-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathHomework.html
More file actions
1199 lines (952 loc) · 91.1 KB
/
Homework.html
File metadata and controls
1199 lines (952 loc) · 91.1 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!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=utf-8" />
<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<title></title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<div class="code">
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="var">HoTT</span> <span class="id" title="var">Join</span> <span class="id" title="var">UnivalenceAxiom</span>.<br/>
<br/>
<br/>
</div>
<div class="doc">
<i>Note</i>: we will use the <span class="inlinecode"><span class="id" title="var">Admitted</span></span> command whenever we do not
give the term for a certain type. We then associate the
following meanings to assertion commands:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="keyword">Fact</span></span> denotes a fact proven in the course or the HoTT book
</li>
<li> <span class="inlinecode"><span class="id" title="var">Proposition</span></span> denotes either a proposition assumed to be true
in the subject, or a partially answered optional question
</li>
<li> <span class="inlinecode"><span class="id" title="keyword">Lemma</span></span>/<span class="inlinecode"><span class="id" title="keyword">Theorem</span></span> denote a partially answered mandatory question
</li>
</ul>
<div class="paragraph"> </div>
We also follow the convention to name <span class="inlinecode"><span class="id" title="var">q_i_j</span></span> the answer to
question j of exercise i.
<div class="paragraph"> </div>
Whenever we use a lemma implemented in the HoTT library,
we will be careful to state which lemma of the course/HoTT book
it corresponds to.
<div class="paragraph"> </div>
<a name="lab1"></a><h1 class="section">Prelude</h1>
<div class="paragraph"> </div>
Notations to match more closely those of the course/subject.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Notation</span> "'ua'" := <span class="id" title="var">equiv_path</span>.<br/>
<span class="id" title="keyword">Notation</span> "'ua^-1'" := <span class="id" title="var">path_universe</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> "2" := <span class="id" title="var">Bool</span>.<br/>
<span class="id" title="keyword">Notation</span> "'Σ' A" := (<span class="id" title="var">Susp</span> <span class="id" title="var">A</span>) (<span class="id" title="var">format</span> "'Σ' A", <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10).<br/>
<span class="id" title="keyword">Notation</span> "'S^' n" := (<span class="id" title="var">Sphere</span> <span class="id" title="var">n</span>) (<span class="id" title="var">format</span> "'S^' n", <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10).<br/>
<span class="id" title="keyword">Notation</span> "'S*^' n" := (<span class="id" title="var">psphere</span> <span class="id" title="var">n</span>) (<span class="id" title="var">format</span> "'S*^' n", <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10).<br/>
<span class="id" title="keyword">Infix</span> "×" := <span class="id" title="var">prod</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 30).<br/>
<span class="id" title="keyword">Infix</span> "∗" := <span class="id" title="var">Join</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 30).<br/>
<span class="id" title="keyword">Infix</span> "≃" := <span class="id" title="var">Equiv</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50).<br/>
<span class="id" title="keyword">Infix</span> "≃*" := <span class="id" title="var">pEquiv</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50).<br/>
<span class="id" title="keyword">Notation</span> "| A |" := (<span class="id" title="var">merely</span> <span class="id" title="var">A</span>) (<span class="id" title="var">format</span> "| A |", <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70).<br/>
<br/>
</div>
<div class="doc">
<a name="lab2"></a><h3 class="section">Spans</h3>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Inductive</span> <span class="id" title="var">Span</span> :=<br/>
| <span class="id" title="var">span</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="keyword">Type</span>} (<span class="id" title="var">f</span> : <span class="id" title="var">B</span> → <span class="id" title="var">A</span>) (<span class="id" title="var">g</span> : <span class="id" title="var">B</span> → <span class="id" title="var">C</span>).<br/>
<br/>
<span class="id" title="keyword">Record</span> <span class="id" title="var">EqSpan'</span><br/>
{<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="keyword">Type</span>} (<span class="id" title="var">f</span> : <span class="id" title="var">B</span> → <span class="id" title="var">A</span>) (<span class="id" title="var">g</span> : <span class="id" title="var">B</span> → <span class="id" title="var">C</span>) <br/>
{<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="keyword">Type</span>} (<span class="id" title="var">f'</span> : <span class="id" title="var">B'</span> → <span class="id" title="var">A'</span>) (<span class="id" title="var">g'</span> : <span class="id" title="var">B'</span> → <span class="id" title="var">C'</span>)<br/>
:= {<br/>
<span class="id" title="var">equiv_left</span> : <span class="id" title="var">A</span> ≃ <span class="id" title="var">A'</span>;<br/>
<span class="id" title="var">equiv_apex</span> : <span class="id" title="var">B</span> ≃ <span class="id" title="var">B'</span>;<br/>
<span class="id" title="var">equiv_right</span> : <span class="id" title="var">C</span> ≃ <span class="id" title="var">C'</span>;<br/>
<span class="id" title="var">coherence_left</span> : <span class="id" title="var">equiv_left</span> <span class="id" title="var">o</span> <span class="id" title="var">f</span> == <span class="id" title="var">f'</span> <span class="id" title="var">o</span> <span class="id" title="var">equiv_apex</span>;<br/>
<span class="id" title="var">coherence_right</span> : <span class="id" title="var">equiv_right</span> <span class="id" title="var">o</span> <span class="id" title="var">g</span> == <span class="id" title="var">g'</span> <span class="id" title="var">o</span> <span class="id" title="var">equiv_apex</span>;<br/>
}.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">EqSpan</span> '(<span class="id" title="var">span</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">f</span> <span class="id" title="var">g</span>) '(<span class="id" title="var">span</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">f'</span> <span class="id" title="var">g'</span>) :=<br/>
<span class="id" title="var">EqSpan'</span> <span class="id" title="var">f</span> <span class="id" title="var">g</span> <span class="id" title="var">f'</span> <span class="id" title="var">g'</span>.<br/>
<br/>
<span class="id" title="keyword">Proposition</span> <span class="id" title="var">path_span</span> {<span class="id" title="var">s</span> <span class="id" title="var">s'</span>} :<br/>
<span class="id" title="var">EqSpan</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> ≃ (<span class="id" title="var">s</span> = <span class="id" title="var">s'</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab3"></a><h1 class="section">Exercise 1</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Exercise_1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab4"></a><h2 class="section">Question 1</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
<span class="id" title="keyword">Context</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>).<br/>
<br/>
<span class="id" title="keyword">Let</span> φ : 2 ∗ <span class="id" title="var">A</span> → <span class="id" title="var">Σ</span> <span class="id" title="var">A</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* By recursion on joins, that is on pushouts *)</span><br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Join_rec</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="comment">(* φ(left(b)) :≡ … *)</span><br/>
* <span class="id" title="tactic">exact</span> (<br/>
<span class="id" title="keyword">fun</span> <span class="id" title="var">b</span> ⇒ <span class="id" title="keyword">match</span> <span class="id" title="var">b</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">false</span> ⇒ <span class="id" title="var">North</span><br/>
| <span class="id" title="var">true</span> ⇒ <span class="id" title="var">South</span><br/>
<span class="id" title="keyword">end</span>).<br/>
<span class="comment">(* φ(right(a)) :≡ … *)</span><br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <span class="id" title="var">South</span>).<br/>
<span class="comment">(* ap_φ(quo(b,a)) :≡ … *)</span><br/>
* <span class="id" title="tactic">intros</span> <span class="id" title="var">b</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">simpl</span>.<br/>
<span class="comment">(* By recursion/case on b *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">b</span>.<br/>
- <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>. <span class="comment">(* <span class="inlinecode"><span class="id" title="var">idpath</span></span> is refl *)</span><br/>
- <span class="id" title="tactic">exact</span> (<span class="id" title="var">merid</span> <span class="id" title="var">a</span>).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> ψ : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> → 2 ∗ <span class="id" title="var">A</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* By recursion on suspensions *)</span><br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Susp_rec</span>.<br/>
<span class="comment">(* ψ(N) :≡ … *)</span><br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">joinl</span> <span class="id" title="var">false</span>).<br/>
<span class="comment">(* ψ(S) :≡ … *)</span><br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">joinl</span> <span class="id" title="var">true</span>).<br/>
<span class="comment">(* ψ(merid_a) :≡ … *)</span><br/>
* <span class="comment">(* <span class="inlinecode"><span class="id" title="var">jglue</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> is quo(b,a) *)</span><br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">exact</span> ((<span class="id" title="var">jglue</span> <span class="id" title="var">false</span> <span class="id" title="var">a</span>) @ (<span class="id" title="var">jglue</span> <span class="id" title="var">true</span> <span class="id" title="var">a</span>)^).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
For this first proof, we will comment every step in order to
document how "informal" reasoning translates into tactics and lemmas
of the HoTT library. In subsequent proofs, we will only document
parts that use new tactics/lemmas.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_1_1</span> :<br/>
2 ∗ <span class="id" title="var">A</span> ≃ <span class="id" title="var">Σ</span> <span class="id" title="var">A</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
<span class="comment">(* We choose φ as the underlying map of the equivalence *)</span><br/>
* <span class="id" title="tactic">exact</span> φ.<br/>
<span class="comment">(* We show that φ is indeed an equivalence, with ψ as inverse *)</span><br/>
* <span class="id" title="tactic">apply</span> <span class="id" title="var">isequiv_adjointify</span> <span class="id" title="keyword">with</span> ψ.<br/>
<span class="comment">(* φ ○ ψ ~ id *)</span><br/>
- <span class="id" title="tactic">unfold</span> <span class="id" title="var">Sect</span>.<br/>
<span class="comment">(* By induction on suspensions *)</span><br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Susp_ind</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="comment">(* φ(ψ(N) = N *)</span><br/>
+ <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="comment">(* φ(ψ(S) = S *)</span><br/>
+ <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="comment">(* tr_{merid x}(refl) = refl *)</span><br/>
+ <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>.<br/>
<span class="comment">(* We use an instance of theorem 2.11.3 of the HoTT book *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">transport_paths_FFlr</span>.<br/>
<span class="comment">(* Compute with the definition of ψ *)</span><br/>
<span class="id" title="tactic">unfold</span> ψ. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Susp_rec_beta_merid</span>.<br/>
<span class="comment">(* Lemma 7 and 8 of the course *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">ap_pp</span>, <span class="id" title="var">ap_V</span>.<br/>
<span class="comment">(* Compute with the definition of φ *)</span><br/>
<span class="id" title="tactic">unfold</span> φ. <span class="id" title="tactic">rewrite</span> 2 <span class="id" title="var">Join_rec_beta_jglue</span>.<br/>
<span class="comment">(* The <span class="inlinecode"><span class="id" title="var">hott_simpl</span></span> tactic tries to simplify paths using<br/>
basic homotopies, such as composition with identity and<br/>
inverse paths *)</span><br/>
<span class="id" title="var">hott_simpl</span>.<br/>
<span class="comment">(* ψ ○ φ ~ id *)</span><br/>
- <span class="id" title="tactic">unfold</span> <span class="id" title="var">Sect</span>.<br/>
<span class="comment">(* By induction on joins *)</span><br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Join_ind</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="comment">(* ψ(φ(left(b))) = left(b) *)</span><br/>
+ <span class="id" title="tactic">intros</span> <span class="id" title="var">b</span>.<br/>
<span class="comment">(* By case on b *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">b</span>; <span class="id" title="tactic">simpl</span>.<br/>
** <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
** <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="comment">(* ψ(φ(right(a))) = right(a) *)</span><br/>
+ <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">exact</span> (<span class="id" title="var">jglue</span> <span class="id" title="var">true</span> <span class="id" title="var">a</span>).<br/>
<span class="comment">(* tr_{quo(b,a)}(refl) = quo(1,a) *)</span><br/>
+ <span class="id" title="tactic">intros</span> <span class="id" title="var">b</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">dp_path_transport</span>.<br/>
<span class="comment">(* Again, same instance of theorem 2.11.3 of the HoTT book *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">transport_paths_FFlr</span>.<br/>
<span class="comment">(* Compute with the definition of φ *)</span><br/>
<span class="id" title="tactic">unfold</span> φ. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Join_rec_beta_jglue</span>.<br/>
<span class="comment">(* By case on b *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">b</span>.<br/>
<span class="comment">(* tr_{quo(1,a)}(refl) = quo(1,a) *)</span><br/>
** <span class="id" title="var">hott_simpl</span>.<br/>
<span class="comment">(* tr_{quo(0,a)}(refl) = quo(1,a) *)</span><br/>
** <span class="comment">(* Compute with the definition of ψ *)</span><br/>
<span class="id" title="tactic">unfold</span> ψ. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Susp_rec_beta_merid</span>.<br/>
<span class="comment">(* Unwhiskering on the right with quo(1,a), that is<br/>
post-composing with quo(1,a) on both sides *)</span><br/>
<span class="id" title="tactic">apply</span> (<span class="id" title="var">cancelR</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> (<span class="id" title="var">jglue</span> <span class="id" title="var">true</span> <span class="id" title="var">a</span>)^).<br/>
<span class="id" title="var">hott_simpl</span>.<br/>
<span class="comment">(* Associativity of composition *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">concat_pp_p</span>.<br/>
<span class="id" title="var">hott_simpl</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab5"></a><h2 class="section">Question 2</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_2</span>.<br/>
<br/>
<span class="id" title="keyword">Proposition</span> <span class="id" title="var">q_1_2</span> : <span class="id" title="keyword">∀</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">A</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">C</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_2</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab6"></a><h2 class="section">Question 3</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
<span class="id" title="keyword">Fact</span> <span class="id" title="var">Bool_Sphere_0</span> :<br/>
2 ≃ <span class="id" title="var">S</span>^0.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
We need functoriality of type formers with respect to equivalence,
in order to apply equivalences to their arguments. We could prove
it individually for each type former of interest, but we choose
instead to use univalence to prove it once and forall.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">ae</span> {<span class="id" title="var">X</span> <span class="id" title="var">Y</span>} (<span class="id" title="var">T</span> : <span class="id" title="keyword">Type</span> → <span class="id" title="keyword">Type</span>) :<br/>
<span class="id" title="var">X</span> ≃ <span class="id" title="var">Y</span> → <span class="id" title="var">T</span> <span class="id" title="var">X</span> ≃ <span class="id" title="var">T</span> <span class="id" title="var">Y</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="var">path_universe_uncurried</span></span> is the map given by univalence *)</span><br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">path_universe_uncurried</span> <span class="id" title="tactic">in</span> <span class="id" title="var">e</span>.<br/>
<span class="comment">(* The <span class="inlinecode"><span class="id" title="var">path_induction</span></span> tactic applies path induction to every<br/>
path found in the context *)</span><br/>
<span class="id" title="var">path_induction</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="var">equiv_idmap</span></span> is the identity equivalence *)</span><br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" title="tactic">rewrite</span></span> tactic of the HoTT library is quite weak since
it does not support equivalences, and does not allow selecting
the occurrences of the term we want to rewrite. Thus we will
compose equivalences manually using the <span class="inlinecode"><span class="id" title="tactic">refine</span></span> tactic and
the <span class="inlinecode"><span class="id" title="var">oE</span></span> composition operator.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">equiv_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_1_3</span> :<br/>
<span class="id" title="var">S</span>^1 ∗ <span class="id" title="var">S</span>^1 ≃ <span class="id" title="var">S</span>^3.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* Rewrite on the left of ∗ using functoriality *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">ae</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">X</span> ⇒ <span class="id" title="var">X</span> ∗ <span class="id" title="var">_</span>) <span class="id" title="var">_</span>).<br/>
2: { <span class="comment">(* Rewrite under Σ using functoriality (since S^n is the<br/>
definition of the sphere based on iterated suspensions) *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">ae</span> <span class="id" title="var">Susp</span> <span class="id" title="var">_</span>).<br/>
<span class="comment">(* Rewrite S^0 into 2 *)</span><br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">Bool_Sphere_0</span>^-1. }<br/>
<span class="comment">(* Rewrite on the left of ∗ using functoriality *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">ae</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">X</span> ⇒ <span class="id" title="var">X</span> ∗ <span class="id" title="var">_</span>) <span class="id" title="var">_</span>).<br/>
2: { <span class="comment">(* Rewrite Σ2 into 2 ∗ 2 using <span class="inlinecode"><span class="id" title="var">q_1_1</span></span> *)</span><br/>
<span class="id" title="tactic">exact</span> (<span class="id" title="var">q_1_1</span> <span class="id" title="var">_</span>)^-1. }<br/>
<span class="comment">(* Associativity of ∗ *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">q_1_2</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>).<br/>
<span class="comment">(* Rewrite on the right of ∗ using functoriality *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">ae</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">X</span> ⇒ <span class="id" title="var">_</span> ∗ <span class="id" title="var">X</span>) <span class="id" title="var">_</span>).<br/>
2: { <span class="comment">(* Rewrite 2 ∗ S^1 into ΣS^1 using <span class="inlinecode"><span class="id" title="var">q_1_1</span></span> *)</span><br/>
<span class="id" title="tactic">exact</span> (<span class="id" title="var">q_1_1</span> <span class="id" title="var">_</span>). }<br/>
<span class="comment">(* Rewrite 2 ∗ S^1 into ΣΣS^1 using <span class="inlinecode"><span class="id" title="var">q_1_1</span></span> *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">q_1_1</span> <span class="id" title="var">_</span>).<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Close</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">equiv_scope</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Exercise_1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab7"></a><h1 class="section">Exercise 2</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Exercise_2</span>.<br/>
<br/>
<span class="id" title="keyword">Context</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">P</span> : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> → <span class="id" title="keyword">Type</span>).<br/>
<br/>
</div>
<div class="doc">
<a name="lab8"></a><h2 class="section">Question 1</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> ψ '(<span class="id" title="var">q</span>, <span class="id" title="var">a</span>) := <span class="id" title="var">transport</span> <span class="id" title="var">P</span> (<span class="id" title="var">merid</span> <span class="id" title="var">a</span>) <span class="id" title="var">q</span>.<br/>
<br/>
<span class="id" title="keyword">Proposition</span> <span class="id" title="var">q_2_1</span> :<br/>
{<span class="id" title="var">x</span> : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> & <span class="id" title="var">P</span> <span class="id" title="var">x</span>} ≃ <span class="id" title="var">Pushout</span> <span class="id" title="var">fst</span> ψ.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Exercise_2</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab9"></a><h1 class="section">Exercise 3</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Exercise_3</span>.<br/>
<br/>
<span class="id" title="keyword">Context</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">μ</span> : <span class="id" title="var">A</span> × <span class="id" title="var">A</span> → <span class="id" title="var">A</span>).<br/>
<br/>
<span class="id" title="keyword">Notation</span> "'μ'" := <span class="id" title="var">μ</span>.<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( a , b )" := (<span class="id" title="var">μ</span> (<span class="id" title="var">pair</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>)) (<span class="id" title="var">format</span> "'μ' ( a , b )").<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( a , '_' )" := (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">x</span>)) (<span class="id" title="var">format</span> "'μ' ( a , _ )").<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( '_' , a )" := (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">μ</span>(<span class="id" title="var">x</span>,<span class="id" title="var">a</span>)) (<span class="id" title="var">format</span> "'μ' ( _ , a )").<br/>
<br/>
<span class="id" title="keyword">Context</span> `(<span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <span class="id" title="var">IsEquiv</span> <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">_</span>)).<br/>
<span class="id" title="keyword">Context</span> `(<span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <span class="id" title="var">IsEquiv</span> <span class="id" title="var">μ</span>(<span class="id" title="var">_</span>,<span class="id" title="var">a</span>)).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">P</span> : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> → <span class="id" title="keyword">Type</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Susp_rec</span>; <span class="id" title="tactic">simpl</span>.<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">A</span>.<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">A</span>.<br/>
* <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">exact</span> (<span class="id" title="var">ua</span>^-1 <span class="id" title="var">μ</span>(<span class="id" title="var">_</span>,<span class="id" title="var">a</span>)).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab10"></a><h2 class="section">Question 1</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
<span class="id" title="keyword">Fact</span> <span class="id" title="var">hottbook_lemma_2_10_5</span> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">B</span> : <span class="id" title="var">A</span> → <span class="id" title="keyword">Type</span>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <span class="id" title="var">A</span>) (<span class="id" title="var">p</span> : <span class="id" title="var">x</span> = <span class="id" title="var">y</span>) (<span class="id" title="var">u</span> : <span class="id" title="var">B</span> <span class="id" title="var">x</span>),<br/>
<span class="id" title="var">transport</span> <span class="id" title="var">B</span> <span class="id" title="var">p</span> <span class="id" title="var">u</span> = <span class="id" title="var">ua</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> (<span class="id" title="var">ap</span> <span class="id" title="var">B</span> <span class="id" title="var">p</span>) <span class="id" title="var">u</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_3_1</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <span class="id" title="var">A</span>,<br/>
<span class="id" title="var">transport</span> <span class="id" title="var">P</span> (<span class="id" title="var">merid</span> <span class="id" title="var">b</span>) <span class="id" title="var">a</span> = <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">b</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">hottbook_lemma_2_10_5</span>. <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Susp_rec_beta_merid</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="var">transport_path_universe</span></span> is the propositional computation<br/>
rule of univalence (see section 2.10 of the HoTT book) *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">transport_path_universe</span>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab11"></a><h2 class="section">Question 2</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_2</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> ψ '(<span class="id" title="var">q</span>, <span class="id" title="var">a</span>) := <span class="id" title="var">transport</span> <span class="id" title="var">P</span> (<span class="id" title="var">merid</span> <span class="id" title="var">a</span>) <span class="id" title="var">q</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">path_</span>ψ<span class="id" title="var">_μ</span> :<br/>
ψ = <span class="id" title="var">μ</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="var">equiv_path_arrow</span></span> is function extensionality *)</span><br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">equiv_path_arrow</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">p</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">p</span> <span class="id" title="keyword">as</span> (<span class="id" title="var">a</span>, <span class="id" title="var">b</span>).<br/>
<span class="id" title="tactic">unfold</span> ψ. <span class="id" title="tactic">apply</span> <span class="id" title="var">q_3_1</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_3_2</span> :<br/>
{<span class="id" title="var">x</span> : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> & <span class="id" title="var">P</span> <span class="id" title="var">x</span>} ≃ <span class="id" title="var">Pushout</span> <span class="id" title="var">fst</span> <span class="id" title="var">μ</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> (<span class="id" title="var">q_2_1</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">path_</span>ψ<span class="id" title="var">_μ</span>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_2</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab12"></a><h2 class="section">Question 3</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Let</span> <span class="id" title="var">s</span> := @<span class="id" title="var">span</span> <span class="id" title="var">A</span> (<span class="id" title="var">A</span> × <span class="id" title="var">A</span>) <span class="id" title="var">A</span> <span class="id" title="var">fst</span> <span class="id" title="var">μ</span>.<br/>
<span class="id" title="keyword">Let</span> <span class="id" title="var">s'</span> := @<span class="id" title="var">span</span> <span class="id" title="var">A</span> (<span class="id" title="var">A</span> × <span class="id" title="var">A</span>) <span class="id" title="var">A</span> <span class="id" title="var">fst</span> <span class="id" title="var">snd</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> <span class="id" title="var">ε</span> : <span class="id" title="var">A</span> × <span class="id" title="var">A</span> → <span class="id" title="var">A</span> × <span class="id" title="var">A</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">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">_</span>) <span class="id" title="var">b</span>).<br/>
<span class="id" title="keyword">Let</span> <span class="id" title="var">ε_inv</span> : <span class="id" title="var">A</span> × <span class="id" title="var">A</span> → <span class="id" title="var">A</span> × <span class="id" title="var">A</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">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">_</span>)^-1 <span class="id" title="var">b</span>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_3_3</span> :<br/>
<span class="id" title="var">s</span> = <span class="id" title="var">s'</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* We use the analysis of equality between spans *)</span><br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">path_span</span>. <span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
<span class="comment">(* Left equivalence *)</span><br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="comment">(* Apex equivalence *)</span><br/>
* <span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
- <span class="id" title="tactic">exact</span> <span class="id" title="var">ε</span>.<br/>
- <span class="id" title="tactic">apply</span> <span class="id" title="var">isequiv_adjointify</span> <span class="id" title="keyword">with</span> <span class="id" title="var">ε_inv</span>.<br/>
+ <span class="id" title="tactic">unfold</span> <span class="id" title="var">Sect</span>. <span class="id" title="tactic">intros</span> [<span class="id" title="var">a</span> <span class="id" title="var">b</span>]. <span class="id" title="tactic">unfold</span> <span class="id" title="var">ε</span>, <span class="id" title="var">ε_inv</span>.<br/>
<span class="comment">(* μ(a,_) is an equivalence *)</span><br/>
<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">eisretr</span> <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">_</span>)).<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
+ <span class="id" title="tactic">unfold</span> <span class="id" title="var">Sect</span>. <span class="id" title="tactic">intros</span> [<span class="id" title="var">a</span> <span class="id" title="var">b</span>]. <span class="id" title="tactic">unfold</span> <span class="id" title="var">ε</span>, <span class="id" title="var">ε_inv</span>.<br/>
<span class="comment">(* μ(a,_) is an equivalence *)</span><br/>
<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">eissect</span> <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">_</span>)).<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="comment">(* Right equivalence *)</span><br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="comment">(* Left coherence *)</span><br/>
* <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">equiv_path_arrow</span>. <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="comment">(* Right coherence *)</span><br/>
* <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">equiv_path_arrow</span>. <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab13"></a><h2 class="section">Question 4</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_4</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_3_4</span> :<br/>
{<span class="id" title="var">x</span> : <span class="id" title="var">Σ</span> <span class="id" title="var">A</span> & <span class="id" title="var">P</span> <span class="id" title="var">x</span>} ≃ <span class="id" title="var">A</span> ∗ <span class="id" title="var">A</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">q_3_2</span>).<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="var">equiv_pushout</span></span> gives an analysis of equivalence between pushouts<br/>
as defined in the HoTT library, which will allow us to reduce it<br/>
to the analysis of equality between spans of <span class="inlinecode"><span class="id" title="var">q_3_3</span></span> *)</span><br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> <span class="id" title="var">oE</span> <span class="id" title="var">equiv_pushout</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/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">equiv_idmap</span>.<br/>
<span class="comment">(* Left coherence... *)</span><br/>
* <span class="id" title="tactic">eapply</span> <span class="id" title="var">coherence_left</span>.<br/>
<span class="comment">(* ...and right coherence... *)</span><br/>
* <span class="id" title="tactic">eapply</span> <span class="id" title="var">coherence_right</span>.<br/>
<span class="id" title="var">Unshelve</span>.<br/>
<span class="comment">(* ...that we get from <span class="inlinecode"><span class="id" title="var">q_3_3</span></span> *)</span><br/>
<span class="id" title="tactic">apply</span> (<span class="id" title="var">path_span</span>^-1 <span class="id" title="var">q_3_3</span>).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_4</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Exercise_3</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab14"></a><h1 class="section">Exercise 4</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Exercise_4</span>.<br/>
<br/>
<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">pointed_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Class</span> <span class="id" title="var">IsHType</span> (<span class="id" title="var">carrier</span> : <span class="id" title="var">pType</span>) := {<br/>
<span class="id" title="var">mult</span> : <span class="id" title="var">carrier</span> × <span class="id" title="var">carrier</span> → <span class="id" title="var">carrier</span>;<br/>
<span class="id" title="var">unit_left</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <span class="id" title="var">mult</span> (<span class="id" title="var">point</span> <span class="id" title="var">carrier</span>, <span class="id" title="var">x</span>) = <span class="id" title="var">x</span>;<br/>
<span class="id" title="var">unit_right</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <span class="id" title="var">mult</span> (<span class="id" title="var">x</span>, <span class="id" title="var">point</span> <span class="id" title="var">carrier</span>) = <span class="id" title="var">x</span>;<br/>
}.<br/>
<br/>
<span class="id" title="keyword">Class</span> <span class="id" title="var">IsConnected</span> <span class="id" title="var">A</span> := <span class="id" title="var">connected</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <span class="id" title="var">A</span>, |<span class="id" title="var">x</span> = <span class="id" title="var">y</span>|.<br/>
<br/>
<span class="id" title="keyword">Context</span> (<span class="id" title="var">A</span> : <span class="id" title="var">pType</span>) `{<span class="id" title="var">IsHType</span> <span class="id" title="var">A</span>} `{<span class="id" title="var">IsConnected</span> <span class="id" title="var">A</span>}.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <span class="id" title="var">e</span> := (<span class="id" title="var">point</span> <span class="id" title="var">A</span>).<br/>
<span class="id" title="keyword">Notation</span> "'μ'" := <span class="id" title="var">mult</span>.<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( a , b )" := (<span class="id" title="var">μ</span> (<span class="id" title="var">pair</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>)) (<span class="id" title="var">format</span> "'μ' ( a , b )").<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( a , '_' )" := (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">x</span>)) (<span class="id" title="var">format</span> "'μ' ( a , _ )").<br/>
<span class="id" title="keyword">Notation</span> "'μ' ( '_' , a )" := (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">μ</span>(<span class="id" title="var">x</span>,<span class="id" title="var">a</span>)) (<span class="id" title="var">format</span> "'μ' ( _ , a )").<br/>
<br/>
</div>
<div class="doc">
<a name="lab15"></a><h2 class="section">Question 1</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
</div>
<div class="doc">
I could not find a proper induction principle for the
propositional truncation in the HoTT library, thus we
assume the one seen in course.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Fact</span> <span class="id" title="var">merely_ind</span> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">P</span> : |<span class="id" title="var">A</span>| → <span class="id" title="keyword">Type</span>),<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <span class="id" title="var">A</span>, <span class="id" title="var">P</span> (<span class="id" title="var">tr</span> <span class="id" title="var">x</span>)) →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : |<span class="id" title="var">A</span>|, <span class="id" title="var">P</span> <span class="id" title="var">x</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
A consequence of the induction principle is that
for any type B, assuming |B| amounts to assuming B.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">destruct_merely</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">B</span>,<br/>
|<span class="id" title="var">B</span>| → <span class="id" title="var">B</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">B</span>. <span class="id" title="tactic">apply</span> (<span class="id" title="var">merely_ind</span> <span class="id" title="var">B</span>). <span class="id" title="tactic">exact</span> <span class="id" title="var">idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_4_1</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <span class="id" title="var">A</span>,<br/>
|<span class="id" title="var">μ</span>(<span class="id" title="var">x</span>,<span class="id" title="var">_</span>) = <span class="id" title="var">idmap</span>| × |<span class="id" title="var">μ</span>(<span class="id" title="var">_</span>,<span class="id" title="var">x</span>) = <span class="id" title="var">idmap</span>|.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">split</span>.<br/>
* <span class="id" title="tactic">apply</span> <span class="id" title="var">tr</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">equiv_path_arrow</span>.<br/>
<span class="id" title="tactic">unfold</span> "==". <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>.<br/>
<span class="id" title="tactic">pose</span> <span class="id" title="var">proof</span> (<span class="id" title="var">connected</span> <span class="id" title="var">μ</span>(<span class="id" title="var">x</span>,<span class="id" title="var">a</span>) <span class="id" title="var">a</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">destruct_merely</span>.<br/>
* <span class="id" title="tactic">apply</span> <span class="id" title="var">tr</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">equiv_path_arrow</span>.<br/>
<span class="id" title="tactic">unfold</span> "==". <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>.<br/>
<span class="id" title="tactic">pose</span> <span class="id" title="var">proof</span> (<span class="id" title="var">connected</span> <span class="id" title="var">μ</span>(<span class="id" title="var">a</span>,<span class="id" title="var">x</span>) <span class="id" title="var">a</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">destruct_merely</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab16"></a><h2 class="section">Question 2</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Instance</span> <span class="id" title="var">isequiv_μ_left</span> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <span class="id" title="var">IsEquiv</span> <span class="id" title="var">μ</span>(<span class="id" title="var">x</span>,<span class="id" title="var">_</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">destruct</span> (<span class="id" title="var">q_4_1</span> <span class="id" title="var">x</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hl</span> <span class="id" title="var">Hr</span>].<br/>
<span class="id" title="tactic">apply</span> (<span class="id" title="var">destruct_merely</span> <span class="id" title="var">_</span>) <span class="id" title="tactic">in</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">isequiv_idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Instance</span> <span class="id" title="var">isequiv_μ_right</span> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <span class="id" title="var">IsEquiv</span> <span class="id" title="var">μ</span>(<span class="id" title="var">_</span>,<span class="id" title="var">x</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">destruct</span> (<span class="id" title="var">q_4_1</span> <span class="id" title="var">x</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hl</span> <span class="id" title="var">Hr</span>].<br/>
<span class="id" title="tactic">apply</span> (<span class="id" title="var">destruct_merely</span> <span class="id" title="var">_</span>) <span class="id" title="tactic">in</span> <span class="id" title="var">Hr</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hr</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">isequiv_idmap</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab17"></a><h2 class="section">Question 3</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
</div>
<div class="doc">
Short name for the equivalence obtained in <span class="inlinecode"><span class="id" title="var">q_3_4</span></span>,
instanciated to our H-type <span class="inlinecode"><span class="id" title="var">A</span></span>.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Let</span> <span class="id" title="var">ε</span> := (<span class="id" title="var">q_3_4</span> <span class="id" title="var">A</span> <span class="id" title="var">μ</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>).<br/>
<br/>
</div>
<div class="doc">
First, we need a few definitions to make our operations
live in the pointed universe.
<div class="paragraph"> </div>
<a name="lab18"></a><h3 class="section">Pointed Σ-types</h3>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">psig</span> {<span class="id" title="var">X</span> : <span class="id" title="var">pType</span>}<br/>
(<span class="id" title="var">P</span> : <span class="id" title="var">X</span> → <span class="id" title="keyword">Type</span>) `{<span class="id" title="var">pointP</span> : <span class="id" title="var">IsPointed</span> (<span class="id" title="var">P</span> (<span class="id" title="var">point</span> <span class="id" title="var">X</span>))} : <span class="id" title="var">pType</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">sig</span> <span class="id" title="var">P</span>).<br/>
* <span class="id" title="tactic">apply</span> <span class="id" title="var">ispointed_sigma</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab19"></a><h3 class="section">Pointed joins</h3>
<div class="paragraph"> </div>
Note that we do not use a general definition for pointed joins:
instead we only define the pointed <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">∗</span> <span class="inlinecode"><span class="id" title="var">A</span></span>, so that we can choose
a specific basepoint (this will allow us to prove ε to be a
pointed equivalence, and not just a "mere" equivalence.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">pjoin</span> : <span class="id" title="var">pType</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">A</span> ∗ <span class="id" title="var">A</span>).<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">ε</span> (<span class="id" title="var">North</span>; <span class="id" title="var">e</span>)).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab20"></a><h3 class="section">Fiber sequences</h3>
<div class="paragraph"> </div>
We also need to define fiber sequences, since they are
not implemented in the HoTT library.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">fib</span> {<span class="id" title="var">X</span> <span class="id" title="var">Y</span> : <span class="id" title="var">pType</span>} (<span class="id" title="var">f</span> : <span class="id" title="var">X</span> ->* <span class="id" title="var">Y</span>) : <span class="id" title="var">pType</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">hfiber</span> <span class="id" title="var">f</span> (<span class="id" title="var">point</span> <span class="id" title="var">Y</span>)).<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">point</span> <span class="id" title="var">X</span>; <span class="id" title="var">point_eq</span> <span class="id" title="var">f</span>).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">fib_proj</span> {<span class="id" title="var">X</span> <span class="id" title="var">Y</span> : <span class="id" title="var">pType</span>} (<span class="id" title="var">f</span> : <span class="id" title="var">X</span> ->* <span class="id" title="var">Y</span>) : <span class="id" title="var">fib</span> <span class="id" title="var">f</span> ->* <span class="id" title="var">X</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
* <span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">x</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">x</span> <span class="id" title="var">_</span>]. <span class="id" title="tactic">exact</span> <span class="id" title="var">x</span>.<br/>
* <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">exact</span> <span class="id" title="var">idpath</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">IsFiberSeq</span> {<span class="id" title="var">X</span> <span class="id" title="var">Y</span> <span class="id" title="var">Z</span> : <span class="id" title="var">pType</span>} (<span class="id" title="var">f</span> : <span class="id" title="var">X</span> ->* <span class="id" title="var">Y</span>) (<span class="id" title="var">g</span> : <span class="id" title="var">Y</span> ->* <span class="id" title="var">Z</span>) :=<br/>
{<span class="id" title="var">p</span> : <span class="id" title="var">X</span> = <span class="id" title="var">fib</span> <span class="id" title="var">g</span> & <span class="id" title="var">transport</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">B</span> ⇒ <span class="id" title="var">B</span> ->* <span class="id" title="var">Y</span>) <span class="id" title="var">p</span> <span class="id" title="var">f</span> = <span class="id" title="var">fib_proj</span> <span class="id" title="var">g</span>}.<br/>
<br/>
<span class="id" title="keyword">Record</span> <span class="id" title="var">FiberSeq</span> (<span class="id" title="var">X</span> <span class="id" title="var">Y</span> <span class="id" title="var">Z</span> : <span class="id" title="var">pType</span>) := {<br/>
<span class="id" title="var">fiberseq_proj</span> : <span class="id" title="var">X</span> ->* <span class="id" title="var">Y</span>;<br/>
<span class="id" title="var">fiberseq_map</span> : <span class="id" title="var">Y</span> ->* <span class="id" title="var">Z</span>;<br/>
<span class="id" title="var">isfiberseq</span> : <span class="id" title="var">IsFiberSeq</span> <span class="id" title="var">fiberseq_proj</span> <span class="id" title="var">fiberseq_map</span>;<br/>
}.<br/>
<br/>
</div>
<div class="doc">
One can make a fiber sequence from a fibration, i.e.
a type family <span class="inlinecode"><span class="id" title="var">C</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="keyword">Type</span></span>, whenever <span class="inlinecode"><span class="id" title="var">X</span></span> is pointed
by <span class="inlinecode">∗</span> and <span class="inlinecode"><span class="id" title="var">C</span>(∗)</span> is pointed as well.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Proposition</span> <span class="id" title="var">fibration_fiberseq</span> {<span class="id" title="var">X</span> : <span class="id" title="var">pType</span>}<br/>
(<span class="id" title="var">C</span> : <span class="id" title="var">X</span> → <span class="id" title="keyword">Type</span>) `{<span class="id" title="var">ispointed_C</span> : <span class="id" title="var">IsPointed</span> (<span class="id" title="var">C</span> (<span class="id" title="var">point</span> <span class="id" title="var">X</span>))} :<br/>
@<span class="id" title="var">FiberSeq</span> {| <span class="id" title="var">pointed_type</span> := <span class="id" title="var">C</span> (<span class="id" title="var">point</span> <span class="id" title="var">X</span>);<br/>
<span class="id" title="var">ispointed_type</span> := <span class="id" title="var">ispointed_C</span> |}<br/>
(<span class="id" title="var">psig</span> <span class="id" title="var">C</span>) <span class="id" title="var">X</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
The trick then is to define the fibration <span class="inlinecode"><span class="id" title="var">C</span></span> so that
it coincides with the <span class="inlinecode"><span class="id" title="var">P</span></span> from exercise 3.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">C</span> : <span class="id" title="var">psusp</span> <span class="id" title="var">A</span> → <span class="id" title="keyword">Type</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">Susp_rec</span>; <span class="id" title="tactic">simpl</span>.<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">A</span>.<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">A</span>.<br/>
* <span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">exact</span> (<span class="id" title="var">ua</span>^-1 <span class="id" title="var">μ</span>(<span class="id" title="var">_</span>,<span class="id" title="var">a</span>)).<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Instance</span> <span class="id" title="var">ispointed_C</span> : <span class="id" title="var">IsPointed</span> (<span class="id" title="var">C</span> (<span class="id" title="var">point</span> (<span class="id" title="var">psusp</span> <span class="id" title="var">A</span>))) := <span class="id" title="var">e</span>.<br/>
<br/>
</div>
<div class="doc">
We use C to get a fiber sequence
<div class="paragraph"> </div>
<span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">→∗</span> <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">ΣA</span>)</span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">C</span>(<span class="id" title="var">x</span>)</span> <span class="inlinecode">→∗</span> <span class="inlinecode"><span class="id" title="var">ΣA</span></span>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Let</span> σ := <span class="id" title="var">fibration_fiberseq</span> <span class="id" title="var">C</span>.<br/>
<br/>
</div>
<div class="doc">
Intuitively, it only remains to "rewrite" <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">ΣA</span>)</span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">C</span>(<span class="id" title="var">x</span>)</span>
into <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">∗</span> <span class="inlinecode"><span class="id" title="var">A</span></span> using ε to conclude. But we cannot do this
directly because we only have an equivalence of non-pointed
types, but sequences are defined on pointed types.
<div class="paragraph"> </div>
So we start by building a pointed equivalence from ε.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">ε_pointed</span> :<br/>
<span class="id" title="var">psig</span> <span class="id" title="var">C</span> ≃* <span class="id" title="var">pjoin</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">unshelve</span> <span class="id" title="tactic">econstructor</span>.<br/>
* <span class="id" title="tactic">exact</span> (<span class="id" title="var">pmap_from_point</span> <span class="id" title="var">ε</span> (<span class="id" title="var">North</span>; <span class="id" title="var">e</span>)).<br/>
* <span class="id" title="tactic">apply</span> <span class="id" title="var">isequiv_adjointify</span> <span class="id" title="keyword">with</span> <span class="id" title="var">ε</span>^-1.<br/>
- <span class="id" title="tactic">apply</span> <span class="id" title="var">eisretr</span>.<br/>
- <span class="id" title="tactic">apply</span> <span class="id" title="var">eissect</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
Then we rewrite, and that's it!
<div class="paragraph"> </div>
In fact we have to cheat a little bit: we use the lemma <span class="inlinecode"><span class="id" title="var">path_ptype</span></span>
to turn the pointed equivalence into a path in the universe of
pointed types, which follows from univalence with a bit of work.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Lemma</span> <span class="id" title="var">q_4_3</span> :<br/>
@<span class="id" title="var">FiberSeq</span> <span class="id" title="var">A</span> (<span class="id" title="var">pjoin</span>) (<span class="id" title="var">psusp</span> <span class="id" title="var">A</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<span class="id" title="var">path_ptype</span> <span class="id" title="var">ε_pointed</span>).<br/>
<span class="id" title="tactic">exact</span> σ.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_3</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Exercise_4</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab21"></a><h1 class="section">Exercise 5</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Exercise_5</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab22"></a><h2 class="section">Question 1</h2>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <span class="id" title="var">Question_1</span>.<br/>
<br/>
</div>
<div class="doc">
<span class="inlinecode"><span class="id" title="var">S1</span></span> is the inductive definition of the circle, not to be confused
with <span class="inlinecode"><span class="id" title="var">S</span>^1</span> which is the definition based on iterated suspension
(though the two definitions are equivalent).
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> ψ' : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <span class="id" title="var">S1</span>, <span class="id" title="var">x</span> = <span class="id" title="var">x</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* By induction on circles *)</span><br/>
<span class="id" title="var">srapply</span> <span class="id" title="var">S1_ind</span>; <span class="id" title="tactic">simpl</span>.<br/>
* <span class="id" title="tactic">exact</span> <span class="id" title="var">loop</span>.<br/>
* <span class="comment">(* Lemma 2.11.2 of the HoTT book *)</span><br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">transport_paths_lr</span>.<br/>
<span class="id" title="var">hott_simpl</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <span class="id" title="var">Question_1</span>.<br/>