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
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
|
\chapter[Extensions of \Gallina{}]{Extensions of \Gallina{}\label{Gallina-extension}\index{Gallina}}
{\gallina} is the kernel language of {\Coq}. We describe here extensions of
the Gallina's syntax.
\section{Record types
\comindex{Record}
\label{Record}}
The \verb+Record+ construction is a macro allowing the definition of
records as is done in many programming languages. Its syntax is
described on Figure~\ref{record-syntax}. In fact, the \verb+Record+
macro is more general than the usual record types, since it allows
also for ``manifest'' expressions. In this sense, the \verb+Record+
construction allows to define ``signatures''.
\begin{figure}[h]
\begin{centerframe}
\begin{tabular}{lcl}
{\sentence} & ++= & {\record}\\
& & \\
{\record} & ::= &
{\tt Record} {\ident} \sequence{\binderlet}{} \zeroone{{\tt :} {\sort}} \verb.:=. \\
&& ~~~~\zeroone{\ident}
\verb!{! \zeroone{\nelist{\field}{;}} \verb!}! \verb:.:\\
& & \\
{\field} & ::= & {\name} : {\type} [ {\tt where} {\it notation} ] \\
& $|$ & {\name} {\typecstr} := {\term}
\end{tabular}
\end{centerframe}
\caption{Syntax for the definition of {\tt Record}}
\label{record-syntax}
\end{figure}
\noindent In the expression
\smallskip
{\tt Record} {\ident} {\params} \texttt{:}
{\sort} := {\ident$_0$} \verb+{+
{\ident$_1$} \texttt{:} {\term$_1$};
\dots
{\ident$_n$} \texttt{:} {\term$_n$} \verb+}+.
\smallskip
\noindent the identifier {\ident} is the name of the defined record
and {\sort} is its type. The identifier {\ident$_0$} is the name of
its constructor. If {\ident$_0$} is omitted, the default name {\tt
Build\_{\ident}} is used. If {\sort} is omitted, the default sort is ``{\Type}''.
The identifiers {\ident$_1$}, ..,
{\ident$_n$} are the names of fields and {\term$_1$}, .., {\term$_n$}
their respective types. Remark that the type of {\ident$_i$} may
depend on the previous {\ident$_j$} (for $j<i$). Thus the order of the
fields is important. Finally, {\params} are the parameters of the
record.
More generally, a record may have explicitly defined (a.k.a.
manifest) fields. For instance, {\tt Record} {\ident} {\tt [}
{\params} {\tt ]} \texttt{:} {\sort} := \verb+{+ {\ident$_1$}
\texttt{:} {\type$_1$} \verb+;+ {\ident$_2$} \texttt{:=} {\term$_2$}
\verb+;+ {\ident$_3$} \texttt{:} {\type$_3$} \verb+}+ in which case
the correctness of {\type$_3$} may rely on the instance {\term$_2$} of
{\ident$_2$} and {\term$_2$} in turn may depend on {\ident$_1$}.
\Example
The set of rational numbers may be defined as:
\begin{coq_eval}
Reset Initial.
\end{coq_eval}
\begin{coq_example}
Record Rat : Set := mkRat
{sign : bool;
top : nat;
bottom : nat;
Rat_bottom_cond : 0 <> bottom;
Rat_irred_cond :
forall x y z:nat, (x * y) = top /\ (x * z) = bottom -> x = 1}.
\end{coq_example}
Remark here that the field
\verb+Rat_cond+ depends on the field \verb+bottom+.
%Let us now see the work done by the {\tt Record} macro.
%First the macro generates an inductive definition
%with just one constructor:
%
%\medskip
%\noindent
%{\tt Inductive {\ident} {\binderlet} : {\sort} := \\
%\mbox{}\hspace{0.4cm} {\ident$_0$} : forall ({\ident$_1$}:{\term$_1$}) ..
%({\ident$_n$}:{\term$_n$}), {\ident} {\rm\sl params}.}
%\medskip
Let us now see the work done by the {\tt Record} macro. First the
macro generates an inductive definition with just one constructor:
\begin{quote}
{\tt Inductive {\ident} {\params} :{\sort} :=} \\
\qquad {\tt
{\ident$_0$} ({\ident$_1$}:{\term$_1$}) .. ({\ident$_n$}:{\term$_n$}).}
\end{quote}
To build an object of type {\ident}, one should provide the
constructor {\ident$_0$} with $n$ terms filling the fields of
the record.
As an example, let us define the rational $1/2$:
\begin{coq_example*}
Require Import Arith.
Theorem one_two_irred :
forall x y z:nat, x * y = 1 /\ x * z = 2 -> x = 1.
\end{coq_example*}
\begin{coq_eval}
Lemma mult_m_n_eq_m_1 : forall m n:nat, m * n = 1 -> m = 1.
destruct m; trivial.
intros; apply f_equal with (f := S).
destruct m; trivial.
destruct n; simpl in H.
rewrite <- mult_n_O in H.
discriminate.
rewrite <- plus_n_Sm in H.
discriminate.
Qed.
intros x y z [H1 H2].
apply mult_m_n_eq_m_1 with (n := y); trivial.
\end{coq_eval}
\ldots
\begin{coq_example*}
Qed.
\end{coq_example*}
\begin{coq_example}
Definition half := mkRat true 1 2 (O_S 1) one_two_irred.
\end{coq_example}
\begin{coq_example}
Check half.
\end{coq_example}
The macro generates also, when it is possible, the projection
functions for destructuring an object of type {\ident}. These
projection functions have the same name that the corresponding
fields. If a field is named ``\verb=_='' then no projection is built
for it. In our example:
\begin{coq_example}
Eval compute in half.(top).
Eval compute in half.(bottom).
Eval compute in half.(Rat_bottom_cond).
\end{coq_example}
\begin{coq_eval}
Reset Initial.
\end{coq_eval}
\begin{Warnings}
\item {\tt Warning: {\ident$_i$} cannot be defined.}
It can happen that the definition of a projection is impossible.
This message is followed by an explanation of this impossibility.
There may be three reasons:
\begin{enumerate}
\item The name {\ident$_i$} already exists in the environment (see
Section~\ref{Axiom}).
\item The body of {\ident$_i$} uses an incorrect elimination for
{\ident} (see Sections~\ref{Fixpoint} and~\ref{Caseexpr}).
\item The type of the projections {\ident$_i$} depends on previous
projections which themselves could not be defined.
\end{enumerate}
\end{Warnings}
\begin{ErrMsgs}
\item \errindex{A record cannot be recursive}
The record name {\ident} appears in the type of its fields.
\item During the definition of the one-constructor inductive
definition, all the errors of inductive definitions, as described in
Section~\ref{gal_Inductive_Definitions}, may also occur.
\end{ErrMsgs}
\SeeAlso Coercions and records in Section~\ref{Coercions-and-records}
of the chapter devoted to coercions.
\Rem {\tt Structure} is a synonym of the keyword {\tt Record}.
\Rem Creation of an object of record type can be done by calling {\ident$_0$}
and passing arguments in the correct order.
\begin{coq_example}
Record point := { x : nat; y : nat }.
Definition a := Build_point 5 3.
\end{coq_example}
The following syntax allows to create objects by using named fields. The
fields do not have to be in any particular order, nor do they have to be all
present if the missing ones can be inferred or prompted for (see
Section~\ref{Program}).
\begin{coq_example}
Definition b := {| x := 5; y := 3 |}.
Definition c := {| y := 3; x := 5 |}.
\end{coq_example}
This syntax can also be used for pattern matching.
\begin{coq_example}
Eval compute in (
match b with
| {| y := S n |} => n
| _ => 0
end).
\end{coq_example}
\begin{coq_eval}
Reset Initial.
\end{coq_eval}
\Rem An experimental syntax for projections based on a dot notation is
available. The command to activate it is
\begin{quote}
{\tt Set Printing Projections.}
\end{quote}
\begin{figure}[t]
\begin{centerframe}
\begin{tabular}{lcl}
{\term} & ++= & {\term} {\tt .(} {\qualid} {\tt )}\\
& $|$ & {\term} {\tt .(} {\qualid} \nelist{\termarg}{} {\tt )}\\
& $|$ & {\term} {\tt .(} {@}{\qualid} \nelist{\term}{} {\tt )}
\end{tabular}
\end{centerframe}
\caption{Syntax of \texttt{Record} projections}
\label{fig:projsyntax}
\end{figure}
The corresponding grammar rules are given Figure~\ref{fig:projsyntax}.
When {\qualid} denotes a projection, the syntax {\tt
{\term}.({\qualid})} is equivalent to {\qualid~\term}, the syntax
{\tt {\term}.({\qualid}~{\termarg}$_1$~ \ldots~ {\termarg}$_n$)} to
{\qualid~{\termarg}$_1$ \ldots {\termarg}$_n$~\term}, and the syntax
{\tt {\term}.(@{\qualid}~{\term}$_1$~\ldots~{\term}$_n$)} to
{@\qualid~{\term}$_1$ \ldots {\term}$_n$~\term}. In each case, {\term}
is the object projected and the other arguments are the parameters of
the inductive type.
To deactivate the printing of projections, use
{\tt Unset Printing Projections}.
\section{Variants and extensions of {\mbox{\tt match}}
\label{Extensions-of-match}
\index{match@{\tt match\ldots with\ldots end}}}
\subsection{Multiple and nested pattern-matching
\index{ML-like patterns}
\label{Mult-match}}
The basic version of \verb+match+ allows pattern-matching on simple
patterns. As an extension, multiple nested patterns or disjunction of
patterns are allowed, as in ML-like languages.
The extension just acts as a macro that is expanded during parsing
into a sequence of {\tt match} on simple patterns. Especially, a
construction defined using the extended {\tt match} is generally
printed under its expanded form (see~\texttt{Set Printing Matching} in
section~\ref{SetPrintingMatching}).
\SeeAlso Chapter~\ref{Mult-match-full}.
\subsection{Pattern-matching on boolean values: the {\tt if} expression
\label{if-then-else}
\index{if@{\tt if ... then ... else}}}
For inductive types with exactly two constructors and for
pattern-matchings expressions which do not depend on the arguments of
the constructors, it is possible to use a {\tt if ... then ... else}
notation. For instance, the definition
\begin{coq_example}
Definition not (b:bool) :=
match b with
| true => false
| false => true
end.
\end{coq_example}
\noindent can be alternatively written
\begin{coq_eval}
Reset not.
\end{coq_eval}
\begin{coq_example}
Definition not (b:bool) := if b then false else true.
\end{coq_example}
More generally, for an inductive type with constructors {\tt C$_1$}
and {\tt C$_2$}, we have the following equivalence
\smallskip
{\tt if {\term} \zeroone{\ifitem} then {\term}$_1$ else {\term}$_2$} $\equiv$
\begin{tabular}[c]{l}
{\tt match {\term} \zeroone{\ifitem} with}\\
{\tt \verb!|! C$_1$ \_ {\ldots} \_ \verb!=>! {\term}$_1$} \\
{\tt \verb!|! C$_2$ \_ {\ldots} \_ \verb!=>! {\term}$_2$} \\
{\tt end}
\end{tabular}
Here is an example.
\begin{coq_example}
Check (fun x (H:{x=0}+{x<>0}) =>
match H with
| left _ => true
| right _ => false
end).
\end{coq_example}
Notice that the printing uses the {\tt if} syntax because {\tt sumbool} is
declared as such (see Section~\ref{printing-options}).
\subsection{Irrefutable patterns: the destructuring {\tt let} variants
\index{let in@{\tt let ... in}}
\label{Letin}}
Pattern-matching on terms inhabiting inductive type having only one
constructor can be alternatively written using {\tt let ... in ...}
constructions. There are two variants of them.
\subsubsection{First destructuring {\tt let} syntax}
The expression {\tt let
(}~{\ident$_1$},\ldots,{\ident$_n$}~{\tt ) :=}~{\term$_0$}~{\tt
in}~{\term$_1$} performs case analysis on a {\term$_0$} which must be in
an inductive type with one constructor having itself $n$ arguments. Variables
{\ident$_1$}\ldots{\ident$_n$} are bound to the $n$ arguments of the
constructor in expression {\term$_1$}. For instance, the definition
\begin{coq_example}
Definition fst (A B:Set) (H:A * B) := match H with
| pair x y => x
end.
\end{coq_example}
can be alternatively written
\begin{coq_eval}
Reset fst.
\end{coq_eval}
\begin{coq_example}
Definition fst (A B:Set) (p:A * B) := let (x, _) := p in x.
\end{coq_example}
Notice that reduction is different from regular {\tt let ... in ...}
construction since it happens only if {\term$_0$} is in constructor
form. Otherwise, the reduction is blocked.
The pretty-printing of a definition by matching on a
irrefutable pattern can either be done using {\tt match} or the {\tt
let} construction (see Section~\ref{printing-options}).
If {\term} inhabits an inductive type with one constructor {\tt C},
we have an equivalence between
{\tt let ({\ident}$_1$,\ldots,{\ident}$_n$) \zeroone{\ifitem} := {\term} in {\term}'}
\noindent and
{\tt match {\term} \zeroone{\ifitem} with C {\ident}$_1$ {\ldots} {\ident}$_n$ \verb!=>! {\term}' end}
\subsubsection{Second destructuring {\tt let} syntax\index{let '... in}}
Another destructuring {\tt let} syntax is available for inductive types with
one constructor by giving an arbitrary pattern instead of just a tuple
for all the arguments. For example, the preceding example can be written:
\begin{coq_eval}
Reset fst.
\end{coq_eval}
\begin{coq_example}
Definition fst (A B:Set) (p:A*B) := let 'pair x _ := p in x.
\end{coq_example}
This is useful to match deeper inside tuples and also to use notations
for the pattern, as the syntax {\tt let 'p := t in b} allows arbitrary
patterns to do the deconstruction. For example:
\begin{coq_example}
Definition deep_tuple (A:Set) (x:(A*A)*(A*A)) : A*A*A*A :=
let '((a,b), (c, d)) := x in (a,b,c,d).
Notation " x 'with' p " := (exist _ x p) (at level 20).
Definition proj1_sig' (A:Set) (P:A->Prop) (t:{ x:A | P x }) : A :=
let 'x with p := t in x.
\end{coq_example}
When printing definitions which are written using this construct it
takes precedence over {\tt let} printing directives for the datatype
under consideration (see Section~\ref{printing-options}).
\subsection{Controlling pretty-printing of {\tt match} expressions
\label{printing-options}}
The following commands give some control over the pretty-printing of
{\tt match} expressions.
\subsubsection{Printing nested patterns
\label{SetPrintingMatching}
\comindex{Set Printing Matching}
\comindex{Unset Printing Matching}
\comindex{Test Printing Matching}}
The Calculus of Inductive Constructions knows pattern-matching only
over simple patterns. It is however convenient to re-factorize nested
pattern-matching into a single pattern-matching over a nested pattern.
{\Coq}'s printer try to do such limited re-factorization.
\begin{quote}
{\tt Set Printing Matching.}
\end{quote}
This tells {\Coq} to try to use nested patterns. This is the default
behavior.
\begin{quote}
{\tt Unset Printing Matching.}
\end{quote}
This tells {\Coq} to print only simple pattern-matching problems in
the same way as the {\Coq} kernel handles them.
\begin{quote}
{\tt Test Printing Matching.}
\end{quote}
This tells if the printing matching mode is on or off. The default is
on.
\subsubsection{Printing of wildcard pattern
\comindex{Set Printing Wildcard}
\comindex{Unset Printing Wildcard}
\comindex{Test Printing Wildcard}}
Some variables in a pattern may not occur in the right-hand side of
the pattern-matching clause. There are options to control the
display of these variables.
\begin{quote}
{\tt Set Printing Wildcard.}
\end{quote}
The variables having no occurrences in the right-hand side of the
pattern-matching clause are just printed using the wildcard symbol
``{\tt \_}''.
\begin{quote}
{\tt Unset Printing Wildcard.}
\end{quote}
The variables, even useless, are printed using their usual name. But some
non dependent variables have no name. These ones are still printed
using a ``{\tt \_}''.
\begin{quote}
{\tt Test Printing Wildcard.}
\end{quote}
This tells if the wildcard printing mode is on or off. The default is
to print wildcard for useless variables.
\subsubsection{Printing of the elimination predicate
\comindex{Set Printing Synth}
\comindex{Unset Printing Synth}
\comindex{Test Printing Synth}}
In most of the cases, the type of the result of a matched term is
mechanically synthesizable. Especially, if the result type does not
depend of the matched term.
\begin{quote}
{\tt Set Printing Synth.}
\end{quote}
The result type is not printed when {\Coq} knows that it can
re-synthesize it.
\begin{quote}
{\tt Unset Printing Synth.}
\end{quote}
This forces the result type to be always printed.
\begin{quote}
{\tt Test Printing Synth.}
\end{quote}
This tells if the non-printing of synthesizable types is on or off.
The default is to not print synthesizable types.
\subsubsection{Printing matching on irrefutable pattern
\comindex{Add Printing Let {\ident}}
\comindex{Remove Printing Let {\ident}}
\comindex{Test Printing Let for {\ident}}
\comindex{Print Table Printing Let}}
If an inductive type has just one constructor,
pattern-matching can be written using {\tt let} ... {\tt :=}
... {\tt in}~...
\begin{quote}
{\tt Add Printing Let {\ident}.}
\end{quote}
This adds {\ident} to the list of inductive types for which
pattern-matching is written using a {\tt let} expression.
\begin{quote}
{\tt Remove Printing Let {\ident}.}
\end{quote}
This removes {\ident} from this list.
\begin{quote}
{\tt Test Printing Let for {\ident}.}
\end{quote}
This tells if {\ident} belongs to the list.
\begin{quote}
{\tt Print Table Printing Let.}
\end{quote}
This prints the list of inductive types for which pattern-matching is
written using a {\tt let} expression.
The list of inductive types for which pattern-matching is written
using a {\tt let} expression is managed synchronously. This means that
it is sensible to the command {\tt Reset}.
\subsubsection{Printing matching on booleans
\comindex{Add Printing If {\ident}}
\comindex{Remove Printing If {\ident}}
\comindex{Test Printing If for {\ident}}
\comindex{Print Table Printing If}}
If an inductive type is isomorphic to the boolean type,
pattern-matching can be written using {\tt if} ... {\tt then} ... {\tt
else} ...
\begin{quote}
{\tt Add Printing If {\ident}.}
\end{quote}
This adds {\ident} to the list of inductive types for which
pattern-matching is written using an {\tt if} expression.
\begin{quote}
{\tt Remove Printing If {\ident}.}
\end{quote}
This removes {\ident} from this list.
\begin{quote}
{\tt Test Printing If for {\ident}.}
\end{quote}
This tells if {\ident} belongs to the list.
\begin{quote}
{\tt Print Table Printing If.}
\end{quote}
This prints the list of inductive types for which pattern-matching is
written using an {\tt if} expression.
The list of inductive types for which pattern-matching is written
using an {\tt if} expression is managed synchronously. This means that
it is sensible to the command {\tt Reset}.
\subsubsection{Example}
This example emphasizes what the printing options offer.
\begin{coq_example}
Test Printing Let for prod.
Print fst.
Remove Printing Let prod.
Unset Printing Synth.
Unset Printing Wildcard.
Print fst.
\end{coq_example}
% \subsection{Still not dead old notations}
% The following variant of {\tt match} is inherited from older version
% of {\Coq}.
% \medskip
% \begin{tabular}{lcl}
% {\term} & ::= & {\annotation} {\tt Match} {\term} {\tt with} {\terms} {\tt end}\\
% \end{tabular}
% \medskip
% This syntax is a macro generating a combination of {\tt match} with {\tt
% Fix} implementing a combinator for primitive recursion equivalent to
% the {\tt Match} construction of \Coq\ V5.8. It is provided only for
% sake of compatibility with \Coq\ V5.8. It is recommended to avoid it.
% (see Section~\ref{Matchexpr}).
% There is also a notation \texttt{Case} that is the
% ancestor of \texttt{match}. Again, it is still in the code for
% compatibility with old versions but the user should not use it.
% Explained in RefMan-gal.tex
%% \section{Forced type}
%% In some cases, one may wish to assign a particular type to a term. The
%% syntax to force the type of a term is the following:
%% \medskip
%% \begin{tabular}{lcl}
%% {\term} & ++= & {\term} {\tt :} {\term}\\
%% \end{tabular}
%% \medskip
%% It forces the first term to be of type the second term. The
%% type must be compatible with
%% the term. More precisely it must be either a type convertible to
%% the automatically inferred type (see Chapter~\ref{Cic}) or a type
%% coercible to it, (see \ref{Coercions}). When the type of a
%% whole expression is forced, it is usually not necessary to give the types of
%% the variables involved in the term.
%% Example:
%% \begin{coq_example}
%% Definition ID := forall X:Set, X -> X.
%% Definition id := (fun X x => x):ID.
%% Check id.
%% \end{coq_example}
\section{Advanced recursive functions}
The \emph{experimental} command
\begin{center}
\texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$}
\{decrease\_annot\} : type$_0$ := \term$_0$}
\comindex{Function}
\label{Function}
\end{center}
can be seen as a generalization of {\tt Fixpoint}. It is actually a
wrapper for several ways of defining a function \emph{and other useful
related objects}, namely: an induction principle that reflects the
recursive structure of the function (see \ref{FunInduction}), and its
fixpoint equality. The meaning of this
declaration is to define a function {\it ident}, similarly to {\tt
Fixpoint}. Like in {\tt Fixpoint}, the decreasing argument must be
given (unless the function is not recursive), but it must not
necessary be \emph{structurally} decreasing. The point of the {\tt
\{\}} annotation is to name the decreasing argument \emph{and} to
describe which kind of decreasing criteria must be used to ensure
termination of recursive calls.
The {\tt Function} construction enjoys also the {\tt with} extension
to define mutually recursive definitions. However, this feature does
not work for non structural recursive functions. % VRAI??
See the documentation of {\tt functional induction}
(see Section~\ref{FunInduction}) and {\tt Functional Scheme}
(see Section~\ref{FunScheme} and \ref{FunScheme-examples}) for how to use the
induction principle to easily reason about the function.
\noindent {\bf Remark: } To obtain the right principle, it is better
to put rigid parameters of the function as first arguments. For
example it is better to define plus like this:
\begin{coq_example*}
Function plus (m n : nat) {struct n} : nat :=
match n with
| 0 => m
| S p => S (plus m p)
end.
\end{coq_example*}
\noindent than like this:
\begin{coq_eval}
Reset plus.
\end{coq_eval}
\begin{coq_example*}
Function plus (n m : nat) {struct n} : nat :=
match n with
| 0 => m
| S p => S (plus p m)
end.
\end{coq_example*}
\paragraph[Limitations]{Limitations\label{sec:Function-limitations}}
\term$_0$ must be build as a \emph{pure pattern-matching tree}
(\texttt{match...with}) with applications only \emph{at the end} of
each branch. For now dependent cases are not treated.
\begin{ErrMsgs}
\item \errindex{The recursive argument must be specified}
\item \errindex{No argument name \ident}
\item \errindex{Cannot use mutual definition with well-founded
recursion or measure}
\item \errindex{Cannot define graph for \ident\dots} (warning)
The generation of the graph relation \texttt{(R\_\ident)} used to
compute the induction scheme of \ident\ raised a typing error. Only
the ident is defined, the induction scheme will not be generated.
This error happens generally when:
\begin{itemize}
\item the definition uses pattern matching on dependent types, which
\texttt{Function} cannot deal with yet.
\item the definition is not a \emph{pattern-matching tree} as
explained above.
\end{itemize}
\item \errindex{Cannot define principle(s) for \ident\dots} (warning)
The generation of the graph relation \texttt{(R\_\ident)} succeeded
but the induction principle could not be built. Only the ident is
defined. Please report.
\item \errindex{Cannot build functional inversion principle} (warning)
\texttt{functional inversion} will not be available for the
function.
\end{ErrMsgs}
\SeeAlso{\ref{FunScheme}, \ref{FunScheme-examples}, \ref{FunInduction}}
Depending on the {\tt \{$\ldots$\}} annotation, different definition
mechanisms are used by {\tt Function}. More precise description
given below.
\begin{Variants}
\item \texttt{ Function {\ident} {\binder$_1$}\ldots{\binder$_n$}
: type$_0$ := \term$_0$}
Defines the not recursive function \ident\ as if declared with
\texttt{Definition}. Moreover the following are defined:
\begin{itemize}
\item {\tt\ident\_rect}, {\tt\ident\_rec} and {\tt\ident\_ind},
which reflect the pattern matching structure of \term$_0$ (see the
documentation of {\tt Inductive} \ref{Inductive});
\item The inductive \texttt{R\_\ident} corresponding to the graph of
\ident\ (silently);
\item \texttt{\ident\_complete} and \texttt{\ident\_correct} which are
inversion information linking the function and its graph.
\end{itemize}
\item \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$}
{\tt \{}{\tt struct} \ident$_0${\tt\}} : type$_0$ := \term$_0$}
Defines the structural recursive function \ident\ as if declared
with \texttt{Fixpoint}. Moreover the following are defined:
\begin{itemize}
\item The same objects as above;
\item The fixpoint equation of \ident: \texttt{\ident\_equation}.
\end{itemize}
\item \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$} {\tt
\{}{\tt measure \term$_1$} \ident$_0${\tt\}} : type$_0$ :=
\term$_0$}
\item \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$}
{\tt \{}{\tt wf \term$_1$} \ident$_0${\tt\}} : type$_0$ := \term$_0$}
Defines a recursive function by well founded recursion. \textbf{The
module \texttt{Recdef} of the standard library must be loaded for this
feature}. The {\tt \{\}} annotation is mandatory and must be one of
the following:
\begin{itemize}
\item {\tt \{measure} \term$_1$ \ident$_0${\tt\}} with \ident$_0$
being the decreasing argument and \term$_1$ being a function
from type of \ident$_0$ to \texttt{nat} for which value on the
decreasing argument decreases (for the {\tt lt} order on {\tt
nat}) at each recursive call of \term$_0$, parameters of the
function are bound in \term$_0$;
\item {\tt \{wf} \term$_1$ \ident$_0${\tt\}} with \ident$_0$ being
the decreasing argument and \term$_1$ an ordering relation on
the type of \ident$_0$ (i.e. of type T$_{\ident_0}$
$\to$ T$_{\ident_0}$ $\to$ {\tt Prop}) for which
the decreasing argument decreases at each recursive call of
\term$_0$. The order must be well founded. parameters of the
function are bound in \term$_0$.
\end{itemize}
Depending on the annotation, the user is left with some proof
obligations that will be used to define the function. These proofs
are: proofs that each recursive call is actually decreasing with
respect to the given criteria, and (if the criteria is \texttt{wf}) a
proof that the ordering relation is well founded.
%Completer sur measure et wf
Once proof obligations are discharged, the following objects are
defined:
\begin{itemize}
\item The same objects as with the \texttt{struct};
\item The lemma \texttt{\ident\_tcc} which collects all proof
obligations in one property;
\item The lemmas \texttt{\ident\_terminate} and \texttt{\ident\_F}
which is needed to be inlined during extraction of \ident.
\end{itemize}
%Complete!!
The way this recursive function is defined is the subject of several
papers by Yves Bertot and Antonia Balaa on the one hand, and Gilles Barthe,
Julien Forest, David Pichardie, and Vlad Rusu on the other hand.
%Exemples ok ici
\bigskip
\noindent {\bf Remark: } Proof obligations are presented as several
subgoals belonging to a Lemma {\ident}{\tt\_tcc}. % These subgoals are independent which means that in order to
% abort them you will have to abort each separately.
%The decreasing argument cannot be dependent of another??
%Exemples faux ici
\end{Variants}
\section{Section mechanism
\index{Sections}
\label{Section}}
The sectioning mechanism allows to organize a proof in structured
sections. Then local declarations become available (see
Section~\ref{Simpl-definitions}).
\subsection{\tt Section {\ident}\comindex{Section}}
This command is used to open a section named {\ident}.
%% Discontinued ?
%% \begin{Variants}
%% \comindex{Chapter}
%% \item{\tt Chapter {\ident}}\\
%% Same as {\tt Section {\ident}}
%% \end{Variants}
\subsection{\tt End {\ident}
\comindex{End}}
This command closes the section named {\ident}. When a section is
closed, all local declarations (variables and local definitions) are
{\em discharged}. This means that all global objects defined in the
section are generalized with respect to all variables and local
definitions it depends on in the section. None of the local
declarations (considered as autonomous declarations) survive the end
of the section.
Here is an example :
\begin{coq_example}
Section s1.
Variables x y : nat.
Let y' := y.
Definition x' := S x.
Definition x'' := x' + y'.
Print x'.
End s1.
Print x'.
Print x''.
\end{coq_example}
Notice the difference between the value of {\tt x'} and {\tt x''}
inside section {\tt s1} and outside.
\begin{ErrMsgs}
\item \errindex{This is not the last opened section}
\end{ErrMsgs}
\begin{Remarks}
\item Most commands, like {\tt Hint}, {\tt Notation}, option management, ...
which appear inside a section are canceled when the
section is closed.
% see Section~\ref{LongNames}
%\item Usually all identifiers must be distinct.
%However, a name already used in a closed section (see \ref{Section})
%can be reused. In this case, the old name is no longer accessible.
% Obsolète
%\item A module implicitly open a section. Be careful not to name a
%module with an identifier already used in the module (see \ref{compiled}).
\end{Remarks}
\input{RefMan-mod.v}
\section{Libraries and qualified names}
\subsection{Names of libraries and files
\label{Libraries}
\index{Libraries}
\index{Physical paths}
\index{Logical paths}}
\paragraph{Libraries}
The theories developed in {\Coq} are stored in {\em library files}
which are hierarchically classified into {\em libraries} and {\em
sublibraries}. To express this hierarchy, library names are
represented by qualified identifiers {\qualid}, i.e. as list of
identifiers separated by dots (see Section~\ref{qualid}). For
instance, the library file {\tt Mult} of the standard {\Coq} library
{\tt Arith} has name {\tt Coq.Arith.Mult}. The identifier
that starts the name of a library is called a {\em library root}.
All library files of the standard library of {\Coq} have reserved root
{\tt Coq} but library file names based on other roots can be obtained
by using {\tt coqc} options {\tt -I} or {\tt -R} (see
Section~\ref{coqoptions}). Also, when an interactive {\Coq} session
starts, a library of root {\tt Top} is started, unless option {\tt
-top} or {\tt -notop} is set (see Section~\ref{coqoptions}).
As library files are stored on the file system of the underlying
operating system, a translation from file-system names to {\Coq} names
is needed. In this translation, names in the file system are called
{\em physical} paths while {\Coq} names are contrastingly called {\em
logical} names. Logical names are mapped to physical paths using the
commands {\tt Add LoadPath} or {\tt Add Rec LoadPath} (see
Sections~\ref{AddLoadPath} and~\ref{AddRecLoadPath}).
\subsection{Qualified names
\label{LongNames}
\index{Qualified identifiers}
\index{Absolute names}}
Library files are modules which possibly contain submodules which
eventually contain constructions (axioms, parameters, definitions,
lemmas, theorems, remarks or facts). The {\em absolute name}, or {\em
full name}, of a construction in some library file is a qualified
identifier starting with the logical name of the library file,
followed by the sequence of submodules names encapsulating the
construction and ended by the proper name of the construction.
Typically, the absolute name {\tt Coq.Init.Logic.eq} denotes Leibniz'
equality defined in the module {\tt Logic} in the sublibrary {\tt
Init} of the standard library of \Coq.
The proper name that ends the name of a construction is the {\it short
name} (or sometimes {\it base name}) of the construction (for
instance, the short name of {\tt Coq.Init.Logic.eq} is {\tt eq}). Any
partial suffix of the absolute name is a {\em partially qualified name}
(e.g. {\tt Logic.eq} is a partially qualified name for {\tt
Coq.Init.Logic.eq}). Especially, the short name of a construction is
its shortest partially qualified name.
{\Coq} does not accept two constructions (definition, theorem, ...)
with the same absolute name but different constructions can have the
same short name (or even same partially qualified names as soon as the
full names are different).
Notice that the notion of absolute, partially qualified and
short names also applies to library file names.
\paragraph{Visibility}
{\Coq} maintains a table called {\it name table} which maps partially
qualified names of constructions to absolute names. This table is
updated by the commands {\tt Require} (see \ref{Require}), {\tt
Import} and {\tt Export} (see \ref{Import}) and also each time a new
declaration is added to the context. An absolute name is called {\it
visible} from a given short or partially qualified name when this
latter name is enough to denote it. This means that the short or
partially qualified name is mapped to the absolute name in {\Coq} name
table.
A similar table exists for library file names. It is updated by the
vernacular commands {\tt Add LoadPath} and {\tt Add Rec LoadPath} (or
their equivalent as options of the {\Coq} executables, {\tt -I} and
{\tt -R}).
It may happen that a visible name is hidden by the short name or a
qualified name of another construction. In this case, the name that
has been hidden must be referred to using one more level of
qualification. To ensure that a construction always remains
accessible, absolute names can never be hidden.
Examples:
\begin{coq_eval}
Reset Initial.
\end{coq_eval}
\begin{coq_example}
Check 0.
Definition nat := bool.
Check 0.
Check Datatypes.nat.
Locate nat.
\end{coq_example}
\SeeAlso Command {\tt Locate} in Section~\ref{Locate} and {\tt Locate
Library} in Section~\ref{Locate Library}.
%% \paragraph{The special case of remarks and facts}
%%
%% In contrast with definitions, lemmas, theorems, axioms and parameters,
%% the absolute name of remarks includes the segment of sections in which
%% it is defined. Concretely, if a remark {\tt R} is defined in
%% subsection {\tt S2} of section {\tt S1} in module {\tt M}, then its
%% absolute name is {\tt M.S1.S2.R}. The same for facts, except that the
%% name of the innermost section is dropped from the full name. Then, if
%% a fact {\tt F} is defined in subsection {\tt S2} of section {\tt S1}
%% in module {\tt M}, then its absolute name is {\tt M.S1.F}.
\section{Implicit arguments
\index{Implicit arguments}
\label{Implicit Arguments}}
An implicit argument of a function is an argument which can be
inferred from contextual knowledge. There are different kinds of
implicit arguments that can be considered implicit in different
ways. There are also various commands to control the setting or the
inference of implicit arguments.
\subsection{The different kinds of implicit arguments}
\subsubsection{Implicit arguments inferable from the knowledge of other
arguments of a function}
The first kind of implicit arguments covers the arguments that are
inferable from the knowledge of the type of other arguments of the
function, or of the type of the surrounding context of the
application. Especially, such implicit arguments correspond to
parameters dependent in the type of the function. Typical implicit
arguments are the type arguments in polymorphic functions.
There are several kinds of such implicit arguments.
\paragraph{Strict Implicit Arguments.}
An implicit argument can be either strict or non strict. An implicit
argument is said {\em strict} if, whatever the other arguments of the
function are, it is still inferable from the type of some other
argument. Technically, an implicit argument is strict if it
corresponds to a parameter which is not applied to a variable which
itself is another parameter of the function (since this parameter
may erase its arguments), not in the body of a {\tt match}, and not
itself applied or matched against patterns (since the original
form of the argument can be lost by reduction).
For instance, the first argument of
\begin{quote}
\verb|cons: forall A:Set, A -> list A -> list A|
\end{quote}
in module {\tt List.v} is strict because {\tt list} is an inductive
type and {\tt A} will always be inferable from the type {\tt
list A} of the third argument of {\tt cons}.
On the contrary, the second argument of a term of type
\begin{quote}
\verb|forall P:nat->Prop, forall n:nat, P n -> ex nat P|
\end{quote}
is implicit but not strict, since it can only be inferred from the
type {\tt P n} of the third argument and if {\tt P} is, e.g., {\tt
fun \_ => True}, it reduces to an expression where {\tt n} does not
occur any longer. The first argument {\tt P} is implicit but not
strict either because it can only be inferred from {\tt P n} and {\tt
P} is not canonically inferable from an arbitrary {\tt n} and the
normal form of {\tt P n} (consider e.g. that {\tt n} is {\tt 0} and
the third argument has type {\tt True}, then any {\tt P} of the form
{\tt fun n => match n with 0 => True | \_ => \mbox{\em anything} end} would
be a solution of the inference problem).
\paragraph{Contextual Implicit Arguments.}
An implicit argument can be {\em contextual} or not. An implicit
argument is said {\em contextual} if it can be inferred only from the
knowledge of the type of the context of the current expression. For
instance, the only argument of
\begin{quote}
\verb|nil : forall A:Set, list A|
\end{quote}
is contextual. Similarly, both arguments of a term of type
\begin{quote}
\verb|forall P:nat->Prop, forall n:nat, P n \/ n = 0|
\end{quote}
are contextual (moreover, {\tt n} is strict and {\tt P} is not).
\paragraph{Reversible-Pattern Implicit Arguments.}
There is another class of implicit arguments that can be reinferred
unambiguously if all the types of the remaining arguments are
known. This is the class of implicit arguments occurring in the type
of another argument in position of reversible pattern, which means it
is at the head of an application but applied only to uninstantiated
distinct variables. Such an implicit argument is called {\em
reversible-pattern implicit argument}. A typical example is the
argument {\tt P} of {\tt nat\_rec} in
\begin{quote}
{\tt nat\_rec : forall P : nat -> Set,
P 0 -> (forall n : nat, P n -> P (S n)) -> forall x : nat, P x}.
\end{quote}
({\tt P} is reinferable by abstracting over {\tt n} in the type {\tt P n}).
See Section~\ref{SetReversiblePatternImplicit} for the automatic declaration
of reversible-pattern implicit arguments.
\subsubsection{Implicit arguments inferable by resolution}
This corresponds to a class of non dependent implicit arguments that
are solved based on the structure of their type only.
\subsection{Maximal or non maximal insertion of implicit arguments}
In case a function is partially applied, and the next argument to be
applied is an implicit argument, two disciplines are applicable. In the
first case, the function is considered to have no arguments furtherly:
one says that the implicit argument is not maximally inserted. In
the second case, the function is considered to be implicitly applied
to the implicit arguments it is waiting for: one says that the
implicit argument is maximally inserted.
Each implicit argument can be declared to have to be inserted
maximally or non maximally. This can be governed argument per argument
by the command {\tt Implicit Arguments} (see~\ref{ImplicitArguments})
or globally by the command {\tt Set Maximal Implicit Insertion}
(see~\ref{SetMaximalImplicitInsertion}). See also
Section~\ref{PrintImplicit}.
\subsection{Casual use of implicit arguments}
In a given expression, if it is clear that some argument of a function
can be inferred from the type of the other arguments, the user can
force the given argument to be guessed by replacing it by ``{\tt \_}''. If
possible, the correct argument will be automatically generated.
\begin{ErrMsgs}
\item \errindex{Cannot infer a term for this placeholder}
{\Coq} was not able to deduce an instantiation of a ``{\tt \_}''.
\end{ErrMsgs}
\subsection{Declaration of implicit arguments for a constant
\comindex{Implicit Arguments}}
\label{ImplicitArguments}
In case one wants that some arguments of a given object (constant,
inductive types, constructors, assumptions, local or not) are always
inferred by Coq, one may declare once and for all which are the expected
implicit arguments of this object. There are two ways to do this,
a-priori and a-posteriori.
\subsubsection{Implicit Argument Binders}
In the first setting, one wants to explicitly give the implicit
arguments of a constant as part of its definition. To do this, one has
to surround the bindings of implicit arguments by curly braces:
\begin{coq_eval}
Reset Initial.
\end{coq_eval}
\begin{coq_example}
Definition id {A : Type} (x : A) : A := x.
\end{coq_example}
This automatically declares the argument {\tt A} of {\tt id} as a
maximally inserted implicit argument. One can then do as-if the argument
was absent in every situation but still be able to specify it if needed:
\begin{coq_example}
Definition compose {A B C} (g : B -> C) (f : A -> B) :=
fun x => g (f x).
Goal forall A, compose id id = id (A:=A).
\end{coq_example}
The syntax is supported in all top-level definitions: {\tt Definition},
{\tt Fixpoint}, {\tt Lemma} and so on. For (co-)inductive datatype
declarations, the semantics are the following: an inductive parameter
declared as an implicit argument need not be repeated in the inductive
definition but will become implicit for the constructors of the
inductive only, not the inductive type itself. For example:
\begin{coq_example}
Inductive list {A : Type} : Type :=
| nil : list
| cons : A -> list -> list.
Print list.
\end{coq_example}
One can always specify the parameter if it is not uniform using the
usual implicit arguments disambiguation syntax.
\subsubsection{The Implicit Arguments Vernacular Command}
To set implicit arguments for a constant a-posteriori, one can use the
command:
\begin{quote}
\tt Implicit Arguments {\qualid} [ \nelist{\possiblybracketedident}{} ]
\end{quote}
where the list of {\possiblybracketedident} is the list of parameters
to be declared implicit, each of the identifier of the list being
optionally surrounded by square brackets, then meaning that this
parameter has to be maximally inserted.
After the above declaration is issued, implicit arguments can just (and
have to) be skipped in any expression involving an application of
{\qualid}.
\begin{Variants}
\item {\tt Global Implicit Arguments {\qualid} [ \nelist{\possiblybracketedident}{} ]
\comindex{Global Implicit Arguments}}
Tells to recompute the implicit arguments of {\qualid} after ending of
the current section if any, enforcing the implicit arguments known
from inside the section to be the ones declared by the command.
\item {\tt Local Implicit Arguments {\qualid} [ \nelist{\possiblybracketedident}{} ]
\comindex{Local Implicit Arguments}}
When in a module, tells not to activate the implicit arguments of
{\qualid} declared by this commands to contexts that requires the
module.
\end{Variants}
\Example
\begin{coq_eval}
Reset Initial.
\end{coq_eval}
\begin{coq_example*}
Inductive list (A:Type) : Type :=
| nil : list A
| cons : A -> list A -> list A.
\end{coq_example*}
\begin{coq_example}
Check (cons nat 3 (nil nat)).
Implicit Arguments cons [A].
Implicit Arguments nil [A].
Check (cons 3 nil).
Fixpoint map (A B:Type) (f:A->B) (l:list A) : list B :=
match l with nil => nil | cons a t => cons (f a) (map A B f t) end.
Fixpoint length (A:Type) (l:list A) : nat :=
match l with nil => 0 | cons _ m => S (length A m) end.
Implicit Arguments map [A B].
Implicit Arguments length [[A]]. (* A has to be maximally inserted *)
Check (fun l:list (list nat) => map length l).
\end{coq_example}
\Rem To know which are the implicit arguments of an object, use the command
{\tt Print Implicit} (see \ref{PrintImplicit}).
\Rem If the list of arguments is empty, the command removes the
implicit arguments of {\qualid}.
\subsection{Automatic declaration of implicit arguments for a constant}
{\Coq} can also automatically detect what are the implicit arguments
of a defined object. The command is just
\begin{quote}
{\tt Implicit Arguments {\qualid}
\comindex{Implicit Arguments}}
\end{quote}
The auto-detection is governed by options telling if strict,
contextual, or reversible-pattern implicit arguments must be
considered or not (see
Sections~\ref{SetStrictImplicit},~\ref{SetContextualImplicit},~\ref{SetReversiblePatternImplicit}
and also~\ref{SetMaximalImplicitInsertion}).
\begin{Variants}
\item {\tt Global Implicit Arguments {\qualid}
\comindex{Global Implicit Arguments}}
Tells to recompute the implicit arguments of {\qualid} after ending of
the current section if any.
\item {\tt Local Implicit Arguments {\qualid}
\comindex{Local Implicit Arguments}}
When in a module, tells not to activate the implicit arguments of
{\qualid} computed by this declaration to contexts that requires the
module.
\end{Variants}
\Example
\begin{coq_eval}
Reset Initial.
\end{coq_eval}
\begin{coq_example*}
Inductive list (A:Set) : Set :=
| nil : list A
| cons : A -> list A -> list A.
\end{coq_example*}
\begin{coq_example}
Implicit Arguments cons.
Print Implicit cons.
Implicit Arguments nil.
Print Implicit nil.
Set Contextual Implicit.
Implicit Arguments nil.
Print Implicit nil.
\end{coq_example}
The computation of implicit arguments takes account of the
unfolding of constants. For instance, the variable {\tt p} below has
type {\tt (Transitivity R)} which is reducible to {\tt forall x,y:U, R x
y -> forall z:U, R y z -> R x z}. As the variables {\tt x}, {\tt y} and
{\tt z} appear strictly in body of the type, they are implicit.
\begin{coq_example*}
Variable X : Type.
Definition Relation := X -> X -> Prop.
Definition Transitivity (R:Relation) :=
forall x y:X, R x y -> forall z:X, R y z -> R x z.
Variables (R : Relation) (p : Transitivity R).
Implicit Arguments p.
\end{coq_example*}
\begin{coq_example}
Print p.
Print Implicit p.
\end{coq_example}
\begin{coq_example*}
Variables (a b c : X) (r1 : R a b) (r2 : R b c).
\end{coq_example*}
\begin{coq_example}
Check (p r1 r2).
\end{coq_example}
\subsection{Mode for automatic declaration of implicit arguments
\label{Auto-implicit}
\comindex{Set Implicit Arguments}
\comindex{Unset Implicit Arguments}}
In case one wants to systematically declare implicit the arguments
detectable as such, one may switch to the automatic declaration of
implicit arguments mode by using the command
\begin{quote}
\tt Set Implicit Arguments.
\end{quote}
Conversely, one may unset the mode by using {\tt Unset Implicit
Arguments}. The mode is off by default. Auto-detection of implicit
arguments is governed by options controlling whether strict and
contextual implicit arguments have to be considered or not.
\subsection{Controlling strict implicit arguments
\comindex{Set Strict Implicit}
\comindex{Unset Strict Implicit}
\label{SetStrictImplicit}}
When the mode for automatic declaration of implicit arguments is on,
the default is to automatically set implicit only the strict implicit
arguments plus, for historical reasons, a small subset of the non
strict implicit arguments. To relax this constraint and to
set implicit all non strict implicit arguments by default, use the command
\begin{quote}
\tt Unset Strict Implicit.
\end{quote}
Conversely, use the command {\tt Set Strict Implicit} to
restore the original mode that declares implicit only the strict implicit arguments plus a small subset of the non strict implicit arguments.
In the other way round, to capture exactly the strict implicit arguments and no more than the strict implicit arguments, use the command:
\comindex{Set Strongly Strict Implicit}
\comindex{Unset Strongly Strict Implicit}
\begin{quote}
\tt Set Strongly Strict Implicit.
\end{quote}
Conversely, use the command {\tt Unset Strongly Strict Implicit} to
let the option ``{\tt Strict Implicit}'' decide what to do.
\Rem In versions of {\Coq} prior to version 8.0, the default was to
declare the strict implicit arguments as implicit.
\subsection{Controlling contextual implicit arguments
\comindex{Set Contextual Implicit}
\comindex{Unset Contextual Implicit}
\label{SetContextualImplicit}}
By default, {\Coq} does not automatically set implicit the contextual
implicit arguments. To tell {\Coq} to infer also contextual implicit
argument, use command
\begin{quote}
\tt Set Contextual Implicit.
\end{quote}
Conversely, use command {\tt Unset Contextual Implicit} to
unset the contextual implicit mode.
\subsection{Controlling reversible-pattern implicit arguments
\comindex{Set Reversible Pattern Implicit}
\comindex{Unset Reversible Pattern Implicit}
\label{SetReversiblePatternImplicit}}
By default, {\Coq} does not automatically set implicit the reversible-pattern
implicit arguments. To tell {\Coq} to infer also reversible-pattern implicit
argument, use command
\begin{quote}
\tt Set Reversible Pattern Implicit.
\end{quote}
Conversely, use command {\tt Unset Reversible Pattern Implicit} to
unset the reversible-pattern implicit mode.
\subsection{Controlling the insertion of implicit arguments not followed by explicit arguments
\comindex{Set Maximal Implicit Insertion}
\comindex{Unset Maximal Implicit Insertion}
\label{SetMaximalImplicitInsertion}}
Implicit arguments can be declared to be automatically inserted when a
function is partially applied and the next argument of the function is
an implicit one. In case the implicit arguments are automatically
declared (with the command {\tt Set Implicit Arguments}), the command
\begin{quote}
\tt Set Maximal Implicit Insertion.
\end{quote}
is used to tell to declare the implicit arguments with a maximal
insertion status. By default, automatically declared implicit
arguments are not declared to be insertable maximally. To restore the
default mode for maximal insertion, use command {\tt Unset Maximal
Implicit Insertion}.
\subsection{Explicit applications
\index{Explicitly given implicit arguments}
\label{Implicits-explicitation}
\index{qualid@{\qualid}}}
In presence of non strict or contextual argument, or in presence of
partial applications, the synthesis of implicit arguments may fail, so
one may have to give explicitly certain implicit arguments of an
application. The syntax for this is {\tt (\ident:=\term)} where {\ident}
is the name of the implicit argument and {\term} is its corresponding
explicit term. Alternatively, one can locally deactivate the hiding of
implicit arguments of a function by using the notation
{\tt @{\qualid}~{\term}$_1$..{\term}$_n$}. This syntax extension is
given Figure~\ref{fig:explicitations}.
\begin{figure}
\begin{centerframe}
\begin{tabular}{lcl}
{\term} & ++= & @ {\qualid} \nelist{\term}{}\\
& $|$ & @ {\qualid}\\
& $|$ & {\qualid} \nelist{\textrm{\textsl{argument}}}{}\\
\\
{\textrm{\textsl{argument}}} & ::= & {\term} \\
& $|$ & {\tt ({\ident}:={\term})}\\
\end{tabular}
\end{centerframe}
\caption{Syntax for explicitly giving implicit arguments}
\label{fig:explicitations}
\end{figure}
\noindent {\bf Example (continued): }
\begin{coq_example}
Check (p r1 (z:=c)).
Check (p (x:=a) (y:=b) r1 (z:=c) r2).
\end{coq_example}
\subsection{Displaying what the implicit arguments are
\comindex{Print Implicit}
\label{PrintImplicit}}
To display the implicit arguments associated to an object, and to know
if each of them is to be used maximally or not, use the command
\begin{quote}
\tt Print Implicit {\qualid}.
\end{quote}
\subsection{Explicit displaying of implicit arguments for pretty-printing
\comindex{Set Printing Implicit}
\comindex{Unset Printing Implicit}
\comindex{Set Printing Implicit Defensive}
\comindex{Unset Printing Implicit Defensive}}
By default the basic pretty-printing rules hide the inferable implicit
arguments of an application. To force printing all implicit arguments,
use command
\begin{quote}
{\tt Set Printing Implicit.}
\end{quote}
Conversely, to restore the hiding of implicit arguments, use command
\begin{quote}
{\tt Unset Printing Implicit.}
\end{quote}
By default the basic pretty-printing rules display the implicit arguments that are not detected as strict implicit arguments. This ``defensive'' mode can quickly make the display cumbersome so this can be deactivated by using the command
\begin{quote}
{\tt Unset Printing Implicit Defensive.}
\end{quote}
Conversely, to force the display of non strict arguments, use command
\begin{quote}
{\tt Set Printing Implicit Defensive.}
\end{quote}
\SeeAlso {\tt Set Printing All} in Section~\ref{SetPrintingAll}.
\subsection{Interaction with subtyping}
When an implicit argument can be inferred from the type of more than
one of the other arguments, then only the type of the first of these
arguments is taken into account, and not an upper type of all of
them. As a consequence, the inference of the implicit argument of
``='' fails in
\begin{coq_example*}
Check nat = Prop.
\end{coq_example*}
but succeeds in
\begin{coq_example*}
Check Prop = nat.
\end{coq_example*}
\subsection{Canonical structures
\comindex{Canonical Structure}}
A canonical structure is an instance of a record/structure type that
can be used to solve equations involving implicit arguments. Assume
that {\qualid} denotes an object $(Build\_struc~ c_1~ \ldots~ c_n)$ in the
structure {\em struct} of which the fields are $x_1$, ...,
$x_n$. Assume that {\qualid} is declared as a canonical structure
using the command
\begin{quote}
{\tt Canonical Structure {\qualid}.}
\end{quote}
Then, each time an equation of the form $(x_i~
\_)=_{\beta\delta\iota\zeta}c_i$ has to be solved during the
type-checking process, {\qualid} is used as a solution. Otherwise
said, {\qualid} is canonically used to extend the field $c_i$ into a
complete structure built on $c_i$.
Canonical structures are particularly useful when mixed with
coercions and strict implicit arguments. Here is an example.
\begin{coq_example*}
Require Import Relations.
Require Import EqNat.
Set Implicit Arguments.
Unset Strict Implicit.
Structure Setoid : Type :=
{Carrier :> Set;
Equal : relation Carrier;
Prf_equiv : equivalence Carrier Equal}.
Definition is_law (A B:Setoid) (f:A -> B) :=
forall x y:A, Equal x y -> Equal (f x) (f y).
Axiom eq_nat_equiv : equivalence nat eq_nat.
Definition nat_setoid : Setoid := Build_Setoid eq_nat_equiv.
Canonical Structure nat_setoid.
\end{coq_example*}
Thanks to \texttt{nat\_setoid} declared as canonical, the implicit
arguments {\tt A} and {\tt B} can be synthesized in the next statement.
\begin{coq_example}
Lemma is_law_S : is_law S.
\end{coq_example}
\Rem If a same field occurs in several canonical structure, then
only the structure declared first as canonical is considered.
\begin{Variants}
\item {\tt Canonical Structure {\ident} := {\term} : {\type}.}\\
{\tt Canonical Structure {\ident} := {\term}.}\\
{\tt Canonical Structure {\ident} : {\type} := {\term}.}
These are equivalent to a regular definition of {\ident} followed by
the declaration
{\tt Canonical Structure {\ident}}.
\end{Variants}
\SeeAlso more examples in user contribution \texttt{category}
(\texttt{Rocq/ALGEBRA}).
\subsubsection{Print Canonical Projections.
\comindex{Print Canonical Projections}}
This displays the list of global names that are components of some
canonical structure. For each of them, the canonical structure of
which it is a projection is indicated. For instance, the above example
gives the following output:
\begin{coq_example}
Print Canonical Projections.
\end{coq_example}
\subsection{Implicit types of variables}
\comindex{Implicit Types}
It is possible to bind variable names to a given type (e.g. in a
development using arithmetic, it may be convenient to bind the names
{\tt n} or {\tt m} to the type {\tt nat} of natural numbers). The
command for that is
\begin{quote}
\tt Implicit Types \nelist{\ident}{} : {\type}
\end{quote}
The effect of the command is to automatically set the type of bound
variables starting with {\ident} (either {\ident} itself or
{\ident} followed by one or more single quotes, underscore or digits)
to be {\type} (unless the bound variable is already declared with an
explicit type in which case, this latter type is considered).
\Example
\begin{coq_example}
Require Import List.
Implicit Types m n : nat.
Lemma cons_inj_nat : forall m n l, n :: l = m :: l -> n = m.
intros m n.
Lemma cons_inj_bool : forall (m n:bool) l, n :: l = m :: l -> n = m.
\end{coq_example}
\begin{Variants}
\item {\tt Implicit Type {\ident} : {\type}}\\
This is useful for declaring the implicit type of a single variable.
\end{Variants}
\subsection{Implicit generalization
\label{implicit-generalization}
\comindex{Generalizable Variables}}
Implicit generalization is an automatic elaboration of a statement with
free variables into a closed statement where these variables are
quantified explicitly. Implicit generalization is done inside binders
starting with a \verb|`| and terms delimited by \verb|`{ }| and
\verb|`( )|, always introducing maximally inserted implicit arguments for
the generalized variables. Inside implicit generalization
delimiters, free variables in the current context are automatically
quantified using a product or a lambda abstraction to generate a closed
term. In the following statement for example, the variables \texttt{n}
and \texttt{m} are autamatically generalized and become explicit
arguments of the lemma as we are using \verb|`( )|:
\begin{coq_example}
Generalizable All Variables.
Lemma nat_comm : `(n = n + 0).
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}
One can control the set of generalizable identifiers with the
\texttt{Generalizable} vernacular command to avoid unexpected
generalizations when mistyping identifiers. There are three variants of
the command:
\begin{quote}
{\tt Generalizable (All|No) Variable(s)? ({\ident$_1$ \ident$_n$})?.}
\end{quote}
\begin{Variants}
\item {\tt Generalizable All Variables.} All variables are candidate for
generalization if they appear free in the context under a
generalization delimiter. This may result in confusing errors in
case of typos. In such cases, the context will probably contain some
unexpected generalized variable.
\item {\tt Generalizable No Variables.} Disable implicit generalization
entirely. This is the default behavior.
\item {\tt Generalizable Variable(s)? {\ident$_1$ \ident$_n$}.}
Allow generalization of the given identifiers only. Calling this
command multiple times adds to the allowed identifiers.
\item {\tt Global Generalizable} Allows to export the choice of
generalizable variables.
\end{Variants}
One can also use implicit generalization for binders, in which case the
generalized variables are added as binders and set maximally implicit.
\begin{coq_example*}
Definition id `(x : A) : A := x.
\end{coq_example*}
\begin{coq_example}
Print id.
\end{coq_example}
The generalizing binders \verb|`{ }| and \verb|`( )| work similarly to
their explicit counterparts, only binding the generalized variables
implicitly, as maximally-inserted arguments. In these binders, the
binding name for the bound object is optional, whereas the type is
mandatory, dually to regular binders.
\section{Coercions
\label{Coercions}
\index{Coercions}}
Coercions can be used to implicitly inject terms from one {\em class} in
which they reside into another one. A {\em class} is either a sort
(denoted by the keyword {\tt Sortclass}), a product type (denoted by the
keyword {\tt Funclass}), or a type constructor (denoted by its name),
e.g. an inductive type or any constant with a type of the form
\texttt{forall} $(x_1:A_1) .. (x_n:A_n),~s$ where $s$ is a sort.
Then the user is able to apply an
object that is not a function, but can be coerced to a function, and
more generally to consider that a term of type A is of type B provided
that there is a declared coercion between A and B. The main command is
\comindex{Coercion}
\begin{quote}
\tt Coercion {\qualid} : {\class$_1$} >-> {\class$_2$}.
\end{quote}
which declares the construction denoted by {\qualid} as a
coercion between {\class$_1$} and {\class$_2$}.
More details and examples, and a description of the commands related
to coercions are provided in Chapter~\ref{Coercions-full}.
\section[Printing constructions in full]{Printing constructions in full\label{SetPrintingAll}
\comindex{Set Printing All}
\comindex{Unset Printing All}}
Coercions, implicit arguments, the type of pattern-matching, but also
notations (see Chapter~\ref{Addoc-syntax}) can obfuscate the behavior
of some tactics (typically the tactics applying to occurrences of
subterms are sensitive to the implicit arguments). The command
\begin{quote}
{\tt Set Printing All.}
\end{quote}
deactivates all high-level printing features such as coercions,
implicit arguments, returned type of pattern-matching, notations and
various syntactic sugar for pattern-matching or record projections.
Otherwise said, {\tt Set Printing All} includes the effects
of the commands {\tt Set Printing Implicit}, {\tt Set Printing
Coercions}, {\tt Set Printing Synth}, {\tt Unset Printing Projections}
and {\tt Unset Printing Notations}. To reactivate the high-level
printing features, use the command
\begin{quote}
{\tt Unset Printing All.}
\end{quote}
\section[Printing universes]{Printing universes\label{PrintingUniverses}
\comindex{Set Printing Universes}
\comindex{Unset Printing Universes}}
The following command:
\begin{quote}
{\tt Set Printing Universes}
\end{quote}
activates the display of the actual level of each occurrence of
{\Type}. See Section~\ref{Sorts} for details. This wizard option, in
combination with \texttt{Set Printing All} (see
section~\ref{SetPrintingAll}) can help to diagnose failures to unify
terms apparently identical but internally different in the Calculus of
Inductive Constructions. To reactivate the display of the actual level
of the occurrences of {\Type}, use
\begin{quote}
{\tt Unset Printing Universes.}
\end{quote}
\comindex{Print Universes}
The constraints on the internal level of the occurrences of {\Type}
(see Section~\ref{Sorts}) can be printed using the command
\begin{quote}
{\tt Print Universes.}
\end{quote}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "Reference-Manual"
%%% End:
|