">pP</span>|<span class="id" title="var">pQ</span>]. <span class="id" title="tactic">left</span>; <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">pPl</span>. <span class="id" title="tactic">right</span>; <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">pQl</span>.<br/>
* <span class="id" title="tactic">intros</span> [<span class="id" title="var">pP</span>|<span class="id" title="var">pQ</span>]. <span class="id" title="tactic">left</span>; <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">pPr</span>. <span class="id" title="tactic">right</span>; <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">pQr</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeOr</span> := @<span class="id" title="var">mergeOr</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mOr</span> := (@<span class="id" title="var">bin</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">full_operators</span> <span class="id" title="var">Disj</span>).<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeFormOr</span> := @<span class="id" title="var">mOr</span>.<br/>
<br/>
</div>
<div class="doc">
The same development for existential quantifiaction. Note that the P argument is a 1-ary predicate.
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeExists</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) (<span class="id" title="var">P</span>:<span class="id" title="var">naryProp</span> 1) (<span class="id" title="var">fP</span>:<span class="id" title="var">form</span>) : <span class="id" title="var">representsP</span> <span class="id" title="var">fP</span> <span class="id" title="var">rho</span> <span class="id" title="var">P</span> -> @<span class="id" title="var">representsP</span> <span class="id" title="var">tr</span> 0 (∃ <span class="id" title="var">fP</span>) <span class="id" title="var">rho</span> (<span class="id" title="tactic">exists</span> <span class="id" title="var">q</span>:<span class="id" title="var">D</span>, <span class="id" title="var">P</span> <span class="id" title="var">q</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">pR</span>. <span class="id" title="tactic">split</span>.<br/>
* <span class="id" title="tactic">intros</span> [<span class="id" title="var">q</span> <span class="id" title="var">Pq</span>]. <span class="id" title="tactic">exists</span> <span class="id" title="var">q</span>. <span class="id" title="tactic">destruct</span> (<span class="id" title="var">pR</span> <span class="id" title="var">q</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">pRl</span> <span class="id" title="var">pRr</span>]. <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">pRl</span>.<br/>
* <span class="id" title="tactic">intros</span> [<span class="id" title="var">q</span> <span class="id" title="var">Pq</span>]. <span class="id" title="tactic">exists</span> <span class="id" title="var">q</span>. <span class="id" title="tactic">destruct</span> (<span class="id" title="var">pR</span> <span class="id" title="var">q</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">pRl</span> <span class="id" title="var">pRr</span>]. <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">pRr</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeExists</span> := @<span class="id" title="var">mergeExists</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mExists</span> := (@<span class="id" title="var">quant</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">full_operators</span> <span class="id" title="var">Ex</span>).<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeFormExists</span> := @<span class="id" title="var">mExists</span>.<br/>
<br/>
</div>
<div class="doc">
The same development for implication. Note that implication is handled in the main reification logic since P -> Q is just syntactic sugar for (forall _:P, Q)
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeImpl</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) (<span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <span class="id" title="var">naryProp</span> 0) (<span class="id" title="var">fP</span> <span class="id" title="var">fQ</span> : <span class="id" title="var">form</span>) : <span class="id" title="var">representsP</span> <span class="id" title="var">fP</span> <span class="id" title="var">rho</span> <span class="id" title="var">P</span> -> <span class="id" title="var">representsP</span> <span class="id" title="var">fQ</span> <span class="id" title="var">rho</span> <span class="id" title="var">Q</span> -> @<span class="id" title="var">representsP</span> <span class="id" title="var">tr</span> 0 (<span class="id" title="var">fP</span> --> <span class="id" title="var">fQ</span>) <span class="id" title="var">rho</span> (<span class="id" title="var">P</span> -> <span class="id" title="var">Q</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">HP</span> <span class="id" title="var">HQ</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">HP</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">pPl</span> <span class="id" title="var">pPr</span>]. <span class="id" title="tactic">destruct</span> <span class="id" title="var">HQ</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">pQl</span> <span class="id" title="var">pQr</span>]. <span class="id" title="tactic">split</span>.<br/>
* <span class="id" title="tactic">intros</span> <span class="id" title="var">PQ</span> <span class="id" title="var">pP</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">pQl</span>, <span class="id" title="var">PQ</span>, <span class="id" title="var">pPr</span>, <span class="id" title="var">pP</span>.<br/>
* <span class="id" title="var">cbn</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">pPQ</span> <span class="id" title="var">pP</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">pQr</span>, <span class="id" title="var">pPQ</span>, <span class="id" title="var">pPl</span>, <span class="id" title="var">pP</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeImpl</span> := @<span class="id" title="var">mergeImpl</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mImpl</span> := (@<span class="id" title="var">bin</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">full_operators</span> <span class="id" title="var">Impl</span>).<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeFormImpl</span> := @<span class="id" title="var">mImpl</span>.<br/>
</div>
<div class="doc">
The same development for forall. Since forall-quantification in Coq is a part of the proper syntax (product types), this will again be handled by the main reification
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeForall</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) (<span class="id" title="var">Q</span>:<span class="id" title="var">naryProp</span> 1) (<span class="id" title="var">phi</span>:<span class="id" title="var">form</span>) : <span class="id" title="var">representsP</span> <span class="id" title="var">phi</span> <span class="id" title="var">rho</span> <span class="id" title="var">Q</span> -> @<span class="id" title="var">representsP</span> <span class="id" title="var">tr</span> 0 (∀ <span class="id" title="var">phi</span>) <span class="id" title="var">rho</span> (<span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>:<span class="id" title="var">D</span>, <span class="id" title="var">Q</span> <span class="id" title="var">x</span>).<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">H</span>. <span class="id" title="var">cbn</span>. <span class="id" title="tactic">split</span>;<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">HH</span> <span class="id" title="var">d</span>; <span class="id" title="tactic">specialize</span> (<span class="id" title="var">HH</span> <span class="id" title="var">d</span>); <span class="id" title="tactic">specialize</span> (<span class="id" title="var">H</span> <span class="id" title="var">d</span>); <span class="id" title="var">cbn</span> <span class="id" title="tactic">in</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>, <span class="id" title="var">HH</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeForall</span> := @<span class="id" title="var">mergeForall</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mForall</span> := (@<span class="id" title="var">quant</span> <span class="id" title="var">fs</span> <span class="id" title="var">ps</span> <span class="id" title="var">full_operators</span> <span class="id" title="keyword">All</span>).<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeFormForall</span> := @<span class="id" title="var">mForall</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeIff</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) (<span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <span class="id" title="var">naryProp</span> 0) (<span class="id" title="var">fP</span> <span class="id" title="var">fQ</span> : <span class="id" title="var">form</span>) : <span class="id" title="var">representsP</span> <span class="id" title="var">fP</span> <span class="id" title="var">rho</span> <span class="id" title="var">P</span> -> <span class="id" title="var">representsP</span> <span class="id" title="var">fQ</span> <span class="id" title="var">rho</span> <span class="id" title="var">Q</span> -> @<span class="id" title="var">representsP</span> <span class="id" title="var">_</span> 0 ((<span class="id" title="var">fP</span>--><span class="id" title="var">fQ</span>)∧(<span class="id" title="var">fQ</span>--><span class="id" title="var">fP</span>)) <span class="id" title="var">rho</span> (<span class="id" title="var">P</span> <-> <span class="id" title="var">Q</span>).<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span>. <span class="id" title="var">cbn</span>. <span class="id" title="var">cbn</span> <span class="id" title="tactic">in</span> <span class="id" title="var">H1</span>,<span class="id" title="var">H2</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H2</span>, <span class="id" title="var">H1</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Defined</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mIff</span> := (<span class="id" title="keyword">fun</span> <span class="id" title="var">fP</span> <span class="id" title="var">fQ</span> : <span class="id" title="var">form</span> => <span class="id" title="var">mAnd</span> (<span class="id" title="var">mImpl</span> <span class="id" title="var">fP</span> <span class="id" title="var">fQ</span>) (<span class="id" title="var">mImpl</span> <span class="id" title="var">fQ</span> <span class="id" title="var">fP</span>)).<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeFormIff</span> := @<span class="id" title="var">mIff</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeIff</span> := <span class="id" title="var">mergeIff</span>.<br/>
<br/>
</div>
<div class="doc">
The same development for truth. Note that truth has no canonical representative in the tarski semantics. It is equivalent to (falsity -> falsity), but not computationally equal. So using True in noProof mode would fail.
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mergeTrue</span> (<span class="id" title="var">rho</span>:<span class="id" title="var">nat</span> -> <span class="id" title="var">D</span>) : @<span class="id" title="var">representsP</span> <span class="id" title="var">_</span> 0 (<span class="id" title="var">fal</span>--><span class="id" title="var">fal</span>) <span class="id" title="var">rho</span> (<span class="id" title="var">True</span>).<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="var">cbn</span>. <span class="id" title="tactic">tauto</span>. <span class="id" title="keyword">Defined</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mTrue</span> := (<span class="id" title="var">mImpl</span> <span class="id" title="var">mFalse</span> <span class="id" title="var">mFalse</span>).<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeFormTrue</span> := @<span class="id" title="var">mTrue</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qMergeTrue</span> := <span class="id" title="var">mergeTrue</span>.<br/>
<br/>
</div>
<div class="doc">
Notation for matching constructs like Coq.Init.Logic.and prop1 prop2 -> (x:="and", l:=<span class="inlinecode"><span class="id" title="var">term1</span>,</span> <span class="inlinecode"><span class="id" title="var">term2</span></span>)
</div>
<div class="code">
<span class="id" title="keyword">Notation</span> <span class="id" title="var">baseLogicConn</span> <span class="id" title="var">x</span> <span class="id" title="var">l</span>:= (<span class="id" title="var">tInd</span> {| <span class="id" title="var">inductive_mind</span> := (<span class="id" title="var">MPfile</span> (["Logic"; "Init"; "Coq"]), <span class="id" title="var">x</span>); <span class="id" title="var">inductive_ind</span> := 0 |} <span class="id" title="var">l</span>).<br/>
</div>
<div class="doc">
We now define reification helpers for each of the primitives (except implication&forall). These helpers recursively apply the main reification function to the subterms and assemble the resulting term/prood
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyFalse</span> : <span class="id" title="var">baseConnectiveReifier</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nil</span> => <span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormFalse</span> ([<span class="id" title="var">tct</span>]), <span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFalse</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">envTerm</span>])) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "False applied to terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyAnd</span> : <span class="id" title="var">baseConnectiveReifier</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">x</span>; <span class="id" title="var">y</span>] => <br/>
<span class="id" title="var">xr</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">x</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;;<span class="id" title="var">yr</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">y</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;; <span class="id" title="keyword">let</span> '((<span class="id" title="var">xt</span>,<span class="id" title="var">xp</span>),(<span class="id" title="var">yt</span>,<span class="id" title="var">yp</span>)) := (<span class="id" title="var">xr</span>,<span class="id" title="var">yr</span>) <span class="id" title="tactic">in</span><br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormAnd</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">xt</span>;<span class="id" title="var">yt</span>]), <span class="id" title="var">tApp</span> <span class="id" title="var">qMergeAnd</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">envTerm</span>;<span class="id" title="var">x</span>;<span class="id" title="var">y</span>;<span class="id" title="var">xt</span>;<span class="id" title="var">yt</span>;<span class="id" title="var">xp</span>;<span class="id" title="var">yp</span>])) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "And applied to != 2 terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyOr</span> : <span class="id" title="var">baseConnectiveReifier</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">x</span>; <span class="id" title="var">y</span>] => <br/>
<span class="id" title="var">xr</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">x</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;;<span class="id" title="var">yr</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">y</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;; <span class="id" title="keyword">let</span> '((<span class="id" title="var">xt</span>,<span class="id" title="var">xp</span>),(<span class="id" title="var">yt</span>,<span class="id" title="var">yp</span>)) := (<span class="id" title="var">xr</span>,<span class="id" title="var">yr</span>) <span class="id" title="tactic">in</span><br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormOr</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">xt</span>;<span class="id" title="var">yt</span>]),<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeOr</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">envTerm</span>;<span class="id" title="var">x</span>;<span class="id" title="var">y</span>;<span class="id" title="var">xt</span>;<span class="id" title="var">yt</span>;<span class="id" title="var">xp</span>;<span class="id" title="var">yp</span>])) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "Or applied to != 2 terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyExist</span>:<span class="id" title="var">baseConnectiveReifier</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">_</span>; <span class="id" title="var">P</span>] => <br/>
<span class="id" title="var">rr</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">P</span> 1 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;; <span class="id" title="keyword">let</span> '(<span class="id" title="var">rt</span>,<span class="id" title="var">rp</span>) := <span class="id" title="var">rr</span> <span class="id" title="tactic">in</span> <br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormExists</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">rt</span>]), <span class="id" title="var">tApp</span> <span class="id" title="var">qMergeExists</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">envTerm</span>;<span class="id" title="var">P</span>;<span class="id" title="var">rt</span>;<span class="id" title="var">rp</span>])) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "Exist applied to wrong terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyIff</span> : <span class="id" title="var">baseConnectiveReifier</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">x</span>; <span class="id" title="var">y</span>] => <br/>
<span class="id" title="var">xr</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">x</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;;<span class="id" title="var">yr</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">y</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;; <span class="id" title="keyword">let</span> '((<span class="id" title="var">xt</span>,<span class="id" title="var">xp</span>),(<span class="id" title="var">yt</span>,<span class="id" title="var">yp</span>)) := (<span class="id" title="var">xr</span>,<span class="id" title="var">yr</span>) <span class="id" title="tactic">in</span><br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormIff</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">xt</span>;<span class="id" title="var">yt</span>]), <span class="id" title="var">tApp</span> <span class="id" title="var">qMergeIff</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">envTerm</span>;<span class="id" title="var">x</span>;<span class="id" title="var">y</span>;<span class="id" title="var">xt</span>;<span class="id" title="var">yt</span>;<span class="id" title="var">xp</span>;<span class="id" title="var">yp</span>])) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "Iff applied to != 2 terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyTrue</span> : <span class="id" title="var">baseConnectiveReifier</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">l</span> <span class="id" title="var">fuel</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nil</span> =><br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormTrue</span> ([<span class="id" title="var">tct</span>]), <span class="id" title="var">tApp</span> <span class="id" title="var">qMergeTrue</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">envTerm</span>]))| <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "True applied to terms" <span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
The main handler. Given the name of the base type, call the respective helper. If nothing matches, call the extension point
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyBase</span> (<span class="id" title="var">s</span>:<span class="id" title="var">string</span>): <span class="id" title="var">baseConnectiveReifier</span> <br/>
:= <span class="id" title="keyword">match</span> <span class="id" title="var">s</span> <span class="id" title="keyword">with</span> "and" => <span class="id" title="var">reifyAnd</span> | "or" => <span class="id" title="var">reifyOr</span> | "ex" => <span class="id" title="var">reifyExist</span> | "False" => <span class="id" title="var">reifyFalse</span> | "True" => <span class="id" title="var">reifyTrue</span> |<br/>
<span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">baseLogicConnHelper</span> <span class="id" title="keyword">with</span> <br/>
<span class="id" title="var">None</span> =><span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> ("Unknown connective "++<span class="id" title="var">s</span>) | <br/>
<span class="id" title="var">Some</span> <span class="id" title="var">k</span> => <span class="id" title="var">k</span> <span class="id" title="var">s</span> <span class="id" title="keyword">end</span> <span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
The same development for no-proof mode. Similar, except we don't build proofs
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">baseConnectiveReifierNP</span> := <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">nat</span> -> <span class="id" title="var">Ast.term</span> -> (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) -> <br/>
(<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">nat</span> -> <span class="id" title="var">Ast.term</span> -> (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) -> <span class="id" title="var">FailureMonad</span> ( <span class="id" title="var">Ast.term</span>)) -> <br/>
(<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">Ast.term</span> -> (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">Ast.term</span>)) -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">Ast.term</span>).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyFalseNP</span> : <span class="id" title="var">baseConnectiveReifierNP</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nil</span> => <span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormFalse</span> ([<span class="id" title="var">tct</span>])) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "False applied to terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyAndNP</span> : <span class="id" title="var">baseConnectiveReifierNP</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">x</span>; <span class="id" title="var">y</span>] => <br/>
<span class="id" title="var">xt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">x</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;;<span class="id" title="var">yt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">y</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;; <br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormAnd</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">xt</span>;<span class="id" title="var">yt</span>])) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "And applied to != 2 terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyOrNP</span> : <span class="id" title="var">baseConnectiveReifierNP</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">x</span>; <span class="id" title="var">y</span>] => <br/>
<span class="id" title="var">xt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">x</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;;<span class="id" title="var">yt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">y</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;; <br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormOr</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">xt</span>;<span class="id" title="var">yt</span>])) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "Or applied to != 2 terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyExistNP</span>:<span class="id" title="var">baseConnectiveReifierNP</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">_</span>; <span class="id" title="var">P</span>] => <br/>
<span class="id" title="var">rt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">P</span> 1 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;;<br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormExists</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">rt</span>])) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "Exist applied to wrong terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyIffNP</span> : <span class="id" title="var">baseConnectiveReifierNP</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">x</span>; <span class="id" title="var">y</span>] => <br/>
<span class="id" title="var">xt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">x</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;;<span class="id" title="var">yt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">y</span> 0 <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span>;;<br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormIff</span> ([<span class="id" title="var">tct</span>;<span class="id" title="var">xt</span>;<span class="id" title="var">yt</span>])) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "Iff applied to != 2 terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyTrueNP</span> : <span class="id" title="var">baseConnectiveReifierNP</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">l</span> <span class="id" title="var">fuel</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nil</span> =><br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qMergeFormTrue</span> ([<span class="id" title="var">tct</span>]))| <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "True applied to terms" <span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
The helper is a bit more interesting, since it calls the extension point, building a recursive "dummy proof" helper and discarding the resulting proof
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyBaseNP</span> (<span class="id" title="var">s</span>:<span class="id" title="var">string</span>): <span class="id" title="var">baseConnectiveReifierNP</span> <br/>
:= <span class="id" title="keyword">match</span> <span class="id" title="var">s</span> <span class="id" title="keyword">with</span> "and" => <span class="id" title="var">reifyAndNP</span> | "or" => <span class="id" title="var">reifyOrNP</span> | "ex" => <span class="id" title="var">reifyExistNP</span> | "False" => <span class="id" title="var">reifyFalseNP</span> | "True" => <span class="id" title="var">reifyTrueNP</span> | <br/>
<span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">baseLogicConnHelper</span> <span class="id" title="keyword">with</span> <br/>
<span class="id" title="var">None</span> =><span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> ("Unknown connective "++<span class="id" title="var">s</span>) | <br/>
<span class="id" title="var">Some</span> <span class="id" title="var">k</span> => <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">fuel</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <span class="id" title="var">fPR</span> <span class="id" title="var">fTR</span> => <span class="id" title="var">both</span> <- <span class="id" title="var">k</span> <span class="id" title="var">s</span> <span class="id" title="var">tct</span> <span class="id" title="var">lst</span> <span class="id" title="var">fuel</span> <span class="id" title="var">envTerm</span> <span class="id" title="var">env</span> <br/>
(<span class="id" title="keyword">fun</span> <span class="id" title="var">P</span> <span class="id" title="var">n</span> <span class="id" title="var">et</span> <span class="id" title="var">e</span> => <span class="id" title="var">rr</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">P</span> <span class="id" title="var">n</span> <span class="id" title="var">et</span> <span class="id" title="var">e</span>;; <span class="id" title="var">ret</span> (<span class="id" title="var">rr</span>, <span class="id" title="var">noProofDummy</span>)) <br/>
(<span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">t</span> <span class="id" title="var">eT</span> <span class="id" title="var">env</span> => <span class="id" title="var">rr</span> <- <span class="id" title="var">fTR</span> <span class="id" title="var">tct</span> <span class="id" title="var">t</span> <span class="id" title="var">eT</span> <span class="id" title="var">env</span>;; <span class="id" title="var">ret</span> (<span class="id" title="var">rr</span>, <span class="id" title="var">noProofDummy</span>));; <br/>
<span class="id" title="keyword">let</span> '(<span class="id" title="var">trm</span>,<span class="id" title="var">_</span>) := <span class="id" title="var">both</span> <span class="id" title="tactic">in</span> <span class="id" title="var">ret</span> <span class="id" title="var">trm</span> <span class="id" title="keyword">end</span> <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">TarskiMerging</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">ReificationHelpers</span>.<br/>
</div>
<div class="doc">
**ReificationHelpers
These functions reify atomic functions/relations Definition of the types. The arguments are:
<ul class="doclist">
<li> tct:Ast.term - quoted instance of tarski_reflector
</li>
<li> av:list Ast.term - the arguments of an atomic relation/function to be recursively reified
</li>
<li> envTerm
</li>
<li> recursive helper, to which av is passed one-by-one
</li>
<li> the return is again either a pair of term&proof, or just the term if in no proof mode
</li>
</ul>
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">termReifier</span> := <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">Ast.term</span> -> (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">prod</span> <span class="id" title="var">Ast.term</span> <span class="id" title="var">Ast.term</span>)) -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">prod</span> <span class="id" title="var">Ast.term</span> <span class="id" title="var">Ast.term</span>).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">termReifierNP</span> := <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">Ast.term</span> -> (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">Ast.term</span>)) -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">Ast.term</span>).<br/>
</div>
<div class="doc">
Args: typeClassTerm -> vector of args -> env term -> recursion for arg vector -> prod (term, proof that term represents input
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">applyRecursively</span> (<span class="id" title="var">lt</span> : <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) (<span class="id" title="var">IH</span> : <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">prod</span> <span class="id" title="var">Ast.term</span> <span class="id" title="var">Ast.term</span>)) : <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">prod</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>)) :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">lt</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nil</span> => <span class="id" title="var">ret</span> (<span class="id" title="var">nil</span>,<span class="id" title="var">nil</span>)<br/>
| <span class="id" title="var">t</span>::<span class="id" title="var">tr</span> => <span class="id" title="var">IHt</span> <- <span class="id" title="var">IH</span> <span class="id" title="var">t</span>;; <span class="id" title="var">atrIH</span> <- <span class="id" title="var">applyRecursively</span> <span class="id" title="var">tr</span> <span class="id" title="var">IH</span>;;<span class="id" title="keyword">let</span> '(<span class="id" title="var">rep</span>,<span class="id" title="var">prf</span>) := <span class="id" title="var">IHt</span> <span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> '(<span class="id" title="var">replist</span>,<span class="id" title="var">fulllist</span>) := <span class="id" title="var">atrIH</span> <span class="id" title="tactic">in</span> <span class="id" title="var">ret</span> (<span class="id" title="var">rep</span>::<span class="id" title="var">replist</span>, <span class="id" title="var">rep</span>::<span class="id" title="var">t</span>::<span class="id" title="var">prf</span>::<span class="id" title="var">fulllist</span>) <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">applyRecursivelyNP</span> (<span class="id" title="var">lt</span> : <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) (<span class="id" title="var">IH</span> : <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">Ast.term</span>)) : <span class="id" title="var">FailureMonad</span> ((<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) ) :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">lt</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nil</span> => <span class="id" title="var">ret</span> (<span class="id" title="var">nil</span>)<br/>
| <span class="id" title="var">t</span>::<span class="id" title="var">tr</span> => <span class="id" title="var">rep</span> <- <span class="id" title="var">IH</span> <span class="id" title="var">t</span>;; <span class="id" title="var">replist</span> <- <span class="id" title="var">applyRecursivelyNP</span> <span class="id" title="var">tr</span> <span class="id" title="var">IH</span>;;<span class="id" title="var">ret</span> (<span class="id" title="var">rep</span>::<span class="id" title="var">replist</span>) <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyTerm</span> (<span class="id" title="var">c</span>:<span class="id" title="var">Ast.term</span>) : <span class="id" title="var">termReifier</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">av</span> <span class="id" title="var">env</span> <span class="id" title="var">IH</span> => <span class="id" title="var">pr</span> <- <span class="id" title="var">applyRecursively</span> <span class="id" title="var">av</span> <span class="id" title="var">IH</span>;; <span class="id" title="keyword">let</span> '(<span class="id" title="var">trm</span>,<span class="id" title="var">fullarg</span>) := <span class="id" title="var">pr</span> <span class="id" title="tactic">in</span><br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qConstructTerm</span> (<span class="id" title="var">tct</span>::<span class="id" title="var">c</span>::<span class="id" title="var">trm</span>), <span class="id" title="var">tApp</span> <span class="id" title="var">qMergeTerm</span> (<span class="id" title="var">tct</span>::<span class="id" title="var">c</span>::<span class="id" title="var">env</span>::<span class="id" title="var">fullarg</span>)).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyForm</span> (<span class="id" title="var">c</span>:<span class="id" title="var">Ast.term</span>) : <span class="id" title="var">termReifier</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">av</span> <span class="id" title="var">env</span> <span class="id" title="var">IH</span> => <span class="id" title="var">pr</span> <- <span class="id" title="var">applyRecursively</span> <span class="id" title="var">av</span> <span class="id" title="var">IH</span>;; <span class="id" title="keyword">let</span> '(<span class="id" title="var">trm</span>,<span class="id" title="var">fullarg</span>) := <span class="id" title="var">pr</span> <span class="id" title="tactic">in</span><br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qConstructForm</span> (<span class="id" title="var">tct</span>::<span class="id" title="var">c</span>::<span class="id" title="var">trm</span>), <span class="id" title="var">tApp</span> <span class="id" title="var">qMergeForm</span> (<span class="id" title="var">tct</span>::<span class="id" title="var">c</span>::<span class="id" title="var">env</span>::<span class="id" title="var">fullarg</span>)).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyTermNP</span> (<span class="id" title="var">c</span>:<span class="id" title="var">Ast.term</span>) : <span class="id" title="var">termReifierNP</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">av</span> <span class="id" title="var">env</span> <span class="id" title="var">IH</span> => <span class="id" title="var">trm</span> <- <span class="id" title="var">applyRecursivelyNP</span> <span class="id" title="var">av</span> <span class="id" title="var">IH</span>;; <br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qConstructTerm</span> (<span class="id" title="var">tct</span>::<span class="id" title="var">c</span>::<span class="id" title="var">trm</span>)).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">reifyFormNP</span> (<span class="id" title="var">c</span>:<span class="id" title="var">Ast.term</span>) : <span class="id" title="var">termReifierNP</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">av</span> <span class="id" title="var">env</span> <span class="id" title="var">IH</span> => <span class="id" title="var">trm</span> <- <span class="id" title="var">applyRecursivelyNP</span> <span class="id" title="var">av</span> <span class="id" title="var">IH</span>;; <br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qConstructForm</span> (<span class="id" title="var">tct</span>::<span class="id" title="var">c</span>::<span class="id" title="var">trm</span>)).<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">ReificationHelpers</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">EnvHelpers</span>.<br/>
</div>
<div class="doc">
<a id="lab6"></a><h2 class="section">EnvHelpers</h2>
Useful functions for extending the environment. A 3-ary predicate P(k1,k2,k3) (for example) is represented by a term t where t in environ (k3:k2:k1:env) represents P (k1,k2,k3). Appends something to the start of the env.
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">appendZero</span> (<span class="id" title="var">env</span>:<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) (<span class="id" title="var">zv</span>:<span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) : (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) := <br/>
<span class="id" title="keyword">fun</span> (<span class="id" title="var">t</span>:<span class="id" title="var">Ast.term</span>) => <span class="id" title="keyword">match</span> <span class="id" title="var">t</span> <span class="id" title="keyword">with</span> <span class="id" title="var">tRel</span> <span class="id" title="var">n</span> => (<span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span> 0 => <span class="id" title="var">zv</span> | <span class="id" title="var">S</span> <span class="id" title="var">n</span> => <span class="id" title="var">env</span> (<span class="id" title="var">tRel</span> <span class="id" title="var">n</span>) <span class="id" title="keyword">end</span>) | <span class="id" title="var">_</span> => <span class="id" title="var">k</span> <- <span class="id" title="var">lowerRelIndex</span> 0 (<span class="id" title="tactic">fail</span> "tRel 0 used when lowering") <span class="id" title="var">t</span>;; (<span class="id" title="var">env</span> <span class="id" title="var">k</span>) <span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
Appends something to the start of the env. At the same time, it expects the terms given to the env helper to be one level deeper, so it "unshifts" the tRels first
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">appendAndLift</span> (<span class="id" title="var">env</span>:<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) (<span class="id" title="var">zv</span>:<span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) : (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) := <br/>
<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">t</span> <span class="id" title="keyword">with</span> <span class="id" title="var">tRel</span> <span class="id" title="var">n</span> => (<span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span> 0 => <span class="id" title="var">zv</span> | <span class="id" title="var">S</span> <span class="id" title="var">n</span> => <span class="id" title="var">k</span> <- <span class="id" title="var">env</span> (<span class="id" title="var">tRel</span> <span class="id" title="var">n</span>);;<span class="id" title="var">ret</span> (<span class="id" title="var">S</span> <span class="id" title="var">k</span>) <span class="id" title="keyword">end</span>) | <span class="id" title="var">_</span> => <span class="id" title="var">k</span> <- <span class="id" title="var">lowerRelIndex</span> 0 (<span class="id" title="tactic">fail</span> "tRel 0 used when lowering") <span class="id" title="var">t</span>;; <span class="id" title="var">v</span> <- <span class="id" title="var">env</span> <span class="id" title="var">k</span>;;<span class="id" title="var">ret</span> (<span class="id" title="var">S</span> <span class="id" title="var">v</span>) <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qD</span> := @<span class="id" title="var">D</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qScons</span> := @<span class="id" title="var">scons</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">raiseEnvTerm</span> (<span class="id" title="var">tct</span>:<span class="id" title="var">Ast.term</span>) (<span class="id" title="var">d</span>:<span class="id" title="var">Ast.term</span>) (<span class="id" title="var">env</span>:<span class="id" title="var">Ast.term</span>) : <span class="id" title="var">Ast.term</span> := <span class="id" title="var">tApp</span> (<span class="id" title="var">qScons</span>) ([<span class="id" title="var">tApp</span> <span class="id" title="var">qD</span> ([<span class="id" title="var">tct</span>]);<span class="id" title="var">d</span>;<span class="id" title="var">env</span>]).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">unboundEnv</span> := (<span class="id" title="keyword">fun</span> <span class="id" title="var">a</span>:<span class="id" title="var">Ast.term</span> => @<span class="id" title="tactic">fail</span> <span class="id" title="var">nat</span> ("unbound " ++ <span class="id" title="var">string_of_term</span> <span class="id" title="var">a</span>)).<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">EnvHelpers</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">EnvConstructor</span>.<br/>
</div>
<div class="doc">
<a id="lab7"></a><h2 class="section">EnvConstructor</h2>
Constructs the environment used to help with free variables in terms Useful quotations
</div>
<div class="code">
<span class="id" title="var">Existing</span> <span class="id" title="keyword">Instance</span> <span class="id" title="var">config.default_checker_flags</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qFs</span> := @<span class="id" title="var">fs</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qLocalVar</span> := @<span class="id" title="var">var</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qI_f</span> := @<span class="id" title="var">i_f</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qI_P</span> := @<span class="id" title="var">i_P</span>.<br/>
<span class="id" title="keyword">Context</span> {<span class="id" title="var">tr</span> : <span class="id" title="var">tarski_reflector</span>}.<br/>
<span class="id" title="keyword">Context</span> {<span class="id" title="var">te</span> : <span class="id" title="var">tarski_reflector_extensions</span> <span class="id" title="var">tr</span>}.<br/>
</div>
<div class="doc">
Given a term in the semantic, construct an environment that contains all atoms that appear in the term (and are not otherwise representable)
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">findUBRecursively</span> (<span class="id" title="var">lt</span> : <span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) (<span class="id" title="var">IH</span> : <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>)) : <span class="id" title="var">FailureMonad</span> ((<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) ) :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">lt</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nil</span> => <span class="id" title="var">ret</span> (<span class="id" title="var">nil</span>)<br/>
| <span class="id" title="var">t</span>::<span class="id" title="var">tr</span> => <span class="id" title="var">rep</span> <- <span class="id" title="var">IH</span> <span class="id" title="var">t</span>;; <span class="id" title="var">replist</span> <- <span class="id" title="var">findUBRecursively</span> <span class="id" title="var">tr</span> <span class="id" title="var">IH</span>;;<span class="id" title="var">ret</span> (<span class="id" title="var">rep</span>++<span class="id" title="var">replist</span>) <span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
Finds all free variables in a term
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">findUnboundVariablesTerm</span> (<span class="id" title="var">fuel</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">t</span>:<span class="id" title="var">Ast.term</span>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">fuel</span>}: (<span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>)) := <span class="id" title="keyword">match</span> <span class="id" title="var">fuel</span> <span class="id" title="keyword">with</span> <br/>
0 => <span class="id" title="tactic">fail</span> "Out of fuel" <br/>
| <span class="id" title="var">S</span> <span class="id" title="var">fuel</span> => <span class="id" title="keyword">let</span> <span class="id" title="var">ffail</span> := <span class="id" title="var">orelse</span> (<span class="id" title="keyword">match</span> @<span class="id" title="var">termReifierVarHelper</span> <span class="id" title="var">_</span> <span class="id" title="var">te</span> <span class="id" title="keyword">with</span> <span class="id" title="var">None</span> => <span class="id" title="tactic">fail</span> "Fallthrough" | <span class="id" title="var">Some</span> <span class="id" title="var">k</span> => <span class="id" title="var">k</span> <span class="id" title="var">fuel</span> <span class="id" title="var">t</span> (<span class="id" title="var">findUnboundVariablesTerm</span> <span class="id" title="var">fuel</span>) <span class="id" title="keyword">end</span>)<br/>
(<span class="id" title="keyword">match</span> <span class="id" title="var">t</span> <span class="id" title="keyword">with</span> <span class="id" title="var">tRel</span> <span class="id" title="var">_</span> => <span class="id" title="var">ret</span> <span class="id" title="var">nil</span> | <span class="id" title="var">_</span> => <span class="id" title="var">ret</span> ([<span class="id" title="var">t</span>]) <span class="id" title="keyword">end</span>) <span class="id" title="tactic">in</span> </div>
<div class="doc">
tRel are things introduced by forall/exists and so on, which we do not add to the environment
</div>
<div class="code">
<span class="id" title="keyword">match</span> <span class="id" title="var">t</span> <span class="id" title="keyword">with</span><br/>
<span class="id" title="var">tApp</span> <span class="id" title="var">arg</span> <span class="id" title="var">l</span> => <span class="id" title="keyword">if</span> <span class="id" title="var">Checker.eq_term</span> <span class="id" title="var">init_graph</span> <span class="id" title="var">arg</span> <span class="id" title="var">qI_f</span> <span class="id" title="keyword">then</span> <span class="id" title="keyword">match</span> <span class="id" title="var">popNElements</span> <span class="id" title="var">l</span> 4 <span class="id" title="keyword">with</span> <br/>
<span class="id" title="var">Some</span> ([<span class="id" title="var">fnc</span>;<span class="id" title="var">v</span>]) => <span class="id" title="var">vr</span> <- <span class="id" title="var">recoverVector</span> <span class="id" title="var">v</span>;;<span class="id" title="var">findUBRecursively</span> <span class="id" title="var">vr</span> (<span class="id" title="var">findUnboundVariablesTerm</span> <span class="id" title="var">fuel</span>)<br/>
| <span class="id" title="var">_</span> => <span class="id" title="var">ffail</span> <span class="id" title="keyword">end</span> <span class="id" title="keyword">else</span> <span class="id" title="var">ffail</span><br/>
| <span class="id" title="var">_</span> => <span class="id" title="var">ffail</span><br/>
<span class="id" title="keyword">end</span> <br/>
<span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Notation</span> <span class="id" title="var">baseLogicConn</span> <span class="id" title="var">x</span> <span class="id" title="var">l</span>:= (<span class="id" title="var">tInd</span> {| <span class="id" title="var">inductive_mind</span> := (<span class="id" title="var">MPfile</span> (["Logic"; "Init"; "Coq"]), <span class="id" title="var">x</span>); <span class="id" title="var">inductive_ind</span> := 0 |} <span class="id" title="var">l</span>).<br/>
</div>
<div class="doc">
Find the unbound variables for our basic forms
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">findUBFalse</span> :<span class="id" title="var">baseConnectiveVars</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nil</span> => <br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">nil</span>) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "False applied to terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">findUBAnd</span> :<span class="id" title="var">baseConnectiveVars</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">x</span>; <span class="id" title="var">y</span>] => <br/>
<span class="id" title="var">xt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">x</span> 0;;<span class="id" title="var">yt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">y</span> 0;; <br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">xt</span>++<span class="id" title="var">yt</span>) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "And applied to != 2 terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">findUBOr</span> :<span class="id" title="var">baseConnectiveVars</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">x</span>; <span class="id" title="var">y</span>] => <br/>
<span class="id" title="var">xt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">x</span> 0;;<span class="id" title="var">yt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">y</span> 0;; <br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">xt</span>++<span class="id" title="var">yt</span>) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "Or applied to != 2 terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">findUBExists</span>:<span class="id" title="var">baseConnectiveVars</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">_</span>; <span class="id" title="var">P</span>] => <span class="id" title="var">fPR</span> <span class="id" title="var">P</span> 1 | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "Exist applied to wrong terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">findUBTrue</span> : <span class="id" title="var">baseConnectiveVars</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">lst</span> <span class="id" title="var">fuel</span> <span class="id" title="var">tct</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nil</span> => <span class="id" title="var">ret</span> <span class="id" title="var">nil</span> | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "True applied to terms" <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">findUBIff</span> :<span class="id" title="var">baseConnectiveVars</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">lst</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">fPR</span> <span class="id" title="var">_</span> => <span class="id" title="keyword">match</span> <span class="id" title="var">lst</span> <span class="id" title="keyword">with</span> [<span class="id" title="var">x</span>; <span class="id" title="var">y</span>] => <br/>
<span class="id" title="var">xt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">x</span> 0;;<span class="id" title="var">yt</span> <- <span class="id" title="var">fPR</span> <span class="id" title="var">y</span> 0;; <br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">xt</span>++<span class="id" title="var">yt</span>) | <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> "Iff applied to != 2 terms" <span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">findUBBase</span> (<span class="id" title="var">s</span>:<span class="id" title="var">string</span>) : <span class="id" title="var">baseConnectiveVars</span> <br/>
:= <span class="id" title="keyword">match</span> <span class="id" title="var">s</span> <span class="id" title="keyword">with</span> "and" => <span class="id" title="var">findUBAnd</span> | "or" => <span class="id" title="var">findUBOr</span> | "ex" => <span class="id" title="var">findUBExists</span> | "False" => <span class="id" title="var">findUBFalse</span> | "True" => <span class="id" title="var">findUBFalse</span> | <span class="id" title="var">_</span> => <br/>
<span class="id" title="keyword">match</span> @<span class="id" title="var">baseLogicVarHelper</span> <span class="id" title="var">tr</span> <span class="id" title="var">te</span> <span class="id" title="keyword">with</span> <span class="id" title="var">None</span> => <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> => <span class="id" title="tactic">fail</span> ("Unknown connective "++<span class="id" title="var">s</span>) | <span class="id" title="var">Some</span> <span class="id" title="var">k</span> => <span class="id" title="var">k</span> <span class="id" title="var">s</span> <span class="id" title="keyword">end</span> <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="var">MetaCoq</span> <span class="id" title="var">Quote</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">qIff</span> := @<span class="id" title="var">iff</span>.<br/>
</div>
<div class="doc">
Checks whether a term is the type of theory terms in Coq.
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">maybeD</span> : <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">Ast.term</span> -> <span class="id" title="var">bool</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">tct</span> <span class="id" title="var">mD</span> => <span class="id" title="keyword">if</span> @<span class="id" title="var">isD</span> <span class="id" title="var">tr</span> <span class="id" title="var">mD</span> <span class="id" title="keyword">then</span> <span class="id" title="var">true</span> <span class="id" title="keyword">else</span> <span class="id" title="var">Checker.eq_term</span> <span class="id" title="var">init_graph</span> <span class="id" title="var">mD</span> (<span class="id" title="var">tApp</span> <span class="id" title="var">qD</span> ([<span class="id" title="var">tct</span>])).<br/>
</div>
<div class="doc">
Finds the unbound variables in a form
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">findUnboundVariablesForm</span> (<span class="id" title="var">tct</span>:<span class="id" title="var">Ast.term</span>) (<span class="id" title="var">fuel</span>:<span class="id" title="var">nat</span>) (<span class="id" title="var">t</span>:<span class="id" title="var">Ast.term</span>) (<span class="id" title="var">frees</span>:<span class="id" title="var">nat</span>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">fuel</span>}: (<span class="id" title="var">FailureMonad</span> (<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>)) := <br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">ffail</span> := <span class="id" title="tactic">fail</span> ("Cannot introspect form "++ <span class="id" title="var">string_of_term</span> <span class="id" title="var">t</span>) <span class="id" title="tactic">in</span> <span class="id" title="keyword">match</span> <span class="id" title="var">fuel</span> <span class="id" title="keyword">with</span> 0 => <span class="id" title="tactic">fail</span> "Out of fuel" | <span class="id" title="var">S</span> <span class="id" title="var">fuel</span> => <br/>
<span class="id" title="keyword">match</span> (<span class="id" title="var">frees</span>,<span class="id" title="var">t</span>) <span class="id" title="keyword">with</span><br/>
(0,(<span class="id" title="var">baseLogicConn</span> <span class="id" title="var">name</span> <span class="id" title="var">nil</span>)) => <span class="id" title="var">findUBBase</span> <span class="id" title="var">name</span> <span class="id" title="var">nil</span> <span class="id" title="var">fuel</span> <span class="id" title="var">tct</span> (<span class="id" title="var">findUnboundVariablesForm</span> <span class="id" title="var">tct</span> <span class="id" title="var">fuel</span>) (<span class="id" title="var">findUnboundVariablesTerm</span> <span class="id" title="var">fuel</span>)<br/>
| (0,(<span class="id" title="var">tApp</span> (<span class="id" title="var">baseLogicConn</span> <span class="id" title="var">name</span> <span class="id" title="var">nil</span>) <span class="id" title="var">lst</span>)) => <span class="id" title="var">findUBBase</span> <span class="id" title="var">name</span> <span class="id" title="var">lst</span> <span class="id" title="var">fuel</span> <span class="id" title="var">tct</span> (<span class="id" title="var">findUnboundVariablesForm</span> <span class="id" title="var">tct</span> <span class="id" title="var">fuel</span>) (<span class="id" title="var">findUnboundVariablesTerm</span> <span class="id" title="var">fuel</span>)<br/>
| (0,(<span class="id" title="var">tApp</span> <span class="id" title="var">arg</span> <span class="id" title="var">lst</span>)) => <span class="id" title="keyword">if</span> <span class="id" title="var">Checker.eq_term</span> <span class="id" title="var">init_graph</span> <span class="id" title="var">arg</span> <span class="id" title="var">qI_P</span> <span class="id" title="keyword">then</span> <span class="id" title="keyword">match</span> <span class="id" title="var">popNElements</span> <span class="id" title="var">lst</span> 4 <span class="id" title="keyword">with</span><br/>
<span class="id" title="var">Some</span> ([<span class="id" title="var">fnc</span>;<span class="id" title="var">v</span>]) => <span class="id" title="var">vr</span> <- <span class="id" title="var">recoverVector</span> <span class="id" title="var">v</span>;;<span class="id" title="var">findUBRecursively</span> <span class="id" title="var">vr</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> => <span class="id" title="var">findUnboundVariablesTerm</span> <span class="id" title="var">fuel</span> <span class="id" title="var">t</span>)<br/>
| <span class="id" title="var">_</span> => <span class="id" title="var">ffail</span> <span class="id" title="keyword">end</span> <span class="id" title="keyword">else</span> <span class="id" title="keyword">if</span> <span class="id" title="var">Checker.eq_term</span> <span class="id" title="var">init_graph</span> <span class="id" title="var">arg</span> <span class="id" title="var">qIff</span> <span class="id" title="keyword">then</span> <span class="id" title="var">findUBIff</span> <span class="id" title="var">lst</span> <span class="id" title="var">fuel</span> <span class="id" title="var">tct</span> (<span class="id" title="var">findUnboundVariablesForm</span> <span class="id" title="var">tct</span> <span class="id" title="var">fuel</span>) (<span class="id" title="var">findUnboundVariablesTerm</span> <span class="id" title="var">fuel</span>) <span class="id" title="keyword">else</span> <span class="id" title="var">ffail</span> <br/>
| (0,<span class="id" title="var">tProd</span> <span class="id" title="var">x</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span>) => <span class="id" title="keyword">if</span> <span class="id" title="var">maybeD</span> <span class="id" title="var">tct</span> (<span class="id" title="var">P</span>) <span class="id" title="keyword">then</span><br/>
<span class="id" title="var">findUnboundVariablesForm</span> <span class="id" title="var">tct</span> <span class="id" title="var">fuel</span> (<span class="id" title="var">tLambda</span> <span class="id" title="var">x</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span>) 1<br/>
<span class="id" title="keyword">else</span> <br/>
<span class="id" title="var">rP</span> <- <span class="id" title="var">findUnboundVariablesForm</span> <span class="id" title="var">tct</span> <span class="id" title="var">fuel</span> <span class="id" title="var">P</span> 0;;<br/>
<span class="id" title="var">Ql</span> <- <span class="id" title="var">lowerRelIndex</span> 0 (<span class="id" title="tactic">fail</span> "Used var of implication precondition") <span class="id" title="var">Q</span>;;<br/>
<span class="id" title="var">rQ</span> <- <span class="id" title="var">findUnboundVariablesForm</span> <span class="id" title="var">tct</span> <span class="id" title="var">fuel</span> <span class="id" title="var">Ql</span> 0;;<br/>
<span class="id" title="var">ret</span> (<span class="id" title="var">rP</span>++<span class="id" title="var">rQ</span>)<br/>
| (<span class="id" title="var">S</span> <span class="id" title="var">n</span>,<span class="id" title="var">tLambda</span> <span class="id" title="var">x</span> <span class="id" title="var">T</span> <span class="id" title="var">P</span>) => <span class="id" title="keyword">if</span> <span class="id" title="var">maybeD</span> <span class="id" title="var">tct</span> <span class="id" title="var">T</span> <span class="id" title="keyword">then</span><br/>
<span class="id" title="var">findUnboundVariablesForm</span> <span class="id" title="var">tct</span> <span class="id" title="var">fuel</span> <span class="id" title="var">P</span> <span class="id" title="var">n</span><br/>
<span class="id" title="keyword">else</span> <span class="id" title="var">ffail</span><br/>
| <span class="id" title="var">_</span> => <span class="id" title="var">ffail</span> <span class="id" title="keyword">end</span> <span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
Builds the actual env out of a list of free variables
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">createEnvTerms</span> (<span class="id" title="var">tct</span>:<span class="id" title="var">Ast.term</span>) (<span class="id" title="var">l</span>:<span class="id" title="var">list</span> <span class="id" title="var">Ast.term</span>) (<span class="id" title="var">base</span>:<span class="id" title="var">Ast.term</span>) : <span class="id" title="var">prod</span> (<span class="id" title="var">Ast.term</span>) (<span class="id" title="var">Ast.term</span> -> <span class="id" title="var">FailureMonad</span> <span class="id" title="var">nat</span>) := <span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span> <br/>
<span class="id" title="var">nil</span> => (<span class="id" title="var">base</span>,<span class="id" title="var">unboundEnv</span>)<br/>
| <span class="id" title="var">x</span>::<span class="id" title="var">xr</span> => <span class="id" title="keyword">let</span> '(<span class="id" title="var">envTerm</span>,<span class="id" title="var">env</span>) := <span class="id" title="var">createEnvTerms</span> <span class="id" title="var">tct</span> <span class="id" title="var">xr</span> <span class="id" title="var">base</span> <span class="id" title="tactic">in</span> (<span class="id" title="var">raiseEnvTerm</span> <span class="id" title="var">tct</span> <span class="id" title="var">x</span> <span class="id" title="var">envTerm</span>, <span class="id" title="keyword">fun</span> <span class="id" title="var">a</span>:<span class="id" title="var">Ast.term</span> => <span class="id" title="keyword">if</span> <span class="id" title="var">Checker.eq_term</span> <span class="id" title="var">init_graph</span> <span class="id" title="var">x</span> <span class="id" title="var">a</span> <span class="id" title="keyword">then</span> <span class="id" title="var">ret</span> 0 <span class="id" title="keyword">else</span> <span class="id" title="var">v</span> <- <span class="id" title="var">env</span> <span class="id" title="var">a</span>;;<span class="id" title="var">ret</span> (<span class="id" title="var">S</span> <span class="id" title="var">v</span>)) <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">EnvConstructor</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">MainReificationFunctions</span>.<br/>
</div>
<div class="doc">
<a id="lab8"></a><h2 class="section">MainReificationFunctions</h2>
Constructs the actual reifications, putting everything together
</div>
<div class="code">
<span class="id" title="keyword">Context</span> {<span class="id" title="var">tr</span> : <span class="id" title="var">tarski_reflector</span>}.<br/>
<span class="id" title="keyword">Context</span> {<span class="id" title="var">te</span> : <span class="id" title="var">tarski_reflector_extensions</span> <span class="id" title="var">tr</span>}.<br/>
<span class="id" title="var">Existing</span> <span class="id" title="keyword">Instance</span> <span class="id" title="var">config.default_checker_flags</span>.<br/>
</div>
<div class="doc">
<a id="lab9"></a><h3 class="section">Terms</h3>
Constructs the representation for terms. Arguments:
<ul class="doclist">
<li> tct -- the quoted instance of tarski_reflector
</li>
<li> fuel -- fuel
</li>
<li> t -- the term to reify
</li>
<li> termEnv -- the env term
</li>
<li> env -- the env helper
</li>
<li> returns the pair of reification/proof
</li>