27:
339:
450:
2162:
699:
1100:
1861:
945:
1316:
776:
1422:
1499:
1009:
863:
251:
1944:
2294:
1371:
2026:
130:
477:
365:
1673:
1623:
1125:
614:
166:
2243:
1722:
1534:
1226:
1964:
1893:
1442:
796:
2205:
1148:
229:
2182:
2066:
2046:
1984:
1802:
1782:
1762:
1742:
1693:
1643:
1597:
1574:
1554:
1246:
1191:
1168:
582:
555:
535:
198:
2071:
619:
1023:
1807:
868:
1255:
714:
1376:
483:
19:
This article is about Kleene's fixed-point theorem in lattice theory. For the fixed-point theorem in computability theory, see
1447:
334:{\displaystyle \bot \sqsubseteq f(\bot )\sqsubseteq f(f(\bot ))\sqsubseteq \cdots \sqsubseteq f^{n}(\bot )\sqsubseteq \cdots }
2395:
950:
804:
2422:
20:
1898:
2248:
1321:
1989:
177:
133:
66:
2326:
2417:
445:{\displaystyle {\textrm {lfp}}(f)=\sup \left(\left\{f^{n}(\bot )\mid n\in \mathbb {N} \right\}\right)}
103:
458:
243:
1011:
holds, and because f is monotone (property of Scott-continuous functions), the result holds as well.
2305:
90:
1652:
1606:
1108:
2379:
2373:
2391:
587:
503:
502:
who proves it for additive functions. Moreover, Kleene Fixed-Point
Theorem can be extended to
495:
491:
345:
201:
173:
139:
2210:
1698:
2383:
1504:
1249:
1196:
169:
1949:
1866:
1427:
781:
2187:
1130:
211:
2167:
2051:
2031:
1969:
1787:
1767:
1747:
1727:
1678:
1628:
1582:
1559:
1539:
1231:
1176:
1153:
567:
540:
520:
183:
82:
2028:. We now do the induction step: From the induction hypothesis and the monotonicity of
2411:
499:
352:
2157:{\displaystyle f^{i}(\bot )\sqsubseteq k~\implies ~f^{i+1}(\bot )\sqsubseteq f(k).}
78:
26:
2350:
74:
58:
2387:
694:{\displaystyle f^{n}(\bot )\sqsubseteq f^{n+1}(\bot ),n\in \mathbb {N} _{0}}
1646:
205:
1095:{\displaystyle \mathbb {M} =\{\bot ,f(\bot ),f(f(\bot )),\ldots \}.}
1017:
As a corollary of the Lemma we have the following directed ω-chain:
1856:{\displaystyle \forall i\in \mathbb {N} :f^{i}(\bot )\sqsubseteq k}
25:
486:
does not consider how fixed points can be computed by iterating
2372:
Stoltenberg-Hansen, V.; Lindstrom, I.; Griffor, E. R. (1994).
2327:"A lattice-theoretical fixpoint theorem and its applications"
940:{\displaystyle f(f^{n-1}(\bot ))\sqsubseteq f(f^{n}(\bot ))}
1311:{\displaystyle f(\sup(\mathbb {M} ))=\sup(f(\mathbb {M} ))}
771:{\displaystyle f^{0}(\bot )=\bot \sqsubseteq f^{1}(\bot ),}
1417:{\displaystyle \mathbb {M} =f(\mathbb {M} )\cup \{\bot \}}
517:
We first have to show that the ascending Kleene chain of
2351:"Constructive versions of Tarski's fixed point theorems"
1494:{\displaystyle \sup(f(\mathbb {M} ))=\sup(\mathbb {M} )}
2375:
Mathematical Theory of
Domains by V. Stoltenberg-Hansen
1603:
fixed point can be done by showing that any element in
1444:
has no influence in determining the supremum we have:
1004:{\displaystyle f^{n-1}(\bot )\sqsubseteq f^{n}(\bot )}
858:{\displaystyle f^{n}(\bot )\sqsubseteq f^{n+1}(\bot )}
2251:
2213:
2190:
2170:
2074:
2054:
2034:
1992:
1972:
1952:
1901:
1869:
1810:
1790:
1770:
1750:
1730:
1701:
1681:
1655:
1631:
1609:
1585:
1562:
1542:
1507:
1450:
1430:
1379:
1324:
1258:
1234:
1199:
1179:
1156:
1133:
1111:
1026:
953:
871:
807:
784:
717:
622:
590:
570:
543:
523:
461:
368:
254:
214:
186:
142:
106:
2288:
2237:
2199:
2176:
2156:
2060:
2040:
2020:
1986:. As the induction hypothesis, we may assume that
1978:
1958:
1938:
1887:
1855:
1796:
1776:
1756:
1736:
1716:
1687:
1667:
1637:
1617:
1591:
1568:
1548:
1528:
1493:
1436:
1416:
1365:
1310:
1240:
1220:
1185:
1162:
1142:
1119:
1094:
1003:
939:
857:
790:
770:
693:
608:
576:
549:
529:
471:
444:
359:. Expressed in a formula, the theorem states that
333:
223:
192:
160:
124:
1939:{\displaystyle f^{0}(\bot )=\bot \sqsubseteq k,}
1702:
1477:
1451:
1340:
1285:
1265:
388:
1105:From the definition of a dcpo it follows that
8:
2289:{\displaystyle f^{i+1}(\bot )\sqsubseteq k.}
1411:
1405:
1086:
1035:
2048:(again, implied by the Scott-continuity of
801:Assume n > 0. Then we have to show that
2107:
2103:
1366:{\displaystyle f(m)=\sup(f(\mathbb {M} ))}
16:Theorem in order theory and lattice theory
2349:Patrick Cousot and Radhia Cousot (1979).
2256:
2250:
2212:
2189:
2169:
2115:
2079:
2073:
2053:
2033:
2021:{\displaystyle f^{i}(\bot )\sqsubseteq k}
1997:
1991:
1971:
1951:
1906:
1900:
1868:
1832:
1821:
1820:
1809:
1789:
1769:
1749:
1729:
1700:
1680:
1654:
1630:
1611:
1610:
1608:
1584:
1561:
1541:
1506:
1484:
1483:
1464:
1463:
1449:
1429:
1395:
1394:
1381:
1380:
1378:
1353:
1352:
1323:
1298:
1297:
1272:
1271:
1257:
1233:
1198:
1178:
1155:
1132:
1113:
1112:
1110:
1028:
1027:
1025:
986:
958:
952:
919:
882:
870:
834:
812:
806:
783:
750:
722:
716:
685:
681:
680:
649:
627:
621:
589:
569:
542:
522:
463:
462:
460:
429:
428:
404:
370:
369:
367:
310:
253:
213:
185:
141:
105:
947:. By inductive assumption, we know that
557:. To show that, we prove the following:
2378:. Cambridge University Press. pp.
2317:
65:)+1 using Kleene's theorem in the real
498:), this result is often attributed to
490:from some seed (also, it pertains to
1744:). This is done by induction: Assume
1724:is smaller than that same element of
136:(dcpo) with a least element, and let
89:, named after American mathematician
30:Computation of the least fixpoint of
7:
584:is a dcpo with a least element, and
1625:is smaller than any fixed-point of
2271:
2130:
2088:
2068:), we may conclude the following:
2006:
1953:
1924:
1915:
1841:
1811:
1431:
1408:
1071:
1050:
1038:
995:
973:
928:
897:
849:
821:
785:
759:
740:
731:
664:
636:
413:
319:
288:
267:
255:
14:
1784:. We now prove by induction over
1150:What remains now is to show that
208:of the ascending Kleene chain of
125:{\displaystyle (L,\sqsubseteq )}
1675:are smaller than an element of
479:denotes the least fixed point.
472:{\displaystyle {\textrm {lfp}}}
134:directed-complete partial order
2355:Pacific Journal of Mathematics
2331:Pacific Journal of Mathematics
2274:
2268:
2223:
2217:
2148:
2142:
2133:
2127:
2104:
2091:
2085:
2009:
2003:
1918:
1912:
1882:
1870:
1844:
1838:
1711:
1705:
1517:
1511:
1488:
1480:
1471:
1468:
1460:
1454:
1399:
1391:
1360:
1357:
1349:
1343:
1334:
1328:
1305:
1302:
1294:
1288:
1279:
1276:
1268:
1262:
1209:
1203:
1077:
1074:
1068:
1062:
1053:
1047:
998:
992:
976:
970:
934:
931:
925:
912:
903:
900:
894:
875:
852:
846:
824:
818:
762:
756:
734:
728:
667:
661:
639:
633:
600:
506:using transfinite iterations.
416:
410:
382:
376:
322:
316:
294:
291:
285:
279:
270:
264:
152:
119:
107:
1:
2164:Now, by the assumption that
1863:. The base of the induction
1668:{\displaystyle D\subseteq L}
1618:{\displaystyle \mathbb {M} }
1193:is a fixed point, i.e. that
1120:{\displaystyle \mathbb {M} }
484:Tarski's fixed point theorem
1649:, if all elements of a set
98:Kleene Fixed-Point Theorem.
2439:
1170:is the least fixed-point.
616:is Scott-continuous, then
87:Kleene fixed-point theorem
21:Kleene's recursion theorem
18:
2388:10.1017/cbo9781139166386
1966:is the least element of
1645:(because by property of
1127:has a supremum, call it
865:. By rearranging we get
609:{\displaystyle f:L\to L}
161:{\displaystyle f:L\to L}
93:, states the following:
2238:{\displaystyle f(k)=k,}
1764:is some fixed-point of
1717:{\displaystyle \sup(D)}
2325:Alfred Tarski (1955).
2290:
2239:
2201:
2178:
2158:
2062:
2042:
2022:
1980:
1960:
1940:
1889:
1857:
1798:
1778:
1758:
1738:
1718:
1689:
1669:
1639:
1619:
1593:
1570:
1550:
1530:
1529:{\displaystyle f(m)=m}
1495:
1438:
1418:
1367:
1312:
1242:
1222:
1221:{\displaystyle f(m)=m}
1187:
1164:
1144:
1121:
1096:
1005:
941:
859:
792:
772:
695:
610:
578:
551:
531:
473:
446:
335:
236:ascending Kleene chain
225:
194:
162:
126:
70:
2291:
2245:and from that we get
2240:
2202:
2179:
2159:
2063:
2043:
2023:
1981:
1961:
1959:{\displaystyle \bot }
1941:
1890:
1888:{\displaystyle (i=0)}
1858:
1799:
1779:
1759:
1739:
1719:
1690:
1670:
1640:
1620:
1594:
1571:
1551:
1531:
1496:
1439:
1437:{\displaystyle \bot }
1419:
1368:
1313:
1243:
1223:
1188:
1165:
1145:
1122:
1097:
1006:
942:
860:
798:is the least element.
793:
791:{\displaystyle \bot }
773:
696:
611:
579:
552:
532:
474:
447:
336:
226:
195:
163:
127:
29:
2423:Fixed-point theorems
2306:fixed-point theorems
2249:
2211:
2188:
2184:is a fixed-point of
2168:
2072:
2052:
2032:
1990:
1970:
1950:
1899:
1867:
1808:
1788:
1768:
1748:
1728:
1699:
1679:
1653:
1629:
1607:
1583:
1560:
1540:
1505:
1448:
1428:
1377:
1322:
1256:
1232:
1197:
1177:
1173:First, we show that
1154:
1131:
1109:
1024:
951:
869:
805:
782:
715:
620:
588:
568:
541:
521:
459:
366:
252:
212:
184:
140:
104:
69:with the usual order
711:Assume n = 0. Then
91:Stephen Cole Kleene
2286:
2235:
2200:{\displaystyle f,}
2197:
2174:
2154:
2058:
2038:
2018:
1976:
1956:
1936:
1885:
1853:
1794:
1774:
1754:
1734:
1714:
1685:
1665:
1635:
1615:
1589:
1566:
1546:
1526:
1501:. It follows that
1491:
1434:
1414:
1363:
1308:
1238:
1218:
1183:
1160:
1143:{\displaystyle m.}
1140:
1117:
1092:
1001:
937:
855:
788:
768:
708:We use induction:
691:
606:
574:
547:
527:
504:monotone functions
492:monotone functions
469:
442:
331:
224:{\displaystyle f.}
221:
190:
158:
122:
71:
2177:{\displaystyle k}
2110:
2102:
2061:{\displaystyle f}
2041:{\displaystyle f}
1979:{\displaystyle L}
1895:obviously holds:
1797:{\displaystyle i}
1777:{\displaystyle f}
1757:{\displaystyle k}
1737:{\displaystyle L}
1688:{\displaystyle L}
1638:{\displaystyle f}
1592:{\displaystyle m}
1569:{\displaystyle f}
1556:a fixed-point of
1549:{\displaystyle m}
1241:{\displaystyle f}
1186:{\displaystyle m}
1163:{\displaystyle m}
577:{\displaystyle L}
550:{\displaystyle L}
530:{\displaystyle f}
496:complete lattices
466:
373:
202:least fixed point
193:{\displaystyle f}
2430:
2402:
2401:
2369:
2363:
2362:
2346:
2340:
2338:
2322:
2295:
2293:
2292:
2287:
2267:
2266:
2244:
2242:
2241:
2236:
2206:
2204:
2203:
2198:
2183:
2181:
2180:
2175:
2163:
2161:
2160:
2155:
2126:
2125:
2108:
2100:
2084:
2083:
2067:
2065:
2064:
2059:
2047:
2045:
2044:
2039:
2027:
2025:
2024:
2019:
2002:
2001:
1985:
1983:
1982:
1977:
1965:
1963:
1962:
1957:
1945:
1943:
1942:
1937:
1911:
1910:
1894:
1892:
1891:
1886:
1862:
1860:
1859:
1854:
1837:
1836:
1824:
1803:
1801:
1800:
1795:
1783:
1781:
1780:
1775:
1763:
1761:
1760:
1755:
1743:
1741:
1740:
1735:
1723:
1721:
1720:
1715:
1694:
1692:
1691:
1686:
1674:
1672:
1671:
1666:
1644:
1642:
1641:
1636:
1624:
1622:
1621:
1616:
1614:
1598:
1596:
1595:
1590:
1575:
1573:
1572:
1567:
1555:
1553:
1552:
1547:
1535:
1533:
1532:
1527:
1500:
1498:
1497:
1492:
1487:
1467:
1443:
1441:
1440:
1435:
1423:
1421:
1420:
1415:
1398:
1384:
1372:
1370:
1369:
1364:
1356:
1317:
1315:
1314:
1309:
1301:
1275:
1250:Scott-continuous
1247:
1245:
1244:
1239:
1227:
1225:
1224:
1219:
1192:
1190:
1189:
1184:
1169:
1167:
1166:
1161:
1149:
1147:
1146:
1141:
1126:
1124:
1123:
1118:
1116:
1101:
1099:
1098:
1093:
1031:
1010:
1008:
1007:
1002:
991:
990:
969:
968:
946:
944:
943:
938:
924:
923:
893:
892:
864:
862:
861:
856:
845:
844:
817:
816:
797:
795:
794:
789:
777:
775:
774:
769:
755:
754:
727:
726:
700:
698:
697:
692:
690:
689:
684:
660:
659:
632:
631:
615:
613:
612:
607:
583:
581:
580:
575:
556:
554:
553:
548:
536:
534:
533:
528:
478:
476:
475:
470:
468:
467:
464:
451:
449:
448:
443:
441:
437:
433:
432:
409:
408:
375:
374:
371:
340:
338:
337:
332:
315:
314:
230:
228:
227:
222:
199:
197:
196:
191:
170:Scott-continuous
167:
165:
164:
159:
131:
129:
128:
123:
53:
51:
50:
47:
44:
2438:
2437:
2433:
2432:
2431:
2429:
2428:
2427:
2408:
2407:
2406:
2405:
2398:
2371:
2370:
2366:
2348:
2347:
2343:
2324:
2323:
2319:
2314:
2301:
2252:
2247:
2246:
2209:
2208:
2186:
2185:
2166:
2165:
2111:
2075:
2070:
2069:
2050:
2049:
2030:
2029:
1993:
1988:
1987:
1968:
1967:
1948:
1947:
1902:
1897:
1896:
1865:
1864:
1828:
1806:
1805:
1786:
1785:
1766:
1765:
1746:
1745:
1726:
1725:
1697:
1696:
1677:
1676:
1651:
1650:
1627:
1626:
1605:
1604:
1599:is in fact the
1581:
1580:
1579:The proof that
1558:
1557:
1538:
1537:
1503:
1502:
1446:
1445:
1426:
1425:
1375:
1374:
1320:
1319:
1254:
1253:
1230:
1229:
1195:
1194:
1175:
1174:
1152:
1151:
1129:
1128:
1107:
1106:
1022:
1021:
982:
954:
949:
948:
915:
878:
867:
866:
830:
808:
803:
802:
780:
779:
746:
718:
713:
712:
679:
645:
623:
618:
617:
586:
585:
566:
565:
539:
538:
519:
518:
512:
457:
456:
400:
399:
395:
391:
364:
363:
306:
250:
249:
210:
209:
204:, which is the
182:
181:
172:(and therefore
138:
137:
102:
101:
48:
45:
42:
41:
39:
24:
17:
12:
11:
5:
2436:
2434:
2426:
2425:
2420:
2410:
2409:
2404:
2403:
2396:
2364:
2341:
2316:
2315:
2313:
2310:
2309:
2308:
2300:
2297:
2285:
2282:
2279:
2276:
2273:
2270:
2265:
2262:
2259:
2255:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2196:
2193:
2173:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2124:
2121:
2118:
2114:
2106:
2099:
2096:
2093:
2090:
2087:
2082:
2078:
2057:
2037:
2017:
2014:
2011:
2008:
2005:
2000:
1996:
1975:
1955:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1909:
1905:
1884:
1881:
1878:
1875:
1872:
1852:
1849:
1846:
1843:
1840:
1835:
1831:
1827:
1823:
1819:
1816:
1813:
1793:
1773:
1753:
1733:
1713:
1710:
1707:
1704:
1684:
1664:
1661:
1658:
1634:
1613:
1588:
1565:
1545:
1525:
1522:
1519:
1516:
1513:
1510:
1490:
1486:
1482:
1479:
1476:
1473:
1470:
1466:
1462:
1459:
1456:
1453:
1433:
1413:
1410:
1407:
1404:
1401:
1397:
1393:
1390:
1387:
1383:
1373:. Also, since
1362:
1359:
1355:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1307:
1304:
1300:
1296:
1293:
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1278:
1274:
1270:
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1261:
1237:
1217:
1214:
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1208:
1205:
1202:
1182:
1159:
1139:
1136:
1115:
1103:
1102:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1030:
1015:
1014:
1013:
1012:
1000:
997:
994:
989:
985:
981:
978:
975:
972:
967:
964:
961:
957:
936:
933:
930:
927:
922:
918:
914:
911:
908:
905:
902:
899:
896:
891:
888:
885:
881:
877:
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854:
851:
848:
843:
840:
837:
833:
829:
826:
823:
820:
815:
811:
799:
787:
767:
764:
761:
758:
753:
749:
745:
742:
739:
736:
733:
730:
725:
721:
702:
701:
688:
683:
678:
675:
672:
669:
666:
663:
658:
655:
652:
648:
644:
641:
638:
635:
630:
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605:
602:
599:
596:
593:
573:
546:
526:
511:
508:
453:
452:
440:
436:
431:
427:
424:
421:
418:
415:
412:
407:
403:
398:
394:
390:
387:
384:
381:
378:
342:
341:
330:
327:
324:
321:
318:
313:
309:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
232:
231:
220:
217:
189:
157:
154:
151:
148:
145:
121:
118:
115:
112:
109:
83:lattice theory
15:
13:
10:
9:
6:
4:
3:
2:
2435:
2424:
2421:
2419:
2416:
2415:
2413:
2399:
2393:
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2385:
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2377:
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2368:
2365:
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2356:
2352:
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2342:
2336:
2332:
2328:
2321:
2318:
2311:
2307:
2303:
2302:
2298:
2296:
2283:
2280:
2277:
2263:
2260:
2257:
2253:
2232:
2229:
2226:
2220:
2214:
2207:we know that
2194:
2191:
2171:
2151:
2145:
2139:
2136:
2122:
2119:
2116:
2112:
2097:
2094:
2080:
2076:
2055:
2035:
2015:
2012:
1998:
1994:
1973:
1933:
1930:
1927:
1921:
1907:
1903:
1879:
1876:
1873:
1850:
1847:
1833:
1829:
1825:
1817:
1814:
1791:
1771:
1751:
1731:
1708:
1682:
1662:
1659:
1656:
1648:
1632:
1602:
1586:
1577:
1563:
1543:
1523:
1520:
1514:
1508:
1474:
1457:
1402:
1388:
1385:
1346:
1337:
1331:
1325:
1291:
1282:
1259:
1251:
1235:
1215:
1212:
1206:
1200:
1180:
1171:
1157:
1137:
1134:
1089:
1083:
1080:
1065:
1059:
1056:
1044:
1041:
1032:
1020:
1019:
1018:
987:
983:
979:
965:
962:
959:
955:
920:
916:
909:
906:
889:
886:
883:
879:
872:
841:
838:
835:
831:
827:
813:
809:
800:
765:
751:
747:
743:
737:
723:
719:
710:
709:
707:
704:
703:
686:
676:
673:
670:
656:
653:
650:
646:
642:
628:
624:
603:
597:
594:
591:
571:
563:
560:
559:
558:
544:
524:
515:
509:
507:
505:
501:
500:Alfred Tarski
497:
493:
489:
485:
480:
438:
434:
425:
422:
419:
405:
401:
396:
392:
385:
379:
362:
361:
360:
358:
354:
353:least element
350:
347:
328:
325:
311:
307:
303:
300:
297:
282:
276:
273:
261:
258:
248:
247:
246:
245:
241:
237:
218:
215:
207:
203:
187:
179:
175:
171:
155:
149:
146:
143:
135:
116:
113:
110:
99:
96:
95:
94:
92:
88:
84:
80:
76:
68:
64:
60:
56:
37:
33:
28:
22:
2418:Order theory
2374:
2367:
2358:
2354:
2344:
2334:
2330:
2320:
1600:
1578:
1424:and because
1172:
1104:
1016:
705:
561:
516:
513:
487:
481:
454:
356:
348:
344:obtained by
343:
239:
235:
233:
97:
86:
75:mathematical
72:
62:
54:
35:
31:
2339:, page 305.
2412:Categories
2397:0521383447
2337:: 285–309.
2312:References
1695:then also
1318:, that is
1228:. Because
537:exists in
2278:⊑
2272:⊥
2137:⊑
2131:⊥
2105:⟹
2095:⊑
2089:⊥
2013:⊑
2007:⊥
1954:⊥
1928:⊑
1925:⊥
1916:⊥
1848:⊑
1842:⊥
1818:∈
1812:∀
1660:⊆
1536:, making
1432:⊥
1409:⊥
1403:∪
1084:…
1072:⊥
1051:⊥
1039:⊥
996:⊥
980:⊑
974:⊥
963:−
929:⊥
907:⊑
898:⊥
887:−
850:⊥
828:⊑
822:⊥
786:⊥
760:⊥
744:⊑
741:⊥
732:⊥
677:∈
665:⊥
643:⊑
637:⊥
601:→
482:Although
426:∈
420:∣
414:⊥
346:iterating
329:⋯
326:⊑
320:⊥
304:⊑
301:⋯
298:⊑
289:⊥
274:⊑
268:⊥
259:⊑
256:⊥
153:→
117:⊑
77:areas of
2361:: 43–57.
2299:See also
1647:supremum
514:Source:
206:supremum
178:function
174:monotone
100:Suppose
67:interval
351:on the
242:is the
180:. Then
73:In the
52:
40:
2394:
2304:Other
2109:
2101:
1946:since
778:since
706:Proof.
562:Lemma.
455:where
200:has a
85:, the
1804:that
1601:least
510:Proof
355:⊥ of
244:chain
168:be a
132:is a
79:order
2392:ISBN
2359:82:1
234:The
81:and
59:atan
38:) =
2384:doi
2335:5:2
1703:sup
1478:sup
1452:sup
1341:sup
1286:sup
1266:sup
1248:is
564:If
494:on
465:lfp
389:sup
372:lfp
238:of
2414::
2390:.
2382:.
2380:24
2357:.
2353:.
2333:.
2329:.
1576:.
1252:,
176:)
49:10
2400:.
2386::
2284:.
2281:k
2275:)
2269:(
2264:1
2261:+
2258:i
2254:f
2233:,
2230:k
2227:=
2224:)
2221:k
2218:(
2215:f
2195:,
2192:f
2172:k
2152:.
2149:)
2146:k
2143:(
2140:f
2134:)
2128:(
2123:1
2120:+
2117:i
2113:f
2098:k
2092:)
2086:(
2081:i
2077:f
2056:f
2036:f
2016:k
2010:)
2004:(
1999:i
1995:f
1974:L
1934:,
1931:k
1922:=
1919:)
1913:(
1908:0
1904:f
1883:)
1880:0
1877:=
1874:i
1871:(
1851:k
1845:)
1839:(
1834:i
1830:f
1826::
1822:N
1815:i
1792:i
1772:f
1752:k
1732:L
1712:)
1709:D
1706:(
1683:L
1663:L
1657:D
1633:f
1612:M
1587:m
1564:f
1544:m
1524:m
1521:=
1518:)
1515:m
1512:(
1509:f
1489:)
1485:M
1481:(
1475:=
1472:)
1469:)
1465:M
1461:(
1458:f
1455:(
1412:}
1406:{
1400:)
1396:M
1392:(
1389:f
1386:=
1382:M
1361:)
1358:)
1354:M
1350:(
1347:f
1344:(
1338:=
1335:)
1332:m
1329:(
1326:f
1306:)
1303:)
1299:M
1295:(
1292:f
1289:(
1283:=
1280:)
1277:)
1273:M
1269:(
1263:(
1260:f
1236:f
1216:m
1213:=
1210:)
1207:m
1204:(
1201:f
1181:m
1158:m
1138:.
1135:m
1114:M
1090:.
1087:}
1081:,
1078:)
1075:)
1069:(
1066:f
1063:(
1060:f
1057:,
1054:)
1048:(
1045:f
1042:,
1036:{
1033:=
1029:M
999:)
993:(
988:n
984:f
977:)
971:(
966:1
960:n
956:f
935:)
932:)
926:(
921:n
917:f
913:(
910:f
904:)
901:)
895:(
890:1
884:n
880:f
876:(
873:f
853:)
847:(
842:1
839:+
836:n
832:f
825:)
819:(
814:n
810:f
766:,
763:)
757:(
752:1
748:f
738:=
735:)
729:(
724:0
720:f
687:0
682:N
674:n
671:,
668:)
662:(
657:1
654:+
651:n
647:f
640:)
634:(
629:n
625:f
604:L
598:L
595::
592:f
572:L
545:L
525:f
488:f
439:)
435:}
430:N
423:n
417:)
411:(
406:n
402:f
397:{
393:(
386:=
383:)
380:f
377:(
357:L
349:f
323:)
317:(
312:n
308:f
295:)
292:)
286:(
283:f
280:(
277:f
271:)
265:(
262:f
240:f
219:.
216:f
188:f
156:L
150:L
147::
144:f
120:)
114:,
111:L
108:(
63:x
61:(
57:+
55:x
46:/
43:1
36:x
34:(
32:f
23:.
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