This file is indexed.

/usr/share/gap/lib/coll.gd is in gap-libs 4r6p5-3.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

   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
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
3042
3043
3044
3045
3046
3047
3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116
3117
3118
3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
3164
3165
3166
3167
3168
3169
3170
3171
3172
3173
3174
3175
3176
3177
3178
3179
3180
3181
3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
3206
3207
3208
3209
3210
3211
3212
3213
3214
3215
3216
3217
3218
3219
3220
3221
3222
3223
3224
3225
3226
3227
3228
3229
3230
3231
3232
#############################################################################
##
#W  coll.gd                     GAP library                  Martin Schönert
#W                                                            & Thomas Breuer
##
##
#Y  Copyright (C)  1997,  Lehrstuhl D für Mathematik,  RWTH Aachen,  Germany
#Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y  Copyright (C) 2002 The GAP Group
##
##  This file declares the operations for collections.
##

#T change the installation of isomorphism and factor maintained methods
#T in the same way as that of subset maintained methods!


#############################################################################
##
##  <#GAPDoc Label="[1]{coll}">
##  A <E>collection</E> in &GAP; consists of elements in the same family
##  (see&nbsp;<Ref Sect="Families"/>).
##  The most important kinds of collections are <E>homogeneous lists</E>
##  (see&nbsp;<Ref Chap="Lists"/>)
##  and <E>domains</E> (see&nbsp;<Ref Chap="Domains"/>).
##  Note that a list is never a domain, and a domain is never a list.
##  A list is a collection if and only if it is nonempty and homogeneous.
##  <P/>
##  Basic operations for collections are <Ref Func="Size"/>
##  and <Ref Func="Enumerator"/>;
##  for <E>finite</E> collections,
##  <Ref Func="Enumerator"/> admits to delegate the other
##  operations for collections
##  (see&nbsp;<Ref Sect="Attributes and Properties for Collections"/>
##  and&nbsp;<Ref Sect="Operations for Collections"/>)
##  to functions for lists (see&nbsp;<Ref Chap="Lists"/>).
##  Obviously, special methods depending on the arguments are needed for
##  the computation of e.g.&nbsp;the intersection of two <E>infinite</E>
##  domains.
##  <#/GAPDoc>
##


#############################################################################
##
#C  IsListOrCollection( <obj> )
##
##  <#GAPDoc Label="IsListOrCollection">
##  <ManSection>
##  <Filt Name="IsListOrCollection" Arg='obj' Type='Category'/>
##
##  <Description>
##  Several functions are defined for both lists and collections,
##  for example <Ref Func="Intersection" Label="for a list"/>,
##  <Ref Func="Iterator"/>,
##  and <Ref Func="Random" Label="for a list or collection"/>.
##  <Ref Func="IsListOrCollection"/> is a supercategory of
##  <Ref Func="IsList"/> and <Ref Func="IsCollection"/>
##  (that is, all lists and collections lie in this category),
##  which is used to describe the arguments of functions such as the ones
##  listed above.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareCategory( "IsListOrCollection", IsObject );


#############################################################################
##
#C  IsCollection( <obj> ) . . . . . . . . . test if an object is a collection
##
##  <#GAPDoc Label="IsCollection">
##  <ManSection>
##  <Filt Name="IsCollection" Arg='obj' Type='Category'/>
##
##  <Description>
##  tests whether an object is a collection.
##  <P/>
##  Some of the functions for lists and collections are described in the
##  chapter about lists,
##  mainly in Section&nbsp;<Ref Sect="Operations for Lists"/>.
##  In the current chapter, we describe those functions for which the
##  <Q>collection aspect</Q> seems to be more important than the
##  <Q>list aspect</Q>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareCategory( "IsCollection", IsListOrCollection );


#############################################################################
##
#A  CollectionsFamily( <Fam> )  . . . . . . . . . . make a collections family
##
##  <#GAPDoc Label="CollectionsFamily">
##  <ManSection>
##  <Attr Name="CollectionsFamily" Arg='Fam'/>
##
##  <Description>
##  For a family <A>Fam</A>, <Ref Func="CollectionsFamily"/> returns the
##  family of all collections over <A>Fam</A>,
##  that is, of all dense lists and domains that consist of objects in
##  <A>Fam</A>.
##  <P/>
##  The <Ref Func="NewFamily"/> call in the standard method of
##  <Ref Func="CollectionsFamily"/> is executed with second argument
##  <Ref Func="IsCollection"/>,
##  since every object in the collections family must be a collection,
##  and with third argument the collections categories of the involved
##  categories in the implied filter of <A>Fam</A>.
##  <P/>
##  Note that families (see&nbsp;<Ref Sect="Families"/>)
##  are used to describe relations between objects.
##  Important such relations are that between an element <M>e</M> and each
##  collection of elements that lie in the same family as <M>e</M>,
##  and that between two collections whose elements lie in the same family.
##  Therefore, all collections of elements in the family <A>Fam</A> form the
##  new family <C>CollectionsFamily( <A>Fam</A> )</C>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "CollectionsFamily", IsFamily );


#############################################################################
##
#C  IsCollectionFamily( <Fam> )  test if an object is a family of collections
##
##  <#GAPDoc Label="IsCollectionFamily">
##  <ManSection>
##  <Filt Name="IsCollectionFamily" Arg='obj' Type='Category'/>
##
##  <Description>
##  is <K>true</K> if <A>Fam</A> is a family of collections,
##  and <K>false</K> otherwise.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareCategoryFamily( "IsCollection" );


#############################################################################
##
#A  ElementsFamily( <Fam> ) . . . . . . . . . . . . fetch the elements family
##
##  <#GAPDoc Label="ElementsFamily">
##  <ManSection>
##  <Attr Name="ElementsFamily" Arg='Fam'/>
##
##  <Description>
##  If <A>Fam</A> is a collections family
##  (see&nbsp;<Ref Func="IsCollectionFamily"/>)
##  then <Ref Func="ElementsFamily"/>
##  returns the family from which <A>Fam</A> was created
##  by <Ref Func="CollectionsFamily"/>.
##  The way a collections family is created, it always has its elements
##  family stored.
##  If <A>Fam</A> is not a collections family then an error is signalled.
##  <P/>
##  <Example><![CDATA[
##  gap> fam:= FamilyObj( (1,2) );;
##  gap> collfam:= CollectionsFamily( fam );;
##  gap> fam = collfam;  fam = ElementsFamily( collfam );
##  false
##  true
##  gap> collfam = FamilyObj( [ (1,2,3) ] );
##  true
##  gap> collfam = FamilyObj( Group( () ) );
##  true
##  gap> collfam = CollectionsFamily( collfam );
##  false
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "ElementsFamily", IsFamily );


#############################################################################
##
#V  CATEGORIES_COLLECTIONS  . . . . . . global list of collections categories
##
##  <ManSection>
##  <Var Name="CATEGORIES_COLLECTIONS"/>
##
##  <Description>
##  </Description>
##  </ManSection>
##
BIND_GLOBAL( "CATEGORIES_COLLECTIONS", [] );


#############################################################################
##
#F  CategoryCollections( <filter> ) . . . . . . . . . .  collections category
##
##  <#GAPDoc Label="CategoryCollections">
##  <ManSection>
##  <Func Name="CategoryCollections" Arg='filter'/>
##
##  <Description>
##  Let <A>filter</A> be a filter that is <K>true</K> for all elements of a
##  family <A>Fam</A>, by the construction of <A>Fam</A>.
##  Then <Ref Func="CategoryCollections"/> returns the
##  <E>collections category</E> of <A>filter</A>.
##  This is a category that is <K>true</K> for all elements in
##  <C>CollectionsFamily( <A>Fam</A> )</C>.
##  <P/>
##  For example, the construction of
##  <Ref Var="PermutationsFamily"/> guarantees that
##  each of its elements lies in the filter
##  <Ref Func="IsPerm"/>,
##  and each collection of permutations (permutation group or dense list of
##  permutations) lies in the category <C>CategoryCollections( IsPerm )</C>.
##  <C>CategoryCollections( IsPerm )</C>.
##  Note that this works only if the collections category is created
##  <E>before</E> the collections family.
##  So it is necessary to construct interesting collections categories
##  immediately after the underlying category has been created.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
BIND_GLOBAL( "CategoryCollections", function ( elms_filter )
    local    pair, super, flags, name, coll_filter;

    # Check whether the collections category is already defined.
    for pair in CATEGORIES_COLLECTIONS do
      if IsIdenticalObj( pair[1], elms_filter ) then
        return pair[2];
      fi;
    od;

    # Find the super category among the known collections categories.
    super := IsCollection;
    flags := WITH_IMPS_FLAGS( FLAGS_FILTER( elms_filter ) );
    for pair in CATEGORIES_COLLECTIONS do
      if IS_SUBSET_FLAGS( flags, FLAGS_FILTER( pair[1] ) ) then
        super := super and pair[2];
      fi;
    od;

    # Construct the name of the category.
    name := "CategoryCollections(";
    APPEND_LIST_INTR( name, SHALLOW_COPY_OBJ( NameFunction(elms_filter) ) );
    APPEND_LIST_INTR( name, ")" );
    CONV_STRING( name );

    # Construct the collections category.
    coll_filter:= NewCategory( name, super );
    ADD_LIST( CATEGORIES_COLLECTIONS, [ elms_filter, coll_filter ] );
    return coll_filter;
end );


#############################################################################
##
#f  DeclareCategoryCollections( <name> )
##
##  binds the collections category of the category that is bound to the
##  global variable with name <name> to the global variable associated to the
##  name <nname>.
##  If <name> is of the form `<initname>Collection' then <nname> is equal to
##  `<initname>CollColl',
##  if <name> is of the form `<initname>Coll' then <nname> is equal to
##  `<initname>CollColl',
##  otherwise we have <nname> equal to `<name>Collection'.
##
BIND_GLOBAL( "DeclareCategoryCollections", function( name )
    local len, coll_name;

    len:= LEN_LIST( name );
    if    3 < len and name{ [ len-3 .. len ] } = "Coll" then
      coll_name:= SHALLOW_COPY_OBJ( name );
      APPEND_LIST_INTR( coll_name, "Coll" );
    elif 9 < len and name{ [ len-9 .. len ] } = "Collection" then
      coll_name:= name{ [ 1 .. len-6 ] };
      APPEND_LIST_INTR( coll_name, "Coll" );
    else
      coll_name:= SHALLOW_COPY_OBJ( name );
      APPEND_LIST_INTR( coll_name, "Collection" );
    fi;

    BIND_GLOBAL( coll_name, CategoryCollections( VALUE_GLOBAL( name ) ) );
end );


#############################################################################
##
#F  DeclareSynonym( <name>, <value> )
#F  DeclareSynonymAttr( <name>, <value> )
##
##  <#GAPDoc Label="DeclareSynonym">
##  <ManSection>
##  <Func Name="DeclareSynonym" Arg='name, value'/>
##  <Func Name="DeclareSynonymAttr" Arg='name, value'/>
##
##  <Description>
##  <Ref Func="DeclareSynonym"/> assigns the string <A>name</A> to a global
##  variable as a synonym for <A>value</A>.
##  Two typical intended usages are to declare an <Q>and-filter</Q>, e.g.
##  <P/>
##  <Log><![CDATA[
##  DeclareSynonym( "IsGroup", IsMagmaWithInverses and IsAssociative );
##  ]]></Log>
##  <P/>
##  and to provide a previously declared global function with an alternative
##  name, e.g.
##  <P/>
##  <Log><![CDATA[
##  DeclareGlobalFunction( "SizeOfSomething" );
##  DeclareSynonym( "OrderOfSomething", SizeOfSomething );
##  ]]></Log>
##  <P/>
##  <E>Note:</E> Before using <Ref Func="DeclareSynonym"/> in the way of this
##  second example,
##  one should determine whether the synonym is really needed.
##  Perhaps an extra index entry in the documentation would be sufficient.
##  <P/>
##  When <A>value</A> is actually an attribute then
##  <Ref Func="DeclareSynonymAttr"/> should be used;
##  this binds also globals variables <C>Set</C><A>name</A> and
##  <C>Has</C><A>name</A> for its setter and tester, respectively.
##  <P/>
##  <Log><![CDATA[
##  DeclareSynonymAttr( "IsField", IsDivisionRing and IsCommutative );
##  DeclareAttribute( "GeneratorsOfDivisionRing", IsDivisionRing );
##  DeclareSynonymAttr( "GeneratorsOfField", GeneratorsOfDivisionRing );
##  ]]></Log>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
BIND_GLOBAL( "DeclareSynonym", function( name, value )
    BIND_GLOBAL( name, value );
end );

BIND_GLOBAL( "DeclareSynonymAttr", function( name, value )
    local nname;
    BIND_GLOBAL( name, value );
    nname:= "Set";
    APPEND_LIST_INTR( nname, name );
    BIND_GLOBAL( nname, Setter( value ) );
    nname:= "Has";
    APPEND_LIST_INTR( nname, name );
    BIND_GLOBAL( nname, Tester( value ) );
end );


#############################################################################
##
#V  SUBSET_MAINTAINED_INFO
##
##  <ManSection>
##  <Var Name="SUBSET_MAINTAINED_INFO"/>
##
##  <Description>
##  is a list of length two.
##  At the first position, a list of lists of the form
##  <C>[ <A>filtsuper</A>, <A>filtsub</A>, <A>opr</A>, <A>testopr</A>, <A>settopr</A> ]</C>
##  is stored,
##  which is used for calls of <C>UseSubsetRelation( <A>super</A>, <A>sub</A> )</C>.
##  At the second position, a corresponding list of lists of the form
##  <C>[ <A>flagsopr</A>, <A>flagssub</A>, <A>rank</A> ]</C>
##  is stored, which is used for choosing an appropriate ordering of the
##  entries when the lists are enlarged in a call to
##  <C>InstallSubsetMaintenance</C>.
##  <P/>
##  The meaning of the entries is as follows.
##  <List>
##  <Mark><A>filtsuper</A> </Mark>
##  <Item>
##      required filter for <A>super</A>,
##  </Item>
##  <Mark><A>filtsub</A> </Mark>
##  <Item>
##      required filter for <A>sub</A>,
##  </Item>
##  <Mark><A>opr</A> </Mark>
##  <Item>
##      operation whose value is inherited from <A>super</A> to <A>sub</A>,
##  </Item>
##  <Mark><A>testopr</A> </Mark>
##  <Item>
##      tester filter of <A>opr</A>,
##  </Item>
##  <Mark><A>settopr</A> </Mark>
##  <Item>
##      setter filter of <A>opr</A>,
##  </Item>
##  <Mark><A>flagsopr</A> </Mark>
##  <Item>
##      list of those true flags of <A>opr</A>
##      that belong neither to categories nor to representations,
##  </Item>
##  <Mark><A>flagssub</A> </Mark>
##  <Item>
##      list of those true flags of <A>filtsub</A>
##      that belong neither to categories nor to representations,
##  </Item>
##  <Mark><A>rank</A> </Mark>
##  <Item>
##      a rational number that denotes the priority of the information
##      in the list; <C>SUBSET_MAINTAINED_INFO</C> is sorted according to
##      decreasing <A>rank</A> value.
##  <!--  We must be careful to choose the right succession of the methods.-->
##  <!--  Note that one method may require a property that is acquired using-->
##  <!--  another method.-->
##  <!--  For that, we give a method a rank that is lower than that of all methods-->
##  <!--  that may yield some of the requirements and that is higher than that of-->
##  <!--  all methods that require <A>opr</A>;-->
##  <!--  if this is not possible then a warning is printed.-->
##  <!--  (Maybe the mechanism has to be changed at some time because of this.-->
##  <!--  Another reason would be the direct installation of methods for-->
##  <!--  <C>UseSubsetRelation</C>, i.e., the ranks of these methods are not affected-->
##  <!--  by the code in <C>InstallSubsetMaintenance</C>.) -->
##  </Item>
##  </List>
##  </Description>
##  </ManSection>
##
BIND_GLOBAL( "SUBSET_MAINTAINED_INFO", [ [], [] ] );


#############################################################################
##
#O  UseSubsetRelation( <super>, <sub> )
##
##  <#GAPDoc Label="UseSubsetRelation">
##  <ManSection>
##  <Oper Name="UseSubsetRelation" Arg='super, sub'/>
##
##  <Description>
##  Methods for this operation transfer possibly useful information from the
##  domain <A>super</A> to its subset <A>sub</A>, and vice versa.
##  <P/>
##  <Ref Oper="UseSubsetRelation"/> is designed to be called automatically
##  whenever substructures of domains are constructed.
##  So the methods must be <E>cheap</E>, and the requirements should be as
##  sharp as possible!
##  <P/>
##  To achieve that <E>all</E> applicable methods are executed, all methods for
##  this operation except the default method must end with <C>TryNextMethod()</C>.
##  This default method deals with the information that is available by
##  the calls of <Ref Func="InstallSubsetMaintenance"/> in the &GAP; library.
##  <P/>
##  <Example><![CDATA[
##  gap> g:= Group( (1,2), (3,4), (5,6) );; h:= Group( (1,2), (3,4) );;
##  gap> IsAbelian( g );  HasIsAbelian( h );
##  true
##  false
##  gap> UseSubsetRelation( g, h );;  HasIsAbelian( h );  IsAbelian( h );
##  true
##  true
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "UseSubsetRelation", [ IsCollection, IsCollection ] );

InstallMethod( UseSubsetRelation,
    "default method that checks maintenances and then returns `true'",
    IsIdenticalObj,
    [ IsCollection, IsCollection ],
    # Make sure that this method is installed with ``real'' rank zero.
    - 2 * RankFilter( IsCollection ),
    function( super, sub )

    local entry;

    for entry in SUBSET_MAINTAINED_INFO[1] do
      if entry[1]( super ) and entry[2]( sub ) and not entry[4]( sub ) then
        entry[5]( sub, entry[3]( super ) );
      fi;
    od;

    return true;
    end );


#############################################################################
##
#F  InstallSubsetMaintenance( <opr>, <super_req>, <sub_req> )
##
##  <#GAPDoc Label="InstallSubsetMaintenance">
##  <ManSection>
##  <Func Name="InstallSubsetMaintenance" Arg='opr, super_req, sub_req'/>
##
##  <Description>
##  <A>opr</A> must be a property or an attribute.
##  The call of <Ref Func="InstallSubsetMaintenance"/> has the effect that
##  for a domain <M>D</M> in the filter <A>super_req</A>,
##  and a domain <M>S</M> in the filter <A>sub_req</A>,
##  the call <C>UseSubsetRelation</C><M>( D, S )</M>
##  (see&nbsp;<Ref Func="UseSubsetRelation"/>)
##  sets a known value of <A>opr</A> for <M>D</M> as value of <A>opr</A> also
##  for <M>S</M>.
##  A typical example for which <Ref Func="InstallSubsetMaintenance"/> is
##  applied is given by <A>opr</A> <C>= IsFinite</C>,
##  <A>super_req</A> <C>= IsCollection and IsFinite</C>,
##  and <A>sub_req</A> <C>= IsCollection</C>.
##  <P/>
##  If <A>opr</A> is a property and the filter <A>super_req</A> lies in the
##  filter <A>opr</A> then we can use also the following inverse implication.
##  If <M>D</M> is in the filter whose intersection with <A>opr</A> is
##  <A>super_req</A> and if <M>S</M> is in the filter <A>sub_req</A>,
##  <M>S</M> is a subset of <M>D</M>, and the value of <A>opr</A> for
##  <M>S</M> is <K>false</K> then the value of <A>opr</A> for <M>D</M> is
##  also <K>false</K>.
##  <!-- This is implemented only for the case <A>super_req</A> = <A>opr</A>
##       and <A>sub_req</A>.-->
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
BIND_GLOBAL( "InstallSubsetMaintenance",
    function( operation, super_req, sub_req )

    local setter,         # setter filter of `operation'
          tester,         # tester filter of `operation'
          upper,
          lower,
          attrprop,       # id `operation' an attribute/property?
          rank,
          filtssub,       # property and attribute flags of `sub_req'
          filtsopr,       # property and attribute flags of `operation'
          triple,         # loop over `SUBSET_MAINTAINED_INFO[2]'
          req,
          flag,
          filt1,
          filt2,
          i;

    setter:= Setter( operation );
    tester:= Tester( operation );

    # Are there methods that may give us some of the requirements?
    upper:= SUM_FLAGS;

    # (We must not call `SUBTR_SET' here because the lists types may be
    # not yet defined.)
    filtssub:= [];
    for flag in TRUES_FLAGS( FLAGS_FILTER( sub_req ) ) do
      if not flag in CATS_AND_REPS then
        ADD_LIST_DEFAULT( filtssub, flag );
      fi;
    od;

    for triple in SUBSET_MAINTAINED_INFO[2] do
      req:= SHALLOW_COPY_OBJ( filtssub );
      INTER_SET( req, triple[1] );
      if LEN_LIST( req ) <> 0 and triple[3] < upper then
        upper:= triple[3];
      fi;
    od;

    # Are there methods that require `operation'?
    lower:= 0;
    attrprop:= true;
    filt1:= FLAGS_FILTER( operation );
    if filt1 = false then

      # `operation' is an attribute.
      filt1:= FLAGS_FILTER( tester );

    else

      # Special treatment of categories, representations (makes sense?),
      # and filters created by `NewFilter'.
      if FLAG2_FILTER( operation ) = 0 then
        attrprop:= false;
      fi;

    fi;

    # (We must not call `SUBTR_SET' here because the lists types may be
    # not yet defined.)
    filtsopr:= [];
    for flag in TRUES_FLAGS( filt1 ) do
      if not flag in CATS_AND_REPS then
        ADD_LIST_DEFAULT( filtsopr, flag );
      fi;
    od;
    for triple in SUBSET_MAINTAINED_INFO[2] do
      req:= SHALLOW_COPY_OBJ( filtsopr );
      INTER_SET( req, triple[2] );
      if LEN_LIST( req ) <> 0 and lower < triple[3] then
        lower:= triple[3];
      fi;
    od;

    # Compute the ``rank'' of the maintenance.
    # (Do we have a cycle?)
    if upper <= lower then
      Print( "#W  warning: cycle in `InstallSubsetMaintenance'\n" );
      rank:= lower;
    else
      rank:= ( upper + lower ) / 2;
    fi;

    filt1:= IsCollection and Tester( super_req ) and super_req and tester;
    filt2:= IsCollection and Tester( sub_req   ) and sub_req;

    # Update the info list.
    i:= LEN_LIST( SUBSET_MAINTAINED_INFO[2] );
    while 0 < i and SUBSET_MAINTAINED_INFO[2][i][3] < rank do
      SUBSET_MAINTAINED_INFO[1][ i+1 ]:= SUBSET_MAINTAINED_INFO[1][ i ];
      SUBSET_MAINTAINED_INFO[2][ i+1 ]:= SUBSET_MAINTAINED_INFO[2][ i ];
      i:= i-1;
    od;
    SUBSET_MAINTAINED_INFO[2][ i+1 ]:= [ filtsopr, filtssub, rank ];
    if attrprop then
      SUBSET_MAINTAINED_INFO[1][ i+1 ]:=
                [ filt1, filt2, operation, tester, setter ];
    else
      SUBSET_MAINTAINED_INFO[1][ i+1 ]:=
                [ filt1, filt2, operation, operation,
                  function( sub, val )
                      SetFeatureObj( sub, operation, val );
                  end ];
    fi;

#T missing in new implementation!
#     # Install the method.
#     if     FLAGS_FILTER( operation ) <> false
#        and IS_EQUAL_FLAGS( FLAGS_FILTER( operation and sub_req ),
#                            FLAGS_FILTER( super_req ) )  then
#         InstallMethod( UseSubsetRelation, infostring, IsIdenticalObj,
#                 [ sub_req, sub_req ], 0,
#             function( super, sub )
#             if tester( sub )  and  not operation( sub )  then
#                 setter( super, false );
#             fi;
#             TryNextMethod();
#         end );
#     fi;
end );


#############################################################################
##
#V  ISOMORPHISM_MAINTAINED_INFO
##
##  <ManSection>
##  <Var Name="ISOMORPHISM_MAINTAINED_INFO"/>
##
##  <Description>
##  is a list of lists of the form
##  <C>[ <A>filtsold</A>, <A>filtsnew</A>, <A>opr</A>, <A>testopr</A>, <A>settopr</A>, <A>old_req</A>,
##  <A>new-req</A> ]</C>
##  which is used for calls of <C>UseIsomorphismRelation( <A>old</A>, <A>new</A> )</C>.
##  This list is enlarged by calls to <C>InstallIsomorphismMaintenance</C>.
##  <P/>
##  The meaning of the entries is as follows.
##  <List>
##  <Mark><A>filtsold</A> </Mark>
##  <Item>
##      required filter for <A>old</A>,
##  </Item>
##  <Mark><A>filtsnew</A> </Mark>
##  <Item>
##      required filter for <A>new</A>,
##  </Item>
##  <Mark><A>opr</A> </Mark>
##  <Item>
##      operation whose value is inherited from <A>old</A> to <A>new</A>,
##  </Item>
##  <Mark><A>testopr</A> </Mark>
##  <Item>
##      tester filter of <A>opr</A>,
##  </Item>
##  <Mark><A>settopr</A> </Mark>
##  <Item>
##      setter filter of <A>opr</A>,
##  </Item>
##  <Mark><A>old-req</A> </Mark>
##  <Item>
##      requirements for <A>old</A> in the <C>InstallIsomorphismMaintenance</C> call,
##  </Item>
##  <Mark><A>new-req</A> </Mark>
##  <Item>
##      requirements for <A>new</A> in the <C>InstallIsomorphismMaintenance</C> call.
##  </Item>
##  </List>
##  </Description>
##  </ManSection>
##
BIND_GLOBAL( "ISOMORPHISM_MAINTAINED_INFO", [] );


#############################################################################
##
#O  UseIsomorphismRelation( <old>, <new> )
##
##  <#GAPDoc Label="UseIsomorphismRelation">
##  <ManSection>
##  <Oper Name="UseIsomorphismRelation" Arg='old, new'/>
##
##  <Description>
##  Methods for this operation transfer possibly useful information from the
##  domain <A>old</A> to the isomorphic domain <A>new</A>.
##  <P/>
##  <Ref Oper="UseIsomorphismRelation"/> is designed to be called
##  automatically whenever isomorphic structures of domains are constructed.
##  So the methods must be <E>cheap</E>, and the requirements should be as
##  sharp as possible!
##  <P/>
##  To achieve that <E>all</E> applicable methods are executed, all methods
##  for this operation except the default method must end with a call to
##  <Ref Func="TryNextMethod"/>.
##  This default method deals with the information that is available by
##  the calls of <Ref Func="InstallIsomorphismMaintenance"/> in the &GAP;
##  library.
##  <P/>
##  <Example><![CDATA[
##  gap> g:= Group( (1,2) );;  h:= Group( [ [ -1 ] ] );;
##  gap> Size( g );  HasSize( h );
##  2
##  false
##  gap> UseIsomorphismRelation( g, h );;  HasSize( h );  Size( h );
##  true
##  2
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "UseIsomorphismRelation", [ IsCollection, IsCollection ] );

InstallMethod( UseIsomorphismRelation,
    "default method that checks maintenances and then returns `true'",
    [ IsCollection, IsCollection ],
    # Make sure that this method is installed with ``real'' rank zero.
    - 2 * RankFilter( IsCollection ),
    function( old, new )
    local entry;

    for entry in ISOMORPHISM_MAINTAINED_INFO do
      if entry[1]( old ) and entry[2]( new ) and not entry[4]( new ) then
        entry[5]( new, entry[3]( old ) );
      fi;
    od;

    return true;
    end );


#############################################################################
##
#F  InstallIsomorphismMaintenanceFunction( <func> )
##
##  <ManSection>
##  <Func Name="InstallIsomorphismMaintenanceFunction" Arg='func'/>
##
##  <Description>
##  <C>InstallIsomorphismMaintenanceFunction</C> installs <A>func</A>, so that
##  <C><A>func</A>( <A>filtsold</A>, <A>filtsnew</A>, <A>opr</A>, <A>testopr</A>, <A>settopr</A>, <A>old_req</A>,
##  <A>new-req</A> )</C> is called for each isomorphism maintenance.
##  More precisely, <A>func</A> is called for each entry in the global list
##  <C>ISOMORPHISM_MAINTAINED_INFO</C>, also to those that are entered into this
##  list after the installation of <A>func</A>.
##  (The mechanism is the same as for attributes, which is installed in the
##  file <C>lib/oper.g</C>.)
##  </Description>
##  </ManSection>
##
BIND_GLOBAL( "ISOM_MAINT_FUNCS", [] );

BIND_GLOBAL( "InstallIsomorphismMaintenanceFunction", function( func )
    local entry;
    for entry in ISOMORPHISM_MAINTAINED_INFO do
      CallFuncList( func, entry );
    od;
    ADD_LIST( ISOM_MAINT_FUNCS, func );
end );

BIND_GLOBAL( "RUN_ISOM_MAINT_FUNCS",
    function( arglist )
    local func;
    for func in ISOM_MAINT_FUNCS do
      CallFuncList( func, arglist );
    od;
    ADD_LIST( ISOMORPHISM_MAINTAINED_INFO, arglist );
end );


#############################################################################
##
#F  InstallIsomorphismMaintenance( <opr>, <old_req>, <new_req> )
##
##  <#GAPDoc Label="InstallIsomorphismMaintenance">
##  <ManSection>
##  <Func Name="InstallIsomorphismMaintenance" Arg='opr, old_req, new_req'/>
##
##  <Description>
##  <A>opr</A> must be a property or an attribute.
##  The call of <Ref Func="InstallIsomorphismMaintenance"/> has the effect
##  that for a domain <M>D</M> in the filter <A>old_req</A>,
##  and a domain <M>E</M> in the filter <A>new_req</A>,
##  the call <C>UseIsomorphismRelation</C><M>( D, E )</M>
##  (see&nbsp;<Ref Func="UseIsomorphismRelation"/>)
##  sets a known value of <A>opr</A> for <M>D</M> as value of <A>opr</A> also
##  for <M>E</M>.
##  A typical example for which <Ref Func="InstallIsomorphismMaintenance"/>
##  is applied is given by <A>opr</A> <C>= Size</C>,
##  <A>old_req</A> <C>= IsCollection</C>,
##  and <A>new_req</A> <C>= IsCollection</C>.
##  <!-- Up to now, there are no dependencies between the maintenances-->
##  <!-- (contrary to the case of subset maintenances),-->
##  <!-- so we do not take care of the succession.-->
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
BIND_GLOBAL( "InstallIsomorphismMaintenance",
    function( opr, old_req, new_req )
    local tester;

    tester:= Tester( opr );

    RUN_ISOM_MAINT_FUNCS(
        [ IsCollection and Tester( old_req ) and old_req and tester,
          IsCollection and Tester( new_req ) and new_req,
          opr,
          tester,
          Setter( opr ),
          old_req,
          new_req ] );
end );


#############################################################################
##
#V  FACTOR_MAINTAINED_INFO
##
##  <ManSection>
##  <Var Name="FACTOR_MAINTAINED_INFO"/>
##
##  <Description>
##  is a list of lists of the form
##  <C>[ <A>filtsnum</A>, <A>filtsden</A>, <A>filtsfac</A>, <A>opr</A>, <A>testopr</A>, <A>settopr</A> ]</C>
##  which is used for calls of <C>UseFactorRelation( <A>num</A>, <A>den</A>, <A>fac</A> )</C>.
##  This list is enlarged by calls to <C>InstallFactorMaintenance</C>.
##  <P/>
##  The meaning of the entries is as follows.
##  <List>
##  <Mark><A>filtsnum</A> </Mark>
##  <Item>
##      required filter for <A>num</A>,
##  </Item>
##  <Mark><A>filtsden</A> </Mark>
##  <Item>
##      required filter for <A>den</A>,
##  </Item>
##  <Mark><A>filtsfac</A> </Mark>
##  <Item>
##      required filter for <A>fac</A>,
##  </Item>
##  <Mark><A>opr</A> </Mark>
##  <Item>
##      operation whose value is inherited from <A>num</A> to <A>fac</A>,
##  </Item>
##  <Mark><A>testopr</A> </Mark>
##  <Item>
##      tester filter of <A>opr</A>,
##  </Item>
##  <Mark><A>settopr</A> </Mark>
##  <Item>
##      setter filter of <A>opr</A>.
##  </Item>
##  </List>
##  </Description>
##  </ManSection>
##
BIND_GLOBAL( "FACTOR_MAINTAINED_INFO", [] );


#############################################################################
##
#O  UseFactorRelation( <numer>, <denom>, <factor> )
##
##  <#GAPDoc Label="UseFactorRelation">
##  <ManSection>
##  <Oper Name="UseFactorRelation" Arg='numer, denom, factor'/>
##
##  <Description>
##  Methods for this operation transfer possibly useful information from the
##  domain <A>numer</A> or its subset <A>denom</A> to the domain
##  <A>factor</A> that is isomorphic to the factor of <A>numer</A> by
##  <A>denom</A>, and vice versa.
##  <A>denom</A> may be <K>fail</K>, for example if <A>factor</A> is just
##  known to be a factor of <A>numer</A> but <A>denom</A> is not available as
##  a &GAP; object;
##  in this case those factor relations are used that are installed without
##  special requirements for <A>denom</A>.
##  <P/>
##  <Ref Oper="UseFactorRelation"/> is designed to be called automatically
##  whenever factor structures of domains are constructed.
##  So the methods must be <E>cheap</E>, and the requirements should be as
##  sharp as possible!
##  <P/>
##  To achieve that <E>all</E> applicable methods are executed, all methods
##  for this operation except the default method must end with a call to
##  <Ref Func="TryNextMethod"/>.
##  This default method deals with the information that is available by
##  the calls of <Ref Func="InstallFactorMaintenance"/> in the &GAP; library.
##  <P/>
##  <Example><![CDATA[
##  gap> g:= Group( (1,2,3,4), (1,2) );; h:= Group( (1,2,3), (1,2) );;
##  gap> IsSolvableGroup( g );  HasIsSolvableGroup( h );
##  true
##  false
##  gap> UseFactorRelation(g, Subgroup( g, [ (1,2)(3,4), (1,3)(2,4) ] ), h);;
##  gap> HasIsSolvableGroup( h );  IsSolvableGroup( h );
##  true
##  true
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "UseFactorRelation",
    [ IsCollection, IsObject, IsCollection ] );

IsIdenticalObjObjObjX := function( F1, F2, F3 )
    return IsIdenticalObj( F1, F2 );
end;

InstallMethod( UseFactorRelation,
    "default method that checks maintenances and then returns `true'",
    true,
    [ IsCollection, IsObject, IsCollection ],
    # Make sure that this method is installed with ``real'' rank zero.
    - 2 * RankFilter( IsCollection )-RankFilter(IsObject),
    function( num, den, fac )

    local entry;

    for entry in FACTOR_MAINTAINED_INFO do
      if entry[1]( num ) and entry[2]( den ) and entry[3]( fac )
                         and not entry[5]( fac ) then
        entry[6]( fac, entry[4]( num ) );
      fi;
    od;

    return true;
    end );


#############################################################################
##
#F  InstallFactorMaintenance( <opr>, <numer_req>, <denom_req>, <factor_req> )
##
##  <#GAPDoc Label="InstallFactorMaintenance">
##  <ManSection>
##  <Func Name="InstallFactorMaintenance"
##   Arg='opr, numer_req, denom_req, factor_req'/>
##
##  <Description>
##  <A>opr</A> must be a property or an attribute.
##  The call of <Ref Func="InstallFactorMaintenance"/> has the effect that
##  for collections <M>N</M>, <M>D</M>, <M>F</M> in the filters
##  <A>numer_req</A>, <A>denom_req</A>, and <A>factor_req</A>, respectively,
##  the call <C>UseFactorRelation</C><M>( N, D, F )</M>
##  (see&nbsp;<Ref Func="UseFactorRelation"/>)
##  sets a known value of <A>opr</A> for <M>N</M> as value of <A>opr</A> also
##  for <M>F</M>.
##  A typical example for which <Ref Func="InstallFactorMaintenance"/> is
##  applied is given by <A>opr</A> <C>= IsFinite</C>,
##  <A>numer_req</A> <C>= IsCollection and IsFinite</C>,
##  <A>denom_req</A> <C>= IsCollection</C>,
##  and <A>factor_req</A> <C>= IsCollection</C>.
##  <P/>
##  For the other direction, if <A>numer_req</A> involves the filter
##  <A>opr</A> then a known <K>false</K> value of <A>opr</A> for <M>F</M>
##  implies a <K>false</K> value for <M>D</M> provided that <M>D</M> lies in
##  the filter obtained from <A>numer_req</A> by removing <A>opr</A>.
##  <P/>
##  Note that an implication of a factor relation holds in particular for the
##  case of isomorphisms.
##  So one need <E>not</E> install an isomorphism maintained method when
##  a factor maintained method is already installed.
##  For example, <Ref Func="UseIsomorphismRelation"/>
##  will transfer a known <Ref Prop="IsFinite"/> value because of the
##  installed factor maintained method.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
BIND_GLOBAL( "InstallFactorMaintenance",
    function( opr, numer_req, denom_req, factor_req )

    local tester;

    # Information that is maintained under taking factors
    # is especially maintained under isomorphisms.
    InstallIsomorphismMaintenance( opr, numer_req, factor_req );

    tester:= Tester( opr );

    ADD_LIST( FACTOR_MAINTAINED_INFO,
        [ IsCollection and Tester( numer_req ) and numer_req and tester,
          Tester( denom_req ) and denom_req,
          IsCollection and Tester( factor_req ) and factor_req,
          opr,
          tester,
          Setter( opr ) ] );

#T not yet available in the new implementation
#     if     FLAGS_FILTER( opr ) <> false
#        and IS_EQUAL_FLAGS( FLAGS_FILTER( opr and factor_req ),
#                            FLAGS_FILTER( numer_req ) )  then
#         InstallMethod( UseFactorRelation, infostring, IsIdenticalObjObjObjX,
#                 [ factor_req, denom_req, factor_req ], 0,
#             function( numer, denom, factor )
#             if tester( factor )  and  not opr( factor )  then
#                 setter( numer, false );
#             fi;
#             TryNextMethod();
#         end );
#     fi;
end );


#############################################################################
##
#O  Iterator( <listorcoll> )  . . . . . . . iterator for a list or collection
##
##  <#GAPDoc Label="Iterator">
##  <ManSection>
##  <Oper Name="Iterator" Arg='listorcoll'/>
##
##  <Description>
##  Iterators provide a possibility to loop over the elements of a
##  (countable) collection or list <A>listorcoll</A>, without repetition.
##  For many collections <M>C</M>,
##  an iterator of <M>C</M> need not store all elements of <M>C</M>,
##  for example it is possible to construct an iterator of some infinite
##  domains, such as the field of rational numbers.
##  <P/>
##  <Ref Func="Iterator"/> returns a mutable <E>iterator</E> <M>iter</M> for
##  its argument.
##  If this argument is a list (which may contain holes),
##  then <M>iter</M> iterates over the elements (but not the holes) of this
##  list in the same order (see&nbsp;<Ref Func="IteratorList"/> for details).
##  If this argument is a collection but not a list then <M>iter</M> iterates
##  over the elements of this collection in an unspecified order,
##  which may change for repeated calls of <Ref Func="Iterator"/>.
##  Because iterators returned by <Ref Func="Iterator"/> are mutable
##  (see&nbsp;<Ref Sect="Mutability and Copyability"/>),
##  each call of <Ref Func="Iterator"/> for the same argument returns a
##  <E>new</E> iterator.
##  Therefore <Ref Func="Iterator"/> is not an attribute
##  (see&nbsp;<Ref Sect="Attributes"/>).
##  <P/>
##  The only operations for iterators are <Ref Func="IsDoneIterator"/>,
##  <Ref Func="NextIterator"/>, and <Ref Func="ShallowCopy"/>.
##  In particular, it is only possible to access the next element of the
##  iterator with <Ref Func="NextIterator"/> if there is one,
##  and this can be checked with <Ref Func="IsDoneIterator"/>
##  For an iterator <M>iter</M>, <Ref Func="ShallowCopy"/> returns a
##  mutable iterator <M>new</M> that iterates over the remaining elements
##  independent of <M>iter</M>;
##  the results of <Ref Func="IsDoneIterator"/> for <M>iter</M> and
##  <M>new</M> are equal,
##  and if <M>iter</M> is mutable then also the results of
##  <Ref Func="NextIterator"/> for <M>iter</M> and <M>new</M> are equal;
##  note that <C>=</C> is not defined for iterators,
##  so the equality of two iterators cannot be checked with <C>=</C>.
##  <P/>
##  When <Ref Func="Iterator"/> is called for a <E>mutable</E> collection
##  <M>C</M> then it is not defined whether <M>iter</M> respects changes to
##  <M>C</M> occurring after the construction of <M>iter</M>,
##  except if the documentation explicitly promises a certain behaviour.
##  The latter is the case if the argument is a mutable list
##  (see&nbsp;<Ref Func="IteratorList"/> for subtleties in this case).
##  <P/>
##  It is possible to have <K>for</K>-loops run over mutable iterators
##  instead of lists.
##  <P/>
##  In some situations, one can construct iterators with a special
##  succession of elements,
##  see&nbsp;<Ref Func="IteratorByBasis"/> for the possibility to loop over
##  the elements of a vector space w.r.t.&nbsp;a given basis.
##  <!-- (also for perm. groups, w.r.t. a given stabilizer chain?)-->
##  <P/>
##  For lists, <Ref Func="Iterator"/> is implemented by
##  <Ref Func="IteratorList"/>.
##  For collections <M>C</M> that are not lists, the default method is
##  <C>IteratorList( Enumerator( </C><M>C</M><C> ) )</C>.
##  Better methods depending on <M>C</M> should be provided if possible.
##  <P/>
##  For random access to the elements of a (possibly infinite) collection,
##  <E>enumerators</E> are used.
##  See&nbsp;<Ref Sect="Enumerators"/> for the facility to compute a list
##  from <M>C</M>, which provides a (partial) mapping from <M>C</M> to the
##  positive integers.
##  <P/>
##  <!--  We wanted to admit an iterator as first argument of <C>Filtered</C>,-->
##  <!--  <C>First</C>, <C>ForAll</C>, <C>ForAny</C>, <C>Number</C>.-->
##  <!--  This is not yet implemented.-->
##  <!--  (Note that the iterator is changed in the call,-->
##  <!--  so the meaning of the operations would be slightly abused,-->
##  <!--  or we must define that these operations first make a shallow copy.)-->
##  <!--  (Additionally, the unspecified order of the elements makes it-->
##  <!--  difficult to define what <C>First</C> and <C>Filtered</C> means for an iterator.)-->
##  <Example><![CDATA[
##  gap> iter:= Iterator( GF(5) );
##  <iterator>
##  gap> l:= [];;
##  gap> for i in iter do Add( l, i ); od; l;
##  [ 0*Z(5), Z(5)^0, Z(5), Z(5)^2, Z(5)^3 ]
##  gap> iter:= Iterator( [ 1, 2, 3, 4 ] );;  l:= [];;
##  gap> for i in iter do
##  >      new:= ShallowCopy( iter );
##  >      for j in new do Add( l, j ); od;
##  >    od; l;
##  [ 2, 3, 4, 3, 4, 4 ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "Iterator", [ IsListOrCollection ] );


#############################################################################
##
#O  IteratorSorted( <C> ) . . . . . . . . . . . set iterator for a collection
#O  IteratorSorted( <list> )  . . . . . . . . . . . . set iterator for a list
##
##  <#GAPDoc Label="IteratorSorted">
##  <ManSection>
##  <Oper Name="IteratorSorted" Arg='listorcoll'/>
##
##  <Description>
##  <Ref Func="IteratorSorted"/> returns a mutable iterator.
##  The argument must be a collection or a list that is not
##  necessarily dense but whose elements lie in the same family
##  (see&nbsp;<Ref Sect="Families"/>).
##  It loops over the different elements in sorted order.
##  <P/>
##  For a collection <M>C</M> that is not a list, the generic method is
##  <C>IteratorList( EnumeratorSorted( </C><A>C</A><C> ) )</C>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "IteratorSorted", [ IsListOrCollection ] );


#############################################################################
##
#C  IsIterator( <obj> ) . . . . . . . . . .  test if an object is an iterator
##
##  <#GAPDoc Label="IsIterator">
##  <ManSection>
##  <Filt Name="IsIterator" Arg='obj' Type='Category'/>
##
##  <Description>
##  Every iterator lies in the category <C>IsIterator</C>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareCategory( "IsIterator", IsObject );


#############################################################################
##
#O  IsDoneIterator( <iter> )  . . . . . . .  test if an iterator is exhausted
##
##  <#GAPDoc Label="IsDoneIterator">
##  <ManSection>
##  <Oper Name="IsDoneIterator" Arg='iter'/>
##
##  <Description>
##  If <A>iter</A> is an iterator for the list or collection <M>C</M> then
##  <C>IsDoneIterator( <A>iter</A> )</C> is <K>true</K> if all elements of
##  <M>C</M> have been returned already by <C>NextIterator( <A>iter</A> )</C>,
##  and <K>false</K> otherwise.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "IsDoneIterator", [ IsIterator ] );


#############################################################################
##
#O  NextIterator( <iter> )  . . . . . . . . . . next element from an iterator
##
##  <#GAPDoc Label="NextIterator">
##  <ManSection>
##  <Oper Name="NextIterator" Arg='iter'/>
##
##  <Description>
##  Let <A>iter</A> be a mutable iterator for the list or collection <M>C</M>.
##  If <C>IsDoneIterator( <A>iter</A> )</C> is <K>false</K> then
##  <Ref Func="NextIterator"/> is applicable to <A>iter</A>,
##  and the result is the next element of <M>C</M>,
##  according to the succession defined by <A>iter</A>.
##  <P/>
##  If <C>IsDoneIterator( <A>iter</A> )</C> is <K>true</K> then it is not
##  defined what happens when <Ref Func="NextIterator"/> is called for
##  <A>iter</A>;
##  that is, it may happen that an error is signalled or that something
##  meaningless is returned, or even that &GAP; crashes.
##  <P/>
##  <Example><![CDATA[
##  gap> iter:= Iterator( [ 1, 2, 3, 4 ] );
##  <iterator>
##  gap> sum:= 0;;
##  gap> while not IsDoneIterator( iter ) do
##  >      sum:= sum + NextIterator( iter );
##  >    od;
##  gap> IsDoneIterator( iter ); sum;
##  true
##  10
##  gap> ir:= Iterator( Rationals );;
##  gap> l:= [];; for i in [1..20] do Add( l, NextIterator( ir ) ); od; l;
##  [ 0, 1, -1, 1/2, 2, -1/2, -2, 1/3, 2/3, 3/2, 3, -1/3, -2/3, -3/2, -3, 
##    1/4, 3/4, 4/3, 4, -1/4 ]
##  gap> for i in ir do
##  >      if DenominatorRat( i ) > 10 then break; fi;
##  >    od;
##  gap> i;
##  1/11
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "NextIterator", [ IsIterator and IsMutable ] );


#############################################################################
##
#F  TrivialIterator( <elm> )
##
##  <#GAPDoc Label="TrivialIterator">
##  <ManSection>
##  <Func Name="TrivialIterator" Arg='elm'/>
##
##  <Description>
##  is a mutable iterator for the collection <C>[ <A>elm</A> ]</C> that
##  consists of exactly one element <A>elm</A>
##  (see&nbsp;<Ref Func="IsTrivial"/>).
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "TrivialIterator" );


#############################################################################
##
#F  IteratorByFunctions( <record> )
##
##  <#GAPDoc Label="IteratorByFunctions">
##  <ManSection>
##  <Func Name="IteratorByFunctions" Arg='record'/>
##
##  <Description>
##  <Ref Func="IteratorByFunctions"/> returns a (mutable) iterator
##  <A>iter</A> for which <Ref Func="NextIterator"/>,
##  <Ref Func="IsDoneIterator"/>,
##  and <Ref Func="ShallowCopy"/>
##  are computed via prescribed functions.
##  <P/>
##  Let <A>record</A> be a record with at least the following components.
##  <List>
##  <Mark><C>NextIterator</C></Mark>
##  <Item>
##      a function taking one argument <A>iter</A>,
##      which returns the next element of <A>iter</A>
##      (see&nbsp;<Ref Func="NextIterator"/>);
##      for that, the components of <A>iter</A> are changed,
##  </Item>
##  <Mark><C>IsDoneIterator</C></Mark>
##  <Item>
##      a function taking one argument <A>iter</A>,
##      which returns the <Ref Func="IsDoneIterator"/> value of <A>iter</A>,
##  </Item>
##  <Mark><C>ShallowCopy</C></Mark>
##  <Item>
##      a function taking one argument <A>iter</A>,
##      which returns a record for which <Ref Func="IteratorByFunctions"/>
##      can be called in order to create a new iterator that is independent
##      of <A>iter</A> but behaves like <A>iter</A> w.r.t. the operations
##      <Ref Func="NextIterator"/> and <Ref Func="IsDoneIterator"/>.
##  </Item>
##  </List>
##  Further (data) components may be contained in <A>record</A> which can be
##  used by these function.
##  <P/>
##  <Ref Func="IteratorByFunctions"/> does <E>not</E> make a shallow copy of
##  <A>record</A>, this record is changed in place
##  (see Section &nbsp;<Ref Sect="Creating Objects"/>).
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "IteratorByFunctions" );


#############################################################################
##
#P  IsEmpty( <C> )  . . . . . . . . . . . . . . test if a collection is empty
#P  IsEmpty( <list> ) . . . . . . . . . . . . . test if a collection is empty
##
##  <#GAPDoc Label="IsEmpty">
##  <ManSection>
##  <Prop Name="IsEmpty" Arg='listorcoll'/>
##
##  <Description>
##  <Ref Func="IsEmpty"/> returns <K>true</K> if the collection or list
##  <A>listorcoll</A> is <E>empty</E> (that is it contains no elements),
##  and <K>false</K> otherwise.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsEmpty", IsListOrCollection );


#############################################################################
##
#P  IsTrivial( <C> )  . . . . . . . . . . . . test if a collection is trivial
##
##  <#GAPDoc Label="IsTrivial">
##  <ManSection>
##  <Prop Name="IsTrivial" Arg='C'/>
##
##  <Description>
##  <Ref Prop="IsTrivial"/> returns <K>true</K> if the collection <A>C</A>
##  consists of exactly one element.
##  <!--  1996/08/08 M.Schönert is this a sensible definition?-->
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsTrivial", IsCollection );

InstallFactorMaintenance( IsTrivial,
    IsCollection and IsTrivial, IsObject, IsCollection );


#############################################################################
##
#P  IsNonTrivial( <C> ) . . . . . . . . .  test if a collection is nontrivial
##
##  <#GAPDoc Label="IsNonTrivial">
##  <ManSection>
##  <Prop Name="IsNonTrivial" Arg='C'/>
##
##  <Description>
##  <Ref Func="IsNonTrivial"/> returns <K>true</K> if the collection <A>C</A>
##  is empty or consists of at least two elements
##  (see&nbsp;<Ref Func="IsTrivial"/>).
##  <P/>
##  <!-- I need this to distinguish trivial rings-with-one from fields!-->
##  <!-- (indication to introduce antifilters?)-->
##  <!--  1996/08/08 M.Schönert is this a sensible definition?-->
##  <Example><![CDATA[
##  gap> IsEmpty( [] ); IsEmpty( [ 1 .. 100 ] ); IsEmpty( Group( (1,2,3) ) );
##  true
##  false
##  false
##  gap> IsFinite( [ 1 .. 100 ] );  IsFinite( Integers );
##  true
##  false
##  gap> IsTrivial( Integers );  IsTrivial( Group( () ) );
##  false
##  true
##  gap> IsNonTrivial( Integers );  IsNonTrivial( Group( () ) );
##  true
##  false
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsNonTrivial", IsCollection );


#############################################################################
##
#P  IsFinite( <C> ) . . . . . . . . . . . . .  test if a collection is finite
##
##  <#GAPDoc Label="IsFinite">
##  <ManSection>
##  <Prop Name="IsFinite" Arg='C'/>
##
##  <Description>
##  <Index Subkey="for a list or collection">finiteness test</Index>
##  <Ref Func="IsFinite"/> returns <K>true</K> if the collection <A>C</A>
##  is finite, and <K>false</K> otherwise.
##  <P/>
##  The default method for <Ref Func="IsFinite"/> checks the size
##  (see&nbsp;<Ref Func="Size"/>) of <A>C</A>.
##  <P/>
##  Methods for <Ref Func="IsFinite"/> may call <Ref Func="Size"/>,
##  but methods for <Ref Func="Size"/> must <E>not</E> call
##  <Ref Func="IsFinite"/>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsFinite", IsCollection );

InstallSubsetMaintenance( IsFinite,
    IsCollection and IsFinite, IsCollection );
InstallFactorMaintenance( IsFinite,
    IsCollection and IsFinite, IsObject, IsCollection );

InstallTrueMethod( IsFinite, IsTrivial );


#############################################################################
##
#P  IsWholeFamily( <C> )  . .  test if a collection contains the whole family
##
##  <#GAPDoc Label="IsWholeFamily">
##  <ManSection>
##  <Prop Name="IsWholeFamily" Arg='C'/>
##
##  <Description>
##  <Ref Prop="IsWholeFamily"/> returns <K>true</K> if the collection
##  <A>C</A> contains the whole family (see&nbsp;<Ref Sect="Families"/>)
##  of its elements.
##  <P/>
##  <Example><![CDATA[
##  gap> IsWholeFamily( Integers )
##  >    ;  # all rationals and cyclotomics lie in the family
##  false
##  gap> IsWholeFamily( Integers mod 3 )
##  >    ;  # all finite field elements in char. 3 lie in this family
##  false
##  gap> IsWholeFamily( Integers mod 4 );
##  true
##  gap> IsWholeFamily( FreeGroup( 2 ) );
##  true
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "IsWholeFamily", IsCollection );


#############################################################################
##
#A  Size( <C> ) . . . . . . . . . . . . . . . . . . . .  size of a collection
#A  Size( <list> )  . . . . . . . . . . . . . . . . . .  size of a collection
##
##  <#GAPDoc Label="Size">
##  <ManSection>
##  <Attr Name="Size" Arg='listorcoll'/>
##
##  <Description>
##  <Index Subkey="of a list or collection">size</Index>
##  <Index Subkey="of a list, collection or domain">order</Index>
##  <Ref Attr="Size"/> returns the size of the list or collection
##  <A>listorcoll</A>, which is either an integer or <Ref Var="infinity"/>.
##  If the argument is a list then the result is its length
##  (see&nbsp;<Ref Func="Length"/>).
##  <P/>
##  The default method for <Ref Attr="Size"/> checks the length of an
##  enumerator of <A>listorcoll</A>.
##  <P/>
##  Methods for <Ref Prop="IsFinite"/> may call <Ref Attr="Size"/>,
##  but methods for <Ref Attr="Size"/> must not call <Ref Prop="IsFinite"/>.
##  <P/>
##  <Example><![CDATA[
##  gap> Size( [1,2,3] );  Size( Group( () ) );  Size( Integers );
##  3
##  1
##  infinity
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "Size", IsListOrCollection );

InstallIsomorphismMaintenance( Size, IsCollection, IsCollection );


#############################################################################
##
#A  Representative( <C> ) . . . . . . . . . . . . one element of a collection
##
##  <#GAPDoc Label="Representative">
##  <ManSection>
##  <Attr Name="Representative" Arg='C'/>
##
##  <Description>
##  <Ref Attr="Representative"/> returns a <E>representative</E>
##  of the collection <A>C</A>.
##  <P/>
##  Note that <Ref Attr="Representative"/> is free in choosing
##  a representative if there are several elements in <A>C</A>.
##  It is not even guaranteed that <Ref Attr="Representative"/> returns
##  the same representative if it is called several times for one collection.
##  The main difference between <Ref Attr="Representative"/> and
##  <Ref Func="Random" Label="for a list or collection"/>
##  is that <Ref Attr="Representative"/> is free
##  to choose a value that is cheap to compute,
##  while <Ref Func="Random" Label="for a list or collection"/>
##  must make an effort to randomly distribute its answers.
##  <P/>
##  If <A>C</A> is a domain then there are methods for
##  <Ref Attr="Representative"/> that try
##  to fetch an element from any known generator list of <A>C</A>,
##  see&nbsp;<Ref Chap="Domains and their Elements"/>.
##  Note that <Ref Attr="Representative"/> does not try to <E>compute</E>
##  generators of <A>C</A>,
##  thus <Ref Attr="Representative"/> may give up and signal an error
##  if <A>C</A> has no generators stored at all.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "Representative", IsListOrCollection );


#############################################################################
##
#A  RepresentativeSmallest( <C> ) . . . . .  smallest element of a collection
##
##  <#GAPDoc Label="RepresentativeSmallest">
##  <ManSection>
##  <Attr Name="RepresentativeSmallest" Arg='C'/>
##
##  <Description>
##  <Index Subkey="of a list or collection">representative</Index>
##  returns the smallest element in the collection <A>C</A>, w.r.t.&nbsp;the
##  ordering <Ref Func="\&lt;"/>.
##  While the operation defaults to comparing all elements,
##  better methods are installed for some collections.
##  <P/>
##  <Example><![CDATA[
##  gap> Representative( Rationals );
##  0
##  gap> Representative( [ -1, -2 .. -100 ] );
##  -1
##  gap> RepresentativeSmallest( [ -1, -2 .. -100 ] );
##  -100
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "RepresentativeSmallest", IsListOrCollection );


#############################################################################
##
#O  Random( <C> ) . . . . . . . . . .  random element of a list or collection
#O  Random( <list> )  . . . . . . . .  random element of a list or collection
#O  Random( <from>, <to> )
##
##  <#GAPDoc Label="Random:coll">
##  <ManSection>
##  <Oper Name="Random" Arg='listorcoll' Label="for a list or collection"/>
##  <Oper Name="Random" Arg='from, to' Label="for lower and upper bound"/>
##
##  <Description>
##  <Index Subkey="of a list or collection">random element</Index>
##  <Ref Oper="Random" Label="for a list or collection"/> returns a
##  (pseudo-)random element of the list or collection <A>listorcoll</A>.
##  <P/>
##  As lists or ranges are restricted in length (<M>2^{28}-1</M> or 
##  <M>2^{60}-1</M> depending on your system), the second form returns a
##  random integer in the range <A>from</A> to <A>to</A> (inclusive) for
##  arbitrary integers <A>from</A> and <A>to</A>.
##  <P/>
##  The distribution of elements returned by
##  <Ref Oper="Random" Label="for a list or collection"/> depends
##  on the argument.
##  For a list, all elements are equally likely.
##  The same holds usually for finite collections that are
##  not lists.
##  For infinite collections some reasonable distribution is used.
##  <P/>
##  See the chapters of the various collections to find out
##  which distribution is being used.
##  <P/>
##  For some collections ensuring a reasonable distribution can be
##  difficult and require substantial runtime.
##  If speed at the cost of equal distribution is desired,
##  the operation <Ref Func="PseudoRandom"/> should be used instead.
##  <P/>
##  Note that <Ref Oper="Random" Label="for a list or collection"/>
##  is of course <E>not</E> an attribute.
##  <P/>
##  <Example><![CDATA[
##  gap> Random(Rationals);
##  4
##  gap> g:= Group( (1,2,3) );;  Random( g );  Random( g );
##  (1,3,2)
##  ()
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "Random", [ IsListOrCollection ] );
DeclareOperation( "Random", [ IS_INT, IS_INT ] );


#############################################################################
##
##  <#GAPDoc Label="[2]{coll}">
##  The method used by &GAP; to obtain random elements may depend on the
##  type object.
##  <P/>
##  Most methods which produce random elements in &GAP; use a global random
##  number generator (see <Ref Var="GlobalMersenneTwister"/>).  
##  This random number generator is (deliberately) initialized to the same
##  values when &GAP; is started, so different runs of &GAP; with the same
##  input will always produce the same result, even if random calculations
##  are involved.
##  <P/>
##  See <Ref Oper="Reset"/> for a description of how to reset the
##  random number generator to a previous state.
##  <P/>
##  
##  <!-- all outdated? (FL)
##  Many random methods in the library are eventually based on the function
##  <Ref Func="RandomList"/>.
##  As <Ref Func="RandomList"/> is restricted to lists of  <M>2^{28}</M>
##  elements, this may create problems for very large collections. Also note
##  that the method used by <Ref Func="RandomList"/> is intended to provide
##  a fast algorithm rather than to produce high quality randomness for
##  statistical purposes.
##  <P/>
##  If you implement your own
##  <Ref Func="Random" Label="for a list or collection"/> methods we recommend
##  that they initialize their seed to a defined value when they are loaded
##  to permit to reproduce calculations even if they involved random
##  elements.
##  -->
##  <#/GAPDoc>
##


#############################################################################
##
#F  RandomList( <list> )
##
##  <#GAPDoc Label="RandomList">
##  <ManSection>
##  <Func Name="RandomList" Arg='list'/>
##
##  <Description>
##  <Index>random seed</Index>
##  For a dense list <A>list</A>,
##  <Ref Func="RandomList"/> returns a (pseudo-)random element with equal
##  distribution.
##  <P/>
##  This function uses the <Ref Var="GlobalMersenneTwister"/> to produce the
##  random elements (a source of high quality random numbers).
##  <P/>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareSynonym( "RandomList", RANDOM_LIST);


#############################################################################
##
#O  PseudoRandom( <C> ) . . . . . . . . pseudo random element of a collection
#O  PseudoRandom( <list> )  . . . . . . . . . pseudo random element of a list
##
##  <#GAPDoc Label="PseudoRandom">
##  <ManSection>
##  <Oper Name="PseudoRandom" Arg='listorcoll'/>
##
##  <Description>
##  <Ref Oper="PseudoRandom"/> returns a pseudo random element
##  of the list or collection <A>listorcoll</A>,
##  which can be roughly described as follows.
##  For a list, <Ref Oper="PseudoRandom"/> returns the same as
##  <Ref Oper="Random" Label="for a list or collection"/>.
##  For collections that are not lists,
##  the elements returned by <Ref Oper="PseudoRandom"/> are
##  <E>not</E> necessarily equally distributed,
##  even for finite collections;
##  the idea is that <Ref Oper="Random" Label="for a list or collection"/>
##  returns elements according to
##  a reasonable distribution, <Ref Oper="PseudoRandom"/> returns elements
##  that are cheap to compute but need not satisfy this strong condition, and
##  <Ref Attr="Representative"/> returns arbitrary elements,
##  probably the same element for each call.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "PseudoRandom", [ IsListOrCollection ] );


#############################################################################
##
#A  PseudoRandomSeed( <C> )
##
##  <ManSection>
##  <Attr Name="PseudoRandomSeed" Arg='C'/>
##
##  <Description>
##  </Description>
##  </ManSection>
##
DeclareAttribute( "PseudoRandomSeed", IsListOrCollection, "mutable" );


#############################################################################
##
#A  Enumerator( <C> ) . . . . . . . . . . .  list of elements of a collection
#A  Enumerator( <list> )  . . . . . . . . . . . .  list of elements of a list
##
##  <#GAPDoc Label="Enumerator">
##  <ManSection>
##  <Attr Name="Enumerator" Arg='listorcoll'/>
##
##  <Description>
##  <Ref Func="Enumerator"/> returns an immutable list <M>enum</M>.
##  If the argument is a list (which may contain holes),
##  then <C>Length( </C><M>enum</M><C> )</C> is the length of this list,
##  and <M>enum</M> contains the elements (and holes) of this list in the
##  same order.
##  If the argument is a collection that is not a list,
##  then <C>Length( </C><M>enum</M><C> )</C> is the number of different
##  elements of <A>C</A>,
##  and <M>enum</M> contains the different elements of the collection in an
##  unspecified order, which may change for repeated calls of
##  <Ref Func="Enumerator"/>.
##  <M>enum[pos]</M> may not execute in constant time
##  (see&nbsp;<Ref Func="IsConstantTimeAccessList"/>),
##  and the size of <M>enum</M> in memory is as small as is feasible.
##  <P/>
##  For lists, the default method is <Ref Func="Immutable"/>.
##  For collections that are not lists, there is no default method.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "Enumerator", IsListOrCollection );


#############################################################################
##
#A  EnumeratorSorted( <C> ) . . . . .  proper set of elements of a collection
#A  EnumeratorSorted( <list> )  . . . . . .  proper set of elements of a list
##
##  <#GAPDoc Label="EnumeratorSorted">
##  <ManSection>
##  <Attr Name="EnumeratorSorted" Arg='listorcoll'/>
##
##  <Description>
##  <Ref Func="EnumeratorSorted"/> returns an immutable list <M>enum</M>.
##  The argument must be a collection or a list <A>listorcoll</A>
##  which may contain holes but whose elements lie in the same family
##  (see&nbsp;<Ref Sect="Families"/>).
##  <C>Length( </C><M>enum</M><C> )</C> is the number of different elements
##  of the argument,
##  and <M>enum</M> contains the different elements in sorted order,
##  w.r.t.&nbsp;<C>&lt;</C>.
##  <M>enum[pos]</M> may not execute in constant time
##  (see&nbsp;<Ref Func="IsConstantTimeAccessList"/>),
##  and the size of <M>enum</M> in memory is as small as is feasible.
##  <P/>
##  <Example><![CDATA[
##  gap> Enumerator( [ 1, 3,, 2 ] );
##  [ 1, 3,, 2 ]
##  gap> enum:= Enumerator( Rationals );;  elm:= enum[ 10^6 ];
##  -69/907
##  gap> Position( enum, elm );
##  1000000
##  gap> IsMutable( enum );  IsSortedList( enum );
##  false
##  false
##  gap> IsConstantTimeAccessList( enum );
##  false
##  gap> EnumeratorSorted( [ 1, 3,, 2 ] );
##  [ 1, 2, 3 ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "EnumeratorSorted", IsListOrCollection );


#############################################################################
##
#F  EnumeratorOfSubset( <list>, <blist>[, <ishomog>] )
##
##  <ManSection>
##  <Func Name="EnumeratorOfSubset" Arg='list, blist[, ishomog]'/>
##
##  <Description>
##  Let <A>list</A> be a list, and <A>blist</A> a Boolean list of the same
##  length (see&nbsp;<Ref Chap="Boolean Lists"/>).
##  <Ref Func="EnumeratorOfSubset"/> returns a list <A>new</A> of length
##  equal to the number of <K>true</K> entries in <A>blist</A>,
##  such that <C><A>new</A>[i]</C>, if bound, equals the entry of <A>list</A>
##  at the <A>i</A>-th <K>true</K> position in <A>blist</A>.
##  <P/>
##  If <A>list</A> is homogeneous then also <A>new</A> is homogeneous.
##  If <A>list</A> is <E>not</E> homogeneous then the third argument
##  <A>ishomog</A> must be present and equal to <K>true</K> or <K>false</K>,
##  saying whether or not <A>new</A> is homogeneous.
##  <P/>
##  This construction is used for example in the situation that <A>list</A>
##  is an enumerator of a large set,
##  and <A>blist</A> describes a union of orbits in an action on this set.
##  </Description>
##  </ManSection>
##
DeclareGlobalFunction( "EnumeratorOfSubset" );


#############################################################################
##
#F  EnumeratorByFunctions( <D>, <record> )
#F  EnumeratorByFunctions( <Fam>, <record> )
##
##  <#GAPDoc Label="EnumeratorByFunctions">
##  <Heading>EnumeratorByFunctions</Heading>
##  <ManSection>
##  <Func Name="EnumeratorByFunctions" Arg='D, record'
##   Label="for a domain and a record"/>
##  <Func Name="EnumeratorByFunctions" Arg='Fam, record'
##   Label="for a family and a record"/>
##
##  <Description>
##  <Ref Func="EnumeratorByFunctions" Label="for a domain and a record"/>
##  returns an immutable, dense, and duplicate-free list <M>enum</M> for
##  which <Ref Func="IsBound" Label="for a list position"/>,
##  element access via <Ref Func="\[\]"/>,
##  <Ref Func="Length"/>, and <Ref Func="Position"/>
##  are computed via prescribed functions.
##  <P/>
##  Let <A>record</A> be a record with at least the following components.
##  <List>
##  <Mark><C>ElementNumber</C></Mark>
##  <Item>
##      a function taking two arguments <A>enum</A> and <A>pos</A>,
##      which returns <C><A>enum</A>[ <A>pos</A> ]</C>
##      (see&nbsp;<Ref Sect="Basic Operations for Lists"/>);
##      it can be assumed that the argument <A>pos</A> is a positive integer,
##      but <A>pos</A> may be larger than the length of <A>enum</A>
##      (in which case an error must be signalled);
##      note that the result must be immutable since <A>enum</A> itself is
##      immutable,
##  </Item>
##  <Mark><C>NumberElement</C></Mark>
##  <Item>
##      a function taking two arguments <A>enum</A> and <A>elm</A>,
##      which returns <C>Position( <A>enum</A>, <A>elm</A> )</C>
##      (see&nbsp;<Ref Func="Position"/>);
##      it cannot be assumed that <A>elm</A> is really contained in
##      <A>enum</A> (and <K>fail</K> must be returned if not);
##      note that for the three argument version of <Ref Func="Position"/>,
##      the method that is available for duplicate-free lists suffices.
##  </Item>
##  </List>
##  <P/>
##  Further (data) components may be contained in <A>record</A>
##  which can be used by these function.
##  <P/>
##  If the first argument is a domain <A>D</A> then <A>enum</A> lists the
##  elements of <A>D</A> (in general <A>enum</A> is <E>not</E> sorted),
##  and methods for <Ref Attr="Length"/>,
##  <Ref Func="IsBound" Label="for a list position"/>,
##  and <Ref Func="PrintObj"/> may use <A>D</A>.
##  <!-- is this really true for Length?-->
##  <P/>
##  If one wants to describe the result without creating a domain then the
##  elements are given implicitly by the functions in <A>record</A>,
##  and the first argument must be a family <A>Fam</A> which will become the
##  family of <A>enum</A>;
##  if <A>enum</A> is not homogeneous then <A>Fam</A> must be
##  <C>ListsFamily</C>,
##  otherwise it must be the collections family of any element in <A>enum</A>.
##  In this case, additionally the following component in <A>record</A> is
##  needed.
##  <P/>
##  <List>
##  <Mark><C>Length</C></Mark>
##  <Item>
##      a function taking the argument <A>enum</A>,
##      which returns the length of <A>enum</A>
##      (see&nbsp;<Ref Func="Length"/>).
##  </Item>
##  </List>
##  <P/>
##  The following components are optional; they are used if they are present
##  but default methods are installed for the case that they are missing.
##  <List>
##  <Mark><C>IsBound\[\]</C></Mark>
##  <Item>
##      a function taking two arguments <A>enum</A> and <A>k</A>,
##      which returns <C>IsBound( <A>enum</A>[ <A>k</A> ] )</C>
##      (see&nbsp;<Ref Sect="Basic Operations for Lists"/>);
##      if this component is missing then <Ref Func="Length"/> is used for
##      computing the result,
##  </Item>
##  <Mark><C>Membership</C></Mark>
##  <Item>
##      a function taking two arguments <A>elm</A> and <A>enum</A>,
##      which returns <K>true</K> is <A>elm</A> is an element of <A>enum</A>,
##      and <K>false</K> otherwise
##      (see&nbsp;<Ref Sect="Basic Operations for Lists"/>);
##      if this component is missing then <C>NumberElement</C> is used
##      for computing the result,
##  </Item>
##  <Mark><C>AsList</C></Mark>
##  <Item>
##      a function taking one argument <A>enum</A>, which returns a list with
##      the property that the access to each of its elements will take
##      roughly the same time
##      (see&nbsp;<Ref Func="IsConstantTimeAccessList"/>);
##      if this component is missing then
##      <Ref Func="ConstantTimeAccessList"/> is used for computing the result,
##  </Item>
##  <Mark><C>ViewObj</C> and <C>PrintObj</C></Mark>
##  <Item>
##      two functions that print what one wants to be printed when
##      <C>View( <A>enum</A> )</C> or <C>Print( <A>enum</A> )</C> is called
##      (see&nbsp;<Ref Sect="View and Print"/>),
##      if the <C>ViewObj</C> component is missing then the <C>PrintObj</C>
##      method is used as a default.
##  </Item>
##  </List>
##  <P/>
##  If the result is known to have additional properties such as being
##  strictly sorted (see&nbsp;<Ref Func="IsSSortedList"/>) then it can be
##  useful to set these properties after the construction of the enumerator,
##  before it is used for the first time.
##  And in the case that a new sorted enumerator of a domain is implemented
##  via <Ref Func="EnumeratorByFunctions" Label="for a domain and a record"/>,
##  and this construction is
##  installed as a method for the operation <Ref Func="Enumerator"/>,
##  then it should be installed also as a method for
##  <Ref Func="EnumeratorSorted"/>.
##  <P/>
##  Note that it is <E>not</E> checked that
##  <Ref Func="EnumeratorByFunctions" Label="for a domain and a record"/>
##  really returns a dense and duplicate-free list.
##  <Ref Func="EnumeratorByFunctions" Label="for a domain and a record"/>
##  does <E>not</E> make a shallow copy of <A>record</A>,
##  this record is changed in place,
##  see&nbsp;<Ref Sect="Creating Objects"/>.
##  <P/>
##  It would be easy to implement a slightly generalized setup for
##  enumerators that need not be duplicate-free (where the three argument
##  version of <Ref Func="Position"/> is supported),
##  but the resulting overhead for the methods seems not to be justified.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "EnumeratorByFunctions" );


#############################################################################
##
#A  UnderlyingCollection( <enum> )
##
##  <ManSection>
##  <Attr Name="UnderlyingCollection" Arg='enum'/>
##
##  <Description>
##  An enumerator of a domain can delegate the task to compute its length to
##  <C>Size</C> for the underlying domain, and <C>ViewObj</C> and <C>PrintObj</C> methods
##  may refer to this domain.
##  </Description>
##  </ManSection>
##
DeclareAttribute( "UnderlyingCollection", IsListOrCollection );


#############################################################################
##
#F  List( <list>[, <func>] )  . . . . . . .  list of elements of a collection
#F  List( <C> )
##
##  <#GAPDoc Label="List:list">
##  <ManSection>
##  <Func Name="List" Arg='list[, func]' Label="for a list (and a function)"/>
##
##  <Description>
##  This function returns a new mutable list <C>new</C> of the same length 
##  as the list <A>list</A> (which may have holes). The entry <C>new[i]</C> 
##  is unbound if <C><A>list</A>[i]</C> is unbound. Otherwise 
##  <C>new[i] = <A>func</A>(<A>list</A>[i])</C>. If the argument <A>func</A> is
##  omitted, its default is <Ref Func="IdFunc"/>, so this function does the
##  same as <Ref Oper="ShallowCopy"/> (see also 
##  <Ref Sect="Duplication of Lists"/>).
##  <P/>
##  <Example><![CDATA[
##  gap> List( [1,2,3], i -> i^2 );
##  [ 1, 4, 9 ]
##  gap> List( [1..10], IsPrime );
##  [ false, true, true, false, true, false, true, false, false, false ]
##  gap> List([,1,,3,4], x-> x > 2);
##  [ , false,, true, true ]
##  ]]></Example>
##  <P/>
##  (See also <Ref Func="List" Label="for a collection"/>.)
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
##  <#GAPDoc Label="List:coll">
##  <ManSection>
##  <Func Name="List" Arg='C' Label="for a collection"/>
##
##  <Description>
##  For a collection <A>C</A> (see&nbsp;<Ref Chap="Collections"/>)
##  that is not a list, <Ref Func="List" Label="for a collection"/> returns
##  a new mutable list <A>new</A> such that <C>Length( <A>new</A> )</C>
##  is the number of different elements of <A>C</A>,
##  and <A>new</A> contains the different elements of <A>C</A> in an
##  unspecified order which may change for repeated calls.
##  <C><A>new</A>[<A>pos</A>]</C> executes in constant time
##  (see&nbsp;<Ref Func="IsConstantTimeAccessList"/>),
##  and the size of <A>new</A> is proportional to its length.
##  The generic method for this case is
##  <C>ShallowCopy( Enumerator( <A>C</A> ) )</C>.
##  <!-- this is not reasonable since <C>ShallowCopy</C> need not guarantee to return-->
##  <!-- a constant time access list-->
##  <P/>
##  <Example><![CDATA[
##  gap> l:= List( Group( (1,2,3) ) );
##  [ (), (1,3,2), (1,2,3) ]
##  gap> IsMutable( l );  IsSortedList( l );  IsConstantTimeAccessList( l );
##  true
##  false
##  true
##  ]]></Example>
##  <P/>
##  (See also <Ref Func="List" Label="for a list (and a function)"/>.)
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "List" );

DeclareOperation( "ListOp", [ IsListOrCollection ] );
DeclareOperation( "ListOp", [ IsListOrCollection, IsFunction ] );


#############################################################################
##
#O  SortedList( <C> )
#O  SortedList( <list> )
##
##  <#GAPDoc Label="SortedList">
##  <ManSection>
##  <Oper Name="SortedList" Arg='listorcoll'/>
##
##  <Description>
##  <Ref Func="SortedList"/> returns a new mutable and dense list <A>new</A>.
##  The argument must be a collection or list <A>listorcoll</A> which may
##  contain holes but whose elements lie in the same family
##  (see&nbsp;<Ref Sect="Families"/>).
##  <C>Length( <A>new</A> )</C> is the number of elements of
##  <A>listorcoll</A>,
##  and <A>new</A> contains the elements in sorted order,
##  w.r.t.&nbsp;<C>&lt;=</C>.
##  <C><A>new</A>[<A>pos</A>]</C> executes in constant time
##  (see&nbsp;<Ref Func="IsConstantTimeAccessList"/>),
##  and the size of <A>new</A> in memory is proportional to its length.
##  <P/>
##  <Example><![CDATA[
##  gap> l:= SortedList( Group( (1,2,3) ) );
##  [ (), (1,2,3), (1,3,2) ]
##  gap> IsMutable( l );  IsSortedList( l );  IsConstantTimeAccessList( l );
##  true
##  true
##  true
##  gap> SortedList( [ 1, 2, 1,, 3, 2 ] );
##  [ 1, 1, 2, 2, 3 ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "SortedList", [ IsListOrCollection ] );


#############################################################################
##
#O  SSortedList( <C> )  . . . . . . . . . . . set of elements of a collection
#O  SSortedList( <list> ) . . . . . . . . . . . . . set of elements of a list
#O  Set( <C> )
##
##  <#GAPDoc Label="SSortedList">
##  <ManSection>
##  <Oper Name="SSortedList" Arg='listorcoll'/>
##  <Oper Name="Set" Arg='C'/>
##
##  <Description>
##  <Ref Func="SSortedList"/> (<Q>strictly sorted list</Q>) returns a new
##  dense, mutable, and duplicate free list <A>new</A>.
##  The argument must be a collection or list <A>listorcoll</A>
##  which may contain holes but whose elements lie in the same family
##  (see&nbsp;<Ref Sect="Families"/>).
##  <C>Length( <A>new</A> )</C> is the number of different elements of
##  <A>listorcoll</A>,
##  and <A>new</A> contains the different elements in strictly sorted order,
##  w.r.t.&nbsp;<Ref Func="\&lt;"/>.
##  <C><A>new</A>[<A>pos</A>]</C> executes in constant time
##  (see&nbsp;<Ref Func="IsConstantTimeAccessList"/>),
##  and the size of <A>new</A> in memory is proportional to its length.
##  <P/>
##  <Ref Func="Set"/> is simply a synonym for <Ref Func="SSortedList"/>.
##  <!-- <P/> -->
##  <!--  For collections that are not lists, the default method is-->
##  <!--  <C>ShallowCopy( EnumeratorSorted( <A>C</A> ) )</C>.-->
##  <P/>
##  <Example><![CDATA[
##  gap> l:= SSortedList( Group( (1,2,3) ) );
##  [ (), (1,2,3), (1,3,2) ]
##  gap> IsMutable( l );  IsSSortedList( l );  IsConstantTimeAccessList( l );
##  true
##  true
##  true
##  gap> SSortedList( [ 1, 2, 1,, 3, 2 ] );
##  [ 1, 2, 3 ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "SSortedList", [ IsListOrCollection ] );
DeclareSynonym( "Set", SSortedList );


#############################################################################
##
#A  AsList( <C> ) . . . . . . . . . . . . .  list of elements of a collection
#A  AsList( <list> )  . . . . . . . . . . . . . .  list of elements of a list
##
##  <#GAPDoc Label="AsList">
##  <ManSection>
##  <Attr Name="AsList" Arg='listorcoll'/>
##
##  <Description>
##  <Ref Func="AsList"/> returns a immutable list <A>imm</A>.
##  If the argument is a list (which may contain holes),
##  then <C>Length( <A>imm</A> )</C> is the <Ref Func="Length"/> value of
##  this list,
##  and <A>imm</A> contains the elements (and holes) of of the list
##  in the same order.
##  If the argument is a collection that is not a list,
##  then <C>Length( <A>imm</A> )</C> is the number of different elements
##  of this collection, and <A>imm</A> contains the different elements of
##  the collection in an unspecified order,
##  which may change for repeated calls of <Ref Func="AsList"/>.
##  <C><A>imm</A>[<A>pos</A>]</C> executes in constant time
##  (see&nbsp;<Ref Func="IsConstantTimeAccessList"/>),
##  and the size of <A>imm</A> in memory is proportional to its length.
##  <P/>
##  If you expect to do many element tests in the resulting list, it might
##  be worth to use a sorted list instead, using <Ref Func="AsSSortedList"/>.
##  <!-- <P/> -->
##  <!--  For both lists and collections, the default method is-->
##  <!--  <C>ConstantTimeAccessList( Enumerator( <A>C</A> ) )</C>.-->
##  <P/>
##  <Example><![CDATA[
##  gap> l:= AsList( [ 1, 3, 3,, 2 ] );
##  [ 1, 3, 3,, 2 ]
##  gap> IsMutable( l );  IsSortedList( l );  IsConstantTimeAccessList( l );
##  false
##  false
##  true
##  gap> AsList( Group( (1,2,3), (1,2) ) );
##  [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "AsList", IsListOrCollection );


#############################################################################
##
#A  AsSortedList( <C> )
#A  AsSortedList( <list> )
##
##  <#GAPDoc Label="AsSortedList">
##  <ManSection>
##  <Attr Name="AsSortedList" Arg='listorcoll'/>
##
##  <Description>
##  <Ref Func="AsSortedList"/> returns a dense and immutable list <A>imm</A>.
##  The argument must be a collection or list <A>listorcoll</A>
##  which may contain holes but whose elements lie in the same family
##  (see&nbsp;<Ref Sect="Families"/>).
##  <C>Length( <A>imm</A> )</C> is the number of elements of the argument,
##  and <A>imm</A> contains the elements in sorted order,
##  w.r.t.&nbsp;<C>&lt;=</C>.
##  <C><A>new</A>[<A>pos</A>]</C> executes in constant time
##  (see&nbsp;<Ref Func="IsConstantTimeAccessList"/>),
##  and the size of <A>imm</A> in memory is proportional to its length.
##  <P/>
##  The only difference to the operation <Ref Func="SortedList"/>
##  is that <Ref Func="AsSortedList"/> returns an <E>immutable</E> list.
##  <P/>
##  <Example><![CDATA[
##  gap> l:= AsSortedList( [ 1, 3, 3,, 2 ] );
##  [ 1, 2, 3, 3 ]
##  gap> IsMutable( l );  IsSortedList( l );  IsConstantTimeAccessList( l );
##  false
##  true
##  true
##  gap> IsSSortedList( l );
##  false
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "AsSortedList", IsListOrCollection );


#############################################################################
##
#A  AsSSortedList( <C> )  . . . . . . . . . . set of elements of a collection
#A  AsSSortedList( <list> ) . . . . . . . . . . . . set of elements of a list
#A  AsSet( <C> )
##
##  <#GAPDoc Label="AsSSortedList">
##  <ManSection>
##  <Attr Name="AsSSortedList" Arg='listorcoll'/>
##  <Attr Name="AsSet" Arg='listorcoll'/>
##
##  <Description>
##  <Index Subkey="of a list or collection">elements</Index>
##  <Ref Func="AsSSortedList"/> (<Q>as strictly sorted list</Q>) returns
##  a dense, immutable, and duplicate free list <A>imm</A>.
##  The argument must be a collection or list <A>listorcoll</A>
##  which may contain holes but whose elements lie in the same family
##  (see&nbsp;<Ref Sect="Families"/>).
##  <C>Length( <A>imm</A> )</C> is the number of different elements
##  of <A>listorcoll</A>,
##  and <A>imm</A> contains the different elements in strictly sorted order,
##  w.r.t.&nbsp;<Ref Func="\&lt;"/>.
##  <C><A>imm</A>[<A>pos</A>]</C> executes in constant time
##  (see&nbsp;<Ref Func="IsConstantTimeAccessList"/>),
##  and the size of <A>imm</A> in memory is proportional to its length.
##  <P/>
##  Because the comparisons required for sorting can be very expensive for
##  some kinds of objects, you should use <Ref Func="AsList"/> instead
##  if you do not require the result to be sorted.
##  <P/>
##  The only difference to the operation <Ref Func="SSortedList"/>
##  is that <Ref Attr="AsSSortedList"/> returns an <E>immutable</E> list.
##  <P/>
##  <Ref Attr="AsSet"/> is simply a synonym for <Ref Attr="AsSSortedList"/>.
##  <P/>
##  In general a function that returns a set of elements is free, in fact
##  encouraged, to return a domain instead of the proper set of its elements.
##  This allows one to keep a given structure, and moreover the
##  representation by a domain object is usually more space efficient.
##  <Ref Attr="AsSSortedList"/> must of course <E>not</E> do this,
##  its only purpose is to create the proper set of elements.
##  <!-- <P/> -->
##  <!--  For both lists and collections, the default method is-->
##  <!--  <C>ConstantTimeAccessList( EnumeratorSorted( <A>C</A> ) )</C>.-->
##  <P/>
##  <Example><![CDATA[
##  gap> l:= AsSSortedList( l );
##  [ 1, 2, 3 ]
##  gap> IsMutable( l );  IsSSortedList( l );  IsConstantTimeAccessList( l );
##  false
##  true
##  true
##  gap> AsSSortedList( Group( (1,2,3), (1,2) ) );
##  [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareAttribute( "AsSSortedList", IsListOrCollection );
DeclareSynonym( "AsSet", AsSSortedList );


#############################################################################
##
#A  AsSSortedListNonstored( <C> )
##
##  <ManSection>
##  <Attr Name="AsSSortedListNonstored" Arg='C'/>
##
##  <Description>
##  returns the <Ref Func="AsSSortedList"/> value of the list or collection
##  <A>C</A> but ensures that this list
##  (nor a permutation or substantial subset) will not be
##  stored in attributes of <A>C</A> unless such a list is already stored.
##  This permits to obtain an element list once
##  without danger of clogging up memory in the long run.
##  <P/>
##  Because of this guarantee of nonstorage, methods for
##  <Ref Func="AsSSortedListNonstored"/> may not default to
##  <Ref Func="AsSSortedList"/>, but only vice versa.
##  </Description>
##  </ManSection>
##
DeclareOperation( "AsSSortedListNonstored", [IsListOrCollection] );


#############################################################################
##
#F  Elements( <C> )
##
##  <#GAPDoc Label="Elements">
##  <ManSection>
##  <Func Name="Elements" Arg='C'/>
##
##  <Description>
##  <Ref Func="Elements"/> does the same as <Ref Func="AsSSortedList"/>,
##  that is, the return value is a strictly sorted list of the elements in
##  the list or collection <A>C</A>.
##  <P/>
##  <Ref Func="Elements"/> is only supported for backwards compatibility.
##  In many situations, the sortedness of the <Q>element list</Q> for a
##  collection is in fact not needed, and one can save a lot of time by
##  asking for a list that is <E>not</E> necessarily sorted,
##  using <Ref Func="AsList"/>.
##  If one is really interested in the strictly sorted list of elements in
##  <A>C</A> then one should use <Ref Func="AsSet"/> or
##  <Ref Func="AsSSortedList"/> instead.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "Elements" );


#############################################################################
##
#F  Sum( <list>[, <init>] ) . . . . . . . . . . sum of the elements of a list
#F  Sum( <C>[, <init>] )  . . . . . . . . sum of the elements of a collection
#F  Sum( <list>, <func>[, <init>] ) . . . . .  sum of images under a function
#F  Sum( <C>, <func>[, <init>] )  . . . . . .  sum of images under a function
##
##  <#GAPDoc Label="Sum">
##  <ManSection>
##  <Func Name="Sum" Arg='listorcoll[, func][, init]'/>
##
##  <Description>
##  Called with one argument, a dense list or collection <A>listorcoll</A>,
##  <Ref Func="Sum"/> returns the sum of the elements of <A>listorcoll</A>
##  (see&nbsp;<Ref Chap="Collections"/>).
##  <P/>
##  Called with a dense list or collection <A>listorcoll</A> and a function
##  <A>func</A>, which must be a function taking one argument,
##  <Ref Func="Sum"/> applies the function <A>func</A>
##  to the elements of <A>listorcoll</A>, and returns the sum of the results.
##  In either case <Ref Func="Sum"/> returns <C>0</C> if the first argument
##  is empty.
##  <P/>
##  The general rules for arithmetic operations apply
##  (see&nbsp;<Ref Sect="Mutability Status and List Arithmetic"/>),
##  so the result is immutable if and only if all summands are immutable.
##  <P/>
##  If <A>listorcoll</A> contains exactly one element then this element
##  (or its image under <A>func</A> if applicable) itself is returned,
##  not a shallow copy of this element.
##  <P/>
##  If an additional initial value <A>init</A> is given,
##  <Ref Func="Sum"/> returns the sum of <A>init</A> and the elements of the
##  first argument resp.&nbsp;of their images under the function <A>func</A>.
##  This is useful for example if the first argument is empty and a different
##  zero than <C>0</C> is desired, in which case <A>init</A> is returned.
##  <P/>
##  <Example><![CDATA[
##  gap> Sum( [ 2, 3, 5, 7, 11, 13, 17, 19 ] );
##  77
##  gap> Sum( [1..10], x->x^2 );
##  385
##  gap> Sum( [ [1,2], [3,4], [5,6] ] );
##  [ 9, 12 ]
##  gap> Sum( GF(8) );
##  0*Z(2)
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "Sum" );


#############################################################################
##
#O  SumOp( <C> )
#O  SumOp( <C>, <func> )
#O  SumOp( <C>, <init> )
#O  SumOp( <C>, <func>, <init> )
##
##  <ManSection>
##  <Oper Name="SumOp" Arg='C'/>
##  <Oper Name="SumOp" Arg='C, func'/>
##  <Oper Name="SumOp" Arg='C, init'/>
##  <Oper Name="SumOp" Arg='C, func, init'/>
##
##  <Description>
##  <C>SumOp</C> is the operation called by <C>Sum</C>
##  if <A>C</A> is not an internal list.
##  </Description>
##  </ManSection>
##
DeclareOperation( "SumOp", [ IsListOrCollection ] );


#############################################################################
##
#F  Product( <list>[, <init>] ) . . . . . . product of the elements of a list
#F  Product( <C>[, <init>] )  . . . . product of the elements of a collection
#F  Product( <list>, <func>[, <init>] ) .  product of images under a function
#F  Product( <C>, <func>[, <init>] )  . .  product of images under a function
##
##  <#GAPDoc Label="Product">
##  <ManSection>
##  <Func Name="Product" Arg='listorcoll[, func][, init]'/>
##
##  <Description>
##  Called with one argument, a dense list or collection <A>listorcoll</A>,
##  <Ref Func="Product"/> returns the product of the elements of
##  <A>listorcoll</A> (see&nbsp;<Ref Chap="Collections"/>).
##  <P/>
##  Called with a dense list or collection <A>listorcoll</A> and a function
##  <A>func</A>, which must be a function taking one argument,
##  <Ref Func="Product"/> applies the function <A>func</A>
##  to the elements of <A>listorcoll</A>, and returns the product of the
##  results.
##  In either case <Ref Func="Product"/> returns <C>1</C> if the first
##  argument is empty.
##  <P/>
##  The general rules for arithmetic operations apply
##  (see&nbsp;<Ref Sect="Mutability Status and List Arithmetic"/>),
##  so the result is immutable if and only if all summands are immutable.
##  <P/>
##  If <A>listorcoll</A> contains exactly one element then this element
##  (or its image under <A>func</A> if applicable) itself is returned,
##  not a shallow copy of this element.
##  <P/>
##  If an additional initial value <A>init</A> is given,
##  <Ref Func="Product"/> returns the product of <A>init</A> and the elements
##  of the first argument resp.&nbsp;of their images under the function
##  <A>func</A>.
##  This is useful for example if the first argument is empty and a different
##  identity than <C>1</C> is desired, in which case <A>init</A> is returned.
##  <P/>
##  <Example><![CDATA[
##  gap> Product( [ 2, 3, 5, 7, 11, 13, 17, 19 ] );
##  9699690
##  gap> Product( [1..10], x->x^2 );
##  13168189440000
##  gap> Product( [ (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) ] );
##  (1,4)(2,3)
##  gap> Product( GF(8) );
##  0*Z(2)
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "Product" );


#############################################################################
##
#O  ProductOp( <C> )
#O  ProductOp( <C>, <func> )
#O  ProductOp( <C>, <init> )
#O  ProductOp( <C>, <func>, <init> )
##
##  <ManSection>
##  <Oper Name="ProductOp" Arg='C'/>
##  <Oper Name="ProductOp" Arg='C, func'/>
##  <Oper Name="ProductOp" Arg='C, init'/>
##  <Oper Name="ProductOp" Arg='C, func, init'/>
##
##  <Description>
##  <C>ProductOp</C> is the operation called by <C>Product</C>
##  if <A>C</A> is not an internal list.
##  </Description>
##  </ManSection>
##
DeclareOperation( "ProductOp", [ IsListOrCollection ] );


#############################################################################
##
#F  Filtered( <list>, <func> )  . . . . extract elements that have a property
#F  Filtered( <C>, <func> ) . . . . . . extract elements that have a property
##
##  <#GAPDoc Label="Filtered">
##  <ManSection>
##  <Func Name="Filtered" Arg='listorcoll, func'/>
##
##  <Description>
##  returns a new list that contains those elements of the list or collection
##  <A>listorcoll</A> (see&nbsp;<Ref Chap="Collections"/>), respectively,
##  for which the unary function <A>func</A> returns <K>true</K>.
##  <P/>
##  If the first argument is a list, the order of the elements in the result
##  is the same as the order of the corresponding elements of this list.
##  If an element for which <A>func</A> returns <K>true</K> appears several
##  times in the list it will also appear the same number of times
##  in the result.
##  The argument list may contain holes,
##  they are ignored by <Ref Func="Filtered"/>.
##  <P/>
##  For each element of <A>listorcoll</A>,
##  <A>func</A> must return either <K>true</K> or <K>false</K>,
##  otherwise an error is signalled.
##  <P/>
##  The result is a new list that is not identical to any other list.
##  The elements of that list however are identical to the corresponding
##  elements of the argument list (see&nbsp;<Ref Sect="Identical Lists"/>).
##  <P/>
##  List assignment using the operator <Ref Func="\{\}"/>
##  (see&nbsp;<Ref Sect="List Assignment"/>) can be used to extract
##  elements of a list according to indices given in another list.
##  <P/>
##  <Example><![CDATA[
##  gap> Filtered( [1..20], IsPrime );
##  [ 2, 3, 5, 7, 11, 13, 17, 19 ]
##  gap> Filtered( [ 1, 3, 4, -4, 4, 7, 10, 6 ], IsPrimePowerInt );
##  [ 3, 4, 4, 7 ]
##  gap> Filtered( [ 1, 3, 4, -4, 4, 7, 10, 6 ],
##  >              n -> IsPrimePowerInt(n) and n mod 2 <> 0 );
##  [ 3, 7 ]
##  gap> Filtered( Group( (1,2), (1,2,3) ), x -> Order( x ) = 2 );
##  [ (2,3), (1,2), (1,3) ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "Filtered" );


#############################################################################
##
#O  FilteredOp( <C>, <func> )
##
##  <ManSection>
##  <Oper Name="FilteredOp" Arg='C, func'/>
##
##  <Description>
##  <C>FilteredOp</C> is the operation called by <C>Filtered</C>
##  if <A>C</A> is not an internal list.
##  </Description>
##  </ManSection>
##
DeclareOperation( "FilteredOp", [ IsListOrCollection, IsFunction ] );


#############################################################################
##
#F  Number( <list> )
#F  Number( <list>, <func> )  . . . . . . count elements that have a property
#F  Number( <C>, <func> ) . . . . . . . . count elements that have a property
##
##  <#GAPDoc Label="Number">
##  <ManSection>
##  <Func Name="Number" Arg='listorcoll[, func]'/>
##
##  <Description>
##  Called with a list <A>listorcoll</A>, <Ref Func="Number"/> returns the
##  number of bound entries in this list.
##  For dense lists <Ref Func="Number"/>, <Ref Func="Length"/>,
##  and <Ref Func="Size"/> return the same value;
##  for lists with holes <Ref Func="Number"/> returns the number of bound
##  entries, <Ref Func="Length"/> returns the largest index of a bound entry,
##  and <Ref Func="Size"/> signals an error.
##  <P/>
##  Called with two arguments, a list or collection <A>listorcoll</A> and a
##  unary function <A>func</A>, <Ref Func="Number"/> returns the number of
##  elements of <A>listorcoll</A> for which <A>func</A> returns <K>true</K>.
##  If an element for which <A>func</A> returns <K>true</K> appears several
##  times in <A>listorcoll</A> it will also be counted the same number of
##  times.
##  <P/>
##  For each element of <A>listorcoll</A>,
##  <A>func</A> must return either <K>true</K> or <K>false</K>,
##  otherwise an error is signalled.
##  <P/>
##  <Ref Func="Filtered"/> allows you to extract the elements of a list
##  that have a certain property.
##  <P/>
##  <Example><![CDATA[
##  gap> Number( [ 2, 3, 5, 7 ] );
##  4
##  gap> Number( [, 2, 3,, 5,, 7,,,, 11 ] );
##  5
##  gap> Number( [1..20], IsPrime );
##  8
##  gap> Number( [ 1, 3, 4, -4, 4, 7, 10, 6 ], IsPrimePowerInt );
##  4
##  gap> Number( [ 1, 3, 4, -4, 4, 7, 10, 6 ],
##  >            n -> IsPrimePowerInt(n) and n mod 2 <> 0 );
##  2
##  gap> Number( Group( (1,2), (1,2,3) ), x -> Order( x ) = 2 );
##  3
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "Number" );


#############################################################################
##
#O  NumberOp( <C>, <func> )
##
##  <ManSection>
##  <Oper Name="NumberOp" Arg='C, func'/>
##
##  <Description>
##  <C>NumberOp</C> is the operation called by <C>Number</C>
##  if <A>C</A> is not an internal list.
##  </Description>
##  </ManSection>
##
DeclareOperation( "NumberOp", [ IsListOrCollection, IsFunction ] );


#############################################################################
##
#F  ForAll( <list>, <func> )
#F  ForAll( <C>, <func> )
##
##  <#GAPDoc Label="ForAll">
##  <ManSection>
##  <Func Name="ForAll" Arg='listorcoll, func'/>
##
##  <Description>
##  tests whether the unary function <A>func</A> returns <K>true</K>
##  for all elements in the list or collection <A>listorcoll</A>.
##  <P/>
##  <Example><![CDATA[
##  gap> ForAll( [1..20], IsPrime );
##  false
##  gap> ForAll( [2,3,4,5,8,9], IsPrimePowerInt );
##  true
##  gap> ForAll( [2..14], n -> IsPrimePowerInt(n) or n mod 2 = 0 );
##  true
##  gap> ForAll( Group( (1,2), (1,2,3) ), i -> SignPerm(i) = 1 );
##  false
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "ForAll" );


#############################################################################
##
#O  ForAllOp( <C>, <func> )
##
##  <ManSection>
##  <Oper Name="ForAllOp" Arg='C, func'/>
##
##  <Description>
##  <C>ForAllOp</C> is the operation called by <C>ForAll</C>
##  if <A>C</A> is not an internal list.
##  </Description>
##  </ManSection>
##
DeclareOperation( "ForAllOp", [ IsListOrCollection, IsFunction ] );


#############################################################################
##
#F  ForAny( <list>, <func> )
#F  ForAny( <C>, <func> )
##
##  <#GAPDoc Label="ForAny">
##  <ManSection>
##  <Func Name="ForAny" Arg='listorcoll, func'/>
##
##  <Description>
##  tests whether the unary function <A>func</A> returns <K>true</K>
##  for at least one element in the list or collection <A>listorcoll</A>.
##  <P/>
##  <Example><![CDATA[
##  gap> ForAny( [1..20], IsPrime );
##  true
##  gap> ForAny( [2,3,4,5,8,9], IsPrimePowerInt );
##  true
##  gap> ForAny( [2..14],
##  >    n -> IsPrimePowerInt(n) and n mod 5 = 0 and not IsPrime(n) );
##  false
##  gap> ForAny( Integers, i ->     i > 0
##  >                           and ForAll( [0,2..4], j -> IsPrime(i+j) ) );
##  true
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "ForAny" );


#############################################################################
##
#O  ForAnyOp( <C>, <func> )
##
##  <ManSection>
##  <Oper Name="ForAnyOp" Arg='C, func'/>
##
##  <Description>
##  <C>ForAnyOp</C> is the operation called by <C>ForAny</C>
##  if <A>C</A> is not an internal list.
##  </Description>
##  </ManSection>
##
DeclareOperation( "ForAnyOp", [ IsListOrCollection, IsFunction ] );


#############################################################################
##
#O  ListX( <arg1>, <arg2>, ... <argn>, <func> )
##
##  <#GAPDoc Label="ListX">
##  <ManSection>
##  <Oper Name="ListX" Arg='arg1, arg2, ... argn, func'/>
##
##  <Description>
##  <Ref Oper="ListX"/> returns a new list constructed from the arguments.
##  <P/>
##  Each of the arguments <A>arg1</A>, <A>arg2</A>, <M>\ldots</M> <A>argn</A>
##  must be one of the following:
##  <List>
##  <Mark>a list or collection</Mark>
##  <Item>
##      this introduces a new for-loop in the sequence of nested
##      for-loops and if-statements;
##  </Item>
##  <Mark>a function returning a list or collection</Mark>
##  <Item>
##      this introduces a new for-loop in the sequence of nested
##      for-loops and if-statements, where the loop-range depends on
##      the values of the outer loop-variables; or
##  </Item>
##  <Mark>a function returning <K>true</K> or <K>false</K></Mark>
##  <Item>
##      this introduces a new if-statement in the sequence of nested
##      for-loops and if-statements.
##  </Item>
##  </List>
##  <P/>
##  The last argument <A>func</A> must be a function,
##  it is applied to the values of the loop-variables
##  and the results are collected.
##  <P/>
##  Thus <C>ListX( <A>list</A>, <A>func</A> )</C> is the same as
##  <C>List( <A>list</A>, <A>func</A> )</C>,
##  and <C>ListX( <A>list</A>, <A>func</A>, x -> x )</C> is the same as
##  <C>Filtered( <A>list</A>, <A>func</A> )</C>.
##  <P/>
##  As a more elaborate example, assume <A>arg1</A> is a list or collection,
##  <A>arg2</A> is a function returning <K>true</K> or <K>false</K>,
##  <A>arg3</A> is a function returning a list or collection, and
##  <A>arg4</A> is another function returning <K>true</K> or <K>false</K>,
##  then
##  <P/>
##  <C><A>result</A> := ListX( <A>arg1</A>, <A>arg2</A>, <A>arg3</A>,
##  <A>arg4</A>, <A>func</A> );</C>
##  <P/>
##  is equivalent to
##  <P/>
##  <Listing><![CDATA[
##  result := [];
##  for v1 in arg1 do
##    if arg2( v1 ) then
##      for v2 in arg3( v1 ) do
##        if arg4( v1, v2 ) then
##          Add( result, func( v1, v2 ) );
##        fi;
##      od;
##    fi;
##  od;
##  ]]></Listing>
##  <P/>
##  The following example shows how <Ref Func="ListX"/> can be used to
##  compute all pairs and all strictly sorted pairs of elements in a list.
##  <P/>
##  <Example><![CDATA[
##  gap> l:= [ 1, 2, 3, 4 ];;
##  gap> pair:= function( x, y ) return [ x, y ]; end;;
##  gap> ListX( l, l, pair );
##  [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 2 ], 
##    [ 2, 3 ], [ 2, 4 ], [ 3, 1 ], [ 3, 2 ], [ 3, 3 ], [ 3, 4 ], 
##    [ 4, 1 ], [ 4, 2 ], [ 4, 3 ], [ 4, 4 ] ]
##  ]]></Example>
##  <P/>
##  In the following example, <Ref Func="\&lt;"/> is the comparison
##  operation:
##  <P/>
##  <Example><![CDATA[
##  gap> ListX( l, l, \<, pair );
##  [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "ListX" );


#############################################################################
##
#O  SetX( <arg1>, <arg2>, ... <func> )
##
##  <#GAPDoc Label="SetX">
##  <ManSection>
##  <Oper Name="SetX" Arg='arg1, arg2, ... func'/>
##
##  <Description>
##  The only difference between <Ref Oper="SetX"/> and <Ref Oper="ListX"/>
##  is that the result list of <Ref Oper="SetX"/> is strictly sorted.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "SetX" );


#############################################################################
##
#O  SumX( <arg1>, <arg2>, ... <func> )
##
##  <#GAPDoc Label="SumX">
##  <ManSection>
##  <Oper Name="SumX" Arg='arg1, arg2, ... func'/>
##
##  <Description>
##  <Ref Oper="SumX"/> returns the sum of the elements in the list obtained
##  by <Ref Oper="ListX"/> when this is called with the same arguments.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "SumX" );


#############################################################################
##
#O  ProductX( <arg1>, <arg2>, ... <func> )
##
##  <#GAPDoc Label="ProductX">
##  <ManSection>
##  <Oper Name="ProductX" Arg='arg1, arg2, ... func'/>
##
##  <Description>
##  <Ref Oper="ProductX"/> returns the product of the elements in the list
##  obtained by <Ref Oper="ListX"/> when this is called with the same
##  arguments.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "ProductX" );


#############################################################################
##
#O  Perform( <list>, <func>)
##
##  <#GAPDoc Label="Perform">
##  <ManSection>
##  <Oper Name="Perform" Arg='list, func'/>
##
##  <Description>
##  <Ref Oper="Perform"/> applies the function <A>func</A> to every element
##  of the list <A>list</A>, discarding any return values.
##  It does not return a value.
##  <P/>
##  <Example><![CDATA[
##  gap> l := [1, 2, 3];; Perform(l, 
##  > function(x) if IsPrimeInt(x) then Print(x,"\n"); fi; end);
##  2
##  3
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "Perform" );


#############################################################################
##
#O  IsSubset( <C1>, <C2> )  . . . . . . . . .  test for subset of collections
##
##  <#GAPDoc Label="IsSubset">
##  <ManSection>
##  <Oper Name="IsSubset" Arg='C1, C2'/>
##
##  <Description>
##  <Index Subkey="for collections">subset test</Index>
##  <Ref Oper="IsSubset"/> returns <K>true</K> if <A>C2</A>,
##  which must be a collection, is a <E>subset</E> of <A>C1</A>,
##  which also must be a collection, and <K>false</K> otherwise.
##  <P/>
##  <A>C2</A> is considered a subset of <A>C1</A> if and only if each element
##  of <A>C2</A> is also an element of <A>C1</A>.
##  That is <Ref Oper="IsSubset"/> behaves as if implemented as
##  <C>IsSubsetSet( AsSSortedList( <A>C1</A> ), AsSSortedList( <A>C2</A> ) )</C>,
##  except that it will also sometimes, but not always,
##  work for infinite collections,
##  and that it will usually work much faster than the above definition.
##  Either argument may also be a proper set
##  (see&nbsp;<Ref Sect="Sorted Lists and Sets"/>).
##  <P/>
##  <Example><![CDATA[
##  gap> IsSubset( Rationals, Integers );
##  true
##  gap> IsSubset( Integers, [ 1, 2, 3 ] );
##  true
##  gap> IsSubset( Group( (1,2,3,4) ), [ (1,2,3) ] );
##  false
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "IsSubset", [ IsListOrCollection, IsListOrCollection ] );


#############################################################################
##
#F  Intersection( <C1>, <C2> ... )  . . . . . . . intersection of collections
#F  Intersection( <list> )  . . . . . . . . . . . intersection of collections
#O  Intersection2( <C1>, <C2> ) . . . . . . . . . intersection of collections
##
##  <#GAPDoc Label="Intersection">
##  <ManSection>
##  <Heading>Intersection</Heading>
##  <Func Name="Intersection" Arg='C1, C2 ...'
##   Label="for various collections"/>
##  <Func Name="Intersection" Arg='list' Label="for a list"/>
##  <Oper Name="Intersection2" Arg='C1, C2'/>
##
##  <Description>
##  <Index Subkey="of collections">intersection</Index>
##  In the first form
##  <Ref Func="Intersection" Label="for various collections"/> returns the
##  intersection of the collections <A>C1</A>, <A>C2</A>, etc.
##  In the second form <A>list</A> must be a <E>nonempty</E> list of
##  collections and <Ref Func="Intersection" Label="for a list"/> returns
##  the intersection of those collections.
##  Each argument or element of <A>list</A> respectively may also be a
##  homogeneous list that is not a proper set,
##  in which case <Ref Func="Intersection" Label="for a list"/> silently
##  applies <Ref Func="Set"/> to it first.
##  <P/>
##  The result of <Ref Func="Intersection" Label="for a list"/> is the set
##  of elements that lie in every of the collections <A>C1</A>, <A>C2</A>,
##  etc.
##  If the result is a list then it is mutable and new, i.e., not identical
##  to any of <A>C1</A>, <A>C2</A>, etc.
##  <P/>
##  Methods can be installed for the operation <Ref Func="Intersection2"/>
##  that takes only two arguments.
##  <Ref Func="Intersection" Label="for a list"/> calls
##  <Ref Func="Intersection2"/>.
##  <P/>
##  Methods for <Ref Func="Intersection2"/> should try to maintain as much
##  structure as possible, for example the intersection of two permutation
##  groups is again a permutation group.
##  <P/>
##  <Example><![CDATA[
##  gap> # this is one of the rare cases where the intersection of two
##  gap> # infinite domains works ('CF' is a shorthand for 'CyclotomicField'):
##  gap> Intersection( CyclotomicField(9), CyclotomicField(12) );
##  CF(3)
##  gap> D12 := Group( (2,6)(3,5), (1,2)(3,6)(4,5) );;
##  gap> Intersection( D12, Group( (1,2), (1,2,3,4,5) ) );
##  Group([ (1,5)(2,4) ])
##  gap> Intersection( D12, [ (1,3)(4,6), (1,2)(3,4) ] )
##  >    ;  # note that the second argument is not a proper set
##  [ (1,3)(4,6) ]
##  gap> # although the result is mathematically a group it is returned as a
##  gap> # proper set because the second argument is not regarded as a group:
##  gap> Intersection( D12, [ (), (1,2)(3,4), (1,3)(4,6), (1,4)(5,6) ] );
##  [ (), (1,3)(4,6) ]
##  gap> Intersection( Group( () ), [1,2,3] );
##  [  ]
##  gap> Intersection( [2,4,6,8,10], [3,6,9,12,15], [5,10,15,20,25] )
##  >    ;  # two or more lists or collections as arguments are legal
##  [  ]
##  gap> Intersection( [ [1,2,4], [2,3,4], [1,3,4] ] )
##  >    ;  # or one list of lists or collections
##  [ 4 ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "Intersection" );

DeclareOperation( "Intersection2",
    [ IsListOrCollection, IsListOrCollection ] );


#############################################################################
##
#F  Union( <C1>, <C2> ... ) . . . . . . . . . . . . . .  union of collections
#F  Union( <list> ) . . . . . . . . . . . . . . . . . .  union of collections
#O  Union2( <C1>, <C2> )  . . . . . . . . . . . . . . .  union of collections
##
##  <#GAPDoc Label="Union">
##  <ManSection>
##  <Heading>Union</Heading>
##  <Func Name="Union" Arg='C1, C2 ...' Label="for various collections"/>
##  <Func Name="Union" Arg='list' Label="for a list"/>
##  <Oper Name="Union2" Arg='C1, C2'/>
##
##  <Description>
##  <Index Subkey="of collections">union</Index>
##  In the first form <Ref Func="Union" Label="for various collections"/>
##  returns the union of the collections <A>C1</A>, <A>C2</A>, etc.
##  In the second form <A>list</A> must be a list of collections
##  and <Ref Func="Union" Label="for a list"/> returns the union of those
##  collections.
##  Each argument or element of <A>list</A> respectively may also be a
##  homogeneous list that is not a proper set,
##  in which case <Ref Func="Union" Label="for a list"/> silently applies
##  <Ref Func="Set"/> to it first.
##  <P/>
##  The result of <Ref Func="Union" Label="for a list"/> is the set of
##  elements that lie in any of the collections <A>C1</A>, <A>C2</A>, etc.
##  If the result is a list then it is mutable and new, i.e., not identical
##  to any of <A>C1</A>, <A>C2</A>, etc.
##  <P/>
##  Methods can be installed for the operation <Ref Oper="Union2"/>
##  that takes only two arguments.
##  <Ref Func="Union" Label="for a list"/> calls <Ref Func="Union2"/>.
##  <P/>
##  <Example><![CDATA[
##  gap> Union( [ (1,2,3), (1,2,3,4) ], Group( (1,2,3), (1,2) ) );
##  [ (), (2,3), (1,2), (1,2,3), (1,2,3,4), (1,3,2), (1,3) ]
##  gap> Union( [2,4,6,8,10], [3,6,9,12,15], [5,10,15,20,25] )
##  >    ;  # two or more lists or collections as arguments are legal
##  [ 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 20, 25 ]
##  gap> Union( [ [1,2,4], [2,3,4], [1,3,4] ] )
##  >    ;  # or one list of lists or collections
##  [ 1, 2, 3, 4 ]
##  gap> Union( [ ] );
##  [  ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareGlobalFunction( "Union" );

DeclareOperation( "Union2", [ IsListOrCollection, IsListOrCollection ] );


#############################################################################
##
#O  Difference( <C1>, <C2> )  . . . . . . . . . . . difference of collections
##
##  <#GAPDoc Label="Difference">
##  <ManSection>
##  <Oper Name="Difference" Arg='C1, C2'/>
##
##  <Description>
##  <Index Subkey="of collections">set difference</Index>
##  <Ref Func="Difference"/> returns the set difference of the collections
##  <A>C1</A> and <A>C2</A>.
##  Either argument may also be a homogeneous list that is not a proper set,
##  in which case <Ref Func="Difference"/> silently applies <Ref Func="Set"/>
##  to it first.
##  <P/>
##  The result of <Ref Func="Difference"/> is the set of elements that lie in
##  <A>C1</A> but not in <A>C2</A>.
##  Note that <A>C2</A> need not be a subset of <A>C1</A>.
##  The elements of <A>C2</A>, however, that are not elements of <A>C1</A>
##  play no role for the result.
##  If the result is a list then it is mutable and new, i.e., not identical
##  to <A>C1</A> or <A>C2</A>.
##  <P/>
##  <Example><![CDATA[
##  gap> Difference( [ (1,2,3), (1,2,3,4) ], Group( (1,2,3), (1,2) ) );
##  [ (1,2,3,4) ]
##  ]]></Example>
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "Difference", [ IsListOrCollection, IsListOrCollection ] );


#############################################################################
##
#P  CanEasilyCompareElements( <obj> )
#F  CanEasilyCompareElementsFamily( <fam> )
#P  CanEasilySortElements( <obj> )
#F  CanEasilySortElementsFamily( <fam> )
##
##  <#GAPDoc Label="CanEasilyCompareElements">
##  <ManSection>
##  <Prop Name="CanEasilyCompareElements" Arg='obj'/>
##  <Func Name="CanEasilyCompareElementsFamily" Arg='fam'/>
##  <Prop Name="CanEasilySortElements" Arg='obj'/>
##  <Func Name="CanEasilySortElementsFamily" Arg='fam'/>
##
##  <Description>
##  For some objects a <Q>normal form</Q> is hard to compute
##  and thus equality of elements of a domain might be expensive to test.
##  Therefore &GAP; provides a (slightly technical) property with which an
##  algorithm can test whether an efficient equality test is available
##  for elements of a certain kind.
##  <P/>
##  <Ref Func="CanEasilyCompareElements"/> indicates whether the elements in
##  the family <A>fam</A> of <A>obj</A> can be easily compared with
##  <Ref Func="\="/>.
##  <P/>
##  The default method for this property is to ask the family of <A>obj</A>,
##  the default method for the family is to return <K>false</K>.
##  <P/>
##  The ability to compare elements may depend on the successful computation
##  of certain information. (For example for finitely presented groups it
##  might depend on the knowledge of a faithful permutation representation.)
##  This information might change over time and thus it might not be a good
##  idea to store a value <K>false</K> too early in a family. Instead the
##  function <Ref Func="CanEasilyCompareElementsFamily"/> should be called
##  for the family of <A>obj</A> which returns <K>false</K> if the value of
##  <Ref Func="CanEasilyCompareElements"/> is not known for the family
##  without computing it. (This is in fact what the above mentioned family
##  dispatch does.)
##  <P/>
##  If a family knows ab initio that it can compare elements this property
##  should be set as implied filter <E>and</E> filter for the family
##  (the 3rd and 4th argument of <Ref Func="NewFamily"/>
##  respectively).
##  This guarantees that code which directly asks the family gets a right
##  answer.
##  <P/>
##  The property <Ref Func="CanEasilySortElements"/> and the function
##  <Ref Func="CanEasilySortElementsFamily"/> behave exactly in the same way,
##  except that they indicate that objects can be compared via
##  <Ref Func="\&lt;"/>.
##  This property implies <Ref Func="CanEasilyCompareElements"/>,
##  as the ordering must be total.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareProperty( "CanEasilyCompareElements", IsObject );
DeclareGlobalFunction( "CanEasilyCompareElementsFamily" );
DeclareProperty( "CanEasilySortElements", IsObject );
DeclareGlobalFunction( "CanEasilySortElementsFamily" );

InstallTrueMethod(CanEasilyCompareElements,CanEasilySortElements);


#############################################################################
##
#O  CanComputeIsSubset( <A>, <B> )
##
##  <#GAPDoc Label="CanComputeIsSubset">
##  <ManSection>
##  <Oper Name="CanComputeIsSubset" Arg='A, B'/>
##
##  <Description>
##  This filter indicates that &GAP; can test (via <Ref Func="IsSubset"/>)
##  whether <A>B</A> is a subset of <A>A</A>.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareOperation( "CanComputeIsSubset", [IsObject,IsObject] );


#############################################################################
##
#F  CanComputeSize( <dom> )
##
##  <#GAPDoc Label="CanComputeSize">
##  <ManSection>
##  <Func Name="CanComputeSize" Arg='dom'/>
##
##  <Description>
##  This filter indicates whether the size of the domain <A>dom</A>
##  (which might be <Ref Var="infinity"/>) can be computed.
##  </Description>
##  </ManSection>
##  <#/GAPDoc>
##
DeclareFilter( "CanComputeSize" );

InstallTrueMethod( CanComputeSize, HasSize );


DeclareOperation( "Randomizer", [IsCollection] );
DeclareOperation( "CheapRandomizer", [IsCollection] );

DeclareAttribute( "RandomizerAttr", IsCollection );
DeclareAttribute( "CheapRandomizerAttr", IsCollection );

# to allow for recusive calls
DeclareGlobalFunction("JoinRanges");

#############################################################################
##
#E