WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 45 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 122 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 9 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. The certificate found is represented by the following graph. 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(1311,1312,[0_1|4]), (1312,1313,[1_1|4]), (1313,1116,[2_1|4]), (1313,1125,[2_1|4]), (1313,1137,[2_1|4]), (1313,1152,[2_1|4]), (1313,1170,[2_1|4]), (1313,1191,[2_1|4]), (1314,1315,[2_1|4]), (1315,1316,[1_1|4]), (1316,1317,[1_1|4]), (1317,1318,[0_1|4]), (1318,1319,[1_1|4]), (1319,1320,[2_1|4]), (1320,1321,[0_1|4]), (1321,1322,[1_1|4]), (1322,1215,[2_1|4]), (1322,1224,[2_1|4]), (1322,1236,[2_1|4]), (1322,1251,[2_1|4]), (1322,1269,[2_1|4]), (1322,1290,[2_1|4]), (1323,1324,[2_1|4]), (1324,1325,[1_1|4]), (1325,1326,[1_1|4]), (1326,1327,[0_1|4]), (1327,1328,[1_1|4]), (1328,1329,[2_1|4]), (1329,1330,[0_1|4]), (1330,1331,[1_1|4]), (1331,1332,[2_1|4]), (1332,1333,[0_1|4]), (1333,1334,[1_1|4]), (1334,1215,[2_1|4]), (1334,1224,[2_1|4]), (1334,1236,[2_1|4]), (1334,1251,[2_1|4]), (1334,1269,[2_1|4]), (1334,1290,[2_1|4]), (1335,1336,[2_1|4]), (1336,1337,[1_1|4]), (1337,1338,[1_1|4]), (1338,1339,[0_1|4]), (1339,1340,[1_1|4]), (1340,1341,[2_1|4]), (1341,1342,[0_1|4]), (1342,1343,[1_1|4]), (1343,1344,[2_1|4]), (1344,1345,[0_1|4]), (1345,1346,[1_1|4]), (1346,1347,[2_1|4]), (1347,1348,[0_1|4]), (1348,1349,[1_1|4]), (1349,1215,[2_1|4]), (1349,1224,[2_1|4]), (1349,1236,[2_1|4]), (1349,1251,[2_1|4]), (1349,1269,[2_1|4]), (1349,1290,[2_1|4]), (1350,1351,[2_1|4]), (1351,1352,[1_1|4]), (1352,1353,[1_1|4]), (1353,1354,[0_1|4]), (1354,1355,[1_1|4]), (1355,1356,[2_1|4]), (1356,1357,[0_1|4]), (1357,1358,[1_1|4]), (1358,1359,[2_1|4]), (1359,1360,[0_1|4]), (1360,1361,[1_1|4]), (1361,1362,[2_1|4]), (1362,1363,[0_1|4]), (1363,1364,[1_1|4]), (1364,1365,[2_1|4]), (1365,1366,[0_1|4]), (1366,1367,[1_1|4]), (1367,1215,[2_1|4]), (1367,1224,[2_1|4]), (1367,1236,[2_1|4]), (1367,1251,[2_1|4]), (1367,1269,[2_1|4]), (1367,1290,[2_1|4]), (1368,1369,[2_1|4]), (1369,1370,[1_1|4]), (1370,1371,[1_1|4]), (1371,1372,[0_1|4]), (1372,1373,[1_1|4]), (1373,1374,[2_1|4]), (1374,1375,[0_1|4]), (1375,1376,[1_1|4]), (1376,1377,[2_1|4]), (1377,1378,[0_1|4]), (1378,1379,[1_1|4]), (1379,1380,[2_1|4]), (1380,1381,[0_1|4]), (1381,1382,[1_1|4]), (1382,1383,[2_1|4]), (1383,1384,[0_1|4]), (1384,1385,[1_1|4]), (1385,1386,[2_1|4]), (1386,1387,[0_1|4]), (1387,1388,[1_1|4]), (1388,1215,[2_1|4]), (1388,1224,[2_1|4]), (1388,1236,[2_1|4]), (1388,1251,[2_1|4]), (1388,1269,[2_1|4]), (1388,1290,[2_1|4]), (1389,1390,[2_1|4]), (1390,1391,[1_1|4]), (1391,1392,[1_1|4]), (1392,1393,[0_1|4]), (1393,1394,[1_1|4]), (1394,1395,[2_1|4]), (1395,1396,[0_1|4]), (1396,1397,[1_1|4]), (1397,1398,[2_1|4]), (1398,1399,[0_1|4]), (1399,1400,[1_1|4]), (1400,1401,[2_1|4]), (1401,1402,[0_1|4]), (1402,1403,[1_1|4]), (1403,1404,[2_1|4]), (1404,1405,[0_1|4]), (1405,1406,[1_1|4]), (1406,1407,[2_1|4]), (1407,1408,[0_1|4]), (1408,1409,[1_1|4]), (1409,1410,[2_1|4]), (1410,1411,[0_1|4]), (1411,1412,[1_1|4]), (1412,1215,[2_1|4]), (1412,1224,[2_1|4]), (1412,1236,[2_1|4]), (1412,1251,[2_1|4]), (1412,1269,[2_1|4]), (1412,1290,[2_1|4]), (1413,1414,[2_1|4]), (1414,1415,[1_1|4]), (1415,1416,[1_1|4]), (1416,1417,[0_1|4]), (1417,1418,[1_1|4]), (1418,1419,[2_1|4]), (1419,1420,[0_1|4]), (1420,1421,[1_1|4]), (1421,1314,[2_1|4]), (1421,1323,[2_1|4]), (1421,1335,[2_1|4]), (1421,1350,[2_1|4]), (1421,1368,[2_1|4]), (1421,1389,[2_1|4]), (1422,1423,[2_1|4]), (1423,1424,[1_1|4]), (1424,1425,[1_1|4]), (1425,1426,[0_1|4]), (1426,1427,[1_1|4]), (1427,1428,[2_1|4]), (1428,1429,[0_1|4]), (1429,1430,[1_1|4]), (1430,1431,[2_1|4]), (1431,1432,[0_1|4]), (1432,1433,[1_1|4]), (1433,1314,[2_1|4]), (1433,1323,[2_1|4]), (1433,1335,[2_1|4]), (1433,1350,[2_1|4]), (1433,1368,[2_1|4]), (1433,1389,[2_1|4]), (1434,1435,[2_1|4]), (1435,1436,[1_1|4]), (1436,1437,[1_1|4]), (1437,1438,[0_1|4]), (1438,1439,[1_1|4]), (1439,1440,[2_1|4]), (1440,1441,[0_1|4]), (1441,1442,[1_1|4]), (1442,1443,[2_1|4]), (1443,1444,[0_1|4]), (1444,1445,[1_1|4]), (1445,1446,[2_1|4]), (1446,1447,[0_1|4]), (1447,1448,[1_1|4]), (1448,1314,[2_1|4]), (1448,1323,[2_1|4]), (1448,1335,[2_1|4]), (1448,1350,[2_1|4]), (1448,1368,[2_1|4]), (1448,1389,[2_1|4]), (1449,1450,[2_1|4]), (1450,1451,[1_1|4]), (1451,1452,[1_1|4]), (1452,1453,[0_1|4]), (1453,1454,[1_1|4]), (1454,1455,[2_1|4]), (1455,1456,[0_1|4]), (1456,1457,[1_1|4]), (1457,1458,[2_1|4]), (1458,1459,[0_1|4]), (1459,1460,[1_1|4]), (1460,1461,[2_1|4]), (1461,1462,[0_1|4]), (1462,1463,[1_1|4]), (1463,1464,[2_1|4]), (1464,1465,[0_1|4]), (1465,1466,[1_1|4]), (1466,1314,[2_1|4]), (1466,1323,[2_1|4]), (1466,1335,[2_1|4]), (1466,1350,[2_1|4]), (1466,1368,[2_1|4]), (1466,1389,[2_1|4]), (1467,1468,[2_1|4]), (1468,1469,[1_1|4]), (1469,1470,[1_1|4]), (1470,1471,[0_1|4]), (1471,1472,[1_1|4]), (1472,1473,[2_1|4]), (1473,1474,[0_1|4]), (1474,1475,[1_1|4]), (1475,1476,[2_1|4]), (1476,1477,[0_1|4]), (1477,1478,[1_1|4]), (1478,1479,[2_1|4]), (1479,1480,[0_1|4]), (1480,1481,[1_1|4]), (1481,1482,[2_1|4]), (1482,1483,[0_1|4]), (1483,1484,[1_1|4]), (1484,1485,[2_1|4]), (1485,1486,[0_1|4]), (1486,1487,[1_1|4]), (1487,1314,[2_1|4]), (1487,1323,[2_1|4]), (1487,1335,[2_1|4]), (1487,1350,[2_1|4]), (1487,1368,[2_1|4]), (1487,1389,[2_1|4]), (1488,1489,[2_1|4]), (1489,1490,[1_1|4]), (1490,1491,[1_1|4]), (1491,1492,[0_1|4]), (1492,1493,[1_1|4]), (1493,1494,[2_1|4]), (1494,1495,[0_1|4]), (1495,1496,[1_1|4]), (1496,1497,[2_1|4]), (1497,1498,[0_1|4]), (1498,1499,[1_1|4]), (1499,1500,[2_1|4]), (1500,1501,[0_1|4]), (1501,1502,[1_1|4]), (1502,1503,[2_1|4]), (1503,1504,[0_1|4]), (1504,1505,[1_1|4]), (1505,1506,[2_1|4]), (1506,1507,[0_1|4]), (1507,1508,[1_1|4]), (1508,1509,[2_1|4]), (1509,1510,[0_1|4]), (1510,1511,[1_1|4]), (1511,1314,[2_1|4]), (1511,1323,[2_1|4]), (1511,1335,[2_1|4]), (1511,1350,[2_1|4]), (1511,1368,[2_1|4]), (1511,1389,[2_1|4]), (1512,1513,[2_1|4]), (1513,1514,[1_1|4]), (1514,1515,[1_1|4]), (1515,1516,[0_1|4]), (1516,1517,[1_1|4]), (1517,1518,[2_1|4]), (1518,1519,[0_1|4]), (1519,1520,[1_1|4]), (1520,1413,[2_1|4]), (1520,1422,[2_1|4]), (1520,1434,[2_1|4]), (1520,1449,[2_1|4]), (1520,1467,[2_1|4]), (1520,1488,[2_1|4]), (1521,1522,[2_1|4]), (1522,1523,[1_1|4]), (1523,1524,[1_1|4]), (1524,1525,[0_1|4]), (1525,1526,[1_1|4]), (1526,1527,[2_1|4]), (1527,1528,[0_1|4]), (1528,1529,[1_1|4]), (1529,1530,[2_1|4]), (1530,1531,[0_1|4]), (1531,1532,[1_1|4]), (1532,1413,[2_1|4]), (1532,1422,[2_1|4]), (1532,1434,[2_1|4]), (1532,1449,[2_1|4]), (1532,1467,[2_1|4]), (1532,1488,[2_1|4]), (1533,1534,[2_1|4]), (1534,1535,[1_1|4]), (1535,1536,[1_1|4]), (1536,1537,[0_1|4]), (1537,1538,[1_1|4]), (1538,1539,[2_1|4]), (1539,1540,[0_1|4]), (1540,1541,[1_1|4]), (1541,1542,[2_1|4]), (1542,1543,[0_1|4]), (1543,1544,[1_1|4]), (1544,1545,[2_1|4]), (1545,1546,[0_1|4]), (1546,1547,[1_1|4]), (1547,1413,[2_1|4]), (1547,1422,[2_1|4]), (1547,1434,[2_1|4]), (1547,1449,[2_1|4]), (1547,1467,[2_1|4]), (1547,1488,[2_1|4]), (1548,1549,[2_1|4]), (1549,1550,[1_1|4]), (1550,1551,[1_1|4]), (1551,1552,[0_1|4]), (1552,1553,[1_1|4]), (1553,1554,[2_1|4]), (1554,1555,[0_1|4]), (1555,1556,[1_1|4]), (1556,1557,[2_1|4]), (1557,1558,[0_1|4]), (1558,1559,[1_1|4]), (1559,1560,[2_1|4]), (1560,1561,[0_1|4]), (1561,1562,[1_1|4]), (1562,1563,[2_1|4]), (1563,1564,[0_1|4]), (1564,1565,[1_1|4]), (1565,1413,[2_1|4]), (1565,1422,[2_1|4]), (1565,1434,[2_1|4]), (1565,1449,[2_1|4]), (1565,1467,[2_1|4]), (1565,1488,[2_1|4]), (1566,1567,[2_1|4]), (1567,1568,[1_1|4]), (1568,1569,[1_1|4]), (1569,1570,[0_1|4]), (1570,1571,[1_1|4]), (1571,1572,[2_1|4]), (1572,1573,[0_1|4]), (1573,1574,[1_1|4]), (1574,1575,[2_1|4]), (1575,1576,[0_1|4]), (1576,1577,[1_1|4]), (1577,1578,[2_1|4]), (1578,1579,[0_1|4]), (1579,1580,[1_1|4]), (1580,1581,[2_1|4]), (1581,1582,[0_1|4]), (1582,1583,[1_1|4]), (1583,1584,[2_1|4]), (1584,1585,[0_1|4]), (1585,1586,[1_1|4]), (1586,1413,[2_1|4]), (1586,1422,[2_1|4]), (1586,1434,[2_1|4]), (1586,1449,[2_1|4]), (1586,1467,[2_1|4]), (1586,1488,[2_1|4]), (1587,1588,[2_1|4]), (1588,1589,[1_1|4]), (1589,1590,[1_1|4]), (1590,1591,[0_1|4]), (1591,1592,[1_1|4]), (1592,1593,[2_1|4]), (1593,1594,[0_1|4]), (1594,1595,[1_1|4]), (1595,1596,[2_1|4]), (1596,1597,[0_1|4]), (1597,1598,[1_1|4]), (1598,1599,[2_1|4]), (1599,1600,[0_1|4]), (1600,1601,[1_1|4]), (1601,1602,[2_1|4]), (1602,1603,[0_1|4]), (1603,1604,[1_1|4]), (1604,1605,[2_1|4]), (1605,1606,[0_1|4]), (1606,1607,[1_1|4]), (1607,1608,[2_1|4]), (1608,1609,[0_1|4]), (1609,1610,[1_1|4]), (1610,1413,[2_1|4]), (1610,1422,[2_1|4]), (1610,1434,[2_1|4]), (1610,1449,[2_1|4]), (1610,1467,[2_1|4]), (1610,1488,[2_1|4])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 0(1(2(1(x1)))) ->^+ 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]. The pumping substitution is [x1 / 1(x1)]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1)))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) Rewrite Strategy: INNERMOST