-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathBasics.html
2349 lines (1887 loc) · 168 KB
/
Basics.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>Basics: Functional Programming</title>
</head>
<script>
<!--
function toggleDisplay(id)
{
var elt = document.getElementById(id);
if (elt.style.display == 'none') {
elt.style.display = 'block';
} else {
elt.style.display = 'none';
}
}
function hideAll(cls)
{
var testClass = new RegExp("(^|s)" + cls + "(s|$)");
var tag = tag || "*";
var elements = document.getElementsByTagName("div");
var current;
var length = elements.length;
for(var i=0; i<length; i++){
current = elements[i];
if(testClass.test(current.className)) {
current.style.display = 'none';
}
}
}
// -->
</script>
<body onLoad="hideAll('proofscript')">
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Basics<span class="subtitle">Functional Programming</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* $Date: 2012-07-21 13:38:33 -0400 (Sat, 21 Jul 2012) $ *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab12"></a><h1 class="section">Enumerated Types</h1>
<div class="paragraph"> </div>
In Coq's programming language, almost nothing is built
in — not even booleans or numbers! Instead, it provides powerful
tools for defining new types of data and functions that process
and transform them.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab13"></a><h2 class="section">Days of the Week</h2>
<div class="paragraph"> </div>
Let's start with a very simple example. The following
declaration tells Coq that we are defining a new set of data
values — a "type." The type is called <span class="inlinecode"><span class="id" type="var">day</span></span>, and its members are
<span class="inlinecode"><span class="id" type="var">monday</span></span>, <span class="inlinecode"><span class="id" type="var">tuesday</span></span>, etc. The lines of the definition can be read
"<span class="inlinecode"><span class="id" type="var">monday</span></span> is a <span class="inlinecode"><span class="id" type="var">day</span></span>, <span class="inlinecode"><span class="id" type="var">tuesday</span></span> is a <span class="inlinecode"><span class="id" type="var">day</span></span>, etc."
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">day</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">monday</span> : <span class="id" type="var">day</span><br/>
| <span class="id" type="var">tuesday</span> : <span class="id" type="var">day</span><br/>
| <span class="id" type="var">wednesday</span> : <span class="id" type="var">day</span><br/>
| <span class="id" type="var">thursday</span> : <span class="id" type="var">day</span><br/>
| <span class="id" type="var">friday</span> : <span class="id" type="var">day</span><br/>
| <span class="id" type="var">saturday</span> : <span class="id" type="var">day</span><br/>
| <span class="id" type="var">sunday</span> : <span class="id" type="var">day</span>.<br/>
<br/>
</div>
<div class="doc">
Having defined <span class="inlinecode"><span class="id" type="var">day</span></span>, we can write functions that operate on
days.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">next_weekday</span> (<span class="id" type="var">d</span>:<span class="id" type="var">day</span>) : <span class="id" type="var">day</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">d</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">monday</span> => <span class="id" type="var">tuesday</span><br/>
| <span class="id" type="var">tuesday</span> => <span class="id" type="var">wednesday</span><br/>
| <span class="id" type="var">wednesday</span> => <span class="id" type="var">thursday</span><br/>
| <span class="id" type="var">thursday</span> => <span class="id" type="var">friday</span><br/>
| <span class="id" type="var">friday</span> => <span class="id" type="var">monday</span><br/>
| <span class="id" type="var">saturday</span> => <span class="id" type="var">monday</span><br/>
| <span class="id" type="var">sunday</span> => <span class="id" type="var">monday</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
One thing to note is that the argument and return types of
this function are explicitly declared. Like most functional
programming languages, Coq can often work out these types even if
they are not given explicitly — i.e., it performs some <i>type
inference</i> — but we'll always include them to make reading
easier.
<div class="paragraph"> </div>
Having defined a function, we should check that it works on
some examples. There are actually three different ways to do this
in Coq. First, we can use the command <span class="inlinecode"><span class="id" type="keyword">Eval</span></span> <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> to evaluate a
compound expression involving <span class="inlinecode"><span class="id" type="var">next_weekday</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Eval</span> <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> (<span class="id" type="var">next_weekday</span> <span class="id" type="var">friday</span>).<br/>
<span class="comment">(* ==> monday : day *)</span><br/>
<span class="id" type="keyword">Eval</span> <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> (<span class="id" type="var">next_weekday</span> (<span class="id" type="var">next_weekday</span> <span class="id" type="var">saturday</span>)).<br/>
<span class="comment">(* ==> tuesday : day *)</span><br/>
<br/>
</div>
<div class="doc">
If you have a computer handy, now would be an excellent
moment to fire up the Coq interpreter under your favorite IDE —
either CoqIde or Proof General — and try this for yourself. Load
this file (<span class="inlinecode"><span class="id" type="var">Basics.v</span></span>) from the book's accompanying Coq sources,
find the above example, submit it to Coq, and observe the
result.
<div class="paragraph"> </div>
The keyword <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> ("simplify") tells Coq precisely how to
evaluate the expression we give it. For the moment, <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> is
the only one we'll need; later on we'll see some alternatives that
are sometimes useful.
<div class="paragraph"> </div>
Second, we can record what we <i>expect</i> the result to be in
the form of a Coq example:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_next_weekday</span>:<br/>
(<span class="id" type="var">next_weekday</span> (<span class="id" type="var">next_weekday</span> <span class="id" type="var">saturday</span>)) = <span class="id" type="var">tuesday</span>.<br/>
<br/>
</div>
<div class="doc">
This declaration does two things: it makes an
assertion (that the second weekday after <span class="inlinecode"><span class="id" type="var">saturday</span></span> is <span class="inlinecode"><span class="id" type="var">tuesday</span></span>),
and it gives the assertion a name that can be used to refer to it
later. Having made the assertion, we can also ask Coq to verify it,
like this:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The details are not important for now (we'll come back to
them in a bit), but essentially this can be read as "The assertion
we've just made can be proved by observing that both sides of the
equality are the same after simplification."
<div class="paragraph"> </div>
Third, we can ask Coq to "extract," from a <span class="inlinecode"><span class="id" type="keyword">Definition</span></span>, a
program in some other, more conventional, programming
language (OCaml, Scheme, or Haskell) with a high-performance
compiler. This facility is very interesting, since it gives us a
way to construct <i>fully certified</i> programs in mainstream
languages. Indeed, this is one of the main uses for which Coq was
developed. We won't have space to dig further into this topic,
but more information can be found in the Coq'Art book by Bertot
and Castéran, as well as the Coq reference manual.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab14"></a><h2 class="section">Booleans</h2>
<div class="paragraph"> </div>
In a similar way, we can define the type <span class="inlinecode"><span class="id" type="var">bool</span></span> of booleans,
with members <span class="inlinecode"><span class="id" type="var">true</span></span> and <span class="inlinecode"><span class="id" type="var">false</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">bool</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">true</span> : <span class="id" type="var">bool</span><br/>
| <span class="id" type="var">false</span> : <span class="id" type="var">bool</span>.<br/>
<br/>
</div>
<div class="doc">
Although we are rolling our own booleans here for the sake
of building up everything from scratch, Coq does, of course,
provide a default implementation of the booleans in its standard
library, together with a multitude of useful functions and
lemmas. (Take a look at <span class="inlinecode"><span class="id" type="var">Coq.Init.Datatypes</span></span> in the Coq library
documentation if you're interested.) Whenever possible, we'll
name our own definitions and theorems so that they exactly
coincide with the ones in the standard library.
<div class="paragraph"> </div>
Functions over booleans can be defined in the same way as
above:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">negb</span> (<span class="id" type="var">b</span>:<span class="id" type="var">bool</span>) : <span class="id" type="var">bool</span> := <br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">b</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">true</span> => <span class="id" type="var">false</span><br/>
| <span class="id" type="var">false</span> => <span class="id" type="var">true</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">andb</span> (<span class="id" type="var">b1</span>:<span class="id" type="var">bool</span>) (<span class="id" type="var">b2</span>:<span class="id" type="var">bool</span>) : <span class="id" type="var">bool</span> := <br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">b1</span> <span class="id" type="keyword">with</span> <br/>
| <span class="id" type="var">true</span> => <span class="id" type="var">b2</span> <br/>
| <span class="id" type="var">false</span> => <span class="id" type="var">false</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">orb</span> (<span class="id" type="var">b1</span>:<span class="id" type="var">bool</span>) (<span class="id" type="var">b2</span>:<span class="id" type="var">bool</span>) : <span class="id" type="var">bool</span> := <br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">b1</span> <span class="id" type="keyword">with</span> <br/>
| <span class="id" type="var">true</span> => <span class="id" type="var">true</span><br/>
| <span class="id" type="var">false</span> => <span class="id" type="var">b2</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
The last two illustrate the syntax for multi-argument
function definitions.
<div class="paragraph"> </div>
The following four "unit tests" constitute a complete
specification — a truth table — for the <span class="inlinecode"><span class="id" type="var">orb</span></span> function:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_orb1</span>: (<span class="id" type="var">orb</span> <span class="id" type="var">true</span> <span class="id" type="var">false</span>) = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_orb2</span>: (<span class="id" type="var">orb</span> <span class="id" type="var">false</span> <span class="id" type="var">false</span>) = <span class="id" type="var">false</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_orb3</span>: (<span class="id" type="var">orb</span> <span class="id" type="var">false</span> <span class="id" type="var">true</span>) = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_orb4</span>: (<span class="id" type="var">orb</span> <span class="id" type="var">true</span> <span class="id" type="var">true</span>) = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<i>A note on notation</i>: We use square brackets to delimit
fragments of Coq code in comments in .v files; this convention,
also used by the <span class="inlinecode"><span class="id" type="var">coqdoc</span></span> documentation tool, keeps them visually
separate from the surrounding text. In the html version of the
files, these pieces of text appear in a <span class="inlinecode"><span class="id" type="var">different</span></span> <span class="inlinecode"><span class="id" type="var">font</span></span>.
<div class="paragraph"> </div>
The following bit of Coq hackery defines a magic value
called <span class="inlinecode"><span class="id" type="var">admit</span></span> that can fill a hole in an incomplete definition or
proof. We'll use it in the definition of <span class="inlinecode"><span class="id" type="var">nandb</span></span> in the following
exercise. In general, your job in the exercises is to replace
<span class="inlinecode"><span class="id" type="var">admit</span></span> or <span class="inlinecode"><span class="id" type="var">Admitted</span></span> with real definitions or proofs.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">admit</span> {<span class="id" type="var">T</span>: <span class="id" type="keyword">Type</span>} : <span class="id" type="var">T</span>. <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab15"></a><h4 class="section">Exercise: 1 star (nandb)</h4>
Complete the definition of the following function, then make
sure that the <span class="inlinecode"><span class="id" type="keyword">Example</span></span> assertions below each can be verified by
Coq.
<div class="paragraph"> </div>
This function should return <span class="inlinecode"><span class="id" type="var">true</span></span> if either or both of
its inputs are <span class="inlinecode"><span class="id" type="var">false</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">nandb</span> (<span class="id" type="var">b1</span>:<span class="id" type="var">bool</span>) (<span class="id" type="var">b2</span>:<span class="id" type="var">bool</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">admit</span>.<br/>
<br/>
</div>
<div class="doc">
Remove "<span class="inlinecode"><span class="id" type="var">Admitted</span>.</span>" and fill in each proof with
"<span class="inlinecode"><span class="id" type="keyword">Proof</span>.</span> <span class="inlinecode"><span class="id" type="tactic">simpl</span>.</span> <span class="inlinecode"><span class="id" type="tactic">reflexivity</span>.</span> <span class="inlinecode"><span class="id" type="keyword">Qed</span>.</span>"
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_nandb1</span>: (<span class="id" type="var">nandb</span> <span class="id" type="var">true</span> <span class="id" type="var">false</span>) = <span class="id" type="var">true</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_nandb2</span>: (<span class="id" type="var">nandb</span> <span class="id" type="var">false</span> <span class="id" type="var">false</span>) = <span class="id" type="var">true</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_nandb3</span>: (<span class="id" type="var">nandb</span> <span class="id" type="var">false</span> <span class="id" type="var">true</span>) = <span class="id" type="var">true</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_nandb4</span>: (<span class="id" type="var">nandb</span> <span class="id" type="var">true</span> <span class="id" type="var">true</span>) = <span class="id" type="var">false</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab16"></a><h4 class="section">Exercise: 1 star (andb3)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">andb3</span> (<span class="id" type="var">b1</span>:<span class="id" type="var">bool</span>) (<span class="id" type="var">b2</span>:<span class="id" type="var">bool</span>) (<span class="id" type="var">b3</span>:<span class="id" type="var">bool</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">admit</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_andb31</span>: (<span class="id" type="var">andb3</span> <span class="id" type="var">true</span> <span class="id" type="var">true</span> <span class="id" type="var">true</span>) = <span class="id" type="var">true</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_andb32</span>: (<span class="id" type="var">andb3</span> <span class="id" type="var">false</span> <span class="id" type="var">true</span> <span class="id" type="var">true</span>) = <span class="id" type="var">false</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_andb33</span>: (<span class="id" type="var">andb3</span> <span class="id" type="var">true</span> <span class="id" type="var">false</span> <span class="id" type="var">true</span>) = <span class="id" type="var">false</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_andb34</span>: (<span class="id" type="var">andb3</span> <span class="id" type="var">true</span> <span class="id" type="var">true</span> <span class="id" type="var">false</span>) = <span class="id" type="var">false</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab17"></a><h2 class="section">Function Types</h2>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="keyword">Check</span></span> command causes Coq to print the type of an
expression. For example, the type of <span class="inlinecode"><span class="id" type="var">negb</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span> is <span class="inlinecode"><span class="id" type="var">bool</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">true</span>.<br/>
<span class="comment">(* ===> true : bool *)</span><br/>
<span class="id" type="keyword">Check</span> (<span class="id" type="var">negb</span> <span class="id" type="var">true</span>).<br/>
<span class="comment">(* ===> negb true : bool *)</span><br/>
<br/>
</div>
<div class="doc">
Functions like <span class="inlinecode"><span class="id" type="var">negb</span></span> itself are also data values, just like
<span class="inlinecode"><span class="id" type="var">true</span></span> and <span class="inlinecode"><span class="id" type="var">false</span></span>. Their types are called <i>function types</i>, and
they are written with arrows.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">negb</span>.<br/>
<span class="comment">(* ===> negb : bool -> bool *)</span><br/>
<br/>
</div>
<div class="doc">
The type of <span class="inlinecode"><span class="id" type="var">negb</span></span>, written <span class="inlinecode"><span class="id" type="var">bool</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">bool</span></span> and pronounced
"<span class="inlinecode"><span class="id" type="var">bool</span></span> arrow <span class="inlinecode"><span class="id" type="var">bool</span></span>," can be read, "Given an input of type
<span class="inlinecode"><span class="id" type="var">bool</span></span>, this function produces an output of type <span class="inlinecode"><span class="id" type="var">bool</span></span>."
Similarly, the type of <span class="inlinecode"><span class="id" type="var">andb</span></span>, written <span class="inlinecode"><span class="id" type="var">bool</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">bool</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">bool</span></span>, can
be read, "Given two inputs, both of type <span class="inlinecode"><span class="id" type="var">bool</span></span>, this function
produces an output of type <span class="inlinecode"><span class="id" type="var">bool</span></span>."
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab18"></a><h2 class="section">Numbers</h2>
<div class="paragraph"> </div>
<i>Technical digression</i>: Coq provides a fairly fancy module system,
to aid in organizing large developments. In this course, we won't
need most of its features, but one of them is useful: if we
enclose a collection of declarations between <span class="inlinecode"><span class="id" type="keyword">Module</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> and <span class="inlinecode"><span class="id" type="keyword">End</span></span>
<span class="inlinecode"><span class="id" type="var">X</span></span> markers, then, in the remainder of the file after the <span class="inlinecode"><span class="id" type="keyword">End</span></span>,
all these definitions will be referred to by names like <span class="inlinecode"><span class="id" type="var">X.foo</span></span>
instead of just <span class="inlinecode"><span class="id" type="var">foo</span></span>. This means that the new definition will
not clash with the unqualified name <span class="inlinecode"><span class="id" type="var">foo</span></span> later, which would
otherwise be an error (a name can only be defined once in a given
scope). Here, we use this feature to introduce the definition of
the type <span class="inlinecode"><span class="id" type="var">nat</span></span> in an inner module so that it does not shadow the
one from the standard library.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">Playground1</span>.<br/>
<br/>
</div>
<div class="doc">
The types we have defined so far are examples of "enumerated
types": their definitions explicitly enumerate a finite set of
elements. A more interesting way of defining a type is to give a
collection of "inductive rules" describing its elements. For
example, we can define the natural numbers as follows:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">nat</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">O</span> : <span class="id" type="var">nat</span><br/>
| <span class="id" type="var">S</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nat</span>.<br/>
<br/>
</div>
<div class="doc">
The clauses of this definition can be read:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">O</span></span> is a natural number (note that this is the letter "<span class="inlinecode"><span class="id" type="var">O</span></span>," not
the numeral "<span class="inlinecode">0</span>").
</li>
<li> <span class="inlinecode"><span class="id" type="var">S</span></span> is a "constructor" that takes a natural number and yields
another one — that is, if <span class="inlinecode"><span class="id" type="var">n</span></span> is a natural number, then <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>
is too.
</li>
</ul>
<div class="paragraph"> </div>
Let's look at this in a little more detail.
<div class="paragraph"> </div>
Every inductively defined set (<span class="inlinecode"><span class="id" type="var">weekday</span></span>, <span class="inlinecode"><span class="id" type="var">nat</span></span>, <span class="inlinecode"><span class="id" type="var">bool</span></span>, etc.) is
actually a set of <i>expressions</i>. The definition of <span class="inlinecode"><span class="id" type="var">nat</span></span> says how
expressions in the set <span class="inlinecode"><span class="id" type="var">nat</span></span> can be constructed:
<div class="paragraph"> </div>
<ul class="doclist">
<li> the expression <span class="inlinecode"><span class="id" type="var">O</span></span> belongs to the set <span class="inlinecode"><span class="id" type="var">nat</span></span>;
</li>
<li> if <span class="inlinecode"><span class="id" type="var">n</span></span> is an expression belonging to the set <span class="inlinecode"><span class="id" type="var">nat</span></span>, then <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>
is also an expression belonging to the set <span class="inlinecode"><span class="id" type="var">nat</span></span>; and
</li>
<li> expressions formed in these two ways are the only ones belonging
to the set <span class="inlinecode"><span class="id" type="var">nat</span></span>.
</li>
</ul>
<div class="paragraph"> </div>
These three conditions are the precise force of the
<span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> declaration. They imply that the expression <span class="inlinecode"><span class="id" type="var">O</span></span>, the
expression <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span></span>, the expression <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span>)</span>, the expression
<span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span>))</span>, and so on all belong to the set <span class="inlinecode"><span class="id" type="var">nat</span></span>, while other
expressions like <span class="inlinecode"><span class="id" type="var">true</span></span>, <span class="inlinecode"><span class="id" type="var">andb</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span> <span class="inlinecode"><span class="id" type="var">false</span></span>, and <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">false</span>)</span> do
not.
<div class="paragraph"> </div>
We can write simple functions that pattern match on natural
numbers just as we did above — for example, predecessor:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">pred</span> (<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">nat</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">n</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="var">O</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">n'</span> => <span class="id" type="var">n'</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
The second branch can be read: "if <span class="inlinecode"><span class="id" type="var">n</span></span> has the form <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span>
for some <span class="inlinecode"><span class="id" type="var">n'</span></span>, then return <span class="inlinecode"><span class="id" type="var">n'</span></span>."
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">Playground1</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">minustwo</span> (<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">nat</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">n</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="var">O</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">O</span> => <span class="id" type="var">O</span><br/>
| <span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n'</span>) => <span class="id" type="var">n'</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Because natural numbers are such a pervasive form of data,
Coq provides a tiny bit of built-in magic for parsing and printing
them: ordinary arabic numerals can be used as an alternative to
the "unary" notation defined by the constructors <span class="inlinecode"><span class="id" type="var">S</span></span> and <span class="inlinecode"><span class="id" type="var">O</span></span>. Coq
prints numbers in arabic form by default:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">O</span>)))).<br/>
<span class="id" type="keyword">Eval</span> <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> (<span class="id" type="var">minustwo</span> 4).<br/>
<br/>
</div>
<div class="doc">
The constructor <span class="inlinecode"><span class="id" type="var">S</span></span> has the type <span class="inlinecode"><span class="id" type="var">nat</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">nat</span></span>, just like the
functions <span class="inlinecode"><span class="id" type="var">minustwo</span></span> and <span class="inlinecode"><span class="id" type="var">pred</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">S</span>.<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">pred</span>.<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">minustwo</span>.<br/>
<br/>
</div>
<div class="doc">
These are all things that can be applied to a number to yield a
number. However, there is a fundamental difference: functions
like <span class="inlinecode"><span class="id" type="var">pred</span></span> and <span class="inlinecode"><span class="id" type="var">minustwo</span></span> come with <i>computation rules</i>
— e.g., the definition of <span class="inlinecode"><span class="id" type="var">pred</span></span> says that <span class="inlinecode"><span class="id" type="var">pred</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> can be
simplified to <span class="inlinecode"><span class="id" type="keyword">match</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode"><span class="id" type="keyword">with</span></span> <span class="inlinecode">|</span> <span class="inlinecode"><span class="id" type="var">O</span></span> <span class="inlinecode">=></span> <span class="inlinecode"><span class="id" type="var">O</span></span> <span class="inlinecode">|</span> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">m'</span></span> <span class="inlinecode">=></span> <span class="inlinecode"><span class="id" type="var">m'</span></span> <span class="inlinecode"><span class="id" type="keyword">end</span></span> — while
the definition of <span class="inlinecode"><span class="id" type="var">S</span></span> has no such behavior attached. Although it
is a function in the sense that it can be applied to an argument,
it does not <i>do</i> anything at all!
<div class="paragraph"> </div>
For most function definitions over numbers, pure pattern
matching is not enough: we also need recursion. For example, to
check that a number <span class="inlinecode"><span class="id" type="var">n</span></span> is even, we may need to recursively check
whether <span class="inlinecode"><span class="id" type="var">n</span>-2</span> is even. To write such functions, we use the
keyword <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">evenb</span> (<span class="id" type="var">n</span>:<span class="id" type="var">nat</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">n</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="var">true</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">O</span> => <span class="id" type="var">false</span><br/>
| <span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n'</span>) => <span class="id" type="var">evenb</span> <span class="id" type="var">n'</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
When Coq checks this definition, it notes that <span class="inlinecode"><span class="id" type="var">evenb</span></span> is
"decreasing on 1st argument." What this means is that we are
performing a <i>structural recursion</i> over the argument <span class="inlinecode"><span class="id" type="var">n</span></span> — i.e.,
that we make recursive calls only on strictly smaller values of
<span class="inlinecode"><span class="id" type="var">n</span></span>. This implies that all calls to <span class="inlinecode"><span class="id" type="var">evenb</span></span> will eventually
terminate. Coq demands that some argument of <i>every</i> <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span>
definition is decreasing.
<div class="paragraph"> </div>
We can define <span class="inlinecode"><span class="id" type="var">oddb</span></span> by a similar <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span> declaration, but here
is a simpler definition that will be a bit easier to work with:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">oddb</span> (<span class="id" type="var">n</span>:<span class="id" type="var">nat</span>) : <span class="id" type="var">bool</span> := <span class="id" type="var">negb</span> (<span class="id" type="var">evenb</span> <span class="id" type="var">n</span>).<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_oddb1</span>: (<span class="id" type="var">oddb</span> (<span class="id" type="var">S</span> <span class="id" type="var">O</span>)) = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_oddb2</span>: (<span class="id" type="var">oddb</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">O</span>))))) = <span class="id" type="var">false</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Naturally, we can also define multi-argument functions by
recursion. (Once again, we use a module to avoid polluting the
namespace.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">Playground2</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">plus</span> (<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>) (<span class="id" type="var">m</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">nat</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">n</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="var">m</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">n'</span> => <span class="id" type="var">S</span> (<span class="id" type="var">plus</span> <span class="id" type="var">n'</span> <span class="id" type="var">m</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Adding three to two now gives us five, as we'd expect.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Eval</span> <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> (<span class="id" type="var">plus</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">O</span>))) (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">O</span>))).<br/>
<br/>
</div>
<div class="doc">
The simplification that Coq performs to reach this conclusion can
be visualized as follows:
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" type="var">plus</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span>)))</span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span>))</span> <br/>
==> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">plus</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span>))</span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span>)))</span> by the second clause of the <span class="inlinecode"><span class="id" type="keyword">match</span></span><br/>
==> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">plus</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span>)</span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span>))))</span> by the second clause of the <span class="inlinecode"><span class="id" type="keyword">match</span></span><br/>
==> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">plus</span></span> <span class="inlinecode"><span class="id" type="var">O</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span>)))))</span> by the second clause of the <span class="inlinecode"><span class="id" type="keyword">match</span></span><br/>
==> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">O</span>))))</span> by the first clause of the <span class="inlinecode"><span class="id" type="keyword">match</span></span><br/>
*)</span><br/>
<br/>
</div>
<div class="doc">
As a notational convenience, if two or more arguments have
the same type, they can be written together. In the following
definition, <span class="inlinecode">(<span class="id" type="var">n</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">nat</span>)</span> means just the same as if we had written
<span class="inlinecode">(<span class="id" type="var">n</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">nat</span>)</span> <span class="inlinecode">(<span class="id" type="var">m</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">nat</span>)</span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">mult</span> (<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">nat</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">n</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="var">O</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">n'</span> => <span class="id" type="var">plus</span> <span class="id" type="var">m</span> (<span class="id" type="var">mult</span> <span class="id" type="var">n'</span> <span class="id" type="var">m</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
You can match two expressions at once by putting a comma
between them:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">minus</span> (<span class="id" type="var">n</span> <span class="id" type="var">m</span>:<span class="id" type="var">nat</span>) : <span class="id" type="var">nat</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">n</span>, <span class="id" type="var">m</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> , <span class="id" type="var">_</span> => <span class="id" type="var">O</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">_</span> , <span class="id" type="var">O</span> => <span class="id" type="var">n</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">n'</span>, <span class="id" type="var">S</span> <span class="id" type="var">m'</span> => <span class="id" type="var">minus</span> <span class="id" type="var">n'</span> <span class="id" type="var">m'</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
The _ in the first line is a <i>wildcard pattern</i>. Writing _ in a
pattern is the same as writing some variable that doesn't get used
on the right-hand side. This avoids the need to invent a bogus
variable name.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">Playground2</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">exp</span> (<span class="id" type="var">base</span> <span class="id" type="var">power</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">nat</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">power</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="var">S</span> <span class="id" type="var">O</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">p</span> => <span class="id" type="var">mult</span> <span class="id" type="var">base</span> (<span class="id" type="var">exp</span> <span class="id" type="var">base</span> <span class="id" type="var">p</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_mult1</span>: (<span class="id" type="var">mult</span> 3 3) = 9.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab19"></a><h4 class="section">Exercise: 1 star (factorial)</h4>
Recall the standard factorial function:
<pre>
factorial(0) = 1
factorial(n) = n * factorial(n-1) (if n>0)
</pre>
Translate this into Coq.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">factorial</span> (<span class="id" type="var">n</span>:<span class="id" type="var">nat</span>) : <span class="id" type="var">nat</span> := <br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">admit</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_factorial1</span>: (<span class="id" type="var">factorial</span> 3) = 6.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_factorial2</span>: (<span class="id" type="var">factorial</span> 5) = (<span class="id" type="var">mult</span> 10 12).<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
We can make numerical expressions a little easier to read and
write by introducing "notations" for addition, multiplication, and
subtraction.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> "x + y" := (<span class="id" type="var">plus</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>) <br/>
(<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 50, <span class="id" type="var">left</span> <span class="id" type="var">associativity</span>) <br/>
: <span class="id" type="var">nat_scope</span>.<br/>
<span class="id" type="keyword">Notation</span> "x - y" := (<span class="id" type="var">minus</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>) <br/>
(<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 50, <span class="id" type="var">left</span> <span class="id" type="var">associativity</span>) <br/>
: <span class="id" type="var">nat_scope</span>.<br/>
<span class="id" type="keyword">Notation</span> "x * y" := (<span class="id" type="var">mult</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>) <br/>
(<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40, <span class="id" type="var">left</span> <span class="id" type="var">associativity</span>) <br/>
: <span class="id" type="var">nat_scope</span>.<br/>
<br/>
<span class="id" type="keyword">Check</span> ((0 + 1) + 1).<br/>
<br/>
</div>
<div class="doc">
Note that these do not change the definitions we've already
made: they are simply instructions to the Coq parser to accept <span class="inlinecode"><span class="id" type="var">x</span></span>
<span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">y</span></span> in place of <span class="inlinecode"><span class="id" type="var">plus</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> and, conversely, to the Coq
pretty-printer to display <span class="inlinecode"><span class="id" type="var">plus</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> as <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">y</span></span>.
<div class="paragraph"> </div>
Each notation-symbol in Coq is active in a <i>notation scope</i>. Coq
tries to guess what scope you mean, so when you write <span class="inlinecode"><span class="id" type="var">S</span>(<span class="id" type="var">O</span>*<span class="id" type="var">O</span>)</span> it
guesses <span class="inlinecode"><span class="id" type="var">nat_scope</span></span>, but when you write the cartesian
product (tuple) type <span class="inlinecode"><span class="id" type="var">bool</span>*<span class="id" type="var">bool</span></span> it guesses <span class="inlinecode"><span class="id" type="var">type_scope</span></span>.
Occasionally you have to help it out with percent-notation by
writing <span class="inlinecode">(<span class="id" type="var">x</span>*<span class="id" type="var">y</span>)%<span class="id" type="var">nat</span></span>, and sometimes in Coq's feedback to you it
will use <span class="inlinecode">%<span class="id" type="var">nat</span></span> to indicate what scope a notation is in.
<div class="paragraph"> </div>
Notation scopes also apply to numeral notation (3,4,5, etc.), so you
may sometimes see <span class="inlinecode">0%<span class="id" type="var">nat</span></span> which means <span class="inlinecode"><span class="id" type="var">O</span></span>, or <span class="inlinecode">0%<span class="id" type="var">Z</span></span> which means the
Integer zero.
<div class="paragraph"> </div>
When we say that Coq comes with nothing built-in, we really
mean it: even equality testing for numbers is a user-defined
operation! The <span class="inlinecode"><span class="id" type="var">beq_nat</span></span> function tests <span class="inlinecode"><span class="id" type="var">nat</span></span>ural numbers for <span class="inlinecode"><span class="id" type="var">eq</span></span>uality,
yielding a <span class="inlinecode"><span class="id" type="var">b</span></span>oolean. Note the use of nested <span class="inlinecode"><span class="id" type="keyword">match</span></span>es (we could
also have used a simultaneous match, as we did in <span class="inlinecode"><span class="id" type="var">minus</span></span>.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">beq_nat</span> (<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">n</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="keyword">match</span> <span class="id" type="var">m</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="var">true</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">m'</span> => <span class="id" type="var">false</span><br/>
<span class="id" type="keyword">end</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">n'</span> => <span class="id" type="keyword">match</span> <span class="id" type="var">m</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="var">false</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">m'</span> => <span class="id" type="var">beq_nat</span> <span class="id" type="var">n'</span> <span class="id" type="var">m'</span><br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Similarly, the <span class="inlinecode"><span class="id" type="var">ble_nat</span></span> function tests <span class="inlinecode"><span class="id" type="var">nat</span></span>ural numbers for
<span class="inlinecode"><span class="id" type="var">l</span></span>ess-or-<span class="inlinecode"><span class="id" type="var">e</span></span>qual, yielding a <span class="inlinecode"><span class="id" type="var">b</span></span>oolean.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">ble_nat</span> (<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">n</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="var">true</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">n'</span> =><br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">m</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> => <span class="id" type="var">false</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">m'</span> => <span class="id" type="var">ble_nat</span> <span class="id" type="var">n'</span> <span class="id" type="var">m'</span><br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_ble_nat1</span>: (<span class="id" type="var">ble_nat</span> 2 2) = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_ble_nat2</span>: (<span class="id" type="var">ble_nat</span> 2 4) = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_ble_nat3</span>: (<span class="id" type="var">ble_nat</span> 4 2) = <span class="id" type="var">false</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab20"></a><h4 class="section">Exercise: 2 stars (blt_nat)</h4>
The <span class="inlinecode"><span class="id" type="var">blt_nat</span></span> function tests <span class="inlinecode"><span class="id" type="var">nat</span></span>ural numbers for <span class="inlinecode"><span class="id" type="var">l</span></span>ess-<span class="inlinecode"><span class="id" type="var">t</span></span>han,
yielding a <span class="inlinecode"><span class="id" type="var">b</span></span>oolean. Instead of making up a new <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span> for
this one, define it in terms of a previously defined function.
<div class="paragraph"> </div>
Note: If you have trouble with the <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> tactic, try using
<span class="inlinecode"><span class="id" type="tactic">compute</span></span>, which is like <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> on steroids. However, there is a
simple, elegant solution for which <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> suffices.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">blt_nat</span> (<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">admit</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_blt_nat1</span>: (<span class="id" type="var">blt_nat</span> 2 2) = <span class="id" type="var">false</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_blt_nat2</span>: (<span class="id" type="var">blt_nat</span> 2 4) = <span class="id" type="var">true</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_blt_nat3</span>: (<span class="id" type="var">blt_nat</span> 4 2) = <span class="id" type="var">false</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab21"></a><h1 class="section">Proof By Simplification</h1>
<div class="paragraph"> </div>
Now that we've defined a few datatypes and functions, let's
turn to the question of how to state and prove properties of their
behavior. Actually, in a sense, we've already started doing this:
each <span class="inlinecode"><span class="id" type="keyword">Example</span></span> in the previous sections makes a precise claim
about the behavior of some function on some particular inputs.
The proofs of these claims were always the same: use the
function's definition to simplify the expressions on both sides of
the <span class="inlinecode">=</span> and notice that they become identical.
<div class="paragraph"> </div>
The same sort of "proof by simplification" can be used to prove
more interesting properties as well. For example, the fact that
<span class="inlinecode">0</span> is a "neutral element" for <span class="inlinecode">+</span> on the left can be proved
just by observing that <span class="inlinecode">0</span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">n</span></span> reduces to <span class="inlinecode"><span class="id" type="var">n</span></span> no matter what
<span class="inlinecode"><span class="id" type="var">n</span></span> is, since the definition of <span class="inlinecode">+</span> is recursive in its first
argument.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_O_n</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>, 0 + <span class="id" type="var">n</span> = <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span> command implicitly simplifies both sides of the
equality before testing to see if they are the same, so we can
shorten the proof a little. (It will be useful later to know that <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span> actually
does somwhat more than <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> — for example, it tries
"unfolding" defined terms, replacing them with their right-hand
sides. The reason for this difference is that, when reflexivity
succeeds, the whole goal is finished and we don't need to look at
whatever expanded expressions <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span> has found; by
contrast, <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> is used in situations where we may have to read
and understand the new goal, so we would not want it blindly
expanding definitions.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_O_n'</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>, 0 + <span class="id" type="var">n</span> = <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The form of this theorem and proof are almost exactly the
same as the examples above: the only differences are that we've
added the quantifier <span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">n</span>:<span class="id" type="var">nat</span></span> and that we've used the
keyword <span class="inlinecode"><span class="id" type="keyword">Theorem</span></span> instead of <span class="inlinecode"><span class="id" type="keyword">Example</span></span>. Indeed, the latter
difference is purely a matter of style; the keywords <span class="inlinecode"><span class="id" type="keyword">Example</span></span> and
<span class="inlinecode"><span class="id" type="keyword">Theorem</span></span> (and a few others, including <span class="inlinecode"><span class="id" type="keyword">Lemma</span></span>, <span class="inlinecode"><span class="id" type="keyword">Fact</span></span>, and
<span class="inlinecode"><span class="id" type="keyword">Remark</span></span>) mean exactly the same thing to Coq.
<div class="paragraph"> </div>
The keywords <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> and <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span> are examples of <i>tactics</i>.
A tactic is a command that is used between <span class="inlinecode"><span class="id" type="keyword">Proof</span></span> and <span class="inlinecode"><span class="id" type="keyword">Qed</span></span> to
tell Coq how it should check the correctness of some claim we are
making. We will see several more tactics in the rest of this
lecture, and yet more in future lectures.
<div class="paragraph"> </div>
<a name="lab22"></a><h4 class="section">Exercise: 1 star, optional (simpl_plus)</h4>
What will Coq print in response to this query?
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* Eval simpl in (forall n:nat, n + 0 = n). *)</span><br/>
<br/>
</div>
<div class="doc">