Termination proof of /tmp/tmpt2CSpN/Test1.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 300 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 290 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 650 ms)

↳8 AND

    ↳9 IDP

        ↳10 IDependencyGraphProof (⇔, 0 ms)

        ↳11 IDP

        ↳12 UsableRulesProof (⇔, 0 ms)

        ↳13 IDP

        ↳14 IDPNonInfProof (⇒, 270 ms)

        ↳15 IDP

        ↳16 IDependencyGraphProof (⇔, 0 ms)

        ↳17 IDP

        ↳18 IDPNonInfProof (⇒, 230 ms)

        ↳19 IDP

        ↳20 IDependencyGraphProof (⇔, 0 ms)

        ↳21 TRUE

    ↳22 IDP

        ↳23 IDependencyGraphProof (⇔, 0 ms)

        ↳24 IDP

        ↳25 UsableRulesProof (⇔, 0 ms)

        ↳26 IDP

        ↳27 IDPNonInfProof (⇒, 350 ms)

        ↳28 IDP

        ↳29 IDependencyGraphProof (⇔, 0 ms)

        ↳30 IDP

        ↳31 IDPNonInfProof (⇒, 140 ms)

        ↳32 IDP

        ↳33 IDependencyGraphProof (⇔, 0 ms)

        ↳34 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: Test1

public class Test1 {
    public static void main(String[] args) {
	rec(args.length, args.length % 5, args.length % 4);
    }

    private static void rec(int x, int y, int z) {
	if (x + y + 3 * z < 0)
	    return;
	else if (x > y)
	    rec(x - 1, y, z);
	else if (y > z)
	    rec (x, y - 2, z);
	else
	    rec (x, y, z - 1);
    }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
Test1.main([Ljava/lang/String;)V: Graph of 44 nodes with 0 SCCs.

Test1.rec(III)V: Graph of 68 nodes with 0 SCCs.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: Test1.rec(III)V
SCC calls the following helper methods: Test1.rec(III)V
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 45 rules for P and 31 rules for R.

P rules:
1351_0_rec_Load(EOS(STATIC_1351), i122, i437, i251, i122) → 1352_0_rec_IntArithmetic(EOS(STATIC_1352), i122, i437, i251, i122, i437)
1352_0_rec_IntArithmetic(EOS(STATIC_1352), i122, i437, i251, i122, i437) → 1354_0_rec_ConstantStackPush(EOS(STATIC_1354), i122, i437, i251, +(i122, i437))
1354_0_rec_ConstantStackPush(EOS(STATIC_1354), i122, i437, i251, i438) → 1355_0_rec_Load(EOS(STATIC_1355), i122, i437, i251, i438, 3)
1355_0_rec_Load(EOS(STATIC_1355), i122, i437, i251, i438, matching1) → 1357_0_rec_IntArithmetic(EOS(STATIC_1357), i122, i437, i251, i438, 3, i251) | =(matching1, 3)
1357_0_rec_IntArithmetic(EOS(STATIC_1357), i122, i437, i251, i438, matching1, i251) → 1358_0_rec_IntArithmetic(EOS(STATIC_1358), i122, i437, i251, i438, *(3, i251)) | =(matching1, 3)
1358_0_rec_IntArithmetic(EOS(STATIC_1358), i122, i437, i251, i438, i439) → 1360_0_rec_GE(EOS(STATIC_1360), i122, i437, i251, +(i438, i439))
1360_0_rec_GE(EOS(STATIC_1360), i122, i437, i251, i444) → 1363_0_rec_GE(EOS(STATIC_1363), i122, i437, i251, i444)
1363_0_rec_GE(EOS(STATIC_1363), i122, i437, i251, i444) → 1366_0_rec_Load(EOS(STATIC_1366), i122, i437, i251) | >=(i444, 0)
1366_0_rec_Load(EOS(STATIC_1366), i122, i437, i251) → 1368_0_rec_Load(EOS(STATIC_1368), i122, i437, i251, i122)
1368_0_rec_Load(EOS(STATIC_1368), i122, i437, i251, i122) → 1373_0_rec_LE(EOS(STATIC_1373), i122, i437, i251, i122, i437)
1373_0_rec_LE(EOS(STATIC_1373), i122, i437, i251, i122, i437) → 1377_0_rec_LE(EOS(STATIC_1377), i122, i437, i251, i122, i437)
1373_0_rec_LE(EOS(STATIC_1373), i122, i437, i251, i122, i437) → 1378_0_rec_LE(EOS(STATIC_1378), i122, i437, i251, i122, i437)
1377_0_rec_LE(EOS(STATIC_1377), i122, i437, i251, i122, i437) → 1379_0_rec_Load(EOS(STATIC_1379), i122, i437, i251) | <=(i122, i437)
1379_0_rec_Load(EOS(STATIC_1379), i122, i437, i251) → 1382_0_rec_Load(EOS(STATIC_1382), i122, i437, i251, i437)
1382_0_rec_Load(EOS(STATIC_1382), i122, i437, i251, i437) → 1385_0_rec_LE(EOS(STATIC_1385), i122, i437, i251, i437, i251)
1385_0_rec_LE(EOS(STATIC_1385), i122, i437, i251, i437, i251) → 1388_0_rec_LE(EOS(STATIC_1388), i122, i437, i251, i437, i251)
1385_0_rec_LE(EOS(STATIC_1385), i122, i437, i251, i437, i251) → 1389_0_rec_LE(EOS(STATIC_1389), i122, i437, i251, i437, i251)
1388_0_rec_LE(EOS(STATIC_1388), i122, i437, i251, i437, i251) → 1392_0_rec_Load(EOS(STATIC_1392), i122, i437, i251) | <=(i437, i251)
1392_0_rec_Load(EOS(STATIC_1392), i122, i437, i251) → 1397_0_rec_Load(EOS(STATIC_1397), i437, i251, i122)
1397_0_rec_Load(EOS(STATIC_1397), i437, i251, i122) → 1401_0_rec_Load(EOS(STATIC_1401), i251, i122, i437)
1401_0_rec_Load(EOS(STATIC_1401), i251, i122, i437) → 1406_0_rec_ConstantStackPush(EOS(STATIC_1406), i122, i437, i251)
1406_0_rec_ConstantStackPush(EOS(STATIC_1406), i122, i437, i251) → 1410_0_rec_IntArithmetic(EOS(STATIC_1410), i122, i437, i251, 1)
1410_0_rec_IntArithmetic(EOS(STATIC_1410), i122, i437, i251, matching1) → 1414_0_rec_InvokeMethod(EOS(STATIC_1414), i122, i437, -(i251, 1)) | =(matching1, 1)
1414_0_rec_InvokeMethod(EOS(STATIC_1414), i122, i437, i453) → 1420_1_rec_InvokeMethod(1420_0_rec_Load(EOS(STATIC_1420), i122, i437, i453), i122, i437, i453)
1420_0_rec_Load(EOS(STATIC_1420), i122, i437, i453) → 1424_0_rec_Load(EOS(STATIC_1424), i122, i437, i453)
1424_0_rec_Load(EOS(STATIC_1424), i122, i437, i453) → 1349_0_rec_Load(EOS(STATIC_1349), i122, i437, i453)
1349_0_rec_Load(EOS(STATIC_1349), i122, i437, i251) → 1351_0_rec_Load(EOS(STATIC_1351), i122, i437, i251, i122)
1389_0_rec_LE(EOS(STATIC_1389), i122, i437, i251, i437, i251) → 1394_0_rec_Load(EOS(STATIC_1394), i122, i437, i251) | >(i437, i251)
1394_0_rec_Load(EOS(STATIC_1394), i122, i437, i251) → 1398_0_rec_Load(EOS(STATIC_1398), i437, i251, i122)
1398_0_rec_Load(EOS(STATIC_1398), i437, i251, i122) → 1403_0_rec_ConstantStackPush(EOS(STATIC_1403), i251, i122, i437)
1403_0_rec_ConstantStackPush(EOS(STATIC_1403), i251, i122, i437) → 1407_0_rec_IntArithmetic(EOS(STATIC_1407), i251, i122, i437, 2)
1407_0_rec_IntArithmetic(EOS(STATIC_1407), i251, i122, i437, matching1) → 1412_0_rec_Load(EOS(STATIC_1412), i251, i122, -(i437, 2)) | =(matching1, 2)
1412_0_rec_Load(EOS(STATIC_1412), i251, i122, i452) → 1415_0_rec_InvokeMethod(EOS(STATIC_1415), i122, i452, i251)
1415_0_rec_InvokeMethod(EOS(STATIC_1415), i122, i452, i251) → 1421_1_rec_InvokeMethod(1421_0_rec_Load(EOS(STATIC_1421), i122, i452, i251), i122, i452, i251)
1421_0_rec_Load(EOS(STATIC_1421), i122, i452, i251) → 1425_0_rec_Load(EOS(STATIC_1425), i122, i452, i251)
1425_0_rec_Load(EOS(STATIC_1425), i122, i452, i251) → 1349_0_rec_Load(EOS(STATIC_1349), i122, i452, i251)
1378_0_rec_LE(EOS(STATIC_1378), i122, i437, i251, i122, i437) → 1381_0_rec_Load(EOS(STATIC_1381), i122, i437, i251) | >(i122, i437)
1381_0_rec_Load(EOS(STATIC_1381), i122, i437, i251) → 1383_0_rec_ConstantStackPush(EOS(STATIC_1383), i437, i251, i122)
1383_0_rec_ConstantStackPush(EOS(STATIC_1383), i437, i251, i122) → 1387_0_rec_IntArithmetic(EOS(STATIC_1387), i437, i251, i122, 1)
1387_0_rec_IntArithmetic(EOS(STATIC_1387), i437, i251, i122, matching1) → 1391_0_rec_Load(EOS(STATIC_1391), i437, i251, -(i122, 1)) | =(matching1, 1)
1391_0_rec_Load(EOS(STATIC_1391), i437, i251, i451) → 1395_0_rec_Load(EOS(STATIC_1395), i251, i451, i437)
1395_0_rec_Load(EOS(STATIC_1395), i251, i451, i437) → 1400_0_rec_InvokeMethod(EOS(STATIC_1400), i451, i437, i251)
1400_0_rec_InvokeMethod(EOS(STATIC_1400), i451, i437, i251) → 1404_1_rec_InvokeMethod(1404_0_rec_Load(EOS(STATIC_1404), i451, i437, i251), i451, i437, i251)
1404_0_rec_Load(EOS(STATIC_1404), i451, i437, i251) → 1409_0_rec_Load(EOS(STATIC_1409), i451, i437, i251)
1409_0_rec_Load(EOS(STATIC_1409), i451, i437, i251) → 1349_0_rec_Load(EOS(STATIC_1349), i451, i437, i251)
R rules:
1360_0_rec_GE(EOS(STATIC_1360), i122, i437, i251, i443) → 1362_0_rec_GE(EOS(STATIC_1362), i122, i437, i251, i443)
1362_0_rec_GE(EOS(STATIC_1362), i122, i437, i251, i443) → 1364_0_rec_Return(EOS(STATIC_1364), i122, i437, i251) | <(i443, 0)
1404_1_rec_InvokeMethod(1364_0_rec_Return(EOS(STATIC_1364), i458, i459, i460), i458, i459, i460) → 1422_0_rec_Return(EOS(STATIC_1422), i458, i459, i460, i458, i459, i460)
1404_1_rec_InvokeMethod(1448_0_rec_Return(EOS(STATIC_1448)), i521, i522, i523) → 1486_0_rec_Return(EOS(STATIC_1486), i521, i522, i523)
1404_1_rec_InvokeMethod(1482_0_rec_Return(EOS(STATIC_1482)), i565, i566, i567) → 1503_0_rec_Return(EOS(STATIC_1503), i565, i566, i567)
1420_1_rec_InvokeMethod(1364_0_rec_Return(EOS(STATIC_1364), i466, i467, i468), i466, i467, i468) → 1437_0_rec_Return(EOS(STATIC_1437), i466, i467, i468, i466, i467, i468)
1420_1_rec_InvokeMethod(1448_0_rec_Return(EOS(STATIC_1448)), i525, i526, i527) → 1488_0_rec_Return(EOS(STATIC_1488), i525, i526, i527)
1420_1_rec_InvokeMethod(1482_0_rec_Return(EOS(STATIC_1482)), i570, i571, i572) → 1505_0_rec_Return(EOS(STATIC_1505), i570, i571, i572)
1421_1_rec_InvokeMethod(1364_0_rec_Return(EOS(STATIC_1364), i476, i477, i478), i476, i477, i478) → 1438_0_rec_Return(EOS(STATIC_1438), i476, i477, i478, i476, i477, i478)
1421_1_rec_InvokeMethod(1448_0_rec_Return(EOS(STATIC_1448)), i531, i532, i533) → 1490_0_rec_Return(EOS(STATIC_1490), i531, i532, i533)
1421_1_rec_InvokeMethod(1482_0_rec_Return(EOS(STATIC_1482)), i576, i577, i578) → 1507_0_rec_Return(EOS(STATIC_1507), i576, i577, i578)
1422_0_rec_Return(EOS(STATIC_1422), i458, i459, i460, i458, i459, i460) → 1427_0_rec_JMP(EOS(STATIC_1427))
1427_0_rec_JMP(EOS(STATIC_1427)) → 1462_0_rec_JMP(EOS(STATIC_1462))
1437_0_rec_Return(EOS(STATIC_1437), i466, i467, i468, i466, i467, i468) → 1448_0_rec_Return(EOS(STATIC_1448))
1438_0_rec_Return(EOS(STATIC_1438), i476, i477, i478, i476, i477, i478) → 1450_0_rec_JMP(EOS(STATIC_1450))
1448_0_rec_Return(EOS(STATIC_1448)) → 1458_0_rec_Return(EOS(STATIC_1458))
1450_0_rec_JMP(EOS(STATIC_1450)) → 1467_0_rec_JMP(EOS(STATIC_1467))
1452_0_rec_Return(EOS(STATIC_1452), i484, i485, i486) → 1462_0_rec_JMP(EOS(STATIC_1462))
1453_0_rec_Return(EOS(STATIC_1453), i489, i491, i492) → 1465_0_rec_Return(EOS(STATIC_1465))
1455_0_rec_Return(EOS(STATIC_1455), i495, i497, i498) → 1467_0_rec_JMP(EOS(STATIC_1467))
1458_0_rec_Return(EOS(STATIC_1458)) → 1465_0_rec_Return(EOS(STATIC_1465))
1462_0_rec_JMP(EOS(STATIC_1462)) → 1480_0_rec_Return(EOS(STATIC_1480))
1465_0_rec_Return(EOS(STATIC_1465)) → 1480_0_rec_Return(EOS(STATIC_1480))
1467_0_rec_JMP(EOS(STATIC_1467)) → 1482_0_rec_Return(EOS(STATIC_1482))
1480_0_rec_Return(EOS(STATIC_1480)) → 1482_0_rec_Return(EOS(STATIC_1482))
1486_0_rec_Return(EOS(STATIC_1486), i521, i522, i523) → 1452_0_rec_Return(EOS(STATIC_1452), i521, i522, i523)
1488_0_rec_Return(EOS(STATIC_1488), i525, i526, i527) → 1453_0_rec_Return(EOS(STATIC_1453), i525, i526, i527)
1490_0_rec_Return(EOS(STATIC_1490), i531, i532, i533) → 1455_0_rec_Return(EOS(STATIC_1455), i531, i532, i533)
1503_0_rec_Return(EOS(STATIC_1503), i565, i566, i567) → 1452_0_rec_Return(EOS(STATIC_1452), i565, i566, i567)
1505_0_rec_Return(EOS(STATIC_1505), i570, i571, i572) → 1453_0_rec_Return(EOS(STATIC_1453), i570, i571, i572)
1507_0_rec_Return(EOS(STATIC_1507), i576, i577, i578) → 1455_0_rec_Return(EOS(STATIC_1455), i576, i577, i578)

Combined rules. Obtained 3 conditional rules for P and 9 conditional rules for R.

P rules:
1351_0_rec_Load(EOS(STATIC_1351), x0, x1, x2, x0) → 1420_1_rec_InvokeMethod(1351_0_rec_Load(EOS(STATIC_1351), x0, x1, -(x2, 1), x0), x0, x1, -(x2, 1)) | &&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1351_0_rec_Load(EOS(STATIC_1351), x0, x1, x2, x0) → 1421_1_rec_InvokeMethod(1351_0_rec_Load(EOS(STATIC_1351), x0, -(x1, 2), x2, x0), x0, -(x1, 2), x2) | &&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1351_0_rec_Load(EOS(STATIC_1351), x0, x1, x2, x0) → 1404_1_rec_InvokeMethod(1351_0_rec_Load(EOS(STATIC_1351), -(x0, 1), x1, x2, -(x0, 1)), -(x0, 1), x1, x2) | &&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2))))
R rules:
1421_1_rec_InvokeMethod(1364_0_rec_Return(EOS(STATIC_1364), x0, x1, x2), x0, x1, x2) → 1482_0_rec_Return(EOS(STATIC_1482))
1404_1_rec_InvokeMethod(1364_0_rec_Return(EOS(STATIC_1364), x0, x1, x2), x0, x1, x2) → 1482_0_rec_Return(EOS(STATIC_1482))
1420_1_rec_InvokeMethod(1364_0_rec_Return(EOS(STATIC_1364), x0, x1, x2), x0, x1, x2) → 1482_0_rec_Return(EOS(STATIC_1482))
1404_1_rec_InvokeMethod(1448_0_rec_Return(EOS(STATIC_1448)), x0, x1, x2) → 1482_0_rec_Return(EOS(STATIC_1482))
1404_1_rec_InvokeMethod(1482_0_rec_Return(EOS(STATIC_1482)), x0, x1, x2) → 1482_0_rec_Return(EOS(STATIC_1482))
1420_1_rec_InvokeMethod(1448_0_rec_Return(EOS(STATIC_1448)), x0, x1, x2) → 1482_0_rec_Return(EOS(STATIC_1482))
1420_1_rec_InvokeMethod(1482_0_rec_Return(EOS(STATIC_1482)), x0, x1, x2) → 1482_0_rec_Return(EOS(STATIC_1482))
1421_1_rec_InvokeMethod(1448_0_rec_Return(EOS(STATIC_1448)), x0, x1, x2) → 1482_0_rec_Return(EOS(STATIC_1482))
1421_1_rec_InvokeMethod(1482_0_rec_Return(EOS(STATIC_1482)), x0, x1, x2) → 1482_0_rec_Return(EOS(STATIC_1482))

Filtered ground terms:

1351_0_rec_Load(x1, x2, x3, x4, x5) → 1351_0_rec_Load(x2, x3, x4, x5)
Cond_1351_0_rec_Load2(x1, x2, x3, x4, x5, x6) → Cond_1351_0_rec_Load2(x1, x3, x4, x5, x6)
Cond_1351_0_rec_Load1(x1, x2, x3, x4, x5, x6) → Cond_1351_0_rec_Load1(x1, x3, x4, x5, x6)
Cond_1351_0_rec_Load(x1, x2, x3, x4, x5, x6) → Cond_1351_0_rec_Load(x1, x3, x4, x5, x6)
1482_0_rec_Return(x1) → 1482_0_rec_Return
1448_0_rec_Return(x1) → 1448_0_rec_Return
1364_0_rec_Return(x1, x2, x3, x4) → 1364_0_rec_Return(x2, x3, x4)

Filtered duplicate args:

1351_0_rec_Load(x1, x2, x3, x4) → 1351_0_rec_Load(x2, x3, x4)
Cond_1351_0_rec_Load(x1, x2, x3, x4, x5) → Cond_1351_0_rec_Load(x1, x3, x4, x5)
Cond_1351_0_rec_Load1(x1, x2, x3, x4, x5) → Cond_1351_0_rec_Load1(x1, x3, x4, x5)
Cond_1351_0_rec_Load2(x1, x2, x3, x4, x5) → Cond_1351_0_rec_Load2(x1, x3, x4, x5)

Filtered unneeded arguments:

1420_1_rec_InvokeMethod(x1, x2, x3, x4) → 1420_1_rec_InvokeMethod(x1)
1421_1_rec_InvokeMethod(x1, x2, x3, x4) → 1421_1_rec_InvokeMethod(x1)
1404_1_rec_InvokeMethod(x1, x2, x3, x4) → 1404_1_rec_InvokeMethod(x1)

Combined rules. Obtained 3 conditional rules for P and 9 conditional rules for R.

P rules:
1351_0_rec_Load(x1, x2, x0) → 1420_1_rec_InvokeMethod(1351_0_rec_Load(x1, -(x2, 1), x0)) | &&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1351_0_rec_Load(x1, x2, x0) → 1421_1_rec_InvokeMethod(1351_0_rec_Load(-(x1, 2), x2, x0)) | &&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1351_0_rec_Load(x1, x2, x0) → 1404_1_rec_InvokeMethod(1351_0_rec_Load(x1, x2, -(x0, 1))) | &&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2))))
R rules:
1421_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1404_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1420_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1404_1_rec_InvokeMethod(1448_0_rec_Return) → 1482_0_rec_Return
1404_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return
1420_1_rec_InvokeMethod(1448_0_rec_Return) → 1482_0_rec_Return
1420_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return
1421_1_rec_InvokeMethod(1448_0_rec_Return) → 1482_0_rec_Return
1421_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return

Performed bisimulation on rules. Used the following equivalence classes: {[1482_0_rec_Return, 1448_0_rec_Return]=1482_0_rec_Return}

Finished conversion. Obtained 6 rules for P and 6 rules for R. System has predefined symbols.

P rules:
1351_0_REC_LOAD(x1, x2, x0) → COND_1351_0_REC_LOAD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
COND_1351_0_REC_LOAD(TRUE, x1, x2, x0) → 1351_0_REC_LOAD(x1, -(x2, 1), x0)
1351_0_REC_LOAD(x1, x2, x0) → COND_1351_0_REC_LOAD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
COND_1351_0_REC_LOAD1(TRUE, x1, x2, x0) → 1351_0_REC_LOAD(-(x1, 2), x2, x0)
1351_0_REC_LOAD(x1, x2, x0) → COND_1351_0_REC_LOAD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
COND_1351_0_REC_LOAD2(TRUE, x1, x2, x0) → 1351_0_REC_LOAD(x1, x2, -(x0, 1))
R rules:
1421_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1404_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1420_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1404_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return
1420_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return
1421_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
1421_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1404_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1420_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1404_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return
1420_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return
1421_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return

The integer pair graph contains the following rules and edges:
(0): 1351_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1351_0_REC_LOAD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], x1[0], x2[0], x0[0])
(1): COND_1351_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1351_0_REC_LOAD(x1[1], x2[1] - 1, x0[1])
(2): 1351_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1351_0_REC_LOAD1(x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2], x1[2], x2[2], x0[2])
(3): COND_1351_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1351_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])
(4): 1351_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1351_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1351_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1351_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)

(0) -> (1), if (x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0] ∧x1[0] →^* x1[1]∧x2[0] →^* x2[1]∧x0[0] →^* x0[1])

(1) -> (0), if (x1[1] →^* x1[0]∧x2[1] - 1 →^* x2[0]∧x0[1] →^* x0[0])

(1) -> (2), if (x1[1] →^* x1[2]∧x2[1] - 1 →^* x2[2]∧x0[1] →^* x0[2])

(1) -> (4), if (x1[1] →^* x1[4]∧x2[1] - 1 →^* x2[4]∧x0[1] →^* x0[4])

(2) -> (3), if (x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2] ∧x1[2] →^* x1[3]∧x2[2] →^* x2[3]∧x0[2] →^* x0[3])

(3) -> (0), if (x1[3] - 2 →^* x1[0]∧x2[3] →^* x2[0]∧x0[3] →^* x0[0])

(3) -> (2), if (x1[3] - 2 →^* x1[2]∧x2[3] →^* x2[2]∧x0[3] →^* x0[2])

(3) -> (4), if (x1[3] - 2 →^* x1[4]∧x2[3] →^* x2[4]∧x0[3] →^* x0[4])

(4) -> (5), if (x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4] ∧x1[4] →^* x1[5]∧x2[4] →^* x2[5]∧x0[4] →^* x0[5])

(5) -> (0), if (x1[5] →^* x1[0]∧x2[5] →^* x2[0]∧x0[5] - 1 →^* x0[0])

(5) -> (2), if (x1[5] →^* x1[2]∧x2[5] →^* x2[2]∧x0[5] - 1 →^* x0[2])

(5) -> (4), if (x1[5] →^* x1[4]∧x2[5] →^* x2[4]∧x0[5] - 1 →^* x0[4])

The set Q consists of the following terms:
1421_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2))
1404_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2))
1420_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2))
1404_1_rec_InvokeMethod(1482_0_rec_Return)
1420_1_rec_InvokeMethod(1482_0_rec_Return)
1421_1_rec_InvokeMethod(1482_0_rec_Return)

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@518944a Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair 1351_0_REC_LOAD(x1, x2, x0) → COND_1351_0_REC_LOAD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:

We consider the chain 1351_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0]), COND_1351_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1351_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1]) which results in the following constraint:
(1)    (&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0]))))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1] ⇒ 1351_0_REC_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧1351_0_REC_LOAD(x1[0], x2[0], x0[0])≥COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])∧(U^Increasing(COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (<=(0, +(+(x0[0], x1[0]), *(3, x2[0])))=TRUE∧>=(x2[0], x1[0])=TRUE∧>=(x1[0], x0[0])=TRUE ⇒ 1351_0_REC_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧1351_0_REC_LOAD(x1[0], x2[0], x0[0])≥COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])∧(U^Increasing(COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧[2]x1[0] + [3]x2[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x2[0] + [(-2)bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)

For Pair COND_1351_0_REC_LOAD(TRUE, x1, x2, x0) → 1351_0_REC_LOAD(x1, -(x2, 1), x0) the following chains were created:

We consider the chain COND_1351_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1351_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1]) which results in the following constraint:
(8)    (COND_1351_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_1351_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1])≥1351_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])∧(U^Increasing(1351_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(1351_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[bni_21] = 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(1351_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[bni_21] = 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(1351_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[bni_21] = 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(1351_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[bni_21] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_22] ≥ 0)

For Pair 1351_0_REC_LOAD(x1, x2, x0) → COND_1351_0_REC_LOAD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:

We consider the chain 1351_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]), COND_1351_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1351_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) which results in the following constraint:
(13)    (&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2]))))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 1351_0_REC_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1351_0_REC_LOAD(x1[2], x2[2], x0[2])≥COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])∧(U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥))

We simplified constraint (13) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(14)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ 1351_0_REC_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1351_0_REC_LOAD(x1[2], x2[2], x0[2])≥COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])∧(U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x2[2] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(20)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(-2)bni_23]x2[2] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

(21)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x2[2] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

For Pair COND_1351_0_REC_LOAD1(TRUE, x1, x2, x0) → 1351_0_REC_LOAD(-(x1, 2), x2, x0) the following chains were created:

We consider the chain COND_1351_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1351_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) which results in the following constraint:
(22)    (COND_1351_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_1351_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3])≥1351_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])∧(U^Increasing(1351_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23)    ((U^Increasing(1351_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_25] = 0∧[4 + (-1)bso_26] ≥ 0)

We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24)    ((U^Increasing(1351_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_25] = 0∧[4 + (-1)bso_26] ≥ 0)

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25)    ((U^Increasing(1351_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_25] = 0∧[4 + (-1)bso_26] ≥ 0)

We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(26)    ((U^Increasing(1351_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_25] = 0∧0 = 0∧0 = 0∧0 = 0∧[4 + (-1)bso_26] ≥ 0)

For Pair 1351_0_REC_LOAD(x1, x2, x0) → COND_1351_0_REC_LOAD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:

We consider the chain 1351_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1351_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1351_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:
(27)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5] ⇒ 1351_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1351_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))

We simplified constraint (27) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(28)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ 1351_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1351_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(30)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(32)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(33)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

(34)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(35)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

(36)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(37)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

(38)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

For Pair COND_1351_0_REC_LOAD2(TRUE, x1, x2, x0) → 1351_0_REC_LOAD(x1, x2, -(x0, 1)) the following chains were created:

We consider the chain COND_1351_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1351_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:
(39)    (COND_1351_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1351_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1351_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))∧(U^Increasing(1351_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥))

We simplified constraint (39) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(40)    ((U^Increasing(1351_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (40) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(41)    ((U^Increasing(1351_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (41) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(42)    ((U^Increasing(1351_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (42) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(43)    ((U^Increasing(1351_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_29] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

1351_0_REC_LOAD(x1, x2, x0) → COND_1351_0_REC_LOAD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
- (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x2[0] + [(-2)bni_19]x1[0] ≥ 0∧[(-1)bso_20] ≥ 0)
COND_1351_0_REC_LOAD(TRUE, x1, x2, x0) → 1351_0_REC_LOAD(x1, -(x2, 1), x0)
- ((U^Increasing(1351_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[bni_21] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_22] ≥ 0)
1351_0_REC_LOAD(x1, x2, x0) → COND_1351_0_REC_LOAD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
- (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(-2)bni_23]x2[2] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)
- (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x2[2] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)
COND_1351_0_REC_LOAD1(TRUE, x1, x2, x0) → 1351_0_REC_LOAD(-(x1, 2), x2, x0)
- ((U^Increasing(1351_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_25] = 0∧0 = 0∧0 = 0∧0 = 0∧[4 + (-1)bso_26] ≥ 0)
1351_0_REC_LOAD(x1, x2, x0) → COND_1351_0_REC_LOAD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
- (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)
- (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)
- (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)
- (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)
COND_1351_0_REC_LOAD2(TRUE, x1, x2, x0) → 1351_0_REC_LOAD(x1, x2, -(x0, 1))
- ((U^Increasing(1351_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_29] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(1421_1_rec_InvokeMethod(x₁)) = [-1]
POL(1364_0_rec_Return(x₁, x₂, x₃)) = [-1]
POL(1482_0_rec_Return) = [-1]
POL(1404_1_rec_InvokeMethod(x₁)) = [-1]
POL(1420_1_rec_InvokeMethod(x₁)) = [-1]
POL(1351_0_REC_LOAD(x₁, x₂, x₃)) = [-1] + [2]x₁
POL(COND_1351_0_REC_LOAD(x₁, x₂, x₃, x₄)) = [-1] + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(<=(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(*(x₁, x₂)) = x₁·x₂
POL(3) = [3]
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]
POL(COND_1351_0_REC_LOAD1(x₁, x₂, x₃, x₄)) = [-1] + [2]x₂
POL(<(x₁, x₂)) = [-1]
POL(2) = [2]
POL(COND_1351_0_REC_LOAD2(x₁, x₂, x₃, x₄)) = [-1] + [2]x₂

The following pairs are in P_>:

COND_1351_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1351_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])

The following pairs are in P_bound:

1351_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])

The following pairs are in P_≥:

1351_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1351_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])
COND_1351_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1351_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])
1351_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1351_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])
1351_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1351_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
COND_1351_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1351_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))

There are no usable rules.

(8) Complex Obligation (AND)

(9) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
1421_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1404_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1420_1_rec_InvokeMethod(1364_0_rec_Return(x0, x1, x2)) → 1482_0_rec_Return
1404_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return
1420_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return
1421_1_rec_InvokeMethod(1482_0_rec_Return) → 1482_0_rec_Return

The integer pair graph contains the following rules and edges:
(0): 1351_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1351_0_REC_LOAD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], x1[0], x2[0], x0[0])
(1): COND_1351_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1351_0_REC_LOAD(x1[1], x2[1] - 1, x0[1])
(2): 1351_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1351_0_REC_LOAD1(x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2], x1[2], x2[2], x0[2])
(4): 1351_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1351_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1351_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1351_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)