summaryrefslogtreecommitdiff
path: root/web/tweb/rcs/plain.tw%
blob: 9f7064aa4340a0da9b85b08002a4fec163d84a35 (plain)
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
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
head	1.1;
access;
symbols;
locks;
comment	@# @;


1.1
date	95.07.12.22.05.45;	author WB;	state Exp;
branches;
next	;


desc
@@


1.1
log
@Initial revision
@
text
@%%%-*- compile-command: "notangle plain.tw > plain.sty"              -*-
%%%=====================================================================
%%% @@Noweb-file{
%%%   author   = "W\l{}odek Bzyl",
%%%   version  = "$Id: plain.tw,v 4.12 1995/07/10 21:32:25 web Exp $",
%%%   date     = "25 April 1995",
%%%   time     = "22:05:47 MET",
%%%   filename = "plain.tw",
%%%   checksum = "45702 2586 10926 101053",
%%%   address  = "Instytut Matematyki,
%%%               Uniwersytet Gda\'nski
%%%               Wita Stwosza 57
%%%               80--952 Gda\'nsk, Poland",
%%%   email    = "matwb@@halina.univ.gda.pl (Internet)",
%%%   abstract = "",
%%% }
%%%=====================================================================

\title{PLAIN.TW}{W\l{}odek Bzyl\hfil matwb@@halina.univ.gda.pl}

\printcontents

@@
\chapter{Introduction}

{\it However, we will not include a verbatim description, because some
parts of that file are too boring, and because the actual macros have
been ``optimized'' with respect to memory space and running time.}
\hfill {\sl D.E.~Knuth about\/ [[plain.tex]] format.}
\bigskip
\bigskip
\noindent
What follows is devoted to the details of the plain \TeX\ format.
This file serves two purposes:
\medskip

\item{(1)} As a documentation of the [[plain.tex]] format.
Weaving and texing this document should produce a handy reference.

\item{(2)}
The division of this web source into `chunks' should ease
creation of other formats tailored to particular applications.
Chunks could be easily modified, removed, added, or replaced.

\medskip
\noindent The change file mechanism
is not needed in case of \TeX\ language. 
Change files are used to incorporate system 
dependent code into source file, but \TeX\ code is already 
system independent. \TeX\ code could be only `format dependent' 
and here change files could be used.
 Another feature of format file is that it evolves with time, yet some
intermediate versions are used for preparation of books, articles etc.
All these versions and configurations must be kept well organized,
otherwise you are lost.
The Revision Control System is the tool that assists with these tasks.
 With the RCS it is possible, with small overhead,
 to preserve \stress{all revisions}
 which evolved from given text document,
 merge changes made by others,
 compare different versions,
 keep log of changes.

This document consists mainly of excerpts from the
{\sl \TeX Book}, but it is organised around the macros as they
appear in the [[plain.tex]] rather than around the
topics as in a user manual. Therefore this document
is not a \stress{user manual}, although many definitions
are contained here.
@@
\chapter{The layout of the format}

<<*>>=
<<Establish standard category code values>>
\catcode`@@ = 11
<<Define commonly used constants>>
<<Provide programming constructs>>
<<Allocate registers>>
<<Assign initial values to parameters>>
<<Set up text fonts>>
<<Set up math fonts>>
<<Provide macros for text formatting>>
<<Provide macros for math formatting>>
<<Prepare page for output>>
<<Read hyphenation patterns>>
<<Initialize the layout>>
\catcode`@@ = 12
<<Identify the format>>
@@
There are 256 characters that \TeX\ might encounter at each step, in
a file or in a line of text typed directly on your terminal. These
256~characters are classified into 16 categories numbered 0 to 15:
\begindisplay \def\\{\hfill}
\hfil\hidewidth\it Category\hidewidth&\it \qquad Meaning\hidewidth&%
\it Default\hfil\cr
\noalign{\smallskip}
\\0&Escape character&[[\]]\cr
\\1&Beginning of group&[[{]]\cr
\\2&End of group&[[}]]\cr
\\3&Math shift&[[$]]\cr
\\4&Alignment tab&[[&]]\cr
\\5&End of line&\<return>\cr
\\6&Parameter&[[#]]\cr
\\7&Superscript&[[^]]\cr
\\8&Subscript&[[_]]\cr
\\9&Ignored character&\<null>\cr
10&Space&\ttt{\char"20}\cr
11&Letter&[[A]], \dots, [[Z]] and [[a]], \dots, [[z]]\cr
12&Other character&none of the above or below\cr
13&Active character&[[~]]\cr
14&Comment character&[[%]]\cr
15&Invalid character&\<delete>\cr
\enddisplay

When [[INITEX]] begins, category 12 (other) has been assigned to all~256
possible characters, except that the 52~letters
[[A...Z]] and [[a...z]] are
category~11 (letter), and assignments equivalent to the
following have been made:
\begindisplay \def\\{\hfill}
[[\catcode `\\ = 0]]&\cr
[[\catcode `\^^M = 5]]&\cr
[[\catcode `\^^@@ = 9]]&\cr
[[\catcode `\  = 10]]&\cr
[[\catcode `\% = 14]]&\cr
[[\catcode `\^^? = 15]]&\cr
\enddisplay
Thus `[[\]]' is already an escape character,
`[[\char"20]]' is a space,
and `[[%]]' is available for comments on the first line of the file;
ASCII \<null> is ignored, ASCII \<return> is an end-of-line character,
and ASCII \<delete> is invalid.
\medskip
Furthermore \<tab> is given category space, \<formfeed>
becomes an active character that will detect runaways on files that
have been divided into ``file pages'' by \<formfeed>
characters. Finally the control sequence [[\active]] is defined to
yield the constant~13.

To re-catcode these special characters---not counting ASCII
\begindisplay \def\\{\hfill}
\\\<null>&       [[^^@@]]\cr
\\\<tab>&        [[^^I]]\cr
\\\<linefeed>&   [[^^J]]\cr
\\\<formfeed>&   [[^^L]]\cr
\\\<return>&     [[^^M]]\cr
\\\<delete>&     [[^^?]]\cr
\enddisplay
---use the control sequence [[\dospecials]]
that lists all the characters whose catcodes should
probably be changed to~12 (other) when copying things verbatim.
Each symbol in the list is preceded by [[\do]], which can be defined
if you want to do something to every item in the list.

<<Establish standard category code values>>=
\catcode`\{=1
\catcode`\}=2
\catcode`\$=3
\catcode`\&=4
\catcode`\#=6
\catcode`\^=7 \catcode`\^^K=7 % uparrow is for superscripts
\catcode`\_=8 \catcode`\^^A=8 % downarrow are for subscripts
\catcode`\^^I=10

\chardef\active=13 \catcode`\~=\active % tilde is active
\catcode`\^^L=\active \outer\def^^L{\par} % ascii form-feed is "\outer\par"

\def\dospecials{\do\ \do\\\do\{\do\}\do\$\do\&%
  \do\#\do\^\do\^^K\do\_\do\^^A\do\%\do\~}
@@
To make the plain macros more efficient in time and space, several
constant values are declared as control sequences.
\stress{If they were changed, anything could happen.}
So be careful!

<<Define commonly used constants>>=
\chardef\@@ne=1
\chardef\tw@@=2
\chardef\thr@@@@=3
\chardef\sixt@@@@n=16
\chardef\@@cclv=255
\mathchardef\@@cclvi=256
\mathchardef\@@m=1000
\mathchardef\@@M=10000
\mathchardef\@@MM=20000
@@
\section{Text fonts}

<<Set up text fonts>>=
<<Provide support for font scaling>>
<<Define text fonts>>
<<Encode special characters, and characters not available on the keyboard>>
<<Provide support for accented characters>>
<<Assign uppercase and lowercase code values>>
<<Assign space factor codes>>
@@
Fonts assigned to [[\preloaded]] are not part of the format,
but they are preloaded so that other format packages can use them.
For example, if another set of macros says [[\font\ninerm=cmr9]],
\TeX\ will not have to reload the font metric information for [[cmr9]].

<<Define text fonts>>=
\font\tenrm=cmr10 % roman text
\font\preloaded=cmr9
\font\preloaded=cmr8
\font\sevenrm=cmr7
\font\preloaded=cmr6
\font\fiverm=cmr5

\font\preloaded=cmss10 % sans serif
\font\preloaded=cmssq8

\font\preloaded=cmssi10 % sans serif italic
\font\preloaded=cmssqi8

\font\tenbf=cmbx10 % boldface extended
\font\preloaded=cmbx9
\font\preloaded=cmbx8
\font\sevenbf=cmbx7
\font\preloaded=cmbx6
\font\fivebf=cmbx5

\font\tentt=cmtt10 % typewriter
\font\preloaded=cmtt9
\font\preloaded=cmtt8

\font\preloaded=cmsltt10 % slanted typewriter

\font\tensl=cmsl10 % slanted roman
\font\preloaded=cmsl9
\font\preloaded=cmsl8

\font\tenit=cmti10 % text italic
\font\preloaded=cmti9
\font\preloaded=cmti8
\font\preloaded=cmti7

\font\preloaded=cmu10 % unslanted text italic

\font\preloaded=cmcsc10 % caps and small caps

\font\preloaded=cmssbx10 % sans serif bold extended

\font\preloaded=cmdunh10 % Dunhill style

\font\preloaded=cmr7 scaled \magstep4 % for titles
\font\preloaded=cmtt10 scaled \magstep2
\font\preloaded=cmssbx10 scaled \magstep2

\font\preloaded=manfnt % METAFONT logo and dragon curve and special symbols
@@
Additional [[\preloaded]] fonts can be specified here.
(And those that were [[\preloaded]] above can be eliminated.)

<<Define text fonts>>=
\let\preloaded=\undefined % preloaded fonts must be declared anew later.
@@
<<Provide support for font scaling>>=
\def\magstephalf{1095 }
\def\magstep#1{\ifcase#1 \@@m\or 1200\or 1440\or 1728\or 2074\or 2488\fi\relax}
\def\magnification{\afterassignment\m@@g\count@@}
\def\m@@g{\mag\count@@
  \hsize6.5truein\vsize8.9truein\dimen\footins8truein}
@@
\subsection{Font encoding}

We usually think of text files as containing characters.  It doesn't
cause any problems most of the time when we use plain ASCII
characters---letters A--Z, a--z, the numerals 0--9 and some
of punctuation characters.  This illusion is broken down when we
start using characters that do not belong to this limited set, for
example, accented characters / mathematical symbols.  Then what we
see on screen may not match what we key in. What gets printed may
not match what we see on screen.  Moreover, what gets shown on screen
and what gets printed depends on what machine we are on and how the
fonts that we are using are set up.
In reality text files contain just numeric codes (in range 0--255)
stored in 8-bit bytes, and the mapping between `character' and numeric
code is quite arbitrary.  This is because there are very many more
characters than the 256 numeric codes possible with 8-bits.
Consequently, there will be a need for more than one possible mapping
or `font encoding', or in other words, there would not be a `standard'
encoding that suits all purposes.
\medskip
When a symbol is built up by forming a box, the [[\leavevmode]] macro
is called first; this starts a new paragraph, if \TeX\ is in vertical
mode, but does nothing if \TeX\ is in horizontal mode or math mode.
[[\chardef]] positions are taken from the fonts [[cmr10]] and [[cmsy10]].

<<Encode special characters, and characters not available on the keyboard>>=
\chardef\%=`\%
\chardef\&=`\&
\chardef\#=`\#
\chardef\$=`\$
\chardef\ss="19
\chardef\ae="1A
\chardef\oe="1B
\chardef\o="1C
\chardef\AE="1D
\chardef\OE="1E
\chardef\O="1F
\chardef\i="10 \chardef\j="11 % dotless letters
\def\aa{\accent23a}
\def\l{\char32l}
\def\L{\leavevmode\setbox0\hbox{L}\hbox to\wd0{\hss\char32L}}

\def\leavevmode{\unhbox\voidb@@x} % begins a paragraph, if necessary
\def\_{\leavevmode \kern.06em \vbox{\hrule width.3em}}
\def\AA{\leavevmode\setbox0\hbox{h}\dimen@@\ht0\advance\dimen@@-1ex%
  \rlap{\raise.67\dimen@@\hbox{\char'27}}A}

\def\mathhexbox#1#2#3{\leavevmode
  \hbox{$\m@@th \mathchar"#1#2#3$}}
\def\dag{\mathhexbox279}
\def\ddag{\mathhexbox27A}
\def\S{\mathhexbox278}
\def\P{\mathhexbox27B}
@@
The accent positions are taken from Computer Modern font family. We
are about to `hard-wire' CM accent encoding into the format. Different
encoding will be necessary if other styles of type are used.

Three alternative control-symbol accents are defined,
suitable for keyboards with extended character sets:
[[\let\^^_=\v]], [[\let\^^S=\u]], [[\let\^^D=\^]].

<<Provide support for accented characters>>=
\def\oalign#1{\leavevmode\vtop{\baselineskip\z@@skip \lineskip.25ex%
  \ialign{##\crcr#1\crcr}}} \def\o@@lign{\lineskiplimit\z@@ \oalign}
\def\ooalign{\lineskiplimit-\maxdimen \oalign} % chars over each other
\def\sh@@ft#1{\dimen\z@@.00#1ex\multiply\dimen\z@@\fontdimen1\font
  \kern-.0156\dimen\z@@} % compensate for slant in lowered accents
\def\d#1{{\o@@lign{\relax#1\crcr\hidewidth\sh@@ft{10}.\hidewidth}}}
\def\b#1{{\o@@lign{\relax#1\crcr\hidewidth\sh@@ft{29}%
    \vbox to.2ex{\hbox{\char22}\vss}\hidewidth}}}
\def\c#1{\setbox\z@@\hbox{#1}\ifdim\ht\z@@=1ex\accent24 #1%
  \else{\ooalign{\unhbox\z@@\crcr\hidewidth\char24\hidewidth}}\fi}
\def\copyright{{\ooalign{\hfil\raise.07ex\hbox{c}\hfil\crcr\mathhexbox20D}}}

\def\dots{\relax\ifmmode\ldots\else$\m@@th\ldots\,$\fi}
\def\TeX{T\kern-.1667em\lower.5ex\hbox{E}\kern-.125emX}

\def\`#1{{\accent18 #1}}
\def\'#1{{\accent19 #1}}
\def\v#1{{\accent20 #1}} \let\^^_=\v
\def\u#1{{\accent21 #1}} \let\^^S=\u
\def\=#1{{\accent22 #1}}
\def\^#1{{\accent94 #1}} \let\^^D=\^
\def\.#1{{\accent95 #1}}
\def\H#1{{\accent"7D #1}}
\def\~#1{{\accent"7E #1}}
\def\"#1{{\accent"7F #1}}
\def\t#1{{\edef\next{\the\font}\the\textfont1\accent"7F\next#1}}
@@
[[INITEX]] sets [[\uccode]]$\,`x=`X$ and
[[\uccode]]$\,`X=`X$ for all letters $x$,
and [[\lccode]]$\,`x=`x$, [[\lccode]]$\,`X=`x$;
all other values are zero.

<<Assign uppercase and lowercase code values>>=
%%% Thats all for English language.
@@
Space factor code affects setting of interword glue.
The space factor is normally 1000, which means that the interword glue
should not be modified. If the space factor $f$ is different from
1000, the interword glue is computed as follows: Take the normal space
glue for the current font, and add the extra space if $f\ge2000$.
Then the stretch component is multiplied by $f/1000$, while the shrink
component is multiplied by $1000/f$.
(Look up the Appendix for the values of normal space, normal stretch,
normal shrink, and extra space for some of CM fonts.)

[[INITEX]] sets space factor codes:
[[\sfcode]]$\,x=1000$ for all $x$,
except that [[\sfcode]]$\,`X=999$ for uppercase letters.

The characters
`[[)]]', `[[']]', and `[[]]]'
does not change space factor.

<<Assign space factor codes>>=
\sfcode`\)=0 \sfcode`\'=0 \sfcode`\]=0
@@
\section{Math fonts}

<<Set up math fonts>>=
<<Define math fonts>>
<<Encode math accents>>
<<Establish spacing around mathematical objects>>
<<Assign math codes>>
<<Assign delimiter codes>>
<<Define font families>>
@@
As was said earlier, the font metric information about preloaded font
will be build into the format.  But, if another set of macros says
[[\font\fiftyfiverm = cmr9 at 55pt]], \TeX\ will \stress{have to
reload again} the font metric information for [[cmr9]].

<<Define math fonts>>=
\font\teni=cmmi10 % math italic
\font\preloaded=cmmi9
\font\preloaded=cmmi8
\font\seveni=cmmi7
\font\preloaded=cmmi6
\font\fivei=cmmi5

\font\tensy=cmsy10 % math symbols
\font\preloaded=cmsy9
\font\preloaded=cmsy8
\font\sevensy=cmsy7
\font\preloaded=cmsy6
\font\fivesy=cmsy5

\font\tenex=cmex10 % math extension

\font\preloaded=cmmib10 % bold math italic
\font\preloaded=cmbsy10 % bold math symbols
@@
\subsection{Mathematical spacing}

Spacing around mathematical object is measured in
[[mu]]---`math units.'
[[1mu]] is equal to $1/18\,$th part of [[\fontdimen 6]] of
the font in family 2.

[[\quad]] spacing does not change with the style of formula, nor does it
depend on the math font families that are being used.
But thin spaces, medium spaces, and thick spaces do get bigger and
smaller as the size of type gets bigger and smaller; this is because
they are defined in terms of \<muglue>.

According to these specifications, thin spaces in plain
\TeX\ do not stretch or shrink; medium spaces can stretch a little, and
they can shrink to zero; thick spaces can stretch a lot, but they never shrink.

\medskip

The following table gives the complete definition of muglue between
mathematical objects.  A formula is converted to a math list, and the
math list consists chiefly of ``atoms'' of eight basic types:
Ord~(ordinary), Op~(large operator), Bin~(binary operation),
Rel~(relation), Open~(opening), Close~(closing), Punct~(punctuation),
and Inner~(a delimited subformula).  Other kinds of atoms, which arise
from commands like [[\overline]] or [[\mathaccent]] or [[\vcenter]],
etc., are all treated as type~Ord; fractions are treated as
type~Inner. The following (non-symmetric) table is used to determine
the spacing between pairs of adjacent atoms:
\medbreak
$$\baselineskip0pt\lineskip0pt
\halign to\hsize
 {\strut\hbox to1.5\parindent{\it#\hfil}& % for the legend "Left atom"
  #\hfil\quad& % for the row labels
  #\hfil\tabskip 0pt plus 20pt& % for the rule at the left
  \hbox to 25pt{\tt\hss#\hss}& % for column 1
  \hbox to 25pt{\tt\hss#\hss}& % for column 2
  \hbox to 25pt{\tt\hss#\hss}& % for column 3
  \hbox to 25pt{\tt\hss#\hss}& % for column 4
  \hbox to 25pt{\tt\hss#\hss}& % for column 5
  \hbox to 25pt{\tt\hss#\hss}& % for column 6
  \hbox to 25pt{\tt\hss#\hss}& % for column 7
  \hbox to 25pt{\tt\hss#\hss}& % for column 8
  #\hfil\tabskip0pt\cr % for the rule at the right
\noalign{\vskip-6pt} % it just happens that there's extra white space
&&&&\multispan7\hss\it Right atom\hss\cr \noalign{\vskip3pt} &&&\rm
Ord&\rm Op&\rm Bin&\rm Rel&\rm Open&\rm Close&\rm Punct&\rm Inner\cr
\noalign{\vskip2pt} \omit&&\multispan{10}\leaders\hrule\hfil\cr
\omit\vbox to 2pt{}&&\vrule&&&&&&&&&\vrule\cr
&Ord&\vrule&0&1&(2)&(3)&0&0&0&(1)&\vrule\cr
&Op&\vrule&1&1&*&(3)&0&0&0&(1)&\vrule\cr
&Bin&\vrule&(2)&(2)&*&*&(2)&*&*&(2)&\vrule\cr
Left&Rel&\vrule&(3)&(3)&*&0&(3)&0&0&(3)&\vrule\cr
atom&Open&\vrule&0&0&*&0&0&0&0&0&\vrule\cr
&Close&\vrule&0&1&(2)&(3)&0&0&0&(1)&\vrule\cr
&Punct&\vrule&(1)&(1)&*&(1)&(1)&(1)&(1)&(1)&\vrule\cr
&Inner&\vrule&(1)&1&(2)&(3)&(1)&0&(1)&(1)&\vrule\cr \omit\vbox to
2pt{}&&\vrule&&&&&&&&&\vrule\cr
\omit&&\multispan{10}\leaders\hrule\hfil\cr}$$
\medbreak
Here [[0]], [[1]], [[2]], and [[3]] stand for
no space, thin space, medium space, and thick space,
respectively.
Thin space, medium space, and thin space are equal to
values of [[\thinmuskim]], [[\medmuskip]], [[\thickmuskip]]
parameters, respectively.
The table entry is parenthesized if the space is to be
inserted only in display and text styles, not in script and
scriptscript styles. For example, many of the entries in the Rel row
and the Rel column are `{\tt(3)}'; this means that thick spaces are
normally inserted before and after relational symbols like `=', but
not in subscripts. Some of the entries in the table are `{\tt*}'; such
cases never arise, because Bin atoms must be preceded and followed by
atoms compatible with the nature of binary operations.
The conversion of math lists to horizontal lists is done
whenever \TeX\ is about to leave math mode, and the inter-atomic
spacing is inserted at that time.

<<Establish spacing around mathematical objects>>=
\thinmuskip=3mu
\medmuskip=4mu plus 2mu minus 4mu
\thickmuskip=5mu plus 5mu
@@
For the positioning of accents over single character the width of
[[\skewchar]] is used.  For most of fonts the default value of
[[\skewchar]] is [[-1]]; but the math italic (family 1)
and math symbol fonts (family 2) have
special [[\skewchar]] values equal to [['177]] and [['60]],
respectively. These are characters
`{\teni\char'177\/}' and `{\tensy\char'60}'.

<<Encode math accents>>=
\skewchar\teni='177 \skewchar\seveni='177 \skewchar\fivei='177
\skewchar\tensy='60 \skewchar\sevensy='60 \skewchar\fivesy='60
@@
A math code is relevant only when the corresponding category code is
11 or~12. When processing in math mode characters of categories~11 and~12,
[[\char]] and [[\chardef]] characters are replaced by their math code.

If we denote 15-bit number by [["uvwz]], then math codes are assigned by
\medskip
[[\mathcode]] \<8-bit number> = [["uvwz]],\quad where
\medskip
\itemitem{[[u]]}  --- the class code (see below for the list)
\itemitem{[[v]]}  --- the font family number (see the font tables at the
                      end of this document)
\itemitem{[[wz]]} --- the position of the character in the font
$$\halign{\indent#\hfil&\quad#\hfil&\quad#\hfil&
\hskip4em#\hfil&\quad#\hfil&\quad#\hfil\cr
\it \kern-2pt Class&\it Meaning&\kern-2pt\it Example&
\it \kern-2pt Class&\it Meaning&\kern-2pt\it Example\cr
\noalign{\vskip2pt}
0&Ordinary&[[/]]&
4&Opening&[[(]]\cr
1&Large operator&[[\sum]]&
5&Closing&[[)]]\cr
2&Binary operation&[[+]]&
6&Punctuation&[[,]]\cr
3&Relation&[[=]]&
7&Variable family&[[x]]\cr
}$$

A [[\mathcode]] can also have the special value \hex{8000}, which
causes the character to behave as if it has catcode~13 (active).  This
feature makes [[']] apostrophe expand to [[\prime]].  The mathcode of
[[']] does not interfere with the use of [[']] in octal constants.
The mathcode of [["8000]] is also assigned to space and underscore.

[[INITEX]] sets up
[[\mathcode]]$\,x=x$ for $x=0..255$, except that
[[\mathcode]]$\,x=x+\hex{7000}$
for each of the ten digits $x={}$[[`0]] to [[`9]];
[[\mathcode]]$\,x=x+\hex{7100}$ for each of the 52~letters.
\TeX\ looks at the mathcode only when it is typesetting a character whose
catcode is 11~(letter) or 12~(other), or when it encounters a character that
is given explicitly as [[\char]]\<number>.

Class 7 is a special case that allows math symbols to change families.
It behaves exactly like class~0, except that the specified family is
replaced by the current value of an integer parameter called [[\fam]],
provided that [[\fam]] is a legal family number (i.e., if it lies between
0 and~15). \TeX\ automatically sets [[\fam=-1]] whenever math mode is entered;
therefore class~7 and class~0 are equivalent unless [[\fam]] has been
given a new value. Plain \TeX\ changes [[\fam]] to~0 when the user
types `[[\rm]]'; this makes it convenient to get roman letters in formulas,
since letters belong to class~7.
The control sequence [[\rm]] is an abbreviation for `[[\fam=0 \tenrm]]';
thus, [[\rm]] causes [[\fam]] to become zero, and it makes
[[\tenrm]] the ``current font.''  In horizontal mode, the [[\fam]]
value is irrelevant and the current font governs the typesetting of
letters; but in math mode, the current font is irrelevant and the
[[\fam]] value governs the letters. The current font affects math mode
only if control space ([[\ ]]) is used or if dimensions are given in
[[ex]] or [[em]] units; it also has an effect if an [[\hbox]] appears
inside a formula, since the contents of an hbox are typeset in
horizontal mode.

<<Assign math codes>>=
\mathcode`\^^@@="2201 % \cdot
\mathcode`\^^A="3223 % \downarrow
\mathcode`\^^B="010B % \alpha
\mathcode`\^^C="010C % \beta
\mathcode`\^^D="225E % \land
\mathcode`\^^E="023A % \lnot
\mathcode`\^^F="3232 % \in
\mathcode`\^^G="0119 % \pi
\mathcode`\^^H="0115 % \lambda
\mathcode`\^^I="010D % \gamma
\mathcode`\^^J="010E % \delta
\mathcode`\^^K="3222 % \uparrow
\mathcode`\^^L="2206 % \pm
\mathcode`\^^M="2208 % \oplus
\mathcode`\^^N="0231 % \infty
\mathcode`\^^O="0140 % \partial
\mathcode`\^^P="321A % \subset
\mathcode`\^^Q="321B % \supset
\mathcode`\^^R="225C % \cap
\mathcode`\^^S="225B % \cup
\mathcode`\^^T="0238 % \forall
\mathcode`\^^U="0239 % \exists
\mathcode`\^^V="220A % \otimes
\mathcode`\^^W="3224 % \leftrightarrow
\mathcode`\^^X="3220 % \leftarrow
\mathcode`\^^Y="3221 % \rightarrow
\mathcode`\^^Z="8000 % \ne
\mathcode`\^^[="2205 % \diamond
\mathcode`\^^\="3214 % \le
\mathcode`\^^]="3215 % \ge
\mathcode`\^^^="3211 % \equiv
\mathcode`\^^_="225F % \lor
\mathcode`\ ="8000 % \space
\mathcode`\!="5021
\mathcode`\'="8000 % ^\prime
\mathcode`\(="4028
\mathcode`\)="5029
\mathcode`\*="2203 % \ast
\mathcode`\+="202B
\mathcode`\,="613B
\mathcode`\-="2200
\mathcode`\.="013A
\mathcode`\/="013D
\mathcode`\:="303A
\mathcode`\;="603B
\mathcode`\<="313C
\mathcode`\=="303D
\mathcode`\>="313E
\mathcode`\?="503F
\mathcode`\[="405B
\mathcode`\\="026E % \backslash
\mathcode`\]="505D
\mathcode`\_="8000 % \_
\mathcode`\{="4266
\mathcode`\|="026A
\mathcode`\}="5267
\mathcode`\^^?="1273 % \smallint

@@
Delimiter codes are used after [[\left]] and [[\right]] commands,
when \TeX\ is looking for a delimiter.

If we denote 24-bit number by [["qrstuv]], then
delimiter codes are assigned by
\medskip
[[\delcode]] \<8-bit number> = [["qrstuv]],\quad where
\medskip
\itemitem{[[q]]}   --- the font family number of
\itemitem{[[rs]]}  --- the position of the the small variant of the delimiter
\itemitem{[[t]]}   --- the font family number of
\itemitem{[[uv]]}  --- the position of the the large variant of the delimiter
\medskip
%%% Assigning delimiters codes to characters completes coding.
%%% The full list consists of:
%%% [[\catcode]], [[\lccode]], [[\uccode]], [[\mathcode]], [[\delcode]].

[[INITEX]] sets all [[\delcode]] values to [[-1]],
which means that no characters are recognized as delimiters
in math formulas,
except [[\delcode]]$\,`.=0$, so that `[[.]]' stands for
``null delimiter''.
[[{]] and [[}]] should \stress{not get} delcodes;
otherwise parameter grouping fails!

<<Assign delimiter codes>>=
\delcode`\(="028300
\delcode`\)="029301
\delcode`\[="05B302
\delcode`\]="05D303
\delcode`\<="26830A
\delcode`\>="26930B
\delcode`\/="02F30E
\delcode`\|="26A30C
\delcode`\\="26E30F
@@
All characters that are typeset in math mode belong to one of sixteen
{\sl families} of fonts, numbered internally from 0 to~15. Each of
these families consists of three fonts: one for text size, one for
script size, and one for scriptscriptsize. The commands [[\textfont]],
[[\scriptfont]], and [[\scriptscriptfont]] are used to specify the
members of each family.  Since there are up to 256~characters per
font, and 3~fonts per family, and 16~families, \TeX\ can access up to
12,288 characters in any one formula (4096 in~each of the three
sizes).

The [[plain.tex]] format uses family~1 for math italic letters, family~2 for
ordinary math symbols, and family~3 for large symbols.
Text italic is put in family~4, slanted roman in family~5, bold roman
in family~6, and typewriter type in family~7.
A macro [[\newfam]] will assign symbolic names to families that aren't
already used.

[[INITEX]] initializes the mathcodes of all letters [[A]] to~[[Z]] and
[[a]] to~[[z]] so that they are symbols of class~7 and family~1; that's
why it is natural to use family~1 for math italics.  Similarly, the
digits [[0]] to~[[9]] are class~7 and family~0.  None of the other
families is treated in any special way by \TeX.

\TeX\ doesn't check to see if the families are sensibly organized.
The only constraint is that the fonts in
families 2 and~3 have special [[\fontdimen]] parameters, which govern
mathematical spacing (see Appendix).
In Computer Modern only [[cmsy]] and [[cmex]] have these
parameters, so their assignment to families 2 and~3 is almost
mandatory.

During the time that a math formula is being read,
\TeX\ remembers each symbol as being ``character position so-and-so in
family number such-and-such,'' but it does not take note of what fonts
are actually in the families until reaching the end of the formula.

<<Define font families>>=
\textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
\def\rm{\fam\z@@\tenrm}
\textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei
\def\mit{\fam\@@ne} \def\oldstyle{\fam\@@ne\teni}
\textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
\def\cal{\fam\tw@@}
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\newfam\itfam \def\it{\fam\itfam\tenit} % \it is family 4
\textfont\itfam=\tenit
\newfam\slfam \def\sl{\fam\slfam\tensl} % \sl is family 5
\textfont\slfam=\tensl
\newfam\bffam \def\bf{\fam\bffam\tenbf} % \bf is family 6
\textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf
\scriptscriptfont\bffam=\fivebf
\newfam\ttfam \def\tt{\fam\ttfam\tentt} % \tt is family 7
\textfont\ttfam=\tentt
@@
\section{Registers allocation}

Here are macros for the automatic allocation of [[\count]],
[[\box]], [[\dimen]], [[\skip]], [[\muskip]], and [[\toks]]
registers, as well as [[\read]] and [[\write]]
stream numbers, [[\fam]] codes, [[\language]] codes,
and [[\insert]] numbers.

The main use of these macros is for registers that are defined by one
macro and used by others, possibly at different nesting levels.

The following counters are reserved:
\begindisplay
  0--9&	 page numbering\cr
    10&	 count allocation\cr
    11&	 dimen allocation\cr
    12&	 skip allocation\cr
    13&	 muskip allocation\cr
    14&  box allocation\cr
    15&  toks allocation\cr
    16&  read file allocation\cr
    17&  write file allocation\cr
    18&  math family allocation\cr
    19&  language allocation\cr
    20&  insert allocation\cr
    21&  the most recently allocated number\cr
    22&  constant -1\cr
\enddisplay
New counters are allocated starting with 23, 24, etc.  Other registers are
allocated starting with 10.  This leaves 0 through 9 for the user to play
with safely, except that counts 0 to 9 are considered to be the page and
subpage numbers (since they are displayed during output). In this scheme,
[[\count]] 10 always contains the number of the highest-numbered counter that
has been allocated, [[\count 14]] the highest-numbered box, etc.
Inserts are given numbers 254, 253, etc., since they require a [[\count]],
[[\dimen]], [[\skip]], and [[\box]] all with the same number;
[[\count]] 20 contains the lowest-numbered insert that has been allocated.
[[\box255]] is reserved for [[\output]];
[[\count255]], [[\dimen255]], and [[\skip255]] can be used freely.

It is recommended that macro designers always use
global assignments with respect to registers numbered 1, 3, 5, 7, 9, and
always non-global assignments with respect to registers 0, 2, 4, 6, 8, 255.
This will prevent ``save stack buildup'' that might otherwise occur.

<<Allocate registers>>=
\count10=22 % allocates \count registers 23, 24, ...
\count11=9 % allocates \dimen registers 10, 11, ...
\count12=9 % allocates \skip registers 10, 11, ...
\count13=9 % allocates \muskip registers 10, 11, ...
\count14=9 % allocates \box registers 10, 11, ...
\count15=9 % allocates \toks registers 10, 11, ...
\count16=-1 % allocates input streams 0, 1, ...
\count17=-1 % allocates output streams 0, 1, ...
\count18=3 % allocates math families 4, 5, ...
\count19=0 % allocates \language codes 1, 2, ...
\count20=255 % allocates insertions 254, 253, ...
\countdef\insc@@unt=20 % the insertion counter
\countdef\allocationnumber=21 % the most recent allocation
\countdef\m@@ne=22 \m@@ne=-1 % a handy constant
\def\wlog{\immediate\write\m@@ne} % write on log file (only)
<<Allocate scratch registers>>
<<Provide user-level register allocation macros>>
<<Define implementation-level register allocation macros>>
<<Initialize register constants>>
@@
Here are abbreviations for the names of scratch registers
that don't need to be allocated.

<<Allocate scratch registers>>=
\countdef\count@@=255
\dimendef\dimen@@=0
\dimendef\dimen@@i=1 % global only
\dimendef\dimen@@ii=2
\skipdef\skip@@=0
\toksdef\toks@@=0
@@
Now, we define [[\newcount]], [[\newbox]], etc.
so that you can say [[\newcount\foo]]
and [[\foo]] will be defined (with [[\countdef]]) to be the next counter.
To find out which counter [[\foo]] is, you can look at [[\allocationnumber]].
Since there's no [[\boxdef]] command, [[\chardef]] is used to define
a [[\newbox]], [[\newinsert]], [[\newfam]], and so on.

<<Provide user-level register allocation macros>>=
\outer\def\newcount{\alloc@@0\count\countdef\insc@@unt}
\outer\def\newdimen{\alloc@@1\dimen\dimendef\insc@@unt}
\outer\def\newskip{\alloc@@2\skip\skipdef\insc@@unt}
\outer\def\newmuskip{\alloc@@3\muskip\muskipdef\@@cclvi}
\outer\def\newbox{\alloc@@4\box\chardef\insc@@unt}
\let\newtoks=\relax % we do this to allow plain.tex to be read in twice
\outer\def\newhelp#1#2{\newtoks#1#1\expandafter{\csname#2\endcsname}}
\outer\def\newtoks{\alloc@@5\toks\toksdef\@@cclvi}
\outer\def\newread{\alloc@@6\read\chardef\sixt@@@@n}
\outer\def\newwrite{\alloc@@7\write\chardef\sixt@@@@n}
\outer\def\newfam{\alloc@@8\fam\chardef\sixt@@@@n}
\outer\def\newlanguage{\alloc@@9\language\chardef\@@cclvi}
@@
<<Define implementation-level register allocation macros>>=
\def\alloc@@#1#2#3#4#5{\global\advance\count1#1by\@@ne
  \ch@@ck#1#4#2% make sure there's still room
  \allocationnumber=\count1#1%
  \global#3#5=\allocationnumber
  \wlog{\string#5=\string#2\the\allocationnumber}}
\outer\def\newinsert#1{\global\advance\insc@@unt by\m@@ne
  \ch@@ck0\insc@@unt\count
  \ch@@ck1\insc@@unt\dimen
  \ch@@ck2\insc@@unt\skip
  \ch@@ck4\insc@@unt\box
  \allocationnumber=\insc@@unt
  \global\chardef#1=\allocationnumber
  \wlog{\string#1=\string\insert\the\allocationnumber}}
\def\ch@@ck#1#2#3{\ifnum\count1#1<#2%
  \else\errmessage{No room for a new #3}\fi}
@@
We finish with the initalization of some constants.
<<Initialize register constants>>=
\newdimen\maxdimen \maxdimen=16383.99999pt % the largest legal <dimen>
\newskip\hideskip \hideskip=-1000pt plus 1fill % negative but can grow
\newskip\centering \centering=0pt plus 1000pt minus 1000pt
\newdimen\p@@ \p@@=1pt % this saves macro space and time
\newdimen\z@@ \z@@=0pt % can be used both for 0pt and 0
\newskip\z@@skip \z@@skip=0pt plus0pt minus0pt
\newbox\voidb@@x % permanently void box register
@@
\section{Parameters}

Let's turn now to \TeX's parameters, which the previous chapters have
introduced one at a time; it will be convenient to assemble them
all together.
\bigskip
An \<integer parameter> is one of the following tokens:
\begindisplay\belowdisplayskip=3pt plus 6pt \abovedisplayskip=3pt plus 1pt%
\openup.15pt
[[\pretolerance]]\qquad badness tolerance before hyphenation\cr
[[\tolerance]]\qquad badness tolerance after hyphenation\cr
[[\hbadness]]\qquad badness above which bad hboxes will be shown\cr
[[\vbadness]]\qquad badness above which bad vboxes will be shown\cr
[[\linepenalty]]\qquad amount added to badness of every line in a paragraph\cr
[[\hyphenpenalty]]\qquad penalty for line break after discretionary hyphen\cr
[[\exhyphenpenalty]]\qquad penalty for line break after explicit hyphen\cr
[[\binoppenalty]]\qquad penalty for line break after binary operation\cr
[[\relpenalty]]\qquad penalty for line break after math relation\cr
[[\clubpenalty]]\qquad penalty for creating a club line at bottom of page\cr
[[\widowpenalty]]\qquad penalty for creating a widow line at top of page\cr
[[\displaywidowpenalty]]\qquad ditto, before a display\cr
[[\brokenpenalty]]\qquad penalty for page break after a hyphenated line\cr
[[\predisplaypenalty]]\qquad penalty for page break just before a display\cr
[[\postdisplaypenalty]]\qquad penalty for page break just after a display\cr
[[\interlinepenalty]]\qquad additional penalty for page break between lines\cr
[[\floatingpenalty]]\qquad penalty for insertions that are split\cr
[[\outputpenalty]]\qquad penalty at the current page break\cr
[[\doublehyphendemerits]]\qquad demerits for consecutive broken lines\cr
[[\finalhyphendemerits]]\qquad demerits for a penultimate broken line\cr
[[\adjdemerits]]\qquad demerits for adjacent incompatible lines\cr
[[\looseness]]\qquad change to the number of lines in a paragraph\cr
[[\pausing]]\qquad positive if pausing after each line is read from a file\cr
[[\holdinginserts]]\qquad positive if insertions
   remain dormant in output box\cr
[[\tracingonline]]\qquad positive if showing diagnostic info on the terminal\cr
[[\tracingmacros]]\qquad positive if showing macros as they are expanded\cr
[[\tracingstats]]\qquad positive if showing statistics about memory usage\cr
[[\tracingparagraphs]]\qquad positive if showing line-break calculations\cr
[[\tracingpages]]\qquad positive if showing page-break calculations\cr
[[\tracingoutput]]\qquad positive if showing boxes that are shipped out\cr
[[\tracinglostchars]]\qquad positive if showing characters not in the font\cr
[[\tracingcommands]]\qquad positive if showing commands
   before they are executed\cr
[[\tracingrestores]]\qquad positive if showing deassignments when groups end\cr
[[\language]]\qquad the current set of hyphenation rules\cr
[[\uchyph]]\qquad positive if hyphenating words
   beginning with capital letters\cr
[[\lefthyphenmin]]\qquad smallest fragment at beginning of hyphenated word\cr
[[\righthyphenmin]]\qquad smallest fragment at end of hyphenated word\cr
[[\globaldefs]]\qquad nonzero if overriding [[\global]] specifications\cr
[[\defaulthyphenchar]]\qquad [[\hyphenchar]] value when a font is loaded\cr
[[\defaultskewchar]]\qquad [[\skewchar]] value when a font is loaded\cr
[[\escapechar]]\qquad escape character in the output of
   control sequence tokens\cr
[[\endlinechar]]\qquad character placed at the right end of an input line\cr
[[\newlinechar]]\qquad character that starts a new output line\cr
[[\maxdeadcycles]]\qquad upper bound on [[\deadcycles]]\cr
[[\hangafter]]\qquad hanging indentation changes after this many lines\cr
[[\fam]]\qquad the current family number\cr
[[\mag]]\qquad magnification ratio, times 1000\cr
[[\delimiterfactor]]\qquad ratio for variable delimiters, times 1000\cr
[[\time]]\qquad current time of day in minutes since midnight\cr
[[\day]]\qquad current day of the month\cr
[[\month]]\qquad current month of the year\cr
[[\year]]\qquad current year of our Lord\cr
[[\showboxbreadth]]\qquad maximum items per level when boxes are shown\cr
[[\showboxdepth]]\qquad maximum level when boxes are shown\cr
[[\errorcontextlines]]\qquad maximum extra context shown when errors occur\cr
\enddisplay
The first few of these parameters have values in units of ``badness'' and
``penalties'' that affect line breaking and page breaking. Then come
demerit-oriented parameters; demerits are essentially given in units of
``badness squared,'' so those parameters tend to have larger values.
By contrast, the next few parameters ([[\looseness]], [[\pausing]], etc.)\
generally have quite small values (either $-1$ or 0 or 1 or~2).
Miscellaneous parameters complete the set.
\bigskip
A \<dimen parameter> is one of the following:
\begindisplay\openup.15pt
[[\hfuzz]]\qquad maximum overrun before overfull hbox messages occur\cr
[[\vfuzz]]\qquad maximum overrun before overfull vbox messages occur\cr
[[\overfullrule]]\qquad width of rules appended to overfull boxes\cr
[[\emergencystretch]]\qquad reduces badnesses on final pass of line-breaking\cr
[[\hsize]]\qquad line width in horizontal mode\cr
[[\vsize]]\qquad page height in vertical mode\cr
[[\maxdepth]]\qquad maximum depth of boxes on main pages\cr
[[\splitmaxdepth]]\qquad maximum depth of boxes on split pages\cr
[[\boxmaxdepth]]\qquad maximum depth of boxes on explicit pages\cr
[[\lineskiplimit]]\qquad threshold where [[\baselineskip]] changes
   to [[\lineskip]]\cr
[[\delimitershortfall]]\qquad maximum space not covered by a delimiter\cr
[[\nulldelimiterspace]]\qquad width of a null delimiter\cr
[[\scriptspace]]\qquad extra space after subscript or superscript\cr
[[\mathsurround]]\qquad kerning before and after math in text\cr
[[\predisplaysize]]\qquad length of text preceding a display\cr
[[\displaywidth]]\qquad length of line for displayed equation\cr
[[\displayindent]]\qquad indentation of line for displayed equation\cr
[[\parindent]]\qquad width of\/ [[\indent]]\cr
[[\hangindent]]\qquad amount of hanging indentation\cr
[[\hoffset]]\qquad horizontal offset in [[\shipout]]\cr
[[\voffset]]\qquad vertical offset in [[\shipout]]\cr
\enddisplay

Before typesetting a delimiter \TeX\ determines the size $f$ of the
formula to be covered as twice the maximum of the height and the depth
of the formula. The size $d$ of the delimiter should be
$$
\min\big\{ f - \hbox{[[\delimitershortfall]]},\>
           \hbox{[[\delimiterfactor]]} \cdot {f \over 1000} \big\}
\leq d
$$
\medskip
And the possibilities for \<glue parameter> are:
\begindisplay\openup.15pt
[[\baselineskip]]\qquad desired glue between baselines\cr
[[\lineskip]]\qquad interline glue if\/ [[\baselineskip]] isn't feasible\cr
[[\parskip]]\qquad extra glue just above paragraphs\cr
[[\abovedisplayskip]]\qquad extra glue just above displays\cr
[[\abovedisplayshortskip]]\qquad ditto, following short lines\cr
[[\belowdisplayskip]]\qquad extra glue just below displays\cr
[[\belowdisplayshortskip]]\qquad ditto, following short lines\cr
[[\leftskip]]\qquad glue at left of justified lines\cr
[[\rightskip]]\qquad glue at right of justified lines\cr
[[\topskip]]\qquad glue at top of main pages\cr
[[\splittopskip]]\qquad glue at top of split pages\cr
[[\tabskip]]\qquad glue between aligned entries\cr
[[\spaceskip]]\qquad glue between words, if nonzero\cr
[[\xspaceskip]]\qquad glue between sentences, if nonzero\cr
[[\parfillskip]]\qquad additional [[\rightskip]] at end of paragraphs\cr
\enddisplay

To above parameters (except [[\parfillskip]]) are assigned values
appropriate for CM family typeset at 12$\,$pt baseline.

\medskip
Finally, there are three permissible \<muglue parameter> tokens:
\begindisplay\openup.15pt
[[\thinmuskip]]\qquad thin space in math formulas\cr
[[\medmuskip]]\qquad medium space in math formulas\cr
[[\thickmuskip]]\qquad thick space in math formulas\cr
\enddisplay
\bigskip
\TeX\ also has parameters that are token lists. Such parameters do not
enter into the definitions of \<number> and such things.
A \<token parameter> is any of:
\begindisplay\openup.15pt
[[\output]]\qquad the user's output routine\cr
[[\everypar]]\qquad tokens to insert when a paragraph begins\cr
[[\everymath]]\qquad tokens to insert when math in text begins\cr
[[\everydisplay]]\qquad tokens to insert when display math begins\cr
[[\everyhbox]]\qquad tokens to insert when an hbox begins\cr
[[\everyvbox]]\qquad tokens to insert when a vbox begins\cr
[[\everyjob]]\qquad tokens to insert when the job begins\cr
[[\everycr]]\qquad tokens
   to insert after every [[\cr]] or nonredundant [[\crcr]]\cr
[[\errhelp]]\qquad tokens that supplement an [[\errmessage]]\cr
\enddisplay
\bigskip
All of numeric parameters are listed below,
but the code is commented out if no special value needs to be set.
[[INITEX]] makes all parameters zero except where noted.

<<Assign initial values to parameters>>=
<<Assign values to integer parameters>>
<<Assign values to dimen parameters>>
<<Assign values to glue parameters>>
<<Assign values to special registers>>
@@
<<Assign values to integer parameters>>=
\pretolerance=100
\tolerance=200 % INITEX sets this to 10000
\hbadness=1000
\vbadness=1000
\linepenalty=10
\hyphenpenalty=50
\exhyphenpenalty=50
\binoppenalty=700
\relpenalty=500
\clubpenalty=150
\widowpenalty=150
\displaywidowpenalty=50
\brokenpenalty=100
\predisplaypenalty=10000
% \postdisplaypenalty=0
% \interlinepenalty=0
% \floatingpenalty=0, set during \insert
% \outputpenalty=0, set before TeX enters \output
\doublehyphendemerits=10000
\finalhyphendemerits=5000
\adjdemerits=10000
% \looseness=0, cleared by TeX after each paragraph
% \pausing=0
% \holdinginserts=0
% \tracingonline=0
% \tracingmacros=0
% \tracingstats=0
% \tracingparagraphs=0
% \tracingpages=0
% \tracingoutput=0
\tracinglostchars=1
% \tracingcommands=0
% \tracingrestores=0
% \language=0
\uchyph=1
% \lefthyphenmin=2 \righthyphenmin=3 set below
% \globaldefs=0
% \maxdeadcycles=25 % INITEX does this
% \hangafter=1 % INITEX does this, also TeX after each paragraph
% \fam=0
% \mag=1000 % INITEX does this
% \escapechar=`\\ % INITEX does this
\defaulthyphenchar=`\-
\defaultskewchar=-1
% \endlinechar=`\^^M % INITEX does this
\newlinechar=-1
\delimiterfactor=901
% \time=now % TeX does this at beginning of job
% \day=now % TeX does this at beginning of job
% \month=now % TeX does this at beginning of job
% \year=now % TeX does this at beginning of job
\showboxbreadth=5
\showboxdepth=3
\errorcontextlines=5
@@
<<Assign values to dimen parameters>>=
\hfuzz=0.1pt
\vfuzz=0.1pt
\overfullrule=5pt
\hsize=6.5in
\vsize=8.9in
\maxdepth=4pt
\splitmaxdepth=\maxdimen
\boxmaxdepth=\maxdimen
% \lineskiplimit=0pt, changed by \normalbaselines
\delimitershortfall=5pt
\nulldelimiterspace=1.2pt
\scriptspace=0.5pt
% \mathsurround=0pt
% \predisplaysize=0pt, set before TeX enters $$
% \displaywidth=0pt, set before TeX enters $$
% \displayindent=0pt, set before TeX enters $$
\parindent=20pt
% \hangindent=0pt, zeroed by TeX after each paragraph
% \hoffset=0pt
% \voffset=0pt
@@
<<Assign values to glue parameters>>=
% \baselineskip=0pt, changed by \normalbaselines
% \lineskip=0pt, changed by \normalbaselines
\parskip=0pt plus 1pt
\abovedisplayskip=12pt plus 3pt minus 9pt
\abovedisplayshortskip=0pt plus 3pt
\belowdisplayskip=12pt plus 3pt minus 9pt
\belowdisplayshortskip=7pt plus 3pt minus 4pt
% \leftskip=0pt
% \rightskip=0pt
\topskip=10pt
\splittopskip=10pt
% \tabskip=0pt
% \spaceskip=0pt
% \xspaceskip=0pt
\parfillskip=0pt plus 1fil
@@
We also define special registers that function like parameters:
<<Assign values to special registers>>=
\newskip\smallskipamount \smallskipamount=3pt plus 1pt minus 1pt
\newskip\medskipamount \medskipamount=6pt plus 2pt minus 2pt
\newskip\bigskipamount \bigskipamount=12pt plus 4pt minus 4pt
\newskip\normalbaselineskip \normalbaselineskip=12pt
\newskip\normallineskip \normallineskip=1pt
\newdimen\normallineskiplimit \normallineskiplimit=0pt
\newdimen\jot \jot=3pt
\newcount\interdisplaylinepenalty \interdisplaylinepenalty=100
\newcount\interfootnotelinepenalty \interfootnotelinepenalty=100
\def\normalbaselines{\lineskip\normallineskip
  \baselineskip\normalbaselineskip \lineskiplimit\normallineskiplimit}
@@
\section{Macros for text}

Here we introduce macros that are used for basic formatting
unrelated to mathematics.

<<Provide macros for text formatting>>=
<<Supply basic macros for text formatting>>
<<Define space macros>>
<<Supply various paragraph shapes>>
<<Define sectioning macros>>
<<Supply ragged setting>>
<<Supply `boxing' macros>>
<<Supply strut>>
<<Provide alignment macros>>
<<Establish spacing after punctuation characters>>
<<Supply various ways to fill space>>
<<Define [[\showhyphens]] macro>>
<<Completing the job>>
@@
[[\]]\<tab> and [[\]]\<return> are defined so that they
expand to [[\]]\<space>; this helps to prevent confusion, since all three
cases look identical when displayed on most computer terminals.

<<Supply basic macros for text formatting>>=
\def\^^M{\ } % control <return> = control <space>
\def\^^I{\ } % same for <tab>
@@
The control sequences [[\endgraf]] and [[\endline]] are made equivalent
to \TeX's primitive [[\par]] and~[[\cr]] operations, since it is often
useful to redefine the meanings of [[\par]] and~[[\cr]] themselves. Then
come the definitions of [[\space]] (a blank space), [[\empty]] (a~list
of no tokens), and [[\null]] (an~empty hbox).

[[\bgroup]] and [[\egroup]] are made to provide ``implicit'' grouping
characters that turn out to be especially useful in macro definitions.

<<Supply basic macros for text formatting>>=
\let\endgraf=\par \let\endline=\cr

\def\space{ }
\def\empty{}
\def\null{\hbox{}}

\let\bgroup={ \let\egroup=}
@@
The [[\obeylines]] macro says `[[\let^^M=\par]]'
instead of `[[\def^^M{\par}]]' because the [[\let]]
technique allows constructions such as
`[[\let\par=\cr]] [[\obeylines]] [[\halign||{...}]]'
in which [[\cr]]'s need not be given within the alignment.

<<Supply basic macros for text formatting>>=
{\catcode`\^^M=\active % these lines must end with %
  \gdef\obeylines{\catcode`\^^M\active \let^^M\par}%
  \global\let^^M\par} % this is in case ^^M appears in a \write
\def\obeyspaces{\catcode`\ \active}
{\obeyspaces\global\let =\space}
@@
The macros [[\lq]], [[\rq]], [[\lbrack]], and [[\rbrack]] are defined,
for people who have difficulty typing quotation marks and/or brackets.

<<Supply basic macros for text formatting>>=
\def\lq{`} \def\rq{'}
\def\lbrack{[} \def\rbrack{]}
@@
The macros [[\enskip]], [[\quad]], and [[\qquad]]
provide spaces that are legitimate breakpoints within a paragraph;
[[\enspace]], [[\thinspace]], and [[\negthinspace]] produce space that cannot
cause a break (although the space will disappear if it occurs just
next to certain kinds of breaks). All six of these spaces are relative
to the current font.

You can get horizontal space that
never disappears by saying `[[\hglue]]\<glue>'; this space is able to
stretch or shrink. Similarly, there's a vertical analog,
`[[\vglue]]\<glue>'.

The [[\nointerlineskip]] macro suppresses interline glue that would
ordinarily be inserted before the next box in vertical mode; this is a
``one shot'' macro, but [[\offinterlineskip]] is more drastic---it sets
things up so that future interline glue will be present, but zero.
There also are macros for potentially breakable vertical spaces:
[[\smallskip]], [[\medskip]], and [[\bigskip]].

<<Define space macros>>=
\def\thinspace{\kern .16667em }
\def\negthinspace{\kern-.16667em }
\def\enspace{\kern.5em }

\def\enskip{\hskip.5em\relax}
\def\quad{\hskip1em\relax}
\def\qquad{\hskip2em\relax}

\def\smallskip{\vskip\smallskipamount}
\def\medskip{\vskip\medskipamount}
\def\bigskip{\vskip\bigskipamount}

\def\nointerlineskip{\prevdepth-1000\p@@}
\def\offinterlineskip{\baselineskip-1000\p@@
  \lineskip\z@@ \lineskiplimit\maxdimen}

\def\topglue{\nointerlineskip\vglue-\topskip\vglue} % for top of page
\def\vglue{\afterassignment\vgl@@\skip@@=}
\def\vgl@@{\par \dimen@@\prevdepth \hrule height\z@@
  \nobreak\vskip\skip@@ \prevdepth\dimen@@}
\def\hglue{\afterassignment\hgl@@\skip@@=}
\def\hgl@@{\leavevmode \count@@\spacefactor \vrule width\z@@
  \nobreak\hskip\skip@@ \spacefactor\count@@}
@@
The following macros introduce penalty markers
that make breaking less, or more, desirable. The [[\break]], [[\nobreak]],
and [[\allowbreak]] macros are intended for use in any mode; the [[~]]~(tie)
and [[\slash]] (hyphen-like~`/') macros are intended for horizontal mode.
The others are intended only for vertical mode, i.e., between paragraphs,
so they begin with [[\par]].

<<Define space macros>>=
\def~{\penalty\@@M \ } % tie
\def\slash{/\penalty\exhyphenpenalty} % a `/' that acts like a `-'

\def\break{\penalty-\@@M}
\def\nobreak{\penalty \@@M}
\def\allowbreak{\penalty \z@@}

\def\filbreak{\par\vfil\penalty-200\vfilneg}
\def\goodbreak{\par\penalty-500 }
\def\eject{\par\break}
\def\supereject{\par\penalty-\@@MM}

\def\removelastskip{\ifdim\lastskip=\z@@\else\vskip-\lastskip\fi}
\def\smallbreak{\par\ifdim\lastskip<\smallskipamount
  \removelastskip\penalty-50\smallskip\fi}
\def\medbreak{\par\ifdim\lastskip<\medskipamount
  \removelastskip\penalty-100\medskip\fi}
\def\bigbreak{\par\ifdim\lastskip<\bigskipamount
  \removelastskip\penalty-200\bigskip\fi}
@@
[[\line]], [[\leftline]], [[\rightline]], and [[\centerline]]
produce boxes of the full line width, while [[\llap]] and [[\rlap]] make
boxes whose effective width is zero. The [[\underbar]] macro puts its
argument into an hbox with a straight line at a fixed distance under it.

[[\underbar]] uses math mode to do its job, although the
operation is essentially non-mathematical in nature. A few of the other
macros below use math mode in similar ways; thus, \TeX's mathematical
abilities prove to be useful even when no mathematical typesetting
is actually being done. A special control sequence [[\m@@th]] is
used to ``turn off'' [[\mathsurround]] when such constructions are
being performed.)

<<Supply `boxing' macros>>=
\def\line{\hbox to\hsize}
\def\leftline#1{\line{#1\hss}}
\def\rightline#1{\line{\hss#1}}
\def\centerline#1{\line{\hss#1\hss}}

\def\rlap#1{\hbox to\z@@{#1\hss}}
\def\llap#1{\hbox to\z@@{\hss#1}}

\def\m@@th{\mathsurround\z@@}
\def\underbar#1{$\setbox\z@@\hbox{#1}\dp\z@@\z@@
  \m@@th \underline{\box\z@@}$}
@@
A [[\strut]] is implemented here as a rule of width zero.
The `[[\relax]]' in this macro and in others below is necessary in case
|\strut| appears first in an alignment entry, because \TeX\ is in a
somewhat unpredictable mode at such times.

<<Supply strut>>=
\newbox\strutbox
\setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width\z@@}
\def\strut{\relax\ifmmode\copy\strutbox\else\unhcopy\strutbox\fi}
@@
The [[\ialign]] macro provides for alignments when it is necessary to
be sure that [[\tabskip]] is initially zero. The [[\hidewidth]] macro can be
used essentially as [[\hfill]] in alignment entries that are permitted to
``stick out'' of their column. There's also [[\multispan]], which permits
alignment entries to span one or more columns.

<<Provide alignment macros>>=
\def\hidewidth{\hskip\hideskip} % for alignment entries that can stick out
\def\ialign{\everycr{}\tabskip\z@@skip\halign} % initialized \halign
\newcount\mscount
\def\multispan#1{\omit \mscount#1\relax
  \loop\ifnum\mscount>\@@ne \sp@@n\repeat}
\def\sp@@n{\span\omit\advance\mscount\m@@ne}
@@
Now we get to the ``tabbing'' macros.
They keep track of the tab positions by
maintaining boxes full of empty boxes having the specified widths.

The macro [[\+]] has been declared `[[\outer]]' here, so that \TeX\ will
be better able to detect runaway arguments and definitions.
A non-[[\outer]] version, called [[\tabalign]], has also been provided
in case it is necessary to use [[\+]] in some ``inner'' place.
You can use [[\tabalign]] just like [[\+]], except after [[\settabs]].

<<Provide alignment macros>>=
\newif\ifus@@ \newif\if@@cr
\newbox\tabs \newbox\tabsyet \newbox\tabsdone

\def\cleartabs{\global\setbox\tabsyet\null \setbox\tabs\null}
\def\settabs{\setbox\tabs\null \futurelet\next\sett@@b}
\let\+=\relax % in case this file is being read in twice
\def\sett@@b{\ifx\next\+\def\nxt{\afterassignment\s@@tt@@b\let\nxt}%
  \else\let\nxt\s@@tcols\fi \let\next\relax \nxt}
\def\s@@tt@@b{\let\nxt\relax \us@@false\m@@ketabbox}
\def\tabalign{\us@@true\m@@ketabbox} % non-\outer version of \+
\outer\def\+{\tabalign}
\def\s@@tcols#1\columns{\count@@#1\dimen@@\hsize
  \loop\ifnum\count@@>\z@@ \@@nother \repeat}
\def\@@nother{\dimen@@ii\dimen@@ \divide\dimen@@ii\count@@
  \setbox\tabs\hbox{\hbox to\dimen@@ii{}\unhbox\tabs}%
  \advance\dimen@@-\dimen@@ii \advance\count@@\m@@ne}

\def\m@@ketabbox{\begingroup
  \global\setbox\tabsyet\copy\tabs
  \global\setbox\tabsdone\null
  \def\cr{\@@crtrue\crcr\egroup\egroup
    \ifus@@\unvbox\z@@\lastbox\fi\endgroup
    \setbox\tabs\hbox{\unhbox\tabsyet\unhbox\tabsdone}}%
  \setbox\z@@\vbox\bgroup\@@crfalse
    \ialign\bgroup&\t@@bbox##\t@@bb@@x\crcr}

\def\t@@bbox{\setbox\z@@\hbox\bgroup}
\def\t@@bb@@x{\if@@cr\egroup % now \box\z@@ holds the column
  \else\hss\egroup \global\setbox\tabsyet\hbox{\unhbox\tabsyet
      \global\setbox\@@ne\lastbox}% now \box\@@ne holds its size
    \ifvoid\@@ne\global\setbox\@@ne\hbox to\wd\z@@{}%
    \else\setbox\z@@\hbox to\wd\@@ne{\unhbox\z@@}\fi
    \global\setbox\tabsdone\hbox{\box\@@ne\unhbox\tabsdone}\fi
  \box\z@@}
@@
Paragraph shapes of a limited but important kind are provided by
[[\item]], [[\itemitem]], and [[\narrower]].
A macro [[\hang]] causes hanging indentation by the normal
amount of [[\parindent]], after the first line;
thus, the entire paragraph will be indented by the same amount
(unless it began with [[\noindent]]).
[[\textindent]][[{stuff}]] is like [[\indent]],
but it puts the `[[stuff]]' into the indentation,
flush right except for an en~space;
it also removes spaces that might
follow the right brace in `[[{stuff}]]'.

<<Supply various paragraph shapes>>=
\def\hang{\hangindent\parindent}
\def\textindent#1{\indent\llap{#1\enspace}\ignorespaces}
\def\item{\par\hang\textindent}
\def\itemitem{\par\indent \hangindent2\parindent \textindent}
\def\narrower{\advance\leftskip\parindent
  \advance\rightskip\parindent}
@@
The [[\beginsection]] macro is intended to mark the beginning of a
new major subdivision in a document; to use it, you say
`[[\beginsection]]\<section title>' followed by a blank line (or~[[\par]]).
The macro first emits glue and penalties, designed to start a new page
if the present page is nearly full; then it makes a [[\bigskip]] and
puts the section title flush left on a line by itself, in boldface type.
The section title is also displayed on the terminal.
After a [[\smallskip]], with page break prohibited, a [[\noindent]]
command is given; this suppresses indentation in the next paragraph,
i.e., in the first paragraph of the new section.
(However, the next ``paragraph'' will be empty if vertical mode
material immediately follows the [[\beginsection]] command.)

Special statements in a mathematical paper are often called theorems,
lemmas, definitions, axioms, postulates, remarks, corollaries, algorithms,
facts, conjectures, or some such things, and they generally are
given special typographic treatment. The [[\proclaim]] macro
puts the title of the proclamation in boldface, then sets the rest
of the paragraph in slanted type. The paragraph is followed by
something similar to [[\medbreak]], except that the amount of penalty
is different so that page breaks are discouraged:

<<Define sectioning macros>>=
\outer\def\beginsection#1\par{\vskip\z@@ plus.3\vsize\penalty-250
  \vskip\z@@ plus-.3\vsize\bigskip\vskip\parskip
  \message{#1}\leftline{\bf#1}\nobreak\smallskip\noindent}
\outer\def\proclaim #1. #2\par{\medbreak
  \noindent{\bf#1.\enspace}{\sl#2\par}%
  \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi}
@@
Ragged-right setting is initiated by restricting the spaces between
words to have a fixed width, and by putting variable space at the
right of each line.
You should not call [[\raggedright]] until your
text font has already been specified.

It is assumed that the ragged-right material will not be in a variety
of different sizes.  If this assumption is not valid, a different
approach should be used: [[\fontdimen]] parameters 3 and~4 of the
fonts you will be using should be set to zero, by saying, e.g.,
`[[\fontdimen3\tenrm=0pt]]'. These parameters specify the
stretchability and shrinkability of interword spaces. A special macro
[[\ttraggedright]] should be used for ragged-right setting in
[[typewriter type]], since the spaces between words are generally
bigger in that style. (Spaces are already unstretchable and
unshrinkable in font~|cmtt|.)

<<Supply ragged setting>>=
\def\raggedright
   {\rightskip\z@@ plus2em \spaceskip.3333em \xspaceskip.5em\relax}
\def\ttraggedright
   {\tt\rightskip\z@@ plus2em\relax} % for use with \tt only
@@
The [[\nonfrenchspacing]] macro is be used to change the default
sfcodes (set by [[INITEX]]) of punctuation marks.  This macro,
when called, affects spacing after punctuation marks.

<<Establish spacing after punctuation characters>>=
\def\frenchspacing{\sfcode`\.\@@m \sfcode`\?\@@m \sfcode`\!\@@m
  \sfcode`\:\@@m \sfcode`\;\@@m \sfcode`\,\@@m}
\def\nonfrenchspacing{\sfcode`\.3000\sfcode`\?3000\sfcode`\!3000%
  \sfcode`\:2000\sfcode`\;1500\sfcode`\,1250 }
@@
[[\rightarrowfill]] and [[\leftarrowfill]] macros use
[[\cleaders]] with a repeatable box consisting of the middle 10 units
of a minus sign, where one unit
is ${1\over18}\em$. The leaders are preceded and followed by $-$ and~$\to$;
there's enough backspacing to compensate for up to 5~units of extra space,
fore and aft, that |\cleaders| might leave blank. In this way a macro is
obtained such that
[[\hbox to 100pt{\rightarrowfill}]]
yields `\hbox to 100pt{\rightarrowfill}'.

The [[\overbrace]] and [[\underbrace]] macros of plain \TeX\ are
constructed by combining characters with rules. Font [[cmex10]]
contains four symbols $\bracelu\;\braceld\;\bracerd\;\braceru$, each
of which has depth zero and height equal to the thickness of a rule
that joins them properly.
Therefore it's easy to define [[\upbracefill]] and [[\downbracefill]]
macros so that you can obtain, e.g.,
\begindisplay
\vbox{\kern4pt
  \hbox to 100pt{\downbracefill}\hbox to 50pt{\upbracefill}
  }
\enddisplay
by saying `[[\hbox to 100pt{\downbracefill}\hbox to 50pt{\upbracefill}]]'
in vertical mode.

<<Supply various ways to fill space>>=
\def\hrulefill{\leaders\hrule\hfill}
\def\dotfill{\cleaders\hbox{$\m@@th \mkern1.5mu.\mkern1.5mu$}\hfill}
\def\rightarrowfill{$\m@@th\smash-\mkern-6mu%
  \cleaders\hbox{$\mkern-2mu\smash-\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill{$\m@@th\mathord\leftarrow\mkern-6mu%
  \cleaders\hbox{$\mkern-2mu\smash-\mkern-2mu$}\hfill
  \mkern-6mu\smash-$}
\mathchardef\braceld="37A \mathchardef\bracerd="37B
\mathchardef\bracelu="37C \mathchardef\braceru="37D
\def\downbracefill{$\m@@th \setbox\z@@\hbox{$\braceld$}%
  \braceld\leaders\vrule height\ht\z@@ depth\z@@\hfill\braceru
  \bracelu\leaders\vrule height\ht\z@@ depth\z@@\hfill\bracerd$}
\def\upbracefill{$\m@@th \setbox\z@@\hbox{$\braceld$}%
  \bracelu\leaders\vrule height\ht\z@@ depth\z@@\hfill\bracerd
  \braceld\leaders\vrule height\ht\z@@ depth\z@@\hfill\braceru$}
@@
If you want to see all of the hyphens that plain \TeX\ will find in some
random text, you can say `[[\showhyphens{]]\<random text>[[}]]' and the
results will appear on your terminal (and in the log file).
The [[\showhyphens]] macro creates an hbox that is intentionally
underfull.

<<Define [[\showhyphens]] macro>>=
\def\showhyphens#1{\setbox0\vbox{\parfillskip\z@@skip\hsize\maxdimen\tenrm
  \pretolerance\m@@ne\tolerance\m@@ne\hbadness0\showboxdepth0\ #1}}
@@
At the end of a \TeX\ manuscript it's usually best to finish
everything off by typing `[[\bye]]'. The `[[\vfill\supereject]]' gets
\TeX\ to flush out all remaining insertions, with blank space filling
the bottom of incomplete pages, and `[[\end]]' sends the computer into
its endgame routine.

<<Completing the job>>=
\outer\def\bye{\par\vfill\supereject\end}
@@
\section{Macros for math}

Most of this section consists of long listings of special symbols
together with their font locations.

<<Provide macros for math formatting>>=
<<Define math space macros>>
<<Define Greek letters>>
<<Define math symbols>>
<<Encode large operators>>
<<Encode binary operations>>
<<Encode relations>>
<<Supply vertical and diagonal dots>>
<<Supply variable-width math accents>>
<<Supply extensible delimiters>>
<<Provide access to delimiters of various sizes>>
<<Supply common math functions>>
<<Provide [[\cases]] and [[\matrix]] macros>>
<<Provide support to typeset displayed equations>>
@@
\TeX\ does automatic spacing of math formulas so that they look right,
and this is almost true. But occasionally you must give \TeX\ some
help. The number of possible math formulas is vast, and \TeX's spacing
rules are rather simple, so it is natural that exceptions should
arise.  The basic elements of space that \TeX\ puts into formulas are
called: thin spaces, medium spaces, thick spaces.
The normal space between words of a paragraph is approximately equal
to two thin spaces.
\bigskip
\noindent You can add your own spacing whenever you want to,
by using the control sequences
$$\halign{\indent#\hfil&\quad#\hfil\cr
|\,|&thin space \ (normally 1/6 of a quad);\cr
|\>|&medium space \ (normally 2/9 of a quad);\cr
|\;|&thick space \ (normally 5/18 of a quad);\cr
|\!|&negative thin space \ (normally $-1/6$ of a quad).\cr}$$

<<Define math space macros>>=
\def\,{\mskip\thinmuskip}
\def\>{\mskip\medmuskip}
\def\;{\mskip\thickmuskip}
\def\!{\mskip-\thinmuskip}
@@
The next job is to define Greek letters and other symbols of type Ord.
Uppercase Greek letters are assigned hexadecimal codes of the form \hex{7xxx},
so that they will change families when [[\fam]] changes.

[[\mathchardef]] defines a control sequence to be a synonym for a math
character (check [[\mathchar]] for the meaning of hex number).

<<Define Greek letters>>=
\mathchardef\alpha="010B
\mathchardef\beta="010C
\mathchardef\gamma="010D
\mathchardef\delta="010E
\mathchardef\epsilon="010F
\mathchardef\zeta="0110
\mathchardef\eta="0111
\mathchardef\theta="0112
\mathchardef\iota="0113
\mathchardef\kappa="0114
\mathchardef\lambda="0115
\mathchardef\mu="0116
\mathchardef\nu="0117
\mathchardef\xi="0118
\mathchardef\pi="0119
\mathchardef\rho="011A
\mathchardef\sigma="011B
\mathchardef\tau="011C
\mathchardef\upsilon="011D
\mathchardef\phi="011E
\mathchardef\chi="011F
\mathchardef\psi="0120
\mathchardef\omega="0121
\mathchardef\varepsilon="0122
\mathchardef\vartheta="0123
\mathchardef\varpi="0124
\mathchardef\varrho="0125
\mathchardef\varsigma="0126
\mathchardef\varphi="0127
\mathchardef\Gamma="7000
\mathchardef\Delta="7001
\mathchardef\Theta="7002
\mathchardef\Lambda="7003
\mathchardef\Xi="7004
\mathchardef\Pi="7005
\mathchardef\Sigma="7006
\mathchardef\Upsilon="7007
\mathchardef\Phi="7008
\mathchardef\Psi="7009
\mathchardef\Omega="700A
@@
<<Define math symbols>>=
\mathchardef\aleph="0240
\def\hbar{{\mathchar'26\mkern-9muh}}
\mathchardef\imath="017B
\mathchardef\jmath="017C
\mathchardef\ell="0160
\mathchardef\wp="017D
\mathchardef\Re="023C
\mathchardef\Im="023D
\mathchardef\partial="0140
\mathchardef\infty="0231
\mathchardef\prime="0230
\mathchardef\emptyset="023B
\mathchardef\nabla="0272
\def\surd{{\mathchar"1270}}
\mathchardef\top="023E
\mathchardef\bot="023F
\def\angle{{\vbox{\ialign{$\m@@th\scriptstyle##$\crcr
      \not\mathrel{\mkern14mu}\crcr
      \noalign{\nointerlineskip}
      \mkern2.5mu\leaders\hrule height.34pt\hfill\mkern2.5mu\crcr}}}}
\mathchardef\triangle="0234
\mathchardef\forall="0238
\mathchardef\exists="0239
\mathchardef\neg="023A \let\lnot=\neg
\mathchardef\flat="015B
\mathchardef\natural="015C
\mathchardef\sharp="015D
\mathchardef\clubsuit="027C
\mathchardef\diamondsuit="027D
\mathchardef\heartsuit="027E
\mathchardef\spadesuit="027F
@@
Integral signs get special treatment so that their limits won't be set
above and below.

<<Encode large operators>>=
\mathchardef\coprod="1360
\mathchardef\bigvee="1357
\mathchardef\bigwedge="1356
\mathchardef\biguplus="1355
\mathchardef\bigcap="1354
\mathchardef\bigcup="1353
\mathchardef\intop="1352 \def\int{\intop\nolimits}
\mathchardef\prod="1351
\mathchardef\sum="1350
\mathchardef\bigotimes="134E
\mathchardef\bigoplus="134C
\mathchardef\bigodot="134A
\mathchardef\ointop="1348 \def\oint{\ointop\nolimits}
\mathchardef\bigsqcup="1346
\mathchardef\smallint="1273
@@
<<Encode binary operations>>=
\mathchardef\triangleleft="212F
\mathchardef\triangleright="212E
\mathchardef\bigtriangleup="2234
\mathchardef\bigtriangledown="2235
\mathchardef\wedge="225E \let\land=\wedge
\mathchardef\vee="225F \let\lor=\vee
\mathchardef\cap="225C
\mathchardef\cup="225B
\mathchardef\ddagger="227A
\mathchardef\dagger="2279
\mathchardef\sqcap="2275
\mathchardef\sqcup="2274
\mathchardef\uplus="225D
\mathchardef\amalg="2271
\mathchardef\diamond="2205
\mathchardef\bullet="220F
\mathchardef\wr="226F
\mathchardef\div="2204
\mathchardef\odot="220C
\mathchardef\oslash="220B
\mathchardef\otimes="220A
\mathchardef\ominus="2209
\mathchardef\oplus="2208
\mathchardef\mp="2207
\mathchardef\pm="2206
\mathchardef\circ="220E
\mathchardef\bigcirc="220D
\mathchardef\setminus="226E % for set difference A\setminus B
\mathchardef\cdot="2201
\mathchardef\ast="2203
\mathchardef\times="2202
\mathchardef\star="213F
@@
Relations are also fairly straightforward, except for the ones that are
constructed from other characters. The [[\mapstochar]] is a character
`$\mapstochar\mskip5mu$' of width zero that is quite useless by itself,
but it combines with right arrows to make [[\mapsto]] `$\mapsto$' and
[[\longmapsto]] `$\longmapsto$'. Similarly, [[\not]] is a relation character of
width zero that puts a slash over the character that follows.  When two
relations are adjacent in a math formula, \TeX\ puts no space between them.

<<Encode relations>>=
\mathchardef\propto="322F
\mathchardef\sqsubseteq="3276
\mathchardef\sqsupseteq="3277
\mathchardef\parallel="326B
\mathchardef\mid="326A
\mathchardef\dashv="3261
\mathchardef\vdash="3260
\mathchardef\nearrow="3225
\mathchardef\searrow="3226
\mathchardef\nwarrow="322D
\mathchardef\swarrow="322E
\mathchardef\Leftrightarrow="322C
\mathchardef\Leftarrow="3228
\mathchardef\Rightarrow="3229
\def\neq{\not=} \let\ne=\neq
\mathchardef\leq="3214 \let\le=\leq
\mathchardef\geq="3215 \let\ge=\geq
\mathchardef\succ="321F
\mathchardef\prec="321E
\mathchardef\approx="3219
\mathchardef\succeq="3217
\mathchardef\preceq="3216
\mathchardef\supset="321B
\mathchardef\subset="321A
\mathchardef\supseteq="3213
\mathchardef\subseteq="3212
\mathchardef\in="3232
\mathchardef\ni="3233 \let\owns=\ni
\mathchardef\gg="321D
\mathchardef\ll="321C
\mathchardef\not="3236
\mathchardef\leftrightarrow="3224
\mathchardef\leftarrow="3220 \let\gets=\leftarrow
\mathchardef\rightarrow="3221 \let\to=\rightarrow
\mathchardef\mapstochar="3237 \def\mapsto{\mapstochar\rightarrow}
\mathchardef\sim="3218
\mathchardef\simeq="3227
\mathchardef\perp="323F
\mathchardef\equiv="3211
\mathchardef\asymp="3210
\mathchardef\smile="315E
\mathchardef\frown="315F
\mathchardef\leftharpoonup="3128
\mathchardef\leftharpoondown="3129
\mathchardef\rightharpoonup="312A
\mathchardef\rightharpoondown="312B

\def\joinrel{\mathrel{\mkern-3mu}}
\def\relbar{\mathrel{\smash-}} % \smash, because - has the same height as +
\def\Relbar{\mathrel=}
\mathchardef\lhook="312C \def\hookrightarrow{\lhook\joinrel\rightarrow}
\mathchardef\rhook="312D \def\hookleftarrow{\leftarrow\joinrel\rhook}
\def\bowtie{\mathrel\triangleright\joinrel\mathrel\triangleleft}
\def\models{\mathrel|\joinrel=}
\def\Longrightarrow{\Relbar\joinrel\Rightarrow}
\def\longrightarrow{\relbar\joinrel\rightarrow}
\def\longleftarrow{\leftarrow\joinrel\relbar}
\def\Longleftarrow{\Leftarrow\joinrel\Relbar}
\def\longmapsto{\mapstochar\longrightarrow}
\def\longleftrightarrow{\leftarrow\joinrel\rightarrow}
\def\Longleftrightarrow{\Leftarrow\joinrel\Rightarrow}
\def\iff{\;\Longleftrightarrow\;}
@@
After defining characters [[\ldotp]] and [[\cdotp]] that act as math
punctuation, [[\ldots]] and [[\cdots]] macros are defined that
give the proper spacing in most circumstances. Vertical and diagonal
dots ([[\vdots]] and [[\ddots]]) are also provided here:

<<Supply vertical and diagonal dots>>=
\mathchardef\ldotp="613A % ldot as a punctuation mark
\mathchardef\cdotp="6201 % cdot as a punctuation mark
\mathchardef\colon="603A % colon as a punctuation mark
\def\ldots{\mathinner{\ldotp\ldotp\ldotp}}
\def\cdots{\mathinner{\cdotp\cdotp\cdotp}}
\def\vdots{\vbox{\baselineskip4\p@@ \lineskiplimit\z@@
    \kern6\p@@\hbox{.}\hbox{.}\hbox{.}}}
\def\ddots{\mathinner{\mkern1mu\raise7\p@@\vbox{\kern7\p@@\hbox{.}}\mkern2mu
    \raise4\p@@\hbox{.}\mkern2mu\raise\p@@\hbox{.}\mkern1mu}}
@@
Most of the math accents are handled entirely by the [[\mathaccent]]
primitive, but a few of the variable-width ones are constructed the
hard way.

<<Supply variable-width math accents>>=
\def\acute{\mathaccent"7013 }
\def\grave{\mathaccent"7012 }
\def\ddot{\mathaccent"707F }
\def\tilde{\mathaccent"707E }
\def\bar{\mathaccent"7016 }
\def\breve{\mathaccent"7015 }
\def\check{\mathaccent"7014 }
\def\hat{\mathaccent"705E }
\def\vec{\mathaccent"017E }
\def\dot{\mathaccent"705F }
\def\widetilde{\mathaccent"0365 }
\def\widehat{\mathaccent"0362 }
\def\overrightarrow#1{\vbox{\m@@th\ialign{##\crcr
      \rightarrowfill\crcr\noalign{\kern-\p@@\nointerlineskip}
      $\hfil\displaystyle{#1}\hfil$\crcr}}}
\def\overleftarrow#1{\vbox{\m@@th\ialign{##\crcr
      \leftarrowfill\crcr\noalign{\kern-\p@@\nointerlineskip}
      $\hfil\displaystyle{#1}\hfil$\crcr}}}
\def\overbrace#1{\mathop{\vbox{\m@@th\ialign{##\crcr\noalign{\kern3\p@@}
      \downbracefill\crcr\noalign{\kern3\p@@\nointerlineskip}
      $\hfil\displaystyle{#1}\hfil$\crcr}}}\limits}
\def\underbrace#1{\mathop{\vtop{\m@@th\ialign{##\crcr
      $\hfil\displaystyle{#1}\hfil$\crcr\noalign{\kern3\p@@\nointerlineskip}
      \upbracefill\crcr\noalign{\kern3\p@@}}}}\limits}
\def\skew#1#2#3{{\muskip\z@@#1mu\divide\muskip\z@@\tw@@ \mkern\muskip\z@@
    #2{\mkern-\muskip\z@@{#3}\mkern\muskip\z@@}\mkern-\muskip\z@@}{}}
@@
Now we come to 24 delimiters that can change their size.
These are denoted explicitly by a \<27-bit number>.

If we denote 27-bit number by [["cqrstuv]], then
delimiter codes are assigned by
\medskip
[[\delimiter "cqrstuv]],\quad where
\medskip
\itemitem{[[c]]}   --- the class
\itemitem{[[q]]}   --- the font family number of
\itemitem{[[rs]]}  --- the position of the the small variant of the delimiter
\itemitem{[[t]]}   --- the font family number of
\itemitem{[[uv]]}  --- the position of the the large variant of the delimiter
\medskip
After [[\left]] and [[\right]] commands the class digit is ignored.
When \TeX\ is not looking for a delimiter the righmost three
digits [[tuv]] are ignored, and the remaining four [[cqrs]]
are treated as a [[\mathchar]].

<<Supply extensible delimiters>>=
\def\lmoustache{\delimiter"437A340 } % top from (, bottom from )
\def\rmoustache{\delimiter"537B341 } % top from ), bottom from (
\def\lgroup{\delimiter"462833A } % extensible ( with sharper tips
\def\rgroup{\delimiter"562933B } % extensible ) with sharper tips
\def\arrowvert{\delimiter"26A33C } % arrow without arrowheads
\def\Arrowvert{\delimiter"26B33D } % double arrow without arrowheads
\def\bracevert{\delimiter"77C33E } % the vertical bar that extends braces
\def\Vert{\delimiter"26B30D } \let\|=\Vert
\def\vert{\delimiter"26A30C }
\def\uparrow{\delimiter"3222378 }
\def\downarrow{\delimiter"3223379 }
\def\updownarrow{\delimiter"326C33F }
\def\Uparrow{\delimiter"322A37E }
\def\Downarrow{\delimiter"322B37F }
\def\Updownarrow{\delimiter"326D377 }
\def\backslash{\delimiter"26E30F } % for double coset G\backslash H
\def\rangle{\delimiter"526930B }
\def\langle{\delimiter"426830A }
\def\rbrace{\delimiter"5267309 } \let\}=\rbrace
\def\lbrace{\delimiter"4266308 } \let\{=\lbrace
\def\rceil{\delimiter"5265307 }
\def\lceil{\delimiter"4264306 }
\def\rfloor{\delimiter"5263305 }
\def\lfloor{\delimiter"4262304 }
@@
In the [[plain.tex]] format and in the Computer Modern math fonts
there is only one radical---the square root.  The meaning of
[[\radical]] is analogous to the [[\delimiter]] commands. Only the
class number is dropped.  Joining the radical character and the
horizontal rule is done by leeting the radical character have a large
depth, and the height which is equal to the rule thickness. The rule
is placed on the baseline and the radical character is placed
below. Then the whole is centered around math axis.

<<Supply extensible delimiters>>=
\def\choose{\atopwithdelims()}
\def\brack{\atopwithdelims[]}
\def\brace{\atopwithdelims\{\}}

\def\sqrt{\radical"270370 }
@@
These macros depend on actual sizes of delimiters.

<<Provide access to delimiters of various sizes>>=
\def\bigl{\mathopen\big}
\def\bigm{\mathrel\big}
\def\bigr{\mathclose\big}
\def\Bigl{\mathopen\Big}
\def\Bigm{\mathrel\Big}
\def\Bigr{\mathclose\Big}
\def\biggl{\mathopen\bigg}
\def\biggm{\mathrel\bigg}
\def\biggr{\mathclose\bigg}
\def\Biggl{\mathopen\Bigg}
\def\Biggm{\mathrel\Bigg}
\def\Biggr{\mathclose\Bigg}
\def\big#1{{\hbox{$\left#1\vbox to8.5\p@@{}\right.\n@@space$}}}
\def\Big#1{{\hbox{$\left#1\vbox to11.5\p@@{}\right.\n@@space$}}}
\def\bigg#1{{\hbox{$\left#1\vbox to14.5\p@@{}\right.\n@@space$}}}
\def\Bigg#1{{\hbox{$\left#1\vbox to17.5\p@@{}\right.\n@@space$}}}
\def\n@@space{\nulldelimiterspace\z@@ \m@@th}
@@
The [[\mathpalette]] operation constructs a formula in all four
styles; it is applied here in the implementation of [[\phantom]],
[[\smash]], [[\root]], and other operations. (Actually [[\phantom]]
and [[\smash]] are not perfect: They assume that the current style is
uncramped.)

These definitions illustrate how other built-up
symbol combinations could be defined to work in all four styles.

The control sequences [[\sp]] and [[\sb]]
are provided for people who can't easily type [[^]] and [[_]];
a ``discretionary times sign'' [[\*]] is defined.

<<Provide macros for math formatting>>=
\def\mathpalette#1#2{\mathchoice{#1\displaystyle{#2}}%
  {#1\textstyle{#2}}{#1\scriptstyle{#2}}{#1\scriptscriptstyle{#2}}}
\newbox\rootbox
\def\root#1\of{\setbox\rootbox\hbox{$\m@@th\scriptscriptstyle{#1}$}
  \mathpalette\r@@@@t}
\def\r@@@@t#1#2{\setbox\z@@\hbox{$\m@@th#1\sqrt{#2}$}
  \dimen@@\ht\z@@ \advance\dimen@@-\dp\z@@
  \mkern5mu\raise.6\dimen@@\copy\rootbox \mkern-10mu \box\z@@}
\newif\ifv@@ \newif\ifh@@
\def\vphantom{\v@@true\h@@false\ph@@nt}
\def\hphantom{\v@@false\h@@true\ph@@nt}
\def\phantom{\v@@true\h@@true\ph@@nt}
\def\ph@@nt{\ifmmode\def\next{\mathpalette\mathph@@nt}%
  \else\let\next\makeph@@nt\fi\next}
\def\makeph@@nt#1{\setbox\z@@\hbox{#1}\finph@@nt}
\def\mathph@@nt#1#2{\setbox\z@@\hbox{$\m@@th#1{#2}$}\finph@@nt}
\def\finph@@nt{\setbox\tw@@\null
  \ifv@@ \ht\tw@@\ht\z@@ \dp\tw@@\dp\z@@\fi
  \ifh@@ \wd\tw@@\wd\z@@\fi \box\tw@@}
\def\mathstrut{\vphantom(}
\def\smash{\relax % \relax, in case this comes first in \halign
  \ifmmode\def\next{\mathpalette\mathsm@@sh}\else\let\next\makesm@@sh
  \fi\next}
\def\makesm@@sh#1{\setbox\z@@\hbox{#1}\finsm@@sh}
\def\mathsm@@sh#1#2{\setbox\z@@\hbox{$\m@@th#1{#2}$}\finsm@@sh}
\def\finsm@@sh{\ht\z@@\z@@ \dp\z@@\z@@ \box\z@@}

\def\cong{\mathrel{\mathpalette\@@vereq\sim}} % congruence sign
\def\@@vereq#1#2{\lower.5\p@@\vbox{\lineskiplimit\maxdimen\lineskip-.5\p@@
    \ialign{$\m@@th#1\hfil##\hfil$\crcr#2\crcr=\crcr}}}
\def\notin{\mathrel{\mathpalette\c@@ncel\in}}
\def\c@@ncel#1#2{\m@@th\ooalign{$\hfil#1\mkern1mu/\hfil$\crcr$#1#2$}}
\def\rightleftharpoons{\mathrel{\mathpalette\rlh@@{}}}
\def\rlh@@#1{\vcenter{\m@@th\hbox{\ooalign{\raise2pt
          \hbox{$#1\rightharpoonup$}\crcr
        $#1\leftharpoondown$}}}}
\def\buildrel#1\over#2{\mathrel{\mathop{\kern\z@@#2}\limits^{#1}}}
\def\doteq{\buildrel\textstyle.\over=}

\let\sp=^ \let\sb=_
\def\*{\discretionary{\thinspace\the\textfont2\char2}{}{}}
@@
The 32 common functions whose names generally appear in roman letters.

<<Supply common math functions>>=
\def\log{\mathop{\rm log}\nolimits}
\def\lg{\mathop{\rm lg}\nolimits}
\def\ln{\mathop{\rm ln}\nolimits}
\def\lim{\mathop{\rm lim}}
\def\limsup{\mathop{\rm lim\,sup}}
\def\liminf{\mathop{\rm lim\,inf}}
\def\sin{\mathop{\rm sin}\nolimits}
\def\arcsin{\mathop{\rm arcsin}\nolimits}
\def\sinh{\mathop{\rm sinh}\nolimits}
\def\cos{\mathop{\rm cos}\nolimits}
\def\arccos{\mathop{\rm arccos}\nolimits}
\def\cosh{\mathop{\rm cosh}\nolimits}
\def\tan{\mathop{\rm tan}\nolimits}
\def\arctan{\mathop{\rm arctan}\nolimits}
\def\tanh{\mathop{\rm tanh}\nolimits}
\def\cot{\mathop{\rm cot}\nolimits}
\def\coth{\mathop{\rm coth}\nolimits}
\def\sec{\mathop{\rm sec}\nolimits}
\def\csc{\mathop{\rm csc}\nolimits}
\def\max{\mathop{\rm max}}
\def\min{\mathop{\rm min}}
\def\sup{\mathop{\rm sup}}
\def\inf{\mathop{\rm inf}}
\def\arg{\mathop{\rm arg}\nolimits}
\def\ker{\mathop{\rm ker}\nolimits}
\def\dim{\mathop{\rm dim}\nolimits}
\def\hom{\mathop{\rm hom}\nolimits}
\def\det{\mathop{\rm det}}
\def\exp{\mathop{\rm exp}\nolimits}
\def\Pr{\mathop{\rm Pr}}
\def\gcd{\mathop{\rm gcd}}
\def\deg{\mathop{\rm deg}\nolimits}

\def\bmod{\nonscript\mskip-\medmuskip\mkern5mu
  \mathbin{\rm mod}\penalty900\mkern5mu\nonscript\mskip-\medmuskip}
\def\pmod#1{\allowbreak\mkern18mu({\rm mod}\,\,#1)}
@@
The definition of [[\matrix]] goes to some pains to ensure that two
$n$-rowed matrices will have the same height and the same depth, unless
at least one of their rows is unusually big. The definition of
[[\bordermatrix]] is even more complicated, but it seems to work
reasonably well; it uses a constant [[\p@@renwd]] that represents the
width of a big extensible left parenthesis.

<<Provide [[\cases]] and [[\matrix]] macros>>=
\def\cases#1{\left\{\,\vcenter{\normalbaselines\m@@th
    \ialign{$##\hfil$&\quad##\hfil\crcr#1\crcr}}\right.}
\def\matrix#1{\null\,\vcenter{\normalbaselines\m@@th
    \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr
      \mathstrut\crcr\noalign{\kern-\baselineskip}
      #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,}
\def\pmatrix#1{\left(\matrix{#1}\right)}
\newdimen\p@@renwd
\setbox0=\hbox{\tenex B} \p@@renwd=\wd0 % width of the big left (
\def\bordermatrix#1{\begingroup \m@@th
  \setbox\z@@\vbox{\def\cr{\crcr\noalign{\kern2\p@@\global\let\cr\endline}}%
    \ialign{$##$\hfil\kern2\p@@\kern\p@@renwd&\thinspace\hfil$##$\hfil
      &&\quad\hfil$##$\hfil\crcr
      \omit\strut\hfil\crcr\noalign{\kern-\baselineskip}%
      #1\crcr\omit\strut\cr}}%
  \setbox\tw@@\vbox{\unvcopy\z@@\global\setbox\@@ne\lastbox}%
  \setbox\tw@@\hbox{\unhbox\@@ne\unskip\global\setbox\@@ne\lastbox}%
  \setbox\tw@@\hbox{$\kern\wd\@@ne\kern-\p@@renwd\left(\kern-\wd\@@ne
    \global\setbox\@@ne\vbox{\box\@@ne\kern2\p@@}%
    \vcenter{\kern-\ht\@@ne\unvbox\z@@\kern-\baselineskip}\,\right)$}%
  \null\;\vbox{\kern\ht\@@ne\box\tw@@}\endgroup}
@@
The value of [[\lineskiplimit]] is assumed to be [[\normallineskiplimit]]
plus the accumulated amount of ``opening up.'' Thus, the [[\vskip]]
instructions in [[\displ@@y]] will compensate for the fact that the
first baseline of an alignment is separated by an opened-up baselineskip
from the last line preceding the display.

<<Provide support to typeset displayed equations>>=
\def\openup{\afterassignment\@@penup\dimen@@=}
\def\@@penup{\advance\lineskip\dimen@@
  \advance\baselineskip\dimen@@
  \advance\lineskiplimit\dimen@@}
\def\eqalign#1{\null\,\vcenter{\openup\jot\m@@th
  \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}
\newif\ifdt@@p
\def\displ@@y{\global\dt@@ptrue\openup\jot\m@@th
  \everycr{\noalign{\ifdt@@p \global\dt@@pfalse \ifdim\prevdepth>-1000\p@@
      \vskip-\lineskiplimit \vskip\normallineskiplimit \fi
      \else \penalty\interdisplaylinepenalty \fi}}}
\def\@@lign{\tabskip\z@@skip\everycr{}} % restore inside \displ@@y
\def\displaylines#1{\displ@@y \tabskip\z@@skip
  \halign{\hbox to\displaywidth{$\@@lign\hfil\displaystyle##\hfil$}\crcr
    #1\crcr}}
\def\eqalignno#1{\displ@@y \tabskip\centering
  \halign to\displaywidth{\hfil$\@@lign\displaystyle{##}$\tabskip\z@@skip
    &$\@@lign\displaystyle{{}##}$\hfil\tabskip\centering
    &\llap{$\@@lign##$}\tabskip\z@@skip\crcr
    #1\crcr}}
\def\leqalignno#1{\displ@@y \tabskip\centering
  \halign to\displaywidth{\hfil$\@@lign\displaystyle{##}$\tabskip\z@@skip
    &$\@@lign\displaystyle{{}##}$\hfil\tabskip\centering
    &\kern-\displaywidth\rlap{$\@@lign##$}\tabskip\displaywidth\crcr
    #1\crcr}}
@@
Below we have an interesting set of
macros that convert [[f''']] into [[f^{\prime\prime\prime}]].

<<Supply common math functions>>=
{\catcode`\'=\active \gdef'{^\bgroup\prim@@s}}
\def\prim@@s{\prime\futurelet\next\pr@@m@@s}
\def\pr@@m@@s{\ifx'\next\let\nxt\pr@@@@@@s \else\ifx^\next\let\nxt\pr@@@@@@t
  \else\let\nxt\egroup\fi\fi \nxt}
\def\pr@@@@@@s#1{\prim@@s} \def\pr@@@@@@t#1#2{#2\egroup}
{\catcode`\^^Z=\active \gdef^^Z{\not=}} % ^^Z is like \ne in math
{\catcode`\_=\active \global\let_=\_} % _ in math is either subscript or \_
@@
\section{Macros for output}

<<Prepare page for output>>=
<<Supply headers and footers>>
<<Supply footnotes>>
<<Supply floating insertions>>
<<Supply raggedbottom setting>>
<<Set up the output routine>>
@@
The [[\makeheadline]] macro constructs a vbox of height and depth
zero. The magic constant $-22.5\pt$ is equal to
$$
\hbox{[[\topskip]]}
       -\hbox{\<height of strut>}-2\,\hbox{[[\baselineskip]]}
$$
i.e., $10\pt-8.5\pt-24\pt$
(assuming default values of [[\topskip]] and the height of the strut);
this places the reference point of the headline exactly $24\pt$ above the
reference point of the top line on the page, unless the headline or
the top line are excessively large.

The [[\advancepageno]] macro normally advances [[\pageno]]
by~$+1$; but if [[\pageno]] is negative (for roman numerals), the advance is
by~$-1$. The new value of [[\pageno]] will be appropriate for the next time
the output routine is called into action.

<<Supply headers and footers>>=
\countdef\pageno=0 \pageno=1 % first page is number 1
\newtoks\headline \headline={\hfil} % headline is normally blank
\newtoks\footline \footline={\hss\tenrm\folio\hss}
% footline is normally a centered page number in font \tenrm
%
\def\makeheadline{\vbox to\z@@{\vskip-22.5\p@@
  \line{\vbox to8.5\p@@{}\the\headline}\vss}\nointerlineskip}
\def\makefootline{\baselineskip24\p@@\line{\the\footline}}
%
\def\folio{\ifnum\pageno<\z@@ \romannumeral-\pageno \else\number\pageno \fi}
\def\nopagenumbers{\footline{\hfil}} % blank out the footline
\def\advancepageno{\ifnum\pageno<\z@@ \global\advance\pageno\m@@ne
  \else\global\advance\pageno\@@ne \fi} % increase |pageno|
@@
Ragged-bottom setting is achieved by inserting
infinite glue, which overpowers the stretchability of [[\topskip]].
This macros assume that [[\topskip = 10pt]]

<<Supply raggedbottom setting>>=
\newif\ifr@@ggedbottom
\def\raggedbottom{\topskip 10\p@@ plus60\p@@ \r@@ggedbottomtrue}
\def\normalbottom{\topskip 10\p@@ \r@@ggedbottomfalse} % undoes \raggedbottom
@@
There are 255 classes of insertions, [[\insert0]] to [[\insert254]], and
they are tied to other registers of the same number. For example,
[[\insert100]] is connected with [[\count100]], [[\dimen100]],
[[\skip100]], and [[\box100]].

\def\n{\thinspace$n$}
For our purposes let's consider a particular class of insertions
called class $n$; we will then be dealing with \TeX's primitive command
\begindisplay
[[\insert]]\n[[{]]\<vertical mode material>[[}]]
\enddisplay
which puts an insertion
item into a horizontal or vertical list. For this class of insertions
\begindisplay
[[\box]]\n\qquad is where the material appears when a page is output;\cr
[[\count]]\n\qquad is the magnification factor for page breaking;\cr
[[\dimen]]\n\qquad is the maximum insertion size per page;\cr
[[\skip]]\n\qquad is the extra space to allocate on a page.\cr
\enddisplay
For example, material inserted with [[\insert100]] will
eventually appear in [[\box100]].
@@
[[\footnote]] macro depends on the value of [[\bigskipamount]], and
[[\parindent]].
Because the value of [[\vsize]] ($8.9\in$)
is greater than [[\dimen\footins]] ($8\in$),
footnotes never fill up the whole page.

<<Supply footnotes>>=
\newinsert\footins
\def\footnote#1{\let\@@sf\empty % parameter #2 (the text) is read later
  \ifhmode\edef\@@sf{\spacefactor\the\spacefactor}\/\fi
  #1\@@sf\vfootnote{#1}}
\def\vfootnote#1{\insert\footins\bgroup
  \interlinepenalty\interfootnotelinepenalty
  \splittopskip\ht\strutbox % top baseline for broken footnotes
  \splitmaxdepth\dp\strutbox \floatingpenalty\@@MM
  \leftskip\z@@skip \rightskip\z@@skip \spaceskip\z@@skip \xspaceskip\z@@skip
  \textindent{#1}\footstrut\futurelet\next\fo@@t}
\def\fo@@t{\ifcat\bgroup\noexpand\next \let\next\f@@@@t
  \else\let\next\f@@t\fi \next}
\def\f@@@@t{\bgroup\aftergroup\@@foot\let\next}
\def\f@@t#1{#1\@@foot}
\def\@@foot{\strut\egroup}
\def\footstrut{\vbox to\splittopskip{}}

\skip\footins=\bigskipamount % space added when footnote is present
\count\footins=1000 % footnote magnification factor (1 to 1)
\dimen\footins=8in % maximum footnotes per page

\def\footnoterule{\kern-3\p@@
  \hrule width 2truein \kern 2.6\p@@} % the \hrule is .4pt high
@@
Here the constant [[12\p@@ = 12pt]] is hard coded into the format.

<<Supply floating insertions>>=
\newinsert\topins
\newif\ifp@@ge \newif\if@@mid
\def\topinsert{\@@midfalse\p@@gefalse\@@ins}
\def\midinsert{\@@midtrue\@@ins}
\def\pageinsert{\@@midfalse\p@@getrue\@@ins}
\skip\topins=\z@@skip % no space added when a topinsert is present
\count\topins=1000 % magnification factor (1 to 1)
\dimen\topins=\maxdimen % no limit per page
\def\@@ins{\par\begingroup\setbox\z@@\vbox\bgroup} % start a \vbox
\def\endinsert{\egroup % finish the \vbox
  \if@@mid \dimen@@\ht\z@@ \advance\dimen@@\dp\z@@ \advance\dimen@@12\p@@
    \advance\dimen@@\pagetotal \advance\dimen@@-\pageshrink
    \ifdim\dimen@@>\pagegoal\@@midfalse\p@@gefalse\fi\fi
  \if@@mid \bigskip\box\z@@\bigbreak
  \else\insert\topins{\penalty100 % floating insertion
    \splittopskip\z@@skip
    \splitmaxdepth\maxdimen \floatingpenalty\z@@
    \ifp@@ge \dimen@@\dp\z@@
    \vbox to\vsize{\unvbox\z@@\kern-\dimen@@}% depth is zero
    \else \box\z@@\nobreak\bigskip\fi}\fi\endgroup}
@@
The value of [[\boxmaxdepth]] is set to [[\maxdepth]] so that the vbox
will be constructed under the assumptions that \TeX's page builder has
used to set up [[\box255]].

The [[\pagecontents]] macro produces a vertical list for
everything that belongs on the main body of the page, namely the contents
of [[\box255]] together with illustrations (inserted at the top)
and footnotes (inserted at the bottom).
[[\topins]] and [[\footins]] are the insertion class numbers for
the two kinds of insertions used in plain \TeX; if more classes of
insertions are added, [[\pagecontents]] should be changed accordingly.
Notice that the boxes are unboxed so that the glue coming from
insertions can help out the glue on the main page. The [[\footnoterule]]
macro places a dividing line between the page and its
footnotes; it makes a net contribution of $0\pt$ to the height of
the vertical list.

The [[\dosupereject]] macro is designed to clear out any insertions
that have been held over, whether they are illustrations or footnotes
or both. The negative [[\kern]] here cancels out the natural space of
the [[\topskip]] glue that goes above the empty [[\line]]; that empty
line box prevents the [[\vfill]] from disappearing into a page break.
The vertical list that results from [[\dosupereject]] is placed on
\TeX's list of things to put out next, just after the straggling
insertions have been reconsidered. Hence another super-eject will
occur, and the process will continue until no insertions remain.

<<Set up the output routine>>=
\output{\plainoutput}
\def\plainoutput{\shipout\vbox{\makeheadline\pagebody\makefootline}%
  \advancepageno
  \ifnum\outputpenalty>-\@@MM \else\dosupereject\fi}

\def\pagebody{\vbox to\vsize{\boxmaxdepth\maxdepth \pagecontents}}

\def\dosupereject{\ifnum\insertpenalties>\z@@ % something is being held over
  \line{}\kern-\topskip\nobreak\vfill\supereject\fi}

\def\pagecontents{\ifvoid\topins\else\unvbox\topins\fi
  \dimen@@=\dp\@@cclv \unvbox\@@cclv % open up \box255
  \ifvoid\footins\else % footnote info is present
    \vskip\skip\footins
    \footnoterule
    \unvbox\footins\fi
  \ifr@@ggedbottom \kern-\dimen@@ \vfil \fi}
@@
\section{Hyphenation}

A discretionary break consists of three sequences of characters
called the {\sl pre-break}, {\sl post-break}, and {\sl no-break\/}
texts. The idea is that if a line break occurs here, the pre-break text
will appear at the end of the current line and the post-break text will
occur at the beginning of the next line; but if no break occurs, the
no-break text will appear in the current line.
The discretionary are specified by writing
\begindisplay
|\discretionary{|\<pre-break text>|}{|\<post-break text>|}{|\<no-break text>|}|
\enddisplay
where the three texts consist entirely of characters, boxes, and kerns.
If a word contains discretionary breaks \TeX\ will not hyphenate it.

Hyphenation exceptions are specified with the statements like
\begindisplay
[[\hyphenation{gal-axy iso-peri-met-ric}]]
\enddisplay
which gives to \TeX\ locations where these words may be hyphenated.

The default values of [[\hyphenchar]] is [[`-]].

Sometimes the typewriter fonts are given
[[\hyphenchar]]\<font name>[[=-1]] which value inhibits hyphenation.

<<Read hyphenation patterns>>=
\lefthyphenmin=2 \righthyphenmin=3 % disallow x- or -xx breaks
\input hyphen
@@
\section{Initalization}

<<Initialize the layout>>=
\normalbaselines\rm % select roman font
\nonfrenchspacing % punctuation affects the spacing
@@
\section{Programming support}

<<Provide programming constructs>>=
<<Supply loops and conditionals>>
<<Define [[\tracingall]] macro>>
@@
The [[\loop...\repeat]] macro provides for iterative operations.
In this macro and several others, the control sequence `[[\next]]' is
given a temporary value that is not going to be needed later; thus,
[[\next]] acts like a ``scratch control sequence.''

The macro [[\newif]] to be used for definitions of new conditionals.
For example, [[\newif\iffoo]] creates [[\footrue]], [[\foofalse]] to
go with [[\iffoo]].

<<Supply loops and conditionals>>=
\def\loop#1\repeat{\def\body{#1}\iterate}
\def\iterate{\body \let\next\iterate \else\let\next\relax\fi \next}
\let\repeat=\fi % this makes \loop...\if...\repeat skippable

\outer\def\newif#1{\count@@\escapechar \escapechar\m@@ne
  \expandafter\expandafter\expandafter
   \edef\@@if#1{true}{\let\noexpand#1=\noexpand\iftrue}%
  \expandafter\expandafter\expandafter
   \edef\@@if#1{false}{\let\noexpand#1=\noexpand\iffalse}%
  \@@if#1{false}\escapechar\count@@} % the condition starts out false
\def\@@if#1#2{\csname\expandafter\if@@\string#1#2\endcsname}
{\uccode`1=`i \uccode`2=`f \uppercase{\gdef\if@@12{}}} % `if' is required
@@
<<Define [[\tracingall]] macro>>=
\def\tracingall{\tracingonline\@@ne\tracingcommands\tw@@\tracingstats\tw@@
  \tracingpages\@@ne\tracingoutput\@@ne\tracinglostchars\@@ne
  \tracingmacros\tw@@\tracingparagraphs\@@ne\tracingrestores\@@ne
  \showboxbreadth\maxdimen\showboxdepth\maxdimen\errorstopmode}
@@
The format name and version number are recorded in control sequences,
in order to help the people who might have to explain why something
doesn't work.

<<Identify the format>>=
\def\fmtname{plain+W}\def\fmtversion{3.14159+W}
@@
\chapter{Appendices}

\section{Efficiency and memory-space considerations}

One difficulty with large sets of macros is that they take up space.
It would be nice to preload every macro that every \TeX\ user has ever
dreamed up; but there might not be enough room,
because \TeX's memory capacity is finite. You might find it necessary
to hold back and to load only the macros that are really needed.
\medskip
How much memory space does a macro require?
\medskip
\noindent There are four kinds of memory involved: token memory,
name memory, string memory, and character memory. (If any of these
becomes too full, it will be necessary to increase what \TeX\ calls the
macro memory size, the hash size, the number of strings, and/or the
pool size, respectively. The token memory is
most important; a macro takes one cell of token memory for each token
in its definition, including the `|{|' and the `|}|'. For example,
the comparatively short definition
\medskip
[[\def\example#1\two{\four}]]
\medskip
\noindent takes five tokens: |#1|, [[\two]], |{|$_1$, [[\four]], and |}|$_2$.
Each control sequence also takes up one cell of name memory, one cell
of string memory, and as many cells of character memory as there are
characters in the name (seven in the case of |\example|). Character
memory is comparatively cheap; four characters, or in some cases five,
will fit in the same number of bits as a single cell of token memory,
inside the machine. Therefore you don't save much by choosing short
macro names.

\TeX\ will tell you how close you come to exceeding its current
memory capacity if you say [[\tracingstats=1]].
\TeX\ governs fourteen kinds of memory:
\begindisplay
|number of strings|\qquad(names of control sequences and files)\cr
|pool size|\qquad(the characters in such names)\cr
|main memory size|\qquad(boxes, glue, breakpoints, token lists,
        characters, etc.)\cr
|hash size|\qquad(control sequence names)\cr
|font memory|\qquad(font metric data)\cr
|exception dictionary|\qquad(hyphenation exceptions)\cr
|input stack size|\qquad(simultaneous input sources)\cr
|semantic nest size|\qquad(unfinished lists being constructed)\cr
|parameter stack size|\qquad(macro parameters)\cr
|buffer size|\qquad(characters in lines being read from files)\cr
|save size|\qquad(values to restore at group ends)\cr
|text input levels|\qquad(|\input| files and error insertions)\cr
|grouping levels|\qquad(unfinished groups)\cr
|pattern memory|\qquad(hyphenation pattern data)\cr
\enddisplay
The current amount of memory available will also be shown.

One obvious way to keep from loading too many macros is to keep
the macro files short and to |\input| only the ones that you need.

\section{Extensible delimiters}

\TeX\ builds large delimiters by using ``extensible'' characters,
which are specified by giving top, midle, bottom, and repeatable
characters in an \bold{extensible} command.
For example, the extensible left parentheses in [[cmex10]] are
defined by (see Figure~4)
\medskip
[[extensible oct"060": oct"060", 0, oct"100", oct"102";]]
\medskip
\noindent this says that character code [[oct"060"]] specifies
an extensible delimiter constructed from itself as the top piece,
from character number [[oct"100"]] as the bottom piece,
and from character number [[oct"102"]] as the piece which
should be repeated as often as necessary to reach a desired size.
In this particular example there is no middle piece,
but characters like curly braces have a middle piece as well.
A zero value in the top, middle, or bottom position means that
no character should be used in that part of the construction;
but a zero value in the final position means that character
number zero is the repeater. The width of an extensible
character is taken to be the width of the repeater.

Also several characters of various sizes can be linked together
in a series by means of a \bold{charlist} command.
For example (see Figure~4),
\medskip
[[charlist oct"000": oct"020": oct"022": oct"040";]]
\medskip
\noindent is used in the font [[cmex10]] to specify the left
parentheses that \TeX\ uses in displayed math formulas.
\TeX\ follows charlist to make variable-size delimiters
and variable-size accents, as well to link
[[\textstyle]] and the [[\displaystyle]] operators.


\section{Font dimensions}

The main information about font consists of the dimensions of the characters.
These numbers \TeX\ finds in the font metric files.
Except character dimensions, font metric files contain:
values for [[\fontdimen]]
parameters, italic correction of characters, ligature and kerning
programs for characters.
We change the fontdimen parameters with the (global) assignment:
\begindisplay
[[\fontdimen]]\<number>\<font>\<equals>\<dimen>,
\enddisplay
for example, the assignment
[[\fontdimen8\tenex = 0.6pt]]
changes width of the fraction bar from default $0.4\pt$ to $0.6\pt$.
\bigskip
\noindent The first seven fontdimen parameters have the following meaning:
\medskip
\item{1.}  the slant per point
\item{2.}  the interword space; that is used unless [[\spaceskip]]
           is specified
\item{3.}  interword stretch
\item{4.}  interword shrink
\item{5.}  the x-height
\item{6.}  the quad width (for the font in family 2,
           $1/18\,$th quad width is equal to $1\,$mu)
\item{7.}  the extra space; that value is added to the interword space
           used whenever [[\spacefactor]]${}\geq 2000$,
           unless [[\xspaceskip]] is specified.
\bigskip
\noindent For the font in family 2 attributes 8--19 specify positioning
of fractions, subscripts, superscripts.
\medskip
\noindent \stress{fraction numerator attributes: minimum shift up,
from the main baseline, of the baseline of the numerator of
a generalized fraction}
\medskip
\item{8.}  num1: for display style
\item{9.}  num2: for text style or smaller if a fraction
          bar is present
\item{10.} num3: for text style or smaller if no a fraction
          bar is present
\medskip
\noindent
\stress{fraction denominator attributes: minimum shift down, from the main
baseline, of the baseline of the denominator of a generalized fraction}
\medskip
\item{11.} denom1: for display style
\item{12.} denom2: for text style or smaller
\medskip
\noindent
\stress{superscript attributes: minimum shift up, from the main baseline,
of the baseline of the superscript}
\medskip
\item{13.} sup1: for display style
\item{14.} sup2: for text style or smaller, non-cramped
\item{15.} sup3: for text style or smaller, cramped
\medskip
\noindent
\stress{subscript attributes: minimum shift down, from the main baseline,
of the baseline of a subscript}
\medskip
\item{16.} sub1: when no superscript is present
\item{17.} sub2: when superscript is present
\medskip
\noindent
\stress{script adjustment attributes: for use only with non-glyph,
that is, composite objects}
\medskip
\item{18.} sup\_drop: maximum distance of superscript baseline below
          top of nucleus
\item{19.} sub\_drop: minimum distance of subscript baseline below
          bottom of nucleus
\medskip
\noindent
Delimiter span attributes: height plus depth of delimiter
enclosing a generalized fraction.
\medskip
\item{20.} delim1: in display style
\item{21.} delim2: in text style or smaller
\medskip
\noindent
The last parameter, the height of the math axis, specifies the height
above the baseline of the fraction bar, and the centre of large
delimiters and most operators and relations. This position is used in
vertical centering.
\medskip
\item{22.} axis\_height.
\bigbreak
\noindent For the font in family 3 attributes 9--12 determine
extra space added when limits are attached to operators.
The attribute 8 specifies thickness of the rule used for
overlines, underlines, radical extenders, and fraction bars.
From that dimension are derived `clearances' around
fraction bar.
The attribute 13 specifies extra space added above and below
attached limits.
\medskip
\item{8.}  default\_rule\_thickness
\item{9.}  big\_op\_spacing1
\item{10.} big\_op\_spacing2
\item{11.} big\_op\_spacing3
\item{12.} big\_op\_spacing4
\item{13.} big\_op\_spacing5
\medskip
\noindent We have:
big\_op\_spacing1(2) $\leq$ space between upper (lower)
limit and top (bottom) of large operator $\leq$ big\_op\_spacing3(4)
\bigskip
\begingroup
\def \loop #1\repeat%
    {\global\def\body{#1}\iterate}

\newcount \dno

\def \?#1%
    {\ifdim\the\fontdimen\dno#1<10pt \hphantom{0}\fi%
     \the\fontdimen\dno#1}

\ialign{\global\advance \dno by 1\relax#&
        \hfil\number\dno.#\quad&&
        \quad\tentt#\cr
\omit&\omit\tensl fontdimen&cmsy10&cmex10&cmtt10&cmr10&cmti10&cmbx10
                                                    \cr\noalign{\bigskip}
\loop&&
\?\tensy&
\?\tenex&
\?\tentt&
\?\tenrm&
\?\tenit&
\?\tenbf&\cr
\ifnum\dno<7\repeat\cr}

\advance \dno by -1
\vskip-\baselineskip

\ialign{\global\advance \dno by 1\relax#&
        \hfil\number\dno.#\quad&&
        \quad\tentt#\cr
\omit&\omit\hphantom{\tensl fontdimen}&\cr
\loop&&
\?\tensy&
\?\tenex&\cr
\ifnum\dno<13\repeat\cr}

\advance \dno by -1
\vskip-\baselineskip

\ialign{\global\advance \dno by 1\relax#&
        \hfil\number\dno.#\quad&&
        \quad\tentt#\cr
\omit&\omit\hphantom{\tensl fontdimen}&\cr
\loop&&
\?\tensy&\cr
\ifnum\dno<22\repeat\cr}
\endgroup
@@
\section{Font tables}

\noindent {\bf Figure 1.\enspace}[[cmr10]]---family 0
\beginchart\tenrm
\normalchart
\endchart

\medskip\vfill
\noindent {\bf Figure 2.\enspace}[[cmmi10]]---family 1
\beginchart\teni
\normalchart
\endchart

\medskip\vfill
\noindent {\bf Figure 3.\enspace}[[cmsy10]]---family 2
\beginchart\tensy
\normalchart
\endchart

\medskip\vfill
\noindent {\bf Figure 4.\enspace}[[cmex10]]---family 3
\beginchart\tenex
\normalchart
\endchart
@