Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 280 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 300 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 570 ms)
↳8 AND
    ↳9 IDP
        ↳10 IDependencyGraphProof (⇔, 0 ms)
        ↳11 IDP
        ↳12 UsableRulesProof (⇔, 0 ms)
        ↳13 IDP
        ↳14 IDPNonInfProof (⇒, 330 ms)
        ↳15 IDP
        ↳16 IDependencyGraphProof (⇔, 0 ms)
        ↳17 IDP
        ↳18 IDPNonInfProof (⇒, 140 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 (⇒, 330 ms)
        ↳28 IDP
        ↳29 IDependencyGraphProof (⇔, 0 ms)
        ↳30 IDP
        ↳31 IDPNonInfProof (⇒, 140 ms)
        ↳32 AND
            ↳33 IDP
                ↳34 IDependencyGraphProof (⇔, 0 ms)
                ↳35 TRUE
            ↳36 IDP
                ↳37 IDependencyGraphProof (⇔, 0 ms)
                ↳38 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:
1337_0_rec_Load(EOS(STATIC_1337), i102, i431, i245, i102) → 1339_0_rec_IntArithmetic(EOS(STATIC_1339), i102, i431, i245, i102, i431)
1339_0_rec_IntArithmetic(EOS(STATIC_1339), i102, i431, i245, i102, i431) → 1340_0_rec_ConstantStackPush(EOS(STATIC_1340), i102, i431, i245, +(i102, i431))
1340_0_rec_ConstantStackPush(EOS(STATIC_1340), i102, i431, i245, i437) → 1342_0_rec_Load(EOS(STATIC_1342), i102, i431, i245, i437, 3)
1342_0_rec_Load(EOS(STATIC_1342), i102, i431, i245, i437, matching1) → 1344_0_rec_IntArithmetic(EOS(STATIC_1344), i102, i431, i245, i437, 3, i245) | =(matching1, 3)
1344_0_rec_IntArithmetic(EOS(STATIC_1344), i102, i431, i245, i437, matching1, i245) → 1346_0_rec_IntArithmetic(EOS(STATIC_1346), i102, i431, i245, i437, *(3, i245)) | =(matching1, 3)
1346_0_rec_IntArithmetic(EOS(STATIC_1346), i102, i431, i245, i437, i439) → 1348_0_rec_GE(EOS(STATIC_1348), i102, i431, i245, +(i437, i439))
1348_0_rec_GE(EOS(STATIC_1348), i102, i431, i245, i444) → 1351_0_rec_GE(EOS(STATIC_1351), i102, i431, i245, i444)
1351_0_rec_GE(EOS(STATIC_1351), i102, i431, i245, i444) → 1355_0_rec_Load(EOS(STATIC_1355), i102, i431, i245) | >=(i444, 0)
1355_0_rec_Load(EOS(STATIC_1355), i102, i431, i245) → 1358_0_rec_Load(EOS(STATIC_1358), i102, i431, i245, i102)
1358_0_rec_Load(EOS(STATIC_1358), i102, i431, i245, i102) → 1362_0_rec_LE(EOS(STATIC_1362), i102, i431, i245, i102, i431)
1362_0_rec_LE(EOS(STATIC_1362), i102, i431, i245, i102, i431) → 1366_0_rec_LE(EOS(STATIC_1366), i102, i431, i245, i102, i431)
1362_0_rec_LE(EOS(STATIC_1362), i102, i431, i245, i102, i431) → 1367_0_rec_LE(EOS(STATIC_1367), i102, i431, i245, i102, i431)
1366_0_rec_LE(EOS(STATIC_1366), i102, i431, i245, i102, i431) → 1369_0_rec_Load(EOS(STATIC_1369), i102, i431, i245) | <=(i102, i431)
1369_0_rec_Load(EOS(STATIC_1369), i102, i431, i245) → 1372_0_rec_Load(EOS(STATIC_1372), i102, i431, i245, i431)
1372_0_rec_Load(EOS(STATIC_1372), i102, i431, i245, i431) → 1375_0_rec_LE(EOS(STATIC_1375), i102, i431, i245, i431, i245)
1375_0_rec_LE(EOS(STATIC_1375), i102, i431, i245, i431, i245) → 1378_0_rec_LE(EOS(STATIC_1378), i102, i431, i245, i431, i245)
1375_0_rec_LE(EOS(STATIC_1375), i102, i431, i245, i431, i245) → 1379_0_rec_LE(EOS(STATIC_1379), i102, i431, i245, i431, i245)
1378_0_rec_LE(EOS(STATIC_1378), i102, i431, i245, i431, i245) → 1382_0_rec_Load(EOS(STATIC_1382), i102, i431, i245) | <=(i431, i245)
1382_0_rec_Load(EOS(STATIC_1382), i102, i431, i245) → 1386_0_rec_Load(EOS(STATIC_1386), i431, i245, i102)
1386_0_rec_Load(EOS(STATIC_1386), i431, i245, i102) → 1391_0_rec_Load(EOS(STATIC_1391), i245, i102, i431)
1391_0_rec_Load(EOS(STATIC_1391), i245, i102, i431) → 1395_0_rec_ConstantStackPush(EOS(STATIC_1395), i102, i431, i245)
1395_0_rec_ConstantStackPush(EOS(STATIC_1395), i102, i431, i245) → 1400_0_rec_IntArithmetic(EOS(STATIC_1400), i102, i431, i245, 1)
1400_0_rec_IntArithmetic(EOS(STATIC_1400), i102, i431, i245, matching1) → 1403_0_rec_InvokeMethod(EOS(STATIC_1403), i102, i431, -(i245, 1)) | =(matching1, 1)
1403_0_rec_InvokeMethod(EOS(STATIC_1403), i102, i431, i453) → 1410_1_rec_InvokeMethod(1410_0_rec_Load(EOS(STATIC_1410), i102, i431, i453), i102, i431, i453)
1410_0_rec_Load(EOS(STATIC_1410), i102, i431, i453) → 1414_0_rec_Load(EOS(STATIC_1414), i102, i431, i453)
1414_0_rec_Load(EOS(STATIC_1414), i102, i431, i453) → 1335_0_rec_Load(EOS(STATIC_1335), i102, i431, i453)
1335_0_rec_Load(EOS(STATIC_1335), i102, i431, i245) → 1337_0_rec_Load(EOS(STATIC_1337), i102, i431, i245, i102)
1379_0_rec_LE(EOS(STATIC_1379), i102, i431, i245, i431, i245) → 1383_0_rec_Load(EOS(STATIC_1383), i102, i431, i245) | >(i431, i245)
1383_0_rec_Load(EOS(STATIC_1383), i102, i431, i245) → 1388_0_rec_Load(EOS(STATIC_1388), i431, i245, i102)
1388_0_rec_Load(EOS(STATIC_1388), i431, i245, i102) → 1392_0_rec_ConstantStackPush(EOS(STATIC_1392), i245, i102, i431)
1392_0_rec_ConstantStackPush(EOS(STATIC_1392), i245, i102, i431) → 1397_0_rec_IntArithmetic(EOS(STATIC_1397), i245, i102, i431, 2)
1397_0_rec_IntArithmetic(EOS(STATIC_1397), i245, i102, i431, matching1) → 1402_0_rec_Load(EOS(STATIC_1402), i245, i102, -(i431, 2)) | =(matching1, 2)
1402_0_rec_Load(EOS(STATIC_1402), i245, i102, i452) → 1405_0_rec_InvokeMethod(EOS(STATIC_1405), i102, i452, i245)
1405_0_rec_InvokeMethod(EOS(STATIC_1405), i102, i452, i245) → 1412_1_rec_InvokeMethod(1412_0_rec_Load(EOS(STATIC_1412), i102, i452, i245), i102, i452, i245)
1412_0_rec_Load(EOS(STATIC_1412), i102, i452, i245) → 1416_0_rec_Load(EOS(STATIC_1416), i102, i452, i245)
1416_0_rec_Load(EOS(STATIC_1416), i102, i452, i245) → 1335_0_rec_Load(EOS(STATIC_1335), i102, i452, i245)
1367_0_rec_LE(EOS(STATIC_1367), i102, i431, i245, i102, i431) → 1370_0_rec_Load(EOS(STATIC_1370), i102, i431, i245) | >(i102, i431)
1370_0_rec_Load(EOS(STATIC_1370), i102, i431, i245) → 1373_0_rec_ConstantStackPush(EOS(STATIC_1373), i431, i245, i102)
1373_0_rec_ConstantStackPush(EOS(STATIC_1373), i431, i245, i102) → 1376_0_rec_IntArithmetic(EOS(STATIC_1376), i431, i245, i102, 1)
1376_0_rec_IntArithmetic(EOS(STATIC_1376), i431, i245, i102, matching1) → 1380_0_rec_Load(EOS(STATIC_1380), i431, i245, -(i102, 1)) | =(matching1, 1)
1380_0_rec_Load(EOS(STATIC_1380), i431, i245, i451) → 1385_0_rec_Load(EOS(STATIC_1385), i245, i451, i431)
1385_0_rec_Load(EOS(STATIC_1385), i245, i451, i431) → 1389_0_rec_InvokeMethod(EOS(STATIC_1389), i451, i431, i245)
1389_0_rec_InvokeMethod(EOS(STATIC_1389), i451, i431, i245) → 1394_1_rec_InvokeMethod(1394_0_rec_Load(EOS(STATIC_1394), i451, i431, i245), i451, i431, i245)
1394_0_rec_Load(EOS(STATIC_1394), i451, i431, i245) → 1399_0_rec_Load(EOS(STATIC_1399), i451, i431, i245)
1399_0_rec_Load(EOS(STATIC_1399), i451, i431, i245) → 1335_0_rec_Load(EOS(STATIC_1335), i451, i431, i245)
R rules:
1348_0_rec_GE(EOS(STATIC_1348), i102, i431, i245, i443) → 1350_0_rec_GE(EOS(STATIC_1350), i102, i431, i245, i443)
1350_0_rec_GE(EOS(STATIC_1350), i102, i431, i245, i443) → 1353_0_rec_Return(EOS(STATIC_1353), i102, i431, i245) | <(i443, 0)
1394_1_rec_InvokeMethod(1353_0_rec_Return(EOS(STATIC_1353), i458, i459, i460), i458, i459, i460) → 1413_0_rec_Return(EOS(STATIC_1413), i458, i459, i460, i458, i459, i460)
1394_1_rec_InvokeMethod(1440_0_rec_Return(EOS(STATIC_1440)), i519, i520, i521) → 1479_0_rec_Return(EOS(STATIC_1479), i519, i520, i521)
1394_1_rec_InvokeMethod(1475_0_rec_Return(EOS(STATIC_1475)), i560, i561, i562) → 1496_0_rec_Return(EOS(STATIC_1496), i560, i561, i562)
1410_1_rec_InvokeMethod(1353_0_rec_Return(EOS(STATIC_1353), i465, i466, i467), i465, i466, i467) → 1429_0_rec_Return(EOS(STATIC_1429), i465, i466, i467, i465, i466, i467)
1410_1_rec_InvokeMethod(1440_0_rec_Return(EOS(STATIC_1440)), i525, i526, i527) → 1481_0_rec_Return(EOS(STATIC_1481), i525, i526, i527)
1410_1_rec_InvokeMethod(1475_0_rec_Return(EOS(STATIC_1475)), i566, i567, i568) → 1498_0_rec_Return(EOS(STATIC_1498), i566, i567, i568)
1412_1_rec_InvokeMethod(1353_0_rec_Return(EOS(STATIC_1353), i472, i473, i474), i472, i473, i474) → 1430_0_rec_Return(EOS(STATIC_1430), i472, i473, i474, i472, i473, i474)
1412_1_rec_InvokeMethod(1440_0_rec_Return(EOS(STATIC_1440)), i529, i530, i531) → 1483_0_rec_Return(EOS(STATIC_1483), i529, i530, i531)
1412_1_rec_InvokeMethod(1475_0_rec_Return(EOS(STATIC_1475)), i570, i571, i572) → 1500_0_rec_Return(EOS(STATIC_1500), i570, i571, i572)
1413_0_rec_Return(EOS(STATIC_1413), i458, i459, i460, i458, i459, i460) → 1418_0_rec_JMP(EOS(STATIC_1418))
1418_0_rec_JMP(EOS(STATIC_1418)) → 1454_0_rec_JMP(EOS(STATIC_1454))
1429_0_rec_Return(EOS(STATIC_1429), i465, i466, i467, i465, i466, i467) → 1440_0_rec_Return(EOS(STATIC_1440))
1430_0_rec_Return(EOS(STATIC_1430), i472, i473, i474, i472, i473, i474) → 1443_0_rec_JMP(EOS(STATIC_1443))
1440_0_rec_Return(EOS(STATIC_1440)) → 1450_0_rec_Return(EOS(STATIC_1450))
1443_0_rec_JMP(EOS(STATIC_1443)) → 1459_0_rec_JMP(EOS(STATIC_1459))
1445_0_rec_Return(EOS(STATIC_1445), i480, i481, i482) → 1454_0_rec_JMP(EOS(STATIC_1454))
1446_0_rec_Return(EOS(STATIC_1446), i486, i487, i488) → 1457_0_rec_Return(EOS(STATIC_1457))
1447_0_rec_Return(EOS(STATIC_1447), i491, i492, i493) → 1459_0_rec_JMP(EOS(STATIC_1459))
1450_0_rec_Return(EOS(STATIC_1450)) → 1457_0_rec_Return(EOS(STATIC_1457))
1454_0_rec_JMP(EOS(STATIC_1454)) → 1473_0_rec_Return(EOS(STATIC_1473))
1457_0_rec_Return(EOS(STATIC_1457)) → 1473_0_rec_Return(EOS(STATIC_1473))
1459_0_rec_JMP(EOS(STATIC_1459)) → 1475_0_rec_Return(EOS(STATIC_1475))
1473_0_rec_Return(EOS(STATIC_1473)) → 1475_0_rec_Return(EOS(STATIC_1475))
1479_0_rec_Return(EOS(STATIC_1479), i519, i520, i521) → 1445_0_rec_Return(EOS(STATIC_1445), i519, i520, i521)
1481_0_rec_Return(EOS(STATIC_1481), i525, i526, i527) → 1446_0_rec_Return(EOS(STATIC_1446), i525, i526, i527)
1483_0_rec_Return(EOS(STATIC_1483), i529, i530, i531) → 1447_0_rec_Return(EOS(STATIC_1447), i529, i530, i531)
1496_0_rec_Return(EOS(STATIC_1496), i560, i561, i562) → 1445_0_rec_Return(EOS(STATIC_1445), i560, i561, i562)
1498_0_rec_Return(EOS(STATIC_1498), i566, i567, i568) → 1446_0_rec_Return(EOS(STATIC_1446), i566, i567, i568)
1500_0_rec_Return(EOS(STATIC_1500), i570, i571, i572) → 1447_0_rec_Return(EOS(STATIC_1447), i570, i571, i572)

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


P rules:
1337_0_rec_Load(EOS(STATIC_1337), x0, x1, x2, x0) → 1410_1_rec_InvokeMethod(1337_0_rec_Load(EOS(STATIC_1337), x0, x1, -(x2, 1), x0), x0, x1, -(x2, 1)) | &&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1337_0_rec_Load(EOS(STATIC_1337), x0, x1, x2, x0) → 1412_1_rec_InvokeMethod(1337_0_rec_Load(EOS(STATIC_1337), x0, -(x1, 2), x2, x0), x0, -(x1, 2), x2) | &&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1337_0_rec_Load(EOS(STATIC_1337), x0, x1, x2, x0) → 1394_1_rec_InvokeMethod(1337_0_rec_Load(EOS(STATIC_1337), -(x0, 1), x1, x2, -(x0, 1)), -(x0, 1), x1, x2) | &&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2))))
R rules:
1412_1_rec_InvokeMethod(1353_0_rec_Return(EOS(STATIC_1353), x0, x1, x2), x0, x1, x2) → 1475_0_rec_Return(EOS(STATIC_1475))
1394_1_rec_InvokeMethod(1353_0_rec_Return(EOS(STATIC_1353), x0, x1, x2), x0, x1, x2) → 1475_0_rec_Return(EOS(STATIC_1475))
1410_1_rec_InvokeMethod(1353_0_rec_Return(EOS(STATIC_1353), x0, x1, x2), x0, x1, x2) → 1475_0_rec_Return(EOS(STATIC_1475))
1394_1_rec_InvokeMethod(1440_0_rec_Return(EOS(STATIC_1440)), x0, x1, x2) → 1475_0_rec_Return(EOS(STATIC_1475))
1394_1_rec_InvokeMethod(1475_0_rec_Return(EOS(STATIC_1475)), x0, x1, x2) → 1475_0_rec_Return(EOS(STATIC_1475))
1410_1_rec_InvokeMethod(1440_0_rec_Return(EOS(STATIC_1440)), x0, x1, x2) → 1475_0_rec_Return(EOS(STATIC_1475))
1410_1_rec_InvokeMethod(1475_0_rec_Return(EOS(STATIC_1475)), x0, x1, x2) → 1475_0_rec_Return(EOS(STATIC_1475))
1412_1_rec_InvokeMethod(1440_0_rec_Return(EOS(STATIC_1440)), x0, x1, x2) → 1475_0_rec_Return(EOS(STATIC_1475))
1412_1_rec_InvokeMethod(1475_0_rec_Return(EOS(STATIC_1475)), x0, x1, x2) → 1475_0_rec_Return(EOS(STATIC_1475))

Filtered ground terms:



1337_0_rec_Load(x1, x2, x3, x4, x5) → 1337_0_rec_Load(x2, x3, x4, x5)
Cond_1337_0_rec_Load2(x1, x2, x3, x4, x5, x6) → Cond_1337_0_rec_Load2(x1, x3, x4, x5, x6)
Cond_1337_0_rec_Load1(x1, x2, x3, x4, x5, x6) → Cond_1337_0_rec_Load1(x1, x3, x4, x5, x6)
Cond_1337_0_rec_Load(x1, x2, x3, x4, x5, x6) → Cond_1337_0_rec_Load(x1, x3, x4, x5, x6)
1475_0_rec_Return(x1) → 1475_0_rec_Return
1440_0_rec_Return(x1) → 1440_0_rec_Return
1353_0_rec_Return(x1, x2, x3, x4) → 1353_0_rec_Return(x2, x3, x4)

Filtered duplicate args:



1337_0_rec_Load(x1, x2, x3, x4) → 1337_0_rec_Load(x2, x3, x4)
Cond_1337_0_rec_Load(x1, x2, x3, x4, x5) → Cond_1337_0_rec_Load(x1, x3, x4, x5)
Cond_1337_0_rec_Load1(x1, x2, x3, x4, x5) → Cond_1337_0_rec_Load1(x1, x3, x4, x5)
Cond_1337_0_rec_Load2(x1, x2, x3, x4, x5) → Cond_1337_0_rec_Load2(x1, x3, x4, x5)

Filtered unneeded arguments:



1410_1_rec_InvokeMethod(x1, x2, x3, x4) → 1410_1_rec_InvokeMethod(x1)
1412_1_rec_InvokeMethod(x1, x2, x3, x4) → 1412_1_rec_InvokeMethod(x1)
1394_1_rec_InvokeMethod(x1, x2, x3, x4) → 1394_1_rec_InvokeMethod(x1)

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


P rules:
1337_0_rec_Load(x1, x2, x0) → 1410_1_rec_InvokeMethod(1337_0_rec_Load(x1, -(x2, 1), x0)) | &&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1337_0_rec_Load(x1, x2, x0) → 1412_1_rec_InvokeMethod(1337_0_rec_Load(-(x1, 2), x2, x0)) | &&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2))))
1337_0_rec_Load(x1, x2, x0) → 1394_1_rec_InvokeMethod(1337_0_rec_Load(x1, x2, -(x0, 1))) | &&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2))))
R rules:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1440_0_rec_Return) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1440_0_rec_Return) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1412_1_rec_InvokeMethod(1440_0_rec_Return) → 1475_0_rec_Return
1412_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return

Performed bisimulation on rules. Used the following equivalence classes: {[1475_0_rec_Return, 1440_0_rec_Return]=1475_0_rec_Return}


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


P rules:
1337_0_REC_LOAD(x1, x2, x0) → COND_1337_0_REC_LOAD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
COND_1337_0_REC_LOAD(TRUE, x1, x2, x0) → 1337_0_REC_LOAD(x1, -(x2, 1), x0)
1337_0_REC_LOAD(x1, x2, x0) → COND_1337_0_REC_LOAD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
COND_1337_0_REC_LOAD1(TRUE, x1, x2, x0) → 1337_0_REC_LOAD(-(x1, 2), x2, x0)
1337_0_REC_LOAD(x1, x2, x0) → COND_1337_0_REC_LOAD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
COND_1337_0_REC_LOAD2(TRUE, x1, x2, x0) → 1337_0_REC_LOAD(x1, x2, -(x0, 1))
R rules:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1412_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_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:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1412_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return

The integer pair graph contains the following rules and edges:
(0): 1337_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1337_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_1337_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1337_0_REC_LOAD(x1[1], x2[1] - 1, x0[1])
(2): 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_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:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_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@594e41bf 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 1337_0_REC_LOAD(x1, x2, x0) → COND_1337_0_REC_LOAD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:
  • We consider the chain 1337_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1337_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_1337_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1337_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] ⇒ 1337_0_REC_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧1337_0_REC_LOAD(x1[0], x2[0], x0[0])≥COND_1337_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])∧(UIncreasing(COND_1337_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 ⇒ 1337_0_REC_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧1337_0_REC_LOAD(x1[0], x2[0], x0[0])≥COND_1337_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])∧(UIncreasing(COND_1337_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 ⇒ (UIncreasing(COND_1337_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] + [bni_19]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_19]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_19]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_19]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_19]x2[0] ≥ 0∧[(-1)bso_20] ≥ 0)







For Pair COND_1337_0_REC_LOAD(TRUE, x1, x2, x0) → 1337_0_REC_LOAD(x1, -(x2, 1), x0) the following chains were created:
  • We consider the chain COND_1337_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1337_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1]) which results in the following constraint:

    (8)    (COND_1337_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_1337_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1])≥1337_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])∧(UIncreasing(1337_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)    ((UIncreasing(1337_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[bni_21] = 0∧[1 + (-1)bso_22] ≥ 0)



    We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (10)    ((UIncreasing(1337_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[bni_21] = 0∧[1 + (-1)bso_22] ≥ 0)



    We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (11)    ((UIncreasing(1337_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[bni_21] = 0∧[1 + (-1)bso_22] ≥ 0)



    We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (12)    ((UIncreasing(1337_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[bni_21] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_22] ≥ 0)







For Pair 1337_0_REC_LOAD(x1, x2, x0) → COND_1337_0_REC_LOAD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:
  • We consider the chain 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_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] ⇒ 1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥COND_1337_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])∧(UIncreasing(COND_1337_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 ⇒ 1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥COND_1337_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])∧(UIncreasing(COND_1337_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 ⇒ (UIncreasing(COND_1337_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] + [bni_23]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_23]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_23]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_23]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_23]x2[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 ⇒ (UIncreasing(COND_1337_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] + [(-1)bni_23]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_23]x2[2] ≥ 0∧[(-1)bso_24] ≥ 0)







For Pair COND_1337_0_REC_LOAD1(TRUE, x1, x2, x0) → 1337_0_REC_LOAD(-(x1, 2), x2, x0) the following chains were created:
  • We consider the chain COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) which results in the following constraint:

    (22)    (COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3])≥1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])∧(UIncreasing(1337_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)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_25] = 0∧[(-1)bso_26] ≥ 0)



    We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (24)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_25] = 0∧[(-1)bso_26] ≥ 0)



    We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (25)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_25] = 0∧[(-1)bso_26] ≥ 0)



    We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (26)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_25] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_26] ≥ 0)







For Pair 1337_0_REC_LOAD(x1, x2, x0) → COND_1337_0_REC_LOAD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:
  • We consider the chain 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_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] ⇒ 1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1337_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 ⇒ 1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1337_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 ⇒ (UIncreasing(COND_1337_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] + [bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)


    (34)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1337_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] + [bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [(-1)bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [(-1)bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)







For Pair COND_1337_0_REC_LOAD2(TRUE, x1, x2, x0) → 1337_0_REC_LOAD(x1, x2, -(x0, 1)) the following chains were created:
  • We consider the chain COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:

    (39)    (COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))∧(UIncreasing(1337_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)    ((UIncreasing(1337_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)    ((UIncreasing(1337_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)    ((UIncreasing(1337_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)    ((UIncreasing(1337_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.
  • 1337_0_REC_LOAD(x1, x2, x0) → COND_1337_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 ⇒ (UIncreasing(COND_1337_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] + [bni_19]x2[0] ≥ 0∧[(-1)bso_20] ≥ 0)

  • COND_1337_0_REC_LOAD(TRUE, x1, x2, x0) → 1337_0_REC_LOAD(x1, -(x2, 1), x0)
    • ((UIncreasing(1337_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[bni_21] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_22] ≥ 0)

  • 1337_0_REC_LOAD(x1, x2, x0) → COND_1337_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 ⇒ (UIncreasing(COND_1337_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] + [(-1)bni_23]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_23]x2[2] ≥ 0∧[(-1)bso_24] ≥ 0)

  • COND_1337_0_REC_LOAD1(TRUE, x1, x2, x0) → 1337_0_REC_LOAD(-(x1, 2), x2, x0)
    • ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_25] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_26] ≥ 0)

  • 1337_0_REC_LOAD(x1, x2, x0) → COND_1337_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 ⇒ (UIncreasing(COND_1337_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] + [bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [(-1)bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [(-1)bni_27]x2[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 ⇒ (UIncreasing(COND_1337_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] + [bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

  • COND_1337_0_REC_LOAD2(TRUE, x1, x2, x0) → 1337_0_REC_LOAD(x1, x2, -(x0, 1))
    • ((UIncreasing(1337_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 Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(1412_1_rec_InvokeMethod(x1)) = [-1]   
POL(1353_0_rec_Return(x1, x2, x3)) = [-1]   
POL(1475_0_rec_Return) = [-1]   
POL(1394_1_rec_InvokeMethod(x1)) = [-1]   
POL(1410_1_rec_InvokeMethod(x1)) = [-1]   
POL(1337_0_REC_LOAD(x1, x2, x3)) = [-1] + x2   
POL(COND_1337_0_REC_LOAD(x1, x2, x3, x4)) = [-1] + x3   
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_1337_0_REC_LOAD1(x1, x2, x3, x4)) = [-1] + x3   
POL(<(x1, x2)) = [-1]   
POL(2) = [2]   
POL(COND_1337_0_REC_LOAD2(x1, x2, x3, x4)) = [-1] + x3   

The following pairs are in P>:

COND_1337_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1337_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])

The following pairs are in Pbound:

1337_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1337_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])

The following pairs are in P≥:

1337_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1337_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])
1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])
1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_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:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1412_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return

The integer pair graph contains the following rules and edges:
(0): 1337_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1337_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])
(2): 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)

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


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


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


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


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


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



The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(10) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(11) 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:

Integer, Boolean


The ITRS R consists of the following rules:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1412_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return

The integer pair graph contains the following rules and edges:
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(3): COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])
(2): 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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) -> (2), if (x1[3] - 2 →* x1[2]∧x2[3] →* x2[2]∧x0[3] →* x0[2])


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


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


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



The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(12) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(13) 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:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(3): COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])
(2): 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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) -> (2), if (x1[3] - 2 →* x1[2]∧x2[3] →* x2[2]∧x0[3] →* x0[2])


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


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


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



The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(14) 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@594e41bf 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 COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) the following chains were created:
  • We consider the chain COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:

    (1)    (COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))∧(UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥))



    We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (2)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)



    We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (3)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)



    We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (4)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)



    We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (5)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_15] ≥ 0)







For Pair 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) the following chains were created:
  • We consider the chain 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:

    (6)    (&&(<(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] ⇒ 1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (6) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (7)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ 1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (8)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x1[4] + [(-1)bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)



    We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (9)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x1[4] + [(-1)bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)



    We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (10)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x1[4] + [(-1)bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)



    We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (11)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x1[4] + [(-1)bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)



    We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (12)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x1[4] + [(-1)bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)


    (13)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x1[4] + [(-1)bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)



    We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (14)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x1[4] + [(-1)bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)


    (15)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x1[4] + [bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)



    We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (16)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x1[4] + [(-1)bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)


    (17)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x1[4] + [bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)







For Pair COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) the following chains were created:
  • We consider the chain COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) which results in the following constraint:

    (18)    (COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3])≥1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])∧(UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥))



    We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (19)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_18] = 0∧[(-1)bso_19] ≥ 0)



    We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (20)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_18] = 0∧[(-1)bso_19] ≥ 0)



    We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (21)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_18] = 0∧[(-1)bso_19] ≥ 0)



    We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (22)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_19] ≥ 0)







For Pair 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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 chains were created:
  • We consider the chain 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) which results in the following constraint:

    (23)    (&&(&&(<(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] ⇒ 1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥COND_1337_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])∧(UIncreasing(COND_1337_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 (23) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (24)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ 1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥COND_1337_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])∧(UIncreasing(COND_1337_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 (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (25)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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_20 + (-1)Bound*bni_20] + [bni_20]x1[2] + [(-1)bni_20]x2[2] ≥ 0∧[2 + (-1)bso_21] ≥ 0)



    We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (26)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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_20 + (-1)Bound*bni_20] + [bni_20]x1[2] + [(-1)bni_20]x2[2] ≥ 0∧[2 + (-1)bso_21] ≥ 0)



    We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (27)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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_20 + (-1)Bound*bni_20] + [bni_20]x1[2] + [(-1)bni_20]x2[2] ≥ 0∧[2 + (-1)bso_21] ≥ 0)



    We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (28)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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_20 + (-1)Bound*bni_20] + [bni_20]x1[2] + [(-1)bni_20]x2[2] ≥ 0∧[2 + (-1)bso_21] ≥ 0)



    We simplified constraint (28) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (29)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[2] ≥ 0∧[2 + (-1)bso_21] ≥ 0)



    We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (30)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[2] ≥ 0∧[2 + (-1)bso_21] ≥ 0)


    (31)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[2] ≥ 0∧[2 + (-1)bso_21] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))
    • ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_15] ≥ 0)

  • 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x1[4] + [(-1)bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x1[4] + [bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x1[4] + [(-1)bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x1[4] + [bni_16]x2[4] ≥ 0∧[(-1)bso_17] ≥ 0)

  • COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])
    • ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_19] ≥ 0)

  • 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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])
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[2] ≥ 0∧[2 + (-1)bso_21] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[2] ≥ 0∧[2 + (-1)bso_21] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(COND_1337_0_REC_LOAD2(x1, x2, x3, x4)) = [1] + x2 + [-1]x3   
POL(1337_0_REC_LOAD(x1, x2, x3)) = [1] + x1 + [-1]x2   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   
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(COND_1337_0_REC_LOAD1(x1, x2, x3, x4)) = [-1] + [-1]x3 + x2   
POL(2) = [2]   
POL(>=(x1, x2)) = [-1]   

The following pairs are in P>:

1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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 Pbound:

1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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≥:

COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))
1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])

There are no usable rules.

(15) 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:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(3): COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])

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


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



The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(16) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(17) 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


R is empty.

The integer pair graph contains the following rules and edges:
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)

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



The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(18) 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@594e41bf 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 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) the following chains were created:
  • We consider the chain 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:

    (1)    (&&(<(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] ⇒ 1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (2)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ 1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (3)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] + [(-1)bni_8]x1[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)



    We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (4)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] + [(-1)bni_8]x1[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)



    We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (5)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] + [(-1)bni_8]x1[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)



    We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (6)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (7)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)


    (8)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)



    We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (9)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)


    (10)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)



    We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (11)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)


    (12)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)







For Pair COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) the following chains were created:
  • We consider the chain COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:

    (13)    (COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))∧(UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥))



    We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (14)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)



    We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (15)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)



    We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (16)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)



    We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (17)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8 + bni_8] + [bni_8]x0[4] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

  • COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))
    • ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(1337_0_REC_LOAD(x1, x2, x3)) = x3 + [-1]x1   
POL(COND_1337_0_REC_LOAD2(x1, x2, x3, x4)) = [-1] + x4 + [-1]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]   

The following pairs are in P>:

1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])

The following pairs are in Pbound:

1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])

The following pairs are in P≥:

COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))

There are no usable rules.

(19) 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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)


The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(20) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(21) TRUE

(22) 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:

Integer, Boolean


The ITRS R consists of the following rules:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1412_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return

The integer pair graph contains the following rules and edges:
(1): COND_1337_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1337_0_REC_LOAD(x1[1], x2[1] - 1, x0[1])
(2): 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)

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


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


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


(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])


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


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


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



The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(23) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(24) 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:

Integer, Boolean


The ITRS R consists of the following rules:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2)) → 1475_0_rec_Return
1394_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1410_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return
1412_1_rec_InvokeMethod(1475_0_rec_Return) → 1475_0_rec_Return

The integer pair graph contains the following rules and edges:
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(3): COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])
(2): 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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) -> (2), if (x1[3] - 2 →* x1[2]∧x2[3] →* x2[2]∧x0[3] →* x0[2])


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


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


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



The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(25) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(26) 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:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(3): COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])
(2): 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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) -> (2), if (x1[3] - 2 →* x1[2]∧x2[3] →* x2[2]∧x0[3] →* x0[2])


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


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


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



The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(27) 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@594e41bf 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 COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) the following chains were created:
  • We consider the chain COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:

    (1)    (COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))∧(UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥))



    We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (2)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)



    We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (3)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)



    We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (4)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)



    We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (5)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)







For Pair 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) the following chains were created:
  • We consider the chain 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:

    (6)    (&&(<(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] ⇒ 1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (6) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (7)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ 1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (8)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[4] + [(-1)bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)



    We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (9)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[4] + [(-1)bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)



    We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (10)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[4] + [(-1)bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)



    We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (11)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[4] + [(-1)bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)



    We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (12)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[4] + [(-1)bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)


    (13)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x1[4] + [(-1)bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)



    We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (14)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[4] + [(-1)bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)


    (15)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[4] + [bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)



    We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (16)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x1[4] + [bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)


    (17)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x1[4] + [(-1)bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)







For Pair COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) the following chains were created:
  • We consider the chain COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) which results in the following constraint:

    (18)    (COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3])≥1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])∧(UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥))



    We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (19)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)



    We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (20)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)



    We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (21)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)



    We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (22)    ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)







For Pair 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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 chains were created:
  • We consider the chain 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) which results in the following constraint:

    (23)    (&&(&&(<(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] ⇒ 1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥COND_1337_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])∧(UIncreasing(COND_1337_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 (23) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (24)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ 1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1337_0_REC_LOAD(x1[2], x2[2], x0[2])≥COND_1337_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])∧(UIncreasing(COND_1337_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 (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (25)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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_16 + (-1)Bound*bni_16] + [bni_16]x1[2] + [(-1)bni_16]x2[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



    We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (26)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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_16 + (-1)Bound*bni_16] + [bni_16]x1[2] + [(-1)bni_16]x2[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



    We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (27)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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_16 + (-1)Bound*bni_16] + [bni_16]x1[2] + [(-1)bni_16]x2[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



    We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (28)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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_16 + (-1)Bound*bni_16] + [bni_16]x1[2] + [(-1)bni_16]x2[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



    We simplified constraint (28) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (29)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



    We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (30)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)


    (31)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))
    • ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

  • 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[4] + [(-1)bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[4] + [bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x1[4] + [bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x1[4] + [(-1)bni_12]x2[4] ≥ 0∧[(-1)bso_13] ≥ 0)

  • COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])
    • ((UIncreasing(1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

  • 1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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])
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)
    • (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1337_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])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(COND_1337_0_REC_LOAD2(x1, x2, x3, x4)) = [1] + x2 + [-1]x3   
POL(1337_0_REC_LOAD(x1, x2, x3)) = [1] + x1 + [-1]x2   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   
POL(&&(x1, x2)) = 0   
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(COND_1337_0_REC_LOAD1(x1, x2, x3, x4)) = [-1]x3 + x2   
POL(2) = [2]   
POL(>=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])
1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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 Pbound:

1337_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1337_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≥:

COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))
1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])

There are no usable rules.

(28) 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:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(3): COND_1337_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1337_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])

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


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



The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(29) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(30) 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


R is empty.

The integer pair graph contains the following rules and edges:
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)

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



The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(31) 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@594e41bf 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 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]) the following chains were created:
  • We consider the chain 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:

    (1)    (&&(<(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] ⇒ 1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (2)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ 1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1337_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))



    We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (3)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[4] + [(-1)bni_8]x1[4] ≥ 0∧[(-1)bso_9] ≥ 0)



    We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (4)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[4] + [(-1)bni_8]x1[4] ≥ 0∧[(-1)bso_9] ≥ 0)



    We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (5)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[4] + [(-1)bni_8]x1[4] ≥ 0∧[(-1)bso_9] ≥ 0)



    We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (6)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (7)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)


    (8)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)



    We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (9)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)


    (10)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)



    We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (11)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)


    (12)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)







For Pair COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) the following chains were created:
  • We consider the chain COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:

    (13)    (COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))∧(UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥))



    We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (14)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧[1 + (-1)bso_11] ≥ 0)



    We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (15)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧[1 + (-1)bso_11] ≥ 0)



    We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (16)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧[1 + (-1)bso_11] ≥ 0)



    We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (17)    ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)
    • (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (UIncreasing(COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_8] + [bni_8]x0[4] ≥ 0∧[(-1)bso_9] ≥ 0)

  • COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))
    • ((UIncreasing(1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(1337_0_REC_LOAD(x1, x2, x3)) = [-1] + x3 + [-1]x1   
POL(COND_1337_0_REC_LOAD2(x1, x2, x3, x4)) = [-1] + x4 + [-1]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]   

The following pairs are in P>:

COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))

The following pairs are in Pbound:

1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])

The following pairs are in P≥:

1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])

There are no usable rules.

(32) Complex Obligation (AND)

(33) 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


R is empty.

The integer pair graph contains the following rules and edges:
(4): 1337_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1337_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])


The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(34) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(35) TRUE

(36) 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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_1337_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1337_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)


The set Q consists of the following terms:
1412_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1410_1_rec_InvokeMethod(1353_0_rec_Return(x0, x1, x2))
1394_1_rec_InvokeMethod(1475_0_rec_Return)
1410_1_rec_InvokeMethod(1475_0_rec_Return)
1412_1_rec_InvokeMethod(1475_0_rec_Return)

(37) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(38) TRUE