-
Notifications
You must be signed in to change notification settings - Fork 28
/
Copy pathstlc5k.stt
6440 lines (5000 loc) · 267 KB
/
stlc5k.stt
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
Ty : U
= (Ty : U)
(nat top bot : Ty)
(arr prod sum : Ty → Ty → Ty)
→ Ty
nat : Ty = λ _ nat _ _ _ _ _. nat
top : Ty = λ _ _ top _ _ _ _. top
bot : Ty = λ _ _ _ bot _ _ _. bot
arr : Ty → Ty → Ty
= λ A B Ty nat top bot arr prod sum.
arr (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
prod : Ty → Ty → Ty
= λ A B Ty nat top bot arr prod sum.
prod (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
sum : Ty → Ty → Ty
= λ A B Ty nat top bot arr prod sum.
sum (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
Con : U
= (Con : U)
(nil : Con)
(snoc : Con → Ty → Con)
→ Con
nil : Con
= λ Con nil snoc. nil
snoc : Con → Ty → Con
= λ Γ A Con nil snoc. snoc (Γ Con nil snoc) A
Var : Con → Ty → U
= λ Γ A.
(Var : Con → Ty → U)
(vz : {Γ A} → Var (snoc Γ A) A)
(vs : {Γ B A} → Var Γ A → Var (snoc Γ B) A)
→ Var Γ A
vz : {Γ A} → Var (snoc Γ A) A
= λ Var vz vs. vz
vs : {Γ B A} → Var Γ A → Var (snoc Γ B) A
= λ x Var vz vs. vs (x Var vz vs)
Tm : Con → Ty → U
= λ Γ A.
(Tm : Con → Ty → U)
(var : {Γ A} → Var Γ A → Tm Γ A)
(lam : {Γ A B} → Tm (snoc Γ A) B → Tm Γ (arr A B))
(app : {Γ A B} → Tm Γ (arr A B) → Tm Γ A → Tm Γ B)
(tt : {Γ} → Tm Γ top)
(pair : {Γ A B} → Tm Γ A → Tm Γ B → Tm Γ (prod A B))
(fst : {Γ A B} → Tm Γ (prod A B) → Tm Γ A)
(snd : {Γ A B} → Tm Γ (prod A B) → Tm Γ B)
(left : {Γ A B} → Tm Γ A → Tm Γ (sum A B))
(right : {Γ A B} → Tm Γ B → Tm Γ (sum A B))
(case : {Γ A B C} → Tm Γ (sum A B) → Tm Γ (arr A C) → Tm Γ (arr B C) → Tm Γ C)
(zero : {Γ} → Tm Γ nat)
(suc : {Γ} → Tm Γ nat → Tm Γ nat)
(rec : {Γ A} → Tm Γ nat → Tm Γ (arr nat (arr A A)) → Tm Γ A → Tm Γ A)
→ Tm Γ A
var : {Γ A} → Var Γ A → Tm Γ A
= λ x Tm var lam app tt pair fst snd left right case zero suc rec.
var x
lam : {Γ A B} → Tm (snoc Γ A) B → Tm Γ (arr A B)
= λ t Tm var lam app tt pair fst snd left right case zero suc rec.
lam (t Tm var lam app tt pair fst snd left right case zero suc rec)
app : {Γ A B} → Tm Γ (arr A B) → Tm Γ A → Tm Γ B
= λ t u Tm var lam app tt pair fst snd left right case zero suc rec.
app (t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
tt : {Γ} → Tm Γ top
= λ Tm var lam app tt pair fst snd left right case zero suc rec. tt
pair : {Γ A B} → Tm Γ A → Tm Γ B → Tm Γ (prod A B)
= λ t u Tm var lam app tt pair fst snd left right case zero suc rec.
pair (t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
fst : {Γ A B} → Tm Γ (prod A B) → Tm Γ A
= λ t Tm var lam app tt pair fst snd left right case zero suc rec.
fst (t Tm var lam app tt pair fst snd left right case zero suc rec)
snd : {Γ A B} → Tm Γ (prod A B) → Tm Γ B
= λ t Tm var lam app tt pair fst snd left right case zero suc rec.
snd (t Tm var lam app tt pair fst snd left right case zero suc rec)
left : {Γ A B} → Tm Γ A → Tm Γ (sum A B)
= λ t Tm var lam app tt pair fst snd left right case zero suc rec.
left (t Tm var lam app tt pair fst snd left right case zero suc rec)
right : {Γ A B} → Tm Γ B → Tm Γ (sum A B)
= λ t Tm var lam app tt pair fst snd left right case zero suc rec.
right (t Tm var lam app tt pair fst snd left right case zero suc rec)
case : {Γ A B C} → Tm Γ (sum A B) → Tm Γ (arr A C) → Tm Γ (arr B C) → Tm Γ C
= λ t u v Tm var lam app tt pair fst snd left right case zero suc rec.
case (t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
(v Tm var lam app tt pair fst snd left right case zero suc rec)
zero : {Γ} → Tm Γ nat
= λ Tm var lam app tt pair fst snd left right case zero suc rec. zero
suc : {Γ} → Tm Γ nat → Tm Γ nat
= λ t Tm var lam app tt pair fst snd left right case zero suc rec.
suc (t Tm var lam app tt pair fst snd left right case zero suc rec)
rec : {Γ A} → Tm Γ nat → Tm Γ (arr nat (arr A A)) → Tm Γ A → Tm Γ A
= λ t u v Tm var lam app tt pair fst snd left right case zero suc rec.
rec (t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
(v Tm var lam app tt pair fst snd left right case zero suc rec)
v0 : {Γ A} → Tm (snoc Γ A) A
= var vz
v1 : {Γ A B} → Tm (snoc (snoc Γ A) B) A
= var (vs vz)
v2 : {Γ A B C} → Tm (snoc (snoc (snoc Γ A) B) C) A
= var (vs (vs vz))
v3 : {Γ A B C D} → Tm (snoc (snoc (snoc (snoc Γ A) B) C) D) A
= var (vs (vs (vs vz)))
tbool : Ty
= sum top top
true : {Γ} → Tm Γ tbool
= left tt
tfalse : {Γ} → Tm Γ tbool
= right tt
ifthenelse : {Γ A} → Tm Γ (arr tbool (arr A (arr A A)))
= lam (lam (lam (case v2 (lam v2) (lam v1))))
times4 : {Γ A} → Tm Γ (arr (arr A A) (arr A A))
= lam (lam (app v1 (app v1 (app v1 (app v1 v0)))))
add : {Γ} → Tm Γ (arr nat (arr nat nat))
= lam (rec v0
(lam (lam (lam (suc (app v1 v0)))))
(lam v0))
mul : {Γ} → Tm Γ (arr nat (arr nat nat))
= lam (rec v0
(lam (lam (lam (app (app add (app v1 v0)) v0))))
(lam zero))
fact : {Γ} → Tm Γ (arr nat nat)
= lam (rec v0 (lam (lam (app (app mul (suc v1)) v0)))
(suc zero))
Ty1 : U
= (Ty1 : U)
(nat top bot : Ty1)
(arr prod sum : Ty1 → Ty1 → Ty1)
→ Ty1
nat1 : Ty1 = λ _ nat1 _ _ _ _ _. nat1
top1 : Ty1 = λ _ _ top1 _ _ _ _. top1
bot1 : Ty1 = λ _ _ _ bot1 _ _ _. bot1
arr1 : Ty1 → Ty1 → Ty1
= λ A B Ty1 nat1 top1 bot1 arr1 prod sum.
arr1 (A Ty1 nat1 top1 bot1 arr1 prod sum) (B Ty1 nat1 top1 bot1 arr1 prod sum)
prod1 : Ty1 → Ty1 → Ty1
= λ A B Ty1 nat1 top1 bot1 arr1 prod1 sum.
prod1 (A Ty1 nat1 top1 bot1 arr1 prod1 sum) (B Ty1 nat1 top1 bot1 arr1 prod1 sum)
sum1 : Ty1 → Ty1 → Ty1
= λ A B Ty1 nat1 top1 bot1 arr1 prod1 sum1.
sum1 (A Ty1 nat1 top1 bot1 arr1 prod1 sum1) (B Ty1 nat1 top1 bot1 arr1 prod1 sum1)
Con1 : U
= (Con1 : U)
(nil : Con1)
(snoc : Con1 → Ty1 → Con1)
→ Con1
nil1 : Con1
= λ Con1 nil1 snoc. nil1
snoc1 : Con1 → Ty1 → Con1
= λ Γ A Con1 nil1 snoc1. snoc1 (Γ Con1 nil1 snoc1) A
Var1 : Con1 → Ty1 → U
= λ Γ A.
(Var1 : Con1 → Ty1 → U)
(vz : {Γ A} → Var1 (snoc1 Γ A) A)
(vs : {Γ B A} → Var1 Γ A → Var1 (snoc1 Γ B) A)
→ Var1 Γ A
vz1 : {Γ A} → Var1 (snoc1 Γ A) A
= λ Var1 vz1 vs. vz1
vs1 : {Γ B A} → Var1 Γ A → Var1 (snoc1 Γ B) A
= λ x Var1 vz1 vs1. vs1 (x Var1 vz1 vs1)
Tm1 : Con1 → Ty1 → U
= λ Γ A.
(Tm1 : Con1 → Ty1 → U)
(var : {Γ A} → Var1 Γ A → Tm1 Γ A)
(lam : {Γ A B} → Tm1 (snoc1 Γ A) B → Tm1 Γ (arr1 A B))
(app : {Γ A B} → Tm1 Γ (arr1 A B) → Tm1 Γ A → Tm1 Γ B)
(tt : {Γ} → Tm1 Γ top1)
(pair : {Γ A B} → Tm1 Γ A → Tm1 Γ B → Tm1 Γ (prod1 A B))
(fst : {Γ A B} → Tm1 Γ (prod1 A B) → Tm1 Γ A)
(snd : {Γ A B} → Tm1 Γ (prod1 A B) → Tm1 Γ B)
(left : {Γ A B} → Tm1 Γ A → Tm1 Γ (sum1 A B))
(right : {Γ A B} → Tm1 Γ B → Tm1 Γ (sum1 A B))
(case : {Γ A B C} → Tm1 Γ (sum1 A B) → Tm1 Γ (arr1 A C) → Tm1 Γ (arr1 B C) → Tm1 Γ C)
(zero : {Γ} → Tm1 Γ nat1)
(suc : {Γ} → Tm1 Γ nat1 → Tm1 Γ nat1)
(rec : {Γ A} → Tm1 Γ nat1 → Tm1 Γ (arr1 nat1 (arr1 A A)) → Tm1 Γ A → Tm1 Γ A)
→ Tm1 Γ A
var1 : {Γ A} → Var1 Γ A → Tm1 Γ A
= λ x Tm1 var1 lam app tt pair fst snd left right case zero suc rec.
var1 x
lam1 : {Γ A B} → Tm1 (snoc1 Γ A) B → Tm1 Γ (arr1 A B)
= λ t Tm1 var1 lam1 app tt pair fst snd left right case zero suc rec.
lam1 (t Tm1 var1 lam1 app tt pair fst snd left right case zero suc rec)
app1 : {Γ A B} → Tm1 Γ (arr1 A B) → Tm1 Γ A → Tm1 Γ B
= λ t u Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec.
app1 (t Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec)
(u Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec)
tt1 : {Γ} → Tm1 Γ top1
= λ Tm1 var1 lam1 app1 tt1 pair fst snd left right case zero suc rec. tt1
pair1 : {Γ A B} → Tm1 Γ A → Tm1 Γ B → Tm1 Γ (prod1 A B)
= λ t u Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec.
pair1 (t Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec)
(u Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec)
fst1 : {Γ A B} → Tm1 Γ (prod1 A B) → Tm1 Γ A
= λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd left right case zero suc rec.
fst1 (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd left right case zero suc rec)
snd1 : {Γ A B} → Tm1 Γ (prod1 A B) → Tm1 Γ B
= λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left right case zero suc rec.
snd1 (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left right case zero suc rec)
left1 : {Γ A B} → Tm1 Γ A → Tm1 Γ (sum1 A B)
= λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right case zero suc rec.
left1 (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right case zero suc rec)
right1 : {Γ A B} → Tm1 Γ B → Tm1 Γ (sum1 A B)
= λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case zero suc rec.
right1 (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case zero suc rec)
case1 : {Γ A B C} → Tm1 Γ (sum1 A B) → Tm1 Γ (arr1 A C) → Tm1 Γ (arr1 B C) → Tm1 Γ C
= λ t u v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec.
case1 (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
(u Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
(v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
zero1 : {Γ} → Tm1 Γ nat1
= λ Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc rec. zero1
suc1 : {Γ} → Tm1 Γ nat1 → Tm1 Γ nat1
= λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec.
suc1 (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec)
rec1 : {Γ A} → Tm1 Γ nat1 → Tm1 Γ (arr1 nat1 (arr1 A A)) → Tm1 Γ A → Tm1 Γ A
= λ t u v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1.
rec1 (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
(u Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
(v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
v01 : {Γ A} → Tm1 (snoc1 Γ A) A
= var1 vz1
v11 : {Γ A B} → Tm1 (snoc1 (snoc1 Γ A) B) A
= var1 (vs1 vz1)
v21 : {Γ A B C} → Tm1 (snoc1 (snoc1 (snoc1 Γ A) B) C) A
= var1 (vs1 (vs1 vz1))
v31 : {Γ A B C D} → Tm1 (snoc1 (snoc1 (snoc1 (snoc1 Γ A) B) C) D) A
= var1 (vs1 (vs1 (vs1 vz1)))
tbool1 : Ty1
= sum1 top1 top1
true1 : {Γ} → Tm1 Γ tbool1
= left1 tt1
tfalse1 : {Γ} → Tm1 Γ tbool1
= right1 tt1
ifthenelse1 : {Γ A} → Tm1 Γ (arr1 tbool1 (arr1 A (arr1 A A)))
= lam1 (lam1 (lam1 (case1 v21 (lam1 v21) (lam1 v11))))
times41 : {Γ A} → Tm1 Γ (arr1 (arr1 A A) (arr1 A A))
= lam1 (lam1 (app1 v11 (app1 v11 (app1 v11 (app1 v11 v01)))))
add1 : {Γ} → Tm1 Γ (arr1 nat1 (arr1 nat1 nat1))
= lam1 (rec1 v01
(lam1 (lam1 (lam1 (suc1 (app1 v11 v01)))))
(lam1 v01))
mul1 : {Γ} → Tm1 Γ (arr1 nat1 (arr1 nat1 nat1))
= lam1 (rec1 v01
(lam1 (lam1 (lam1 (app1 (app1 add1 (app1 v11 v01)) v01))))
(lam1 zero1))
fact1 : {Γ} → Tm1 Γ (arr1 nat1 nat1)
= lam1 (rec1 v01 (lam1 (lam1 (app1 (app1 mul1 (suc1 v11)) v01)))
(suc1 zero1))
Ty2 : U
= (Ty2 : U)
(nat top bot : Ty2)
(arr prod sum : Ty2 → Ty2 → Ty2)
→ Ty2
nat2 : Ty2 = λ _ nat2 _ _ _ _ _. nat2
top2 : Ty2 = λ _ _ top2 _ _ _ _. top2
bot2 : Ty2 = λ _ _ _ bot2 _ _ _. bot2
arr2 : Ty2 → Ty2 → Ty2
= λ A B Ty2 nat2 top2 bot2 arr2 prod sum.
arr2 (A Ty2 nat2 top2 bot2 arr2 prod sum) (B Ty2 nat2 top2 bot2 arr2 prod sum)
prod2 : Ty2 → Ty2 → Ty2
= λ A B Ty2 nat2 top2 bot2 arr2 prod2 sum.
prod2 (A Ty2 nat2 top2 bot2 arr2 prod2 sum) (B Ty2 nat2 top2 bot2 arr2 prod2 sum)
sum2 : Ty2 → Ty2 → Ty2
= λ A B Ty2 nat2 top2 bot2 arr2 prod2 sum2.
sum2 (A Ty2 nat2 top2 bot2 arr2 prod2 sum2) (B Ty2 nat2 top2 bot2 arr2 prod2 sum2)
Con2 : U
= (Con2 : U)
(nil : Con2)
(snoc : Con2 → Ty2 → Con2)
→ Con2
nil2 : Con2
= λ Con2 nil2 snoc. nil2
snoc2 : Con2 → Ty2 → Con2
= λ Γ A Con2 nil2 snoc2. snoc2 (Γ Con2 nil2 snoc2) A
Var2 : Con2 → Ty2 → U
= λ Γ A.
(Var2 : Con2 → Ty2 → U)
(vz : {Γ A} → Var2 (snoc2 Γ A) A)
(vs : {Γ B A} → Var2 Γ A → Var2 (snoc2 Γ B) A)
→ Var2 Γ A
vz2 : {Γ A} → Var2 (snoc2 Γ A) A
= λ Var2 vz2 vs. vz2
vs2 : {Γ B A} → Var2 Γ A → Var2 (snoc2 Γ B) A
= λ x Var2 vz2 vs2. vs2 (x Var2 vz2 vs2)
Tm2 : Con2 → Ty2 → U
= λ Γ A.
(Tm2 : Con2 → Ty2 → U)
(var : {Γ A} → Var2 Γ A → Tm2 Γ A)
(lam : {Γ A B} → Tm2 (snoc2 Γ A) B → Tm2 Γ (arr2 A B))
(app : {Γ A B} → Tm2 Γ (arr2 A B) → Tm2 Γ A → Tm2 Γ B)
(tt : {Γ} → Tm2 Γ top2)
(pair : {Γ A B} → Tm2 Γ A → Tm2 Γ B → Tm2 Γ (prod2 A B))
(fst : {Γ A B} → Tm2 Γ (prod2 A B) → Tm2 Γ A)
(snd : {Γ A B} → Tm2 Γ (prod2 A B) → Tm2 Γ B)
(left : {Γ A B} → Tm2 Γ A → Tm2 Γ (sum2 A B))
(right : {Γ A B} → Tm2 Γ B → Tm2 Γ (sum2 A B))
(case : {Γ A B C} → Tm2 Γ (sum2 A B) → Tm2 Γ (arr2 A C) → Tm2 Γ (arr2 B C) → Tm2 Γ C)
(zero : {Γ} → Tm2 Γ nat2)
(suc : {Γ} → Tm2 Γ nat2 → Tm2 Γ nat2)
(rec : {Γ A} → Tm2 Γ nat2 → Tm2 Γ (arr2 nat2 (arr2 A A)) → Tm2 Γ A → Tm2 Γ A)
→ Tm2 Γ A
var2 : {Γ A} → Var2 Γ A → Tm2 Γ A
= λ x Tm2 var2 lam app tt pair fst snd left right case zero suc rec.
var2 x
lam2 : {Γ A B} → Tm2 (snoc2 Γ A) B → Tm2 Γ (arr2 A B)
= λ t Tm2 var2 lam2 app tt pair fst snd left right case zero suc rec.
lam2 (t Tm2 var2 lam2 app tt pair fst snd left right case zero suc rec)
app2 : {Γ A B} → Tm2 Γ (arr2 A B) → Tm2 Γ A → Tm2 Γ B
= λ t u Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec.
app2 (t Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec)
(u Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec)
tt2 : {Γ} → Tm2 Γ top2
= λ Tm2 var2 lam2 app2 tt2 pair fst snd left right case zero suc rec. tt2
pair2 : {Γ A B} → Tm2 Γ A → Tm2 Γ B → Tm2 Γ (prod2 A B)
= λ t u Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec.
pair2 (t Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec)
(u Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec)
fst2 : {Γ A B} → Tm2 Γ (prod2 A B) → Tm2 Γ A
= λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd left right case zero suc rec.
fst2 (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd left right case zero suc rec)
snd2 : {Γ A B} → Tm2 Γ (prod2 A B) → Tm2 Γ B
= λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left right case zero suc rec.
snd2 (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left right case zero suc rec)
left2 : {Γ A B} → Tm2 Γ A → Tm2 Γ (sum2 A B)
= λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right case zero suc rec.
left2 (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right case zero suc rec)
right2 : {Γ A B} → Tm2 Γ B → Tm2 Γ (sum2 A B)
= λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case zero suc rec.
right2 (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case zero suc rec)
case2 : {Γ A B C} → Tm2 Γ (sum2 A B) → Tm2 Γ (arr2 A C) → Tm2 Γ (arr2 B C) → Tm2 Γ C
= λ t u v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec.
case2 (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
(u Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
(v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
zero2 : {Γ} → Tm2 Γ nat2
= λ Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc rec. zero2
suc2 : {Γ} → Tm2 Γ nat2 → Tm2 Γ nat2
= λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec.
suc2 (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec)
rec2 : {Γ A} → Tm2 Γ nat2 → Tm2 Γ (arr2 nat2 (arr2 A A)) → Tm2 Γ A → Tm2 Γ A
= λ t u v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2.
rec2 (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
(u Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
(v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
v02 : {Γ A} → Tm2 (snoc2 Γ A) A
= var2 vz2
v12 : {Γ A B} → Tm2 (snoc2 (snoc2 Γ A) B) A
= var2 (vs2 vz2)
v22 : {Γ A B C} → Tm2 (snoc2 (snoc2 (snoc2 Γ A) B) C) A
= var2 (vs2 (vs2 vz2))
v32 : {Γ A B C D} → Tm2 (snoc2 (snoc2 (snoc2 (snoc2 Γ A) B) C) D) A
= var2 (vs2 (vs2 (vs2 vz2)))
tbool2 : Ty2
= sum2 top2 top2
true2 : {Γ} → Tm2 Γ tbool2
= left2 tt2
tfalse2 : {Γ} → Tm2 Γ tbool2
= right2 tt2
ifthenelse2 : {Γ A} → Tm2 Γ (arr2 tbool2 (arr2 A (arr2 A A)))
= lam2 (lam2 (lam2 (case2 v22 (lam2 v22) (lam2 v12))))
times42 : {Γ A} → Tm2 Γ (arr2 (arr2 A A) (arr2 A A))
= lam2 (lam2 (app2 v12 (app2 v12 (app2 v12 (app2 v12 v02)))))
add2 : {Γ} → Tm2 Γ (arr2 nat2 (arr2 nat2 nat2))
= lam2 (rec2 v02
(lam2 (lam2 (lam2 (suc2 (app2 v12 v02)))))
(lam2 v02))
mul2 : {Γ} → Tm2 Γ (arr2 nat2 (arr2 nat2 nat2))
= lam2 (rec2 v02
(lam2 (lam2 (lam2 (app2 (app2 add2 (app2 v12 v02)) v02))))
(lam2 zero2))
fact2 : {Γ} → Tm2 Γ (arr2 nat2 nat2)
= lam2 (rec2 v02 (lam2 (lam2 (app2 (app2 mul2 (suc2 v12)) v02)))
(suc2 zero2))
Ty3 : U
= (Ty3 : U)
(nat top bot : Ty3)
(arr prod sum : Ty3 → Ty3 → Ty3)
→ Ty3
nat3 : Ty3 = λ _ nat3 _ _ _ _ _. nat3
top3 : Ty3 = λ _ _ top3 _ _ _ _. top3
bot3 : Ty3 = λ _ _ _ bot3 _ _ _. bot3
arr3 : Ty3 → Ty3 → Ty3
= λ A B Ty3 nat3 top3 bot3 arr3 prod sum.
arr3 (A Ty3 nat3 top3 bot3 arr3 prod sum) (B Ty3 nat3 top3 bot3 arr3 prod sum)
prod3 : Ty3 → Ty3 → Ty3
= λ A B Ty3 nat3 top3 bot3 arr3 prod3 sum.
prod3 (A Ty3 nat3 top3 bot3 arr3 prod3 sum) (B Ty3 nat3 top3 bot3 arr3 prod3 sum)
sum3 : Ty3 → Ty3 → Ty3
= λ A B Ty3 nat3 top3 bot3 arr3 prod3 sum3.
sum3 (A Ty3 nat3 top3 bot3 arr3 prod3 sum3) (B Ty3 nat3 top3 bot3 arr3 prod3 sum3)
Con3 : U
= (Con3 : U)
(nil : Con3)
(snoc : Con3 → Ty3 → Con3)
→ Con3
nil3 : Con3
= λ Con3 nil3 snoc. nil3
snoc3 : Con3 → Ty3 → Con3
= λ Γ A Con3 nil3 snoc3. snoc3 (Γ Con3 nil3 snoc3) A
Var3 : Con3 → Ty3 → U
= λ Γ A.
(Var3 : Con3 → Ty3 → U)
(vz : {Γ A} → Var3 (snoc3 Γ A) A)
(vs : {Γ B A} → Var3 Γ A → Var3 (snoc3 Γ B) A)
→ Var3 Γ A
vz3 : {Γ A} → Var3 (snoc3 Γ A) A
= λ Var3 vz3 vs. vz3
vs3 : {Γ B A} → Var3 Γ A → Var3 (snoc3 Γ B) A
= λ x Var3 vz3 vs3. vs3 (x Var3 vz3 vs3)
Tm3 : Con3 → Ty3 → U
= λ Γ A.
(Tm3 : Con3 → Ty3 → U)
(var : {Γ A} → Var3 Γ A → Tm3 Γ A)
(lam : {Γ A B} → Tm3 (snoc3 Γ A) B → Tm3 Γ (arr3 A B))
(app : {Γ A B} → Tm3 Γ (arr3 A B) → Tm3 Γ A → Tm3 Γ B)
(tt : {Γ} → Tm3 Γ top3)
(pair : {Γ A B} → Tm3 Γ A → Tm3 Γ B → Tm3 Γ (prod3 A B))
(fst : {Γ A B} → Tm3 Γ (prod3 A B) → Tm3 Γ A)
(snd : {Γ A B} → Tm3 Γ (prod3 A B) → Tm3 Γ B)
(left : {Γ A B} → Tm3 Γ A → Tm3 Γ (sum3 A B))
(right : {Γ A B} → Tm3 Γ B → Tm3 Γ (sum3 A B))
(case : {Γ A B C} → Tm3 Γ (sum3 A B) → Tm3 Γ (arr3 A C) → Tm3 Γ (arr3 B C) → Tm3 Γ C)
(zero : {Γ} → Tm3 Γ nat3)
(suc : {Γ} → Tm3 Γ nat3 → Tm3 Γ nat3)
(rec : {Γ A} → Tm3 Γ nat3 → Tm3 Γ (arr3 nat3 (arr3 A A)) → Tm3 Γ A → Tm3 Γ A)
→ Tm3 Γ A
var3 : {Γ A} → Var3 Γ A → Tm3 Γ A
= λ x Tm3 var3 lam app tt pair fst snd left right case zero suc rec.
var3 x
lam3 : {Γ A B} → Tm3 (snoc3 Γ A) B → Tm3 Γ (arr3 A B)
= λ t Tm3 var3 lam3 app tt pair fst snd left right case zero suc rec.
lam3 (t Tm3 var3 lam3 app tt pair fst snd left right case zero suc rec)
app3 : {Γ A B} → Tm3 Γ (arr3 A B) → Tm3 Γ A → Tm3 Γ B
= λ t u Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec.
app3 (t Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec)
(u Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec)
tt3 : {Γ} → Tm3 Γ top3
= λ Tm3 var3 lam3 app3 tt3 pair fst snd left right case zero suc rec. tt3
pair3 : {Γ A B} → Tm3 Γ A → Tm3 Γ B → Tm3 Γ (prod3 A B)
= λ t u Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec.
pair3 (t Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec)
(u Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec)
fst3 : {Γ A B} → Tm3 Γ (prod3 A B) → Tm3 Γ A
= λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd left right case zero suc rec.
fst3 (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd left right case zero suc rec)
snd3 : {Γ A B} → Tm3 Γ (prod3 A B) → Tm3 Γ B
= λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left right case zero suc rec.
snd3 (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left right case zero suc rec)
left3 : {Γ A B} → Tm3 Γ A → Tm3 Γ (sum3 A B)
= λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right case zero suc rec.
left3 (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right case zero suc rec)
right3 : {Γ A B} → Tm3 Γ B → Tm3 Γ (sum3 A B)
= λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case zero suc rec.
right3 (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case zero suc rec)
case3 : {Γ A B C} → Tm3 Γ (sum3 A B) → Tm3 Γ (arr3 A C) → Tm3 Γ (arr3 B C) → Tm3 Γ C
= λ t u v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec.
case3 (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
(u Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
(v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
zero3 : {Γ} → Tm3 Γ nat3
= λ Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc rec. zero3
suc3 : {Γ} → Tm3 Γ nat3 → Tm3 Γ nat3
= λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec.
suc3 (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec)
rec3 : {Γ A} → Tm3 Γ nat3 → Tm3 Γ (arr3 nat3 (arr3 A A)) → Tm3 Γ A → Tm3 Γ A
= λ t u v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3.
rec3 (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
(u Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
(v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
v03 : {Γ A} → Tm3 (snoc3 Γ A) A
= var3 vz3
v13 : {Γ A B} → Tm3 (snoc3 (snoc3 Γ A) B) A
= var3 (vs3 vz3)
v23 : {Γ A B C} → Tm3 (snoc3 (snoc3 (snoc3 Γ A) B) C) A
= var3 (vs3 (vs3 vz3))
v33 : {Γ A B C D} → Tm3 (snoc3 (snoc3 (snoc3 (snoc3 Γ A) B) C) D) A
= var3 (vs3 (vs3 (vs3 vz3)))
tbool3 : Ty3
= sum3 top3 top3
true3 : {Γ} → Tm3 Γ tbool3
= left3 tt3
tfalse3 : {Γ} → Tm3 Γ tbool3
= right3 tt3
ifthenelse3 : {Γ A} → Tm3 Γ (arr3 tbool3 (arr3 A (arr3 A A)))
= lam3 (lam3 (lam3 (case3 v23 (lam3 v23) (lam3 v13))))
times43 : {Γ A} → Tm3 Γ (arr3 (arr3 A A) (arr3 A A))
= lam3 (lam3 (app3 v13 (app3 v13 (app3 v13 (app3 v13 v03)))))
add3 : {Γ} → Tm3 Γ (arr3 nat3 (arr3 nat3 nat3))
= lam3 (rec3 v03
(lam3 (lam3 (lam3 (suc3 (app3 v13 v03)))))
(lam3 v03))
mul3 : {Γ} → Tm3 Γ (arr3 nat3 (arr3 nat3 nat3))
= lam3 (rec3 v03
(lam3 (lam3 (lam3 (app3 (app3 add3 (app3 v13 v03)) v03))))
(lam3 zero3))
fact3 : {Γ} → Tm3 Γ (arr3 nat3 nat3)
= lam3 (rec3 v03 (lam3 (lam3 (app3 (app3 mul3 (suc3 v13)) v03)))
(suc3 zero3))
Ty4 : U
= (Ty4 : U)
(nat top bot : Ty4)
(arr prod sum : Ty4 → Ty4 → Ty4)
→ Ty4
nat4 : Ty4 = λ _ nat4 _ _ _ _ _. nat4
top4 : Ty4 = λ _ _ top4 _ _ _ _. top4
bot4 : Ty4 = λ _ _ _ bot4 _ _ _. bot4
arr4 : Ty4 → Ty4 → Ty4
= λ A B Ty4 nat4 top4 bot4 arr4 prod sum.
arr4 (A Ty4 nat4 top4 bot4 arr4 prod sum) (B Ty4 nat4 top4 bot4 arr4 prod sum)
prod4 : Ty4 → Ty4 → Ty4
= λ A B Ty4 nat4 top4 bot4 arr4 prod4 sum.
prod4 (A Ty4 nat4 top4 bot4 arr4 prod4 sum) (B Ty4 nat4 top4 bot4 arr4 prod4 sum)
sum4 : Ty4 → Ty4 → Ty4
= λ A B Ty4 nat4 top4 bot4 arr4 prod4 sum4.
sum4 (A Ty4 nat4 top4 bot4 arr4 prod4 sum4) (B Ty4 nat4 top4 bot4 arr4 prod4 sum4)
Con4 : U
= (Con4 : U)
(nil : Con4)
(snoc : Con4 → Ty4 → Con4)
→ Con4
nil4 : Con4
= λ Con4 nil4 snoc. nil4
snoc4 : Con4 → Ty4 → Con4
= λ Γ A Con4 nil4 snoc4. snoc4 (Γ Con4 nil4 snoc4) A
Var4 : Con4 → Ty4 → U
= λ Γ A.
(Var4 : Con4 → Ty4 → U)
(vz : {Γ A} → Var4 (snoc4 Γ A) A)
(vs : {Γ B A} → Var4 Γ A → Var4 (snoc4 Γ B) A)
→ Var4 Γ A
vz4 : {Γ A} → Var4 (snoc4 Γ A) A
= λ Var4 vz4 vs. vz4
vs4 : {Γ B A} → Var4 Γ A → Var4 (snoc4 Γ B) A
= λ x Var4 vz4 vs4. vs4 (x Var4 vz4 vs4)
Tm4 : Con4 → Ty4 → U
= λ Γ A.
(Tm4 : Con4 → Ty4 → U)
(var : {Γ A} → Var4 Γ A → Tm4 Γ A)
(lam : {Γ A B} → Tm4 (snoc4 Γ A) B → Tm4 Γ (arr4 A B))
(app : {Γ A B} → Tm4 Γ (arr4 A B) → Tm4 Γ A → Tm4 Γ B)
(tt : {Γ} → Tm4 Γ top4)
(pair : {Γ A B} → Tm4 Γ A → Tm4 Γ B → Tm4 Γ (prod4 A B))
(fst : {Γ A B} → Tm4 Γ (prod4 A B) → Tm4 Γ A)
(snd : {Γ A B} → Tm4 Γ (prod4 A B) → Tm4 Γ B)
(left : {Γ A B} → Tm4 Γ A → Tm4 Γ (sum4 A B))
(right : {Γ A B} → Tm4 Γ B → Tm4 Γ (sum4 A B))
(case : {Γ A B C} → Tm4 Γ (sum4 A B) → Tm4 Γ (arr4 A C) → Tm4 Γ (arr4 B C) → Tm4 Γ C)
(zero : {Γ} → Tm4 Γ nat4)
(suc : {Γ} → Tm4 Γ nat4 → Tm4 Γ nat4)
(rec : {Γ A} → Tm4 Γ nat4 → Tm4 Γ (arr4 nat4 (arr4 A A)) → Tm4 Γ A → Tm4 Γ A)
→ Tm4 Γ A
var4 : {Γ A} → Var4 Γ A → Tm4 Γ A
= λ x Tm4 var4 lam app tt pair fst snd left right case zero suc rec.
var4 x
lam4 : {Γ A B} → Tm4 (snoc4 Γ A) B → Tm4 Γ (arr4 A B)
= λ t Tm4 var4 lam4 app tt pair fst snd left right case zero suc rec.
lam4 (t Tm4 var4 lam4 app tt pair fst snd left right case zero suc rec)
app4 : {Γ A B} → Tm4 Γ (arr4 A B) → Tm4 Γ A → Tm4 Γ B
= λ t u Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec.
app4 (t Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec)
(u Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec)
tt4 : {Γ} → Tm4 Γ top4
= λ Tm4 var4 lam4 app4 tt4 pair fst snd left right case zero suc rec. tt4
pair4 : {Γ A B} → Tm4 Γ A → Tm4 Γ B → Tm4 Γ (prod4 A B)
= λ t u Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec.
pair4 (t Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec)
(u Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec)
fst4 : {Γ A B} → Tm4 Γ (prod4 A B) → Tm4 Γ A
= λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd left right case zero suc rec.
fst4 (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd left right case zero suc rec)
snd4 : {Γ A B} → Tm4 Γ (prod4 A B) → Tm4 Γ B
= λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left right case zero suc rec.
snd4 (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left right case zero suc rec)
left4 : {Γ A B} → Tm4 Γ A → Tm4 Γ (sum4 A B)
= λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right case zero suc rec.
left4 (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right case zero suc rec)
right4 : {Γ A B} → Tm4 Γ B → Tm4 Γ (sum4 A B)
= λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case zero suc rec.
right4 (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case zero suc rec)
case4 : {Γ A B C} → Tm4 Γ (sum4 A B) → Tm4 Γ (arr4 A C) → Tm4 Γ (arr4 B C) → Tm4 Γ C
= λ t u v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec.
case4 (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
(u Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
(v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
zero4 : {Γ} → Tm4 Γ nat4
= λ Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc rec. zero4
suc4 : {Γ} → Tm4 Γ nat4 → Tm4 Γ nat4
= λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec.
suc4 (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec)
rec4 : {Γ A} → Tm4 Γ nat4 → Tm4 Γ (arr4 nat4 (arr4 A A)) → Tm4 Γ A → Tm4 Γ A
= λ t u v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4.
rec4 (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
(u Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
(v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
v04 : {Γ A} → Tm4 (snoc4 Γ A) A
= var4 vz4
v14 : {Γ A B} → Tm4 (snoc4 (snoc4 Γ A) B) A
= var4 (vs4 vz4)
v24 : {Γ A B C} → Tm4 (snoc4 (snoc4 (snoc4 Γ A) B) C) A
= var4 (vs4 (vs4 vz4))
v34 : {Γ A B C D} → Tm4 (snoc4 (snoc4 (snoc4 (snoc4 Γ A) B) C) D) A
= var4 (vs4 (vs4 (vs4 vz4)))
tbool4 : Ty4
= sum4 top4 top4
true4 : {Γ} → Tm4 Γ tbool4
= left4 tt4
tfalse4 : {Γ} → Tm4 Γ tbool4
= right4 tt4
ifthenelse4 : {Γ A} → Tm4 Γ (arr4 tbool4 (arr4 A (arr4 A A)))
= lam4 (lam4 (lam4 (case4 v24 (lam4 v24) (lam4 v14))))
times44 : {Γ A} → Tm4 Γ (arr4 (arr4 A A) (arr4 A A))
= lam4 (lam4 (app4 v14 (app4 v14 (app4 v14 (app4 v14 v04)))))
add4 : {Γ} → Tm4 Γ (arr4 nat4 (arr4 nat4 nat4))
= lam4 (rec4 v04
(lam4 (lam4 (lam4 (suc4 (app4 v14 v04)))))
(lam4 v04))
mul4 : {Γ} → Tm4 Γ (arr4 nat4 (arr4 nat4 nat4))
= lam4 (rec4 v04
(lam4 (lam4 (lam4 (app4 (app4 add4 (app4 v14 v04)) v04))))
(lam4 zero4))
fact4 : {Γ} → Tm4 Γ (arr4 nat4 nat4)
= lam4 (rec4 v04 (lam4 (lam4 (app4 (app4 mul4 (suc4 v14)) v04)))
(suc4 zero4))
Ty5 : U
= (Ty5 : U)
(nat top bot : Ty5)
(arr prod sum : Ty5 → Ty5 → Ty5)
→ Ty5
nat5 : Ty5 = λ _ nat5 _ _ _ _ _. nat5
top5 : Ty5 = λ _ _ top5 _ _ _ _. top5
bot5 : Ty5 = λ _ _ _ bot5 _ _ _. bot5
arr5 : Ty5 → Ty5 → Ty5
= λ A B Ty5 nat5 top5 bot5 arr5 prod sum.
arr5 (A Ty5 nat5 top5 bot5 arr5 prod sum) (B Ty5 nat5 top5 bot5 arr5 prod sum)
prod5 : Ty5 → Ty5 → Ty5
= λ A B Ty5 nat5 top5 bot5 arr5 prod5 sum.
prod5 (A Ty5 nat5 top5 bot5 arr5 prod5 sum) (B Ty5 nat5 top5 bot5 arr5 prod5 sum)
sum5 : Ty5 → Ty5 → Ty5
= λ A B Ty5 nat5 top5 bot5 arr5 prod5 sum5.
sum5 (A Ty5 nat5 top5 bot5 arr5 prod5 sum5) (B Ty5 nat5 top5 bot5 arr5 prod5 sum5)
Con5 : U
= (Con5 : U)
(nil : Con5)
(snoc : Con5 → Ty5 → Con5)
→ Con5
nil5 : Con5
= λ Con5 nil5 snoc. nil5
snoc5 : Con5 → Ty5 → Con5
= λ Γ A Con5 nil5 snoc5. snoc5 (Γ Con5 nil5 snoc5) A
Var5 : Con5 → Ty5 → U
= λ Γ A.
(Var5 : Con5 → Ty5 → U)
(vz : {Γ A} → Var5 (snoc5 Γ A) A)
(vs : {Γ B A} → Var5 Γ A → Var5 (snoc5 Γ B) A)
→ Var5 Γ A
vz5 : {Γ A} → Var5 (snoc5 Γ A) A
= λ Var5 vz5 vs. vz5
vs5 : {Γ B A} → Var5 Γ A → Var5 (snoc5 Γ B) A
= λ x Var5 vz5 vs5. vs5 (x Var5 vz5 vs5)
Tm5 : Con5 → Ty5 → U
= λ Γ A.
(Tm5 : Con5 → Ty5 → U)
(var : {Γ A} → Var5 Γ A → Tm5 Γ A)
(lam : {Γ A B} → Tm5 (snoc5 Γ A) B → Tm5 Γ (arr5 A B))
(app : {Γ A B} → Tm5 Γ (arr5 A B) → Tm5 Γ A → Tm5 Γ B)
(tt : {Γ} → Tm5 Γ top5)
(pair : {Γ A B} → Tm5 Γ A → Tm5 Γ B → Tm5 Γ (prod5 A B))
(fst : {Γ A B} → Tm5 Γ (prod5 A B) → Tm5 Γ A)
(snd : {Γ A B} → Tm5 Γ (prod5 A B) → Tm5 Γ B)
(left : {Γ A B} → Tm5 Γ A → Tm5 Γ (sum5 A B))
(right : {Γ A B} → Tm5 Γ B → Tm5 Γ (sum5 A B))
(case : {Γ A B C} → Tm5 Γ (sum5 A B) → Tm5 Γ (arr5 A C) → Tm5 Γ (arr5 B C) → Tm5 Γ C)
(zero : {Γ} → Tm5 Γ nat5)
(suc : {Γ} → Tm5 Γ nat5 → Tm5 Γ nat5)
(rec : {Γ A} → Tm5 Γ nat5 → Tm5 Γ (arr5 nat5 (arr5 A A)) → Tm5 Γ A → Tm5 Γ A)
→ Tm5 Γ A
var5 : {Γ A} → Var5 Γ A → Tm5 Γ A
= λ x Tm5 var5 lam app tt pair fst snd left right case zero suc rec.
var5 x
lam5 : {Γ A B} → Tm5 (snoc5 Γ A) B → Tm5 Γ (arr5 A B)
= λ t Tm5 var5 lam5 app tt pair fst snd left right case zero suc rec.
lam5 (t Tm5 var5 lam5 app tt pair fst snd left right case zero suc rec)
app5 : {Γ A B} → Tm5 Γ (arr5 A B) → Tm5 Γ A → Tm5 Γ B
= λ t u Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec.
app5 (t Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec)
(u Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec)
tt5 : {Γ} → Tm5 Γ top5
= λ Tm5 var5 lam5 app5 tt5 pair fst snd left right case zero suc rec. tt5
pair5 : {Γ A B} → Tm5 Γ A → Tm5 Γ B → Tm5 Γ (prod5 A B)
= λ t u Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec.
pair5 (t Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec)
(u Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec)
fst5 : {Γ A B} → Tm5 Γ (prod5 A B) → Tm5 Γ A
= λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd left right case zero suc rec.
fst5 (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd left right case zero suc rec)
snd5 : {Γ A B} → Tm5 Γ (prod5 A B) → Tm5 Γ B
= λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left right case zero suc rec.
snd5 (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left right case zero suc rec)
left5 : {Γ A B} → Tm5 Γ A → Tm5 Γ (sum5 A B)
= λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right case zero suc rec.
left5 (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right case zero suc rec)
right5 : {Γ A B} → Tm5 Γ B → Tm5 Γ (sum5 A B)
= λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case zero suc rec.
right5 (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case zero suc rec)
case5 : {Γ A B C} → Tm5 Γ (sum5 A B) → Tm5 Γ (arr5 A C) → Tm5 Γ (arr5 B C) → Tm5 Γ C
= λ t u v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec.
case5 (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
(u Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
(v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
zero5 : {Γ} → Tm5 Γ nat5
= λ Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc rec. zero5
suc5 : {Γ} → Tm5 Γ nat5 → Tm5 Γ nat5
= λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec.
suc5 (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec)
rec5 : {Γ A} → Tm5 Γ nat5 → Tm5 Γ (arr5 nat5 (arr5 A A)) → Tm5 Γ A → Tm5 Γ A
= λ t u v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5.
rec5 (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
(u Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
(v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
v05 : {Γ A} → Tm5 (snoc5 Γ A) A
= var5 vz5
v15 : {Γ A B} → Tm5 (snoc5 (snoc5 Γ A) B) A
= var5 (vs5 vz5)
v25 : {Γ A B C} → Tm5 (snoc5 (snoc5 (snoc5 Γ A) B) C) A
= var5 (vs5 (vs5 vz5))
v35 : {Γ A B C D} → Tm5 (snoc5 (snoc5 (snoc5 (snoc5 Γ A) B) C) D) A
= var5 (vs5 (vs5 (vs5 vz5)))
tbool5 : Ty5
= sum5 top5 top5
true5 : {Γ} → Tm5 Γ tbool5
= left5 tt5
tfalse5 : {Γ} → Tm5 Γ tbool5
= right5 tt5
ifthenelse5 : {Γ A} → Tm5 Γ (arr5 tbool5 (arr5 A (arr5 A A)))
= lam5 (lam5 (lam5 (case5 v25 (lam5 v25) (lam5 v15))))
times45 : {Γ A} → Tm5 Γ (arr5 (arr5 A A) (arr5 A A))
= lam5 (lam5 (app5 v15 (app5 v15 (app5 v15 (app5 v15 v05)))))
add5 : {Γ} → Tm5 Γ (arr5 nat5 (arr5 nat5 nat5))
= lam5 (rec5 v05
(lam5 (lam5 (lam5 (suc5 (app5 v15 v05)))))
(lam5 v05))
mul5 : {Γ} → Tm5 Γ (arr5 nat5 (arr5 nat5 nat5))
= lam5 (rec5 v05
(lam5 (lam5 (lam5 (app5 (app5 add5 (app5 v15 v05)) v05))))
(lam5 zero5))
fact5 : {Γ} → Tm5 Γ (arr5 nat5 nat5)
= lam5 (rec5 v05 (lam5 (lam5 (app5 (app5 mul5 (suc5 v15)) v05)))
(suc5 zero5))
Ty6 : U
= (Ty6 : U)
(nat top bot : Ty6)
(arr prod sum : Ty6 → Ty6 → Ty6)
→ Ty6
nat6 : Ty6 = λ _ nat6 _ _ _ _ _. nat6
top6 : Ty6 = λ _ _ top6 _ _ _ _. top6
bot6 : Ty6 = λ _ _ _ bot6 _ _ _. bot6
arr6 : Ty6 → Ty6 → Ty6
= λ A B Ty6 nat6 top6 bot6 arr6 prod sum.
arr6 (A Ty6 nat6 top6 bot6 arr6 prod sum) (B Ty6 nat6 top6 bot6 arr6 prod sum)
prod6 : Ty6 → Ty6 → Ty6
= λ A B Ty6 nat6 top6 bot6 arr6 prod6 sum.
prod6 (A Ty6 nat6 top6 bot6 arr6 prod6 sum) (B Ty6 nat6 top6 bot6 arr6 prod6 sum)
sum6 : Ty6 → Ty6 → Ty6
= λ A B Ty6 nat6 top6 bot6 arr6 prod6 sum6.
sum6 (A Ty6 nat6 top6 bot6 arr6 prod6 sum6) (B Ty6 nat6 top6 bot6 arr6 prod6 sum6)
Con6 : U
= (Con6 : U)
(nil : Con6)
(snoc : Con6 → Ty6 → Con6)
→ Con6
nil6 : Con6
= λ Con6 nil6 snoc. nil6
snoc6 : Con6 → Ty6 → Con6
= λ Γ A Con6 nil6 snoc6. snoc6 (Γ Con6 nil6 snoc6) A