1283:
that takes the diagonal of all enumerated 1-place computable partial functions and adds 1 to them is an example of a creative set. Post gave a version of Gödel's
Incompleteness Theorem using his creative sets, where originally Gödel had in some sense constructed a sentence that could be freely
1284:
translated as saying "I am unprovable in this axiomatic theory." However, Gödel's proof did not work from the concept of true sentences, and rather used the concept of a consistent theory, which led to the
463:
685:
256:
345:
399:
172:
205:
1664:
1542:
1291:"The conclusion is unescapable that even for such a fixed, well defined body of mathematical propositions, mathematical thinking is, and must remain, essentially creative."
882:
790:
755:
720:
584:
546:
513:
88:
1726:
1427:
1281:
1066:
1027:
979:
823:
643:
849:
610:
1389:
1369:
1341:
303:
1602:
1582:
1562:
1492:
1472:
1450:
1312:
1219:
950:
276:
144:
1178:
1869:
910:
it contains other numbers, and moreover there is an effective procedure to produce an example of such a number from the index
1850:
408:
2043:
2013:
1911:
1736:
formalization of an effectively calculable partial function, which can neither be proved or disproved. Church used
1759:
1755:
347:
is productive. Not every productive set has a recursively enumerable complement, however, as illustrated below.
648:
210:
17:
2070:
1285:
322:
353:
1818:
1150:
149:
2036:
Recursively enumerable sets and degrees: A study of computable functions and computably generated sets
1173:
is a recursively enumerable set of true sentences, there is at least one true sentence that is not in
177:
1607:
1497:
854:
762:
725:
690:
554:
518:
483:
1740:, Turing an idealized computer, and later Emil Post in his approach, all of which are equivalent.
66:
1669:
1394:
1224:
1042:
1003:
955:
795:
615:
1154:
1110:
108:
932:
The following theorems, due to Myhill (1955), show that in a sense all creative sets are like
1751:
918:, because this would imply that its complement, a productive set, is recursively enumerable.
828:
589:
2053:
2023:
1994:
1960:
1929:
1921:
1892:
1830:
1374:
1354:
91:
1317:
8:
2001:
1899:
95:
1971:(1944), "Recursively enumerable sets of positive integers and their decision problems",
285:
1729:
1587:
1567:
1547:
1477:
1457:
1435:
1297:
1204:
1118:
1101:
1069:
1030:
935:
922:
480:
To see this, we apply the definition of a productive function and show separately that
261:
129:
44:
2039:
2009:
1907:
1883:
1846:
1985:
1980:
1948:
1878:
1838:
1181:, because no recursively enumerable set is productive. The complement of the set
1146:
28:
1288:. After Post completed his version of incompleteness he then added the following:
2049:
2031:
2019:
1990:
1956:
1926:
1917:
1888:
1826:
1737:
402:
1865:"Some remarks on witness functions for nonpolynomial and noncomplete sets in NP"
1162:
1968:
1860:
1743:
1348:
926:
123:
40:
2064:
1952:
1864:
915:
1169:
sentences in the system will be a productive set, which means that whenever
1936:
1825:, Series in Information Processing and Computers, New York: McGraw-Hill,
1763:
1732:
that says the mathematical notion of computable partial functions is the
1344:
1474:), and is universal in the sense that any calculable partial function
47:. They are a standard topic in mathematical logic textbooks such as
1189:
is an example of a productive set whose complement is not creative.
1201:
defined the concept he called a
Creative set. Reiterating, the set
851:, but we have demonstrated the contrary in the previous point. So
1941:
Zeitschrift fĂĽr
Mathematische Logik und Grundlagen der Mathematik
2038:, Perspectives in Mathematical Logic, Berlin: Springer-Verlag,
1351:
showed the existence of a universal computer that computes the
1221:
referenced above and defined as the domain of the function
1087:
be a set of natural numbers. The following are equivalent:
991:
be a set of natural numbers. The following are equivalent:
1758:, and used it to provide potential counterexamples to the
2006:
Theory of recursive functions and effective computability
1843:
1666:, and the diagonal function arises quite naturally as
1672:
1610:
1590:
1570:
1550:
1500:
1480:
1460:
1438:
1397:
1377:
1357:
1320:
1300:
1227:
1207:
1140:
1045:
1006:
958:
938:
921:
Any productive set has a productive function that is
857:
831:
798:
765:
728:
693:
651:
618:
592:
557:
521:
486:
411:
356:
325:
288:
264:
213:
180:
152:
132:
69:
458:{\displaystyle {\bar {K}}=\{i\mid i\not \in W_{i}\}}
1720:
1658:
1596:
1576:
1556:
1536:
1486:
1466:
1444:
1421:
1383:
1363:
1335:
1306:
1275:
1213:
1145:The set of all provable sentences in an effective
1060:
1021:
973:
944:
876:
843:
817:
784:
749:
714:
679:
637:
604:
578:
540:
507:
457:
393:
339:
297:
270:
250:
199:
166:
138:
82:
2062:
1431:the result of applying the instructions coded by
1177:. This can be used to give a rigorous proof of
897:can be recursively enumerable, because whenever
63:For the remainder of this article, assume that
1973:Bulletin of the American Mathematical Society
1185:will not be recursively enumerable, and thus
319:is recursively enumerable and its complement
452:
427:
388:
363:
1859:
1747:
1728:. Ultimately, these ideas are connected to
2008:(2nd ed.), Cambridge, MA: MIT Press,
1834:. Reprinted in 1982 by Dover Publications.
1153:. If the system is suitably complex, like
1984:
1882:
680:{\displaystyle W_{i}\subseteq {\bar {K}}}
327:
251:{\displaystyle f(i)\in A\setminus W_{i}.}
160:
58:
1906:, Mineola, NY: Dover Publications Inc.,
1837:
1801:
1925:. Reprint of the 1967 original, Wiley,
1117:, that is, there is a total computable
465:is productive with productive function
340:{\displaystyle \mathbb {N} \setminus A}
14:
2063:
2000:
1935:
1898:
1797:
1795:
1786:
394:{\displaystyle K=\{i\mid i\in W_{i}\}}
52:
2030:
1817:
1782:
914:. Similarly, no creative set can be
901:contains every number in an r.e. set
48:
1967:
1343:has its own historical development.
1314:defined using the diagonal function
1198:
1179:Gödel's first incompleteness theorem
722:, this leads to a contradiction. So
43:that have important applications in
1792:
1750:) formulated an analogous concept,
107:the corresponding numbering of the
24:
1611:
1516:
1398:
1378:
1358:
1141:Applications in mathematical logic
25:
2082:
1124:on the natural numbers such that
952:and all productive sets are like
331:
232:
167:{\displaystyle i\in \mathbb {N} }
200:{\displaystyle W_{i}\subseteq A}
126:recursive (computable) function
1986:10.1090/S0002-9904-1944-08111-1
1823:Computability and unsolvability
1756:computational complexity theory
1659:{\displaystyle \Phi (e,x)=](x)}
1537:{\displaystyle f(x)=\Phi (e,x)}
877:{\displaystyle i\not \in W_{i}}
785:{\displaystyle i\not \in W_{i}}
750:{\displaystyle i\in {\bar {K}}}
715:{\displaystyle i\in {\bar {K}}}
579:{\displaystyle i\in {\bar {K}}}
541:{\displaystyle i\not \in W_{i}}
508:{\displaystyle i\in {\bar {K}}}
350:The archetypal creative set is
1776:
1746: and Paul Young (
1709:
1703:
1700:
1697:
1691:
1688:
1682:
1676:
1653:
1647:
1644:
1641:
1635:
1632:
1626:
1614:
1531:
1519:
1510:
1504:
1413:
1401:
1330:
1324:
1264:
1258:
1255:
1252:
1246:
1243:
1237:
1231:
1052:
1013:
965:
741:
706:
671:
570:
499:
418:
223:
217:
13:
1:
1811:
1286:second incompleteness theorem
888:
825:, then it would be true that
311:of natural numbers is called
118:of natural numbers is called
1884:10.1016/0304-3975(85)90140-9
1870:Theoretical Computer Science
1604:. Using the above notation
83:{\displaystyle \varphi _{i}}
7:
1760:Berman–Hartmanis conjecture
1721:{\displaystyle d(x)=](x)+1}
1584:codes the instructions for
1422:{\displaystyle \Phi (w,x)=}
1276:{\displaystyle d(x)=](x)+1}
401:, the set representing the
10:
2087:
1347:in a 1936 article on the
1192:
1151:recursively enumerable set
1061:{\displaystyle {\bar {K}}}
1022:{\displaystyle {\bar {K}}}
974:{\displaystyle {\bar {K}}}
818:{\displaystyle i\in W_{i}}
638:{\displaystyle i\in W_{i}}
1939:(1955), "Creative sets",
477:(the identity function).
1953:10.1002/malq.19550010205
1769:
1371:function. The function
1294:The usual creative set
1863:; Young, Paul (1985),
1722:
1660:
1598:
1578:
1558:
1538:
1488:
1468:
1446:
1423:
1385:
1365:
1337:
1308:
1277:
1215:
1155:first-order arithmetic
1111:recursively isomorphic
1062:
1023:
975:
946:
878:
845:
844:{\displaystyle i\in K}
819:
786:
751:
716:
681:
639:
606:
605:{\displaystyle i\in K}
580:
542:
509:
459:
395:
341:
299:
272:
252:
201:
168:
140:
109:recursively enumerable
84:
59:Definition and example
1752:polynomial creativity
1723:
1661:
1599:
1579:
1559:
1539:
1489:
1469:
1447:
1424:
1391:is defined such that
1386:
1384:{\displaystyle \Phi }
1366:
1364:{\displaystyle \Phi }
1338:
1309:
1278:
1216:
1197:The seminal paper of
1063:
1024:
976:
947:
879:
846:
820:
787:
752:
717:
682:
640:
607:
581:
543:
510:
460:
396:
342:
300:
273:
253:
202:
169:
141:
85:
39:are types of sets of
2071:Computability theory
1900:Kleene, Stephen Cole
1839:Enderton, Herbert B.
1670:
1608:
1588:
1568:
1548:
1498:
1478:
1458:
1436:
1395:
1375:
1355:
1336:{\displaystyle d(x)}
1318:
1298:
1225:
1205:
1043:
1004:
956:
936:
855:
829:
796:
763:
726:
691:
649:
616:
590:
555:
519:
484:
409:
354:
323:
286:
262:
211:
178:
150:
130:
96:computable functions
92:admissible numbering
67:
29:computability theory
2002:Rogers, Hartley Jr.
280:productive function
1904:Mathematical logic
1845:, Academic Press,
1804:, pp. 79, 80, 120.
1762:on isomorphism of
1744:Deborah Joseph
1718:
1656:
1594:
1574:
1554:
1534:
1484:
1464:
1442:
1419:
1381:
1361:
1333:
1304:
1273:
1211:
1058:
1019:
971:
942:
893:No productive set
874:
841:
815:
782:
747:
712:
677:
635:
602:
576:
538:
505:
455:
391:
337:
298:{\displaystyle A.}
295:
268:
248:
197:
164:
136:
122:if there exists a
80:
45:mathematical logic
1852:978-0-12-384958-8
1597:{\displaystyle f}
1577:{\displaystyle e}
1557:{\displaystyle x}
1487:{\displaystyle f}
1467:{\displaystyle x}
1445:{\displaystyle w}
1307:{\displaystyle K}
1214:{\displaystyle K}
1055:
1016:
968:
945:{\displaystyle K}
744:
709:
674:
645:, now given that
573:
502:
421:
405:. Its complement
271:{\displaystyle f}
139:{\displaystyle f}
16:(Redirected from
2078:
2056:
2032:Soare, Robert I.
2026:
1997:
1988:
1963:
1924:
1895:
1886:
1877:(2–3): 225–237,
1855:
1833:
1805:
1799:
1790:
1780:
1727:
1725:
1724:
1719:
1665:
1663:
1662:
1657:
1603:
1601:
1600:
1595:
1583:
1581:
1580:
1575:
1563:
1561:
1560:
1555:
1543:
1541:
1540:
1535:
1493:
1491:
1490:
1485:
1473:
1471:
1470:
1465:
1451:
1449:
1448:
1443:
1428:
1426:
1425:
1420:
1390:
1388:
1387:
1382:
1370:
1368:
1367:
1362:
1342:
1340:
1339:
1334:
1313:
1311:
1310:
1305:
1282:
1280:
1279:
1274:
1220:
1218:
1217:
1212:
1147:axiomatic system
1067:
1065:
1064:
1059:
1057:
1056:
1048:
1028:
1026:
1025:
1020:
1018:
1017:
1009:
980:
978:
977:
972:
970:
969:
961:
951:
949:
948:
943:
883:
881:
880:
875:
873:
872:
850:
848:
847:
842:
824:
822:
821:
816:
814:
813:
791:
789:
788:
783:
781:
780:
756:
754:
753:
748:
746:
745:
737:
721:
719:
718:
713:
711:
710:
702:
686:
684:
683:
678:
676:
675:
667:
661:
660:
644:
642:
641:
636:
634:
633:
611:
609:
608:
603:
585:
583:
582:
577:
575:
574:
566:
547:
545:
544:
539:
537:
536:
514:
512:
511:
506:
504:
503:
495:
464:
462:
461:
456:
451:
450:
423:
422:
414:
400:
398:
397:
392:
387:
386:
346:
344:
343:
338:
330:
304:
302:
301:
296:
277:
275:
274:
269:
257:
255:
254:
249:
244:
243:
206:
204:
203:
198:
190:
189:
173:
171:
170:
165:
163:
146:so that for all
145:
143:
142:
137:
89:
87:
86:
81:
79:
78:
21:
2086:
2085:
2081:
2080:
2079:
2077:
2076:
2075:
2061:
2060:
2046:
2016:
1914:
1861:Joseph, Deborah
1853:
1814:
1809:
1808:
1802:Enderton (2010)
1800:
1793:
1781:
1777:
1772:
1738:lambda calculus
1730:Church's thesis
1671:
1668:
1667:
1609:
1606:
1605:
1589:
1586:
1585:
1569:
1566:
1565:
1549:
1546:
1545:
1499:
1496:
1495:
1479:
1476:
1475:
1459:
1456:
1455:
1437:
1434:
1433:
1396:
1393:
1392:
1376:
1373:
1372:
1356:
1353:
1352:
1319:
1316:
1315:
1299:
1296:
1295:
1226:
1223:
1222:
1206:
1203:
1202:
1195:
1157:, then the set
1143:
1047:
1046:
1044:
1041:
1040:
1008:
1007:
1005:
1002:
1001:
960:
959:
957:
954:
953:
937:
934:
933:
909:
891:
868:
864:
856:
853:
852:
830:
827:
826:
809:
805:
797:
794:
793:
776:
772:
764:
761:
760:
736:
735:
727:
724:
723:
701:
700:
692:
689:
688:
666:
665:
656:
652:
650:
647:
646:
629:
625:
617:
614:
613:
591:
588:
587:
565:
564:
556:
553:
552:
532:
528:
520:
517:
516:
494:
493:
485:
482:
481:
446:
442:
413:
412:
410:
407:
406:
403:halting problem
382:
378:
355:
352:
351:
326:
324:
321:
320:
287:
284:
283:
263:
260:
259:
239:
235:
212:
209:
208:
185:
181:
179:
176:
175:
159:
151:
148:
147:
131:
128:
127:
106:
74:
70:
68:
65:
64:
61:
41:natural numbers
33:productive sets
23:
22:
15:
12:
11:
5:
2084:
2074:
2073:
2059:
2058:
2044:
2028:
2014:
1998:
1979:(5): 284–316,
1965:
1933:
1912:
1896:
1857:
1851:
1835:
1813:
1810:
1807:
1806:
1791:
1774:
1773:
1771:
1768:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1593:
1573:
1553:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1483:
1463:
1441:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1380:
1360:
1349:Turing machine
1332:
1329:
1326:
1323:
1303:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1210:
1194:
1191:
1142:
1139:
1138:
1137:
1104:
1095:
1078:
1077:
1054:
1051:
1038:
1015:
1012:
999:
998:is productive.
967:
964:
941:
905:
890:
887:
886:
885:
871:
867:
863:
860:
840:
837:
834:
812:
808:
804:
801:
779:
775:
771:
768:
758:
743:
740:
734:
731:
708:
705:
699:
696:
673:
670:
664:
659:
655:
632:
628:
624:
621:
601:
598:
595:
572:
569:
563:
560:
535:
531:
527:
524:
501:
498:
492:
489:
454:
449:
445:
441:
438:
435:
432:
429:
426:
420:
417:
390:
385:
381:
377:
374:
371:
368:
365:
362:
359:
336:
333:
329:
294:
291:
278:is called the
267:
247:
242:
238:
234:
231:
228:
225:
222:
219:
216:
196:
193:
188:
184:
162:
158:
155:
135:
102:
77:
73:
60:
57:
18:Productive set
9:
6:
4:
3:
2:
2083:
2072:
2069:
2068:
2066:
2055:
2051:
2047:
2045:3-540-15299-7
2041:
2037:
2033:
2029:
2025:
2021:
2017:
2015:0-262-68052-1
2011:
2007:
2003:
1999:
1996:
1992:
1987:
1982:
1978:
1974:
1970:
1969:Post, Emil L.
1966:
1962:
1958:
1954:
1950:
1947:(2): 97–108,
1946:
1942:
1938:
1934:
1931:
1928:
1923:
1919:
1915:
1913:0-486-42533-9
1909:
1905:
1901:
1897:
1894:
1890:
1885:
1880:
1876:
1872:
1871:
1866:
1862:
1858:
1854:
1848:
1844:
1840:
1836:
1832:
1828:
1824:
1820:
1819:Davis, Martin
1816:
1815:
1803:
1798:
1796:
1788:
1787:Rogers (1987)
1784:
1779:
1775:
1767:
1765:
1761:
1757:
1753:
1749:
1745:
1741:
1739:
1735:
1731:
1715:
1712:
1706:
1694:
1685:
1679:
1673:
1650:
1638:
1629:
1623:
1620:
1617:
1591:
1571:
1551:
1528:
1525:
1522:
1513:
1507:
1501:
1481:
1461:
1454:
1439:
1432:
1416:
1410:
1407:
1404:
1350:
1346:
1327:
1321:
1301:
1292:
1289:
1287:
1270:
1267:
1261:
1249:
1240:
1234:
1228:
1208:
1200:
1190:
1188:
1184:
1180:
1176:
1172:
1168:
1164:
1163:Gödel numbers
1160:
1156:
1152:
1148:
1135:
1131:
1127:
1123:
1120:
1116:
1112:
1108:
1105:
1103:
1099:
1096:
1093:
1090:
1089:
1088:
1086:
1082:
1075:
1071:
1049:
1039:
1036:
1032:
1010:
1000:
997:
994:
993:
992:
990:
986:
982:
962:
939:
930:
928:
924:
919:
917:
913:
908:
904:
900:
896:
869:
865:
861:
858:
838:
835:
832:
810:
806:
802:
799:
792:: in fact if
777:
773:
769:
766:
759:
738:
732:
729:
703:
697:
694:
668:
662:
657:
653:
630:
626:
622:
619:
599:
596:
593:
567:
561:
558:
551:
550:
549:
533:
529:
525:
522:
496:
490:
487:
478:
476:
472:
468:
447:
443:
439:
436:
433:
430:
424:
415:
404:
383:
379:
375:
372:
369:
366:
360:
357:
348:
334:
318:
314:
310:
305:
292:
289:
281:
265:
258:The function
245:
240:
236:
229:
226:
220:
214:
194:
191:
186:
182:
156:
153:
133:
125:
121:
117:
112:
110:
105:
101:
97:
93:
75:
71:
56:
54:
53:Rogers (1987)
50:
46:
42:
38:
37:creative sets
34:
30:
19:
2035:
2005:
1976:
1972:
1944:
1940:
1937:Myhill, John
1903:
1874:
1868:
1842:
1822:
1783:Soare (1987)
1778:
1742:
1733:
1494:is given by
1453:to the input
1452:
1430:
1293:
1290:
1196:
1186:
1182:
1174:
1170:
1166:
1158:
1149:is always a
1144:
1133:
1129:
1125:
1121:
1114:
1106:
1097:
1094:is creative.
1091:
1084:
1080:
1079:
1073:
1034:
995:
988:
984:
983:
931:
920:
911:
906:
902:
898:
894:
892:
479:
474:
470:
466:
349:
316:
312:
308:
306:
279:
119:
115:
113:
103:
99:
62:
49:Soare (1987)
36:
32:
26:
1764:NP-complete
1345:Alan Turing
1199:Post (1944)
1070:m-reducible
1031:1-reducible
1812:References
1102:1-complete
889:Properties
586:: suppose
120:productive
1612:Φ
1517:Φ
1399:Φ
1379:Φ
1359:Φ
1119:bijection
1053:¯
1014:¯
966:¯
923:injective
916:decidable
836:∈
803:∈
742:¯
733:∈
707:¯
698:∈
672:¯
663:⊆
623:∈
597:∈
571:¯
562:∈
500:¯
491:∈
434:∣
419:¯
376:∈
370:∣
332:∖
233:∖
227:∈
192:⊆
157:∈
72:φ
2065:Category
2034:(1987),
2004:(1987),
1902:(2002),
1841:(2010),
1821:(1958),
1544:for all
1081:Theorem.
985:Theorem.
862:∉
770:∉
687:we have
526:∉
440:∉
313:creative
2054:0882921
2024:0886890
1995:0010514
1961:0071379
1930:0216930
1922:1950307
1893:0821203
1831:0124208
1734:correct
1193:History
612:, then
94:of the
2052:
2042:
2022:
2012:
1993:
1959:
1920:
1910:
1891:
1849:
1829:
1766:sets.
1564:where
307:A set
114:A set
111:sets.
90:is an
1770:Notes
1754:, in
927:total
207:then
174:, if
124:total
2040:ISBN
2010:ISBN
1908:ISBN
1847:ISBN
1748:1985
1167:true
1132:) =
1083:Let
987:Let
925:and
515:and
473:) =
282:for
98:and
51:and
35:and
1981:doi
1949:doi
1879:doi
1165:of
1161:of
1113:to
1109:is
1100:is
1072:to
1068:is
1033:to
1029:is
315:if
27:In
2067::
2050:MR
2048:,
2020:MR
2018:,
1991:MR
1989:,
1977:50
1975:,
1957:MR
1955:,
1943:,
1927:MR
1918:MR
1916:,
1889:MR
1887:,
1875:39
1873:,
1867:,
1827:MR
1794:^
1785:;
981:.
929:.
548::
55:.
31:,
2057:.
2027:.
1983::
1964:.
1951::
1945:1
1932:.
1881::
1856:.
1789:.
1716:1
1713:+
1710:)
1707:x
1704:(
1701:]
1698:]
1695:x
1692:[
1689:[
1686:=
1683:)
1680:x
1677:(
1674:d
1654:)
1651:x
1648:(
1645:]
1642:]
1639:e
1636:[
1633:[
1630:=
1627:)
1624:x
1621:,
1618:e
1615:(
1592:f
1572:e
1552:x
1532:)
1529:x
1526:,
1523:e
1520:(
1514:=
1511:)
1508:x
1505:(
1502:f
1482:f
1462:x
1440:w
1429:(
1417:=
1414:)
1411:x
1408:,
1405:w
1402:(
1331:)
1328:x
1325:(
1322:d
1302:K
1271:1
1268:+
1265:)
1262:x
1259:(
1256:]
1253:]
1250:x
1247:[
1244:[
1241:=
1238:)
1235:x
1232:(
1229:d
1209:K
1187:T
1183:T
1175:W
1171:W
1159:T
1136:.
1134:K
1130:C
1128:(
1126:f
1122:f
1115:K
1107:C
1098:C
1092:C
1085:C
1076:.
1074:P
1050:K
1037:.
1035:P
1011:K
996:P
989:P
963:K
940:K
912:i
907:i
903:W
899:A
895:A
884:.
870:i
866:W
859:i
839:K
833:i
811:i
807:W
800:i
778:i
774:W
767:i
757:.
739:K
730:i
704:K
695:i
669:K
658:i
654:W
631:i
627:W
620:i
600:K
594:i
568:K
559:i
534:i
530:W
523:i
497:K
488:i
475:i
471:i
469:(
467:f
453:}
448:i
444:W
437:i
431:i
428:{
425:=
416:K
389:}
384:i
380:W
373:i
367:i
364:{
361:=
358:K
335:A
328:N
317:A
309:A
293:.
290:A
266:f
246:.
241:i
237:W
230:A
224:)
221:i
218:(
215:f
195:A
187:i
183:W
161:N
154:i
134:f
116:A
104:i
100:W
76:i
20:)
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