Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 110 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 280 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 380 ms)
↳8 AND
    ↳9 IDP
        ↳10 IDependencyGraphProof (⇔, 0 ms)
        ↳11 IDP
        ↳12 UsableRulesProof (⇔, 0 ms)
        ↳13 IDP
        ↳14 IDPNonInfProof (⇒, 60 ms)
        ↳15 AND
            ↳16 IDP
                ↳17 IDependencyGraphProof (⇔, 0 ms)
                ↳18 TRUE
            ↳19 IDP
                ↳20 IDependencyGraphProof (⇔, 0 ms)
                ↳21 TRUE
    ↳22 IDP
        ↳23 IDependencyGraphProof (⇔, 0 ms)
        ↳24 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Ackermann 
/**
 * The classical Ackermann function.
 *
 * All calls terminate.
 *
 * Julia + BinTerm prove that all calls terminate
 *
 * Note that we have to express the basic cases as m <= 0 and n <= 0
 * in order to prove termination.
 *
 * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A>
 */

public class Ackermann {

    public static int ack(int m, int n) {
	if (m <= 0) return n + 1;
	else if (n <= 0) return ack(m - 1,1);
	else return ack(m - 1,ack(m,n - 1));
    }

    public static void main(String[] args) {
	ack(10,12);
    }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
Ackermann.main([Ljava/lang/String;)V: Graph of 34 nodes with 0 SCCs.

Ackermann.ack(II)I: Graph of 55 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: Ackermann.ack(II)I
SCC calls the following helper methods: Ackermann.ack(II)I
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 38 rules for P and 25 rules for R.


P rules:
383_0_ack_GT(EOS(STATIC_383), i40, i38, i40) → 386_0_ack_GT(EOS(STATIC_386), i40, i38, i40)
386_0_ack_GT(EOS(STATIC_386), i40, i38, i40) → 389_0_ack_Load(EOS(STATIC_389), i40, i38) | >(i40, 0)
389_0_ack_Load(EOS(STATIC_389), i40, i38) → 393_0_ack_GT(EOS(STATIC_393), i40, i38, i38)
393_0_ack_GT(EOS(STATIC_393), i40, i43, i43) → 397_0_ack_GT(EOS(STATIC_397), i40, i43, i43)
393_0_ack_GT(EOS(STATIC_393), i40, i44, i44) → 398_0_ack_GT(EOS(STATIC_398), i40, i44, i44)
397_0_ack_GT(EOS(STATIC_397), i40, i43, i43) → 402_0_ack_Load(EOS(STATIC_402), i40, i43) | <=(i43, 0)
402_0_ack_Load(EOS(STATIC_402), i40, i43) → 408_0_ack_ConstantStackPush(EOS(STATIC_408), i40, i43, i40)
408_0_ack_ConstantStackPush(EOS(STATIC_408), i40, i43, i40) → 416_0_ack_IntArithmetic(EOS(STATIC_416), i40, i43, i40, 1)
416_0_ack_IntArithmetic(EOS(STATIC_416), i40, i43, i40, matching1) → 420_0_ack_ConstantStackPush(EOS(STATIC_420), i40, i43, -(i40, 1)) | &&(>(i40, 0), =(matching1, 1))
420_0_ack_ConstantStackPush(EOS(STATIC_420), i40, i43, i46) → 424_0_ack_InvokeMethod(EOS(STATIC_424), i40, i43, i46, 1)
424_0_ack_InvokeMethod(EOS(STATIC_424), i40, i43, i46, matching1) → 427_1_ack_InvokeMethod(427_0_ack_Load(EOS(STATIC_427), i46, 1), i40, i43, i46, 1) | =(matching1, 1)
427_0_ack_Load(EOS(STATIC_427), i46, matching1) → 431_0_ack_Load(EOS(STATIC_431), i46, 1) | =(matching1, 1)
431_0_ack_Load(EOS(STATIC_431), i46, matching1) → 380_0_ack_Load(EOS(STATIC_380), i46, 1) | =(matching1, 1)
380_0_ack_Load(EOS(STATIC_380), i37, i38) → 383_0_ack_GT(EOS(STATIC_383), i37, i38, i37)
398_0_ack_GT(EOS(STATIC_398), i40, i44, i44) → 403_0_ack_Load(EOS(STATIC_403), i40, i44) | >(i44, 0)
403_0_ack_Load(EOS(STATIC_403), i40, i44) → 410_0_ack_ConstantStackPush(EOS(STATIC_410), i40, i44, i40)
410_0_ack_ConstantStackPush(EOS(STATIC_410), i40, i44, i40) → 417_0_ack_IntArithmetic(EOS(STATIC_417), i40, i44, i40, 1)
417_0_ack_IntArithmetic(EOS(STATIC_417), i40, i44, i40, matching1) → 422_0_ack_Load(EOS(STATIC_422), i40, i44, -(i40, 1)) | &&(>(i40, 0), =(matching1, 1))
422_0_ack_Load(EOS(STATIC_422), i40, i44, i47) → 425_0_ack_Load(EOS(STATIC_425), i44, i47, i40)
425_0_ack_Load(EOS(STATIC_425), i44, i47, i40) → 429_0_ack_ConstantStackPush(EOS(STATIC_429), i47, i40, i44)
429_0_ack_ConstantStackPush(EOS(STATIC_429), i47, i40, i44) → 432_0_ack_IntArithmetic(EOS(STATIC_432), i47, i40, i44, 1)
432_0_ack_IntArithmetic(EOS(STATIC_432), i47, i40, i44, matching1) → 434_0_ack_InvokeMethod(EOS(STATIC_434), i47, i40, -(i44, 1)) | &&(>(i44, 0), =(matching1, 1))
434_0_ack_InvokeMethod(EOS(STATIC_434), i47, i40, i48) → 441_1_ack_InvokeMethod(441_0_ack_Load(EOS(STATIC_441), i40, i48), i47, i40, i48)
441_0_ack_Load(EOS(STATIC_441), i40, i48) → 444_0_ack_Load(EOS(STATIC_444), i40, i48)
441_1_ack_InvokeMethod(446_0_ack_Return(EOS(STATIC_446), i54, matching1, i45), i47, i54, matching2) → 458_0_ack_Return(EOS(STATIC_458), i47, i54, 0, i54, 0, i45) | &&(=(matching1, 0), =(matching2, 0))
441_1_ack_InvokeMethod(503_0_ack_Return(EOS(STATIC_503), i81, matching1, i62), i47, i81, matching2) → 526_0_ack_Return(EOS(STATIC_526), i47, i81, 0, i81, 0, i62) | &&(=(matching1, 0), =(matching2, 0))
441_1_ack_InvokeMethod(544_0_ack_Return(EOS(STATIC_544), i45), i47, i97, i98) → 567_0_ack_Return(EOS(STATIC_567), i47, i97, i98, i45)
441_1_ack_InvokeMethod(572_0_ack_Return(EOS(STATIC_572), i45), i47, i112, i113) → 589_0_ack_Return(EOS(STATIC_589), i47, i112, i113, i45)
444_0_ack_Load(EOS(STATIC_444), i40, i48) → 380_0_ack_Load(EOS(STATIC_380), i40, i48)
458_0_ack_Return(EOS(STATIC_458), i47, i54, matching1, i54, matching2, i45) → 462_0_ack_InvokeMethod(EOS(STATIC_462), i47, i45) | &&(=(matching1, 0), =(matching2, 0))
462_0_ack_InvokeMethod(EOS(STATIC_462), i47, i45) → 506_0_ack_InvokeMethod(EOS(STATIC_506), i47, i45)
506_0_ack_InvokeMethod(EOS(STATIC_506), i47, i62) → 512_1_ack_InvokeMethod(512_0_ack_Load(EOS(STATIC_512), i47, i62), i47, i62)
512_0_ack_Load(EOS(STATIC_512), i47, i62) → 520_0_ack_Load(EOS(STATIC_520), i47, i62)
520_0_ack_Load(EOS(STATIC_520), i47, i62) → 380_0_ack_Load(EOS(STATIC_380), i47, i62)
526_0_ack_Return(EOS(STATIC_526), i47, i81, matching1, i81, matching2, i62) → 458_0_ack_Return(EOS(STATIC_458), i47, i81, 0, i81, 0, i62) | &&(=(matching1, 0), =(matching2, 0))
567_0_ack_Return(EOS(STATIC_567), i47, i97, i98, i45) → 498_0_ack_Return(EOS(STATIC_498), i47, i97, i98, i45)
498_0_ack_Return(EOS(STATIC_498), i47, i68, i69, i62) → 506_0_ack_InvokeMethod(EOS(STATIC_506), i47, i62)
589_0_ack_Return(EOS(STATIC_589), i47, i112, i113, i45) → 498_0_ack_Return(EOS(STATIC_498), i47, i112, i113, i45)
R rules:
383_0_ack_GT(EOS(STATIC_383), matching1, i38, matching2) → 385_0_ack_GT(EOS(STATIC_385), 0, i38, 0) | &&(=(matching1, 0), =(matching2, 0))
385_0_ack_GT(EOS(STATIC_385), matching1, i38, matching2) → 387_0_ack_Load(EOS(STATIC_387), 0, i38) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
387_0_ack_Load(EOS(STATIC_387), matching1, i38) → 391_0_ack_ConstantStackPush(EOS(STATIC_391), 0, i38, i38) | =(matching1, 0)
391_0_ack_ConstantStackPush(EOS(STATIC_391), matching1, i38, i38) → 395_0_ack_IntArithmetic(EOS(STATIC_395), 0, i38, i38, 1) | =(matching1, 0)
395_0_ack_IntArithmetic(EOS(STATIC_395), matching1, i38, i38, matching2) → 400_0_ack_Return(EOS(STATIC_400), 0, i38, +(i38, 1)) | &&(=(matching1, 0), =(matching2, 1))
427_1_ack_InvokeMethod(400_0_ack_Return(EOS(STATIC_400), matching1, matching2, i45), i40, i43, matching3, matching4) → 442_0_ack_Return(EOS(STATIC_442), i40, i43, 0, 1, 0, 1, i45) | &&(&&(&&(=(matching1, 0), =(matching2, 1)), =(matching3, 0)), =(matching4, 1))
427_1_ack_InvokeMethod(544_0_ack_Return(EOS(STATIC_544), i45), i40, i43, i95, matching1) → 564_0_ack_Return(EOS(STATIC_564), i40, i43, i95, 1, i45) | =(matching1, 1)
427_1_ack_InvokeMethod(572_0_ack_Return(EOS(STATIC_572), i45), i40, i43, i110, matching1) → 586_0_ack_Return(EOS(STATIC_586), i40, i43, i110, 1, i45) | =(matching1, 1)
442_0_ack_Return(EOS(STATIC_442), i40, i43, matching1, matching2, matching3, matching4, i45) → 446_0_ack_Return(EOS(STATIC_446), i40, i43, i45) | &&(&&(&&(=(matching1, 0), =(matching2, 1)), =(matching3, 0)), =(matching4, 1))
446_0_ack_Return(EOS(STATIC_446), i40, i43, i45) → 503_0_ack_Return(EOS(STATIC_503), i40, i43, i45)
497_0_ack_Return(EOS(STATIC_497), i40, i43, i66, matching1, i62) → 503_0_ack_Return(EOS(STATIC_503), i40, i43, i62) | =(matching1, 1)
512_1_ack_InvokeMethod(400_0_ack_Return(EOS(STATIC_400), matching1, i86, i45), matching2, i86) → 536_0_ack_Return(EOS(STATIC_536), 0, i86, 0, i86, i45) | &&(=(matching1, 0), =(matching2, 0))
512_1_ack_InvokeMethod(446_0_ack_Return(EOS(STATIC_446), i87, i88, i45), i87, i88) → 537_0_ack_Return(EOS(STATIC_537), i87, i88, i87, i88, i45)
512_1_ack_InvokeMethod(503_0_ack_Return(EOS(STATIC_503), i90, i91, i89), i90, i91) → 540_0_ack_Return(EOS(STATIC_540), i90, i91, i90, i91, i89)
512_1_ack_InvokeMethod(544_0_ack_Return(EOS(STATIC_544), i45), i101, i102) → 568_0_ack_Return(EOS(STATIC_568), i101, i102, i45)
512_1_ack_InvokeMethod(572_0_ack_Return(EOS(STATIC_572), i45), i116, i117) → 591_0_ack_Return(EOS(STATIC_591), i116, i117, i45)
536_0_ack_Return(EOS(STATIC_536), matching1, i86, matching2, i86, i45) → 541_0_ack_Return(EOS(STATIC_541), i45) | &&(=(matching1, 0), =(matching2, 0))
537_0_ack_Return(EOS(STATIC_537), i87, i88, i87, i88, i45) → 544_0_ack_Return(EOS(STATIC_544), i45)
540_0_ack_Return(EOS(STATIC_540), i90, i91, i90, i91, i89) → 537_0_ack_Return(EOS(STATIC_537), i90, i91, i90, i91, i89)
541_0_ack_Return(EOS(STATIC_541), i45) → 544_0_ack_Return(EOS(STATIC_544), i45)
544_0_ack_Return(EOS(STATIC_544), i45) → 572_0_ack_Return(EOS(STATIC_572), i45)
564_0_ack_Return(EOS(STATIC_564), i40, i43, i95, matching1, i45) → 497_0_ack_Return(EOS(STATIC_497), i40, i43, i95, 1, i45) | =(matching1, 1)
568_0_ack_Return(EOS(STATIC_568), i101, i102, i45) → 572_0_ack_Return(EOS(STATIC_572), i45)
586_0_ack_Return(EOS(STATIC_586), i40, i43, i110, matching1, i45) → 497_0_ack_Return(EOS(STATIC_497), i40, i43, i110, 1, i45) | =(matching1, 1)
591_0_ack_Return(EOS(STATIC_591), i116, i117, i45) → 568_0_ack_Return(EOS(STATIC_568), i116, i117, i45)

Combined rules. Obtained 6 conditional rules for P and 9 conditional rules for R.


P rules:
383_0_ack_GT(EOS(STATIC_383), x0, x1, x0) → 427_1_ack_InvokeMethod(383_0_ack_GT(EOS(STATIC_383), -(x0, 1), 1, -(x0, 1)), x0, x1, -(x0, 1), 1) | &&(<=(x1, 0), >(x0, 0))
383_0_ack_GT(EOS(STATIC_383), x0, x1, x0) → 441_1_ack_InvokeMethod(383_0_ack_GT(EOS(STATIC_383), x0, -(x1, 1), x0), -(x0, 1), x0, -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
441_1_ack_InvokeMethod(446_0_ack_Return(EOS(STATIC_446), x0, 0, x2), x3, x0, 0) → 512_1_ack_InvokeMethod(383_0_ack_GT(EOS(STATIC_383), x3, x2, x3), x3, x2)
441_1_ack_InvokeMethod(503_0_ack_Return(EOS(STATIC_503), x0, 0, x2), x3, x0, 0) → 512_1_ack_InvokeMethod(383_0_ack_GT(EOS(STATIC_383), x3, x2, x3), x3, x2)
441_1_ack_InvokeMethod(544_0_ack_Return(EOS(STATIC_544), x0), x1, x2, x3) → 512_1_ack_InvokeMethod(383_0_ack_GT(EOS(STATIC_383), x1, x0, x1), x1, x0)
441_1_ack_InvokeMethod(572_0_ack_Return(EOS(STATIC_572), x0), x1, x2, x3) → 512_1_ack_InvokeMethod(383_0_ack_GT(EOS(STATIC_383), x1, x0, x1), x1, x0)
R rules:
383_0_ack_GT(EOS(STATIC_383), 0, x1, 0) → 400_0_ack_Return(EOS(STATIC_400), 0, x1, +(x1, 1))
427_1_ack_InvokeMethod(400_0_ack_Return(EOS(STATIC_400), 0, 1, x2), x3, x4, 0, 1) → 503_0_ack_Return(EOS(STATIC_503), x3, x4, x2)
512_1_ack_InvokeMethod(544_0_ack_Return(EOS(STATIC_544), x0), x1, x2) → 572_0_ack_Return(EOS(STATIC_572), x0)
512_1_ack_InvokeMethod(572_0_ack_Return(EOS(STATIC_572), x0), x1, x2) → 572_0_ack_Return(EOS(STATIC_572), x0)
512_1_ack_InvokeMethod(446_0_ack_Return(EOS(STATIC_446), x0, x1, x2), x0, x1) → 572_0_ack_Return(EOS(STATIC_572), x2)
512_1_ack_InvokeMethod(503_0_ack_Return(EOS(STATIC_503), x0, x1, x2), x0, x1) → 572_0_ack_Return(EOS(STATIC_572), x2)
512_1_ack_InvokeMethod(400_0_ack_Return(EOS(STATIC_400), 0, x1, x2), 0, x1) → 572_0_ack_Return(EOS(STATIC_572), x2)
427_1_ack_InvokeMethod(544_0_ack_Return(EOS(STATIC_544), x0), x1, x2, x3, 1) → 503_0_ack_Return(EOS(STATIC_503), x1, x2, x0)
427_1_ack_InvokeMethod(572_0_ack_Return(EOS(STATIC_572), x0), x1, x2, x3, 1) → 503_0_ack_Return(EOS(STATIC_503), x1, x2, x0)

Filtered ground terms:



383_0_ack_GT(x1, x2, x3, x4) → 383_0_ack_GT(x2, x3, x4)
572_0_ack_Return(x1, x2) → 572_0_ack_Return(x2)
544_0_ack_Return(x1, x2) → 544_0_ack_Return(x2)
503_0_ack_Return(x1, x2, x3, x4) → 503_0_ack_Return(x2, x3, x4)
446_0_ack_Return(x1, x2, x3, x4) → 446_0_ack_Return(x2, x3, x4)
Cond_383_0_ack_GT1(x1, x2, x3, x4, x5) → Cond_383_0_ack_GT1(x1, x3, x4, x5)
427_1_ack_InvokeMethod(x1, x2, x3, x4, x5) → 427_1_ack_InvokeMethod(x1, x2, x3, x4)
Cond_383_0_ack_GT(x1, x2, x3, x4, x5) → Cond_383_0_ack_GT(x1, x3, x4, x5)
400_0_ack_Return(x1, x2, x3, x4) → 400_0_ack_Return(x3, x4)

Filtered duplicate args:



383_0_ack_GT(x1, x2, x3) → 383_0_ack_GT(x2, x3)
Cond_383_0_ack_GT(x1, x2, x3, x4) → Cond_383_0_ack_GT(x1, x3, x4)
Cond_383_0_ack_GT1(x1, x2, x3, x4) → Cond_383_0_ack_GT1(x1, x3, x4)

Filtered unneeded arguments:



427_1_ack_InvokeMethod(x1, x2, x3, x4) → 427_1_ack_InvokeMethod(x1, x3, x4)
441_1_ack_InvokeMethod(x1, x2, x3, x4) → 441_1_ack_InvokeMethod(x1, x2, x4)
446_0_ack_Return(x1, x2, x3) → 446_0_ack_Return(x2, x3)
503_0_ack_Return(x1, x2, x3) → 503_0_ack_Return(x2, x3)

Combined rules. Obtained 6 conditional rules for P and 9 conditional rules for R.


P rules:
383_0_ack_GT(x1, x0) → 427_1_ack_InvokeMethod(383_0_ack_GT(1, -(x0, 1)), x1, -(x0, 1)) | &&(<=(x1, 0), >(x0, 0))
383_0_ack_GT(x1, x0) → 441_1_ack_InvokeMethod(383_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
441_1_ack_InvokeMethod(446_0_ack_Return(0, x2), x3, 0) → 512_1_ack_InvokeMethod(383_0_ack_GT(x2, x3), x3, x2)
441_1_ack_InvokeMethod(503_0_ack_Return(0, x2), x3, 0) → 512_1_ack_InvokeMethod(383_0_ack_GT(x2, x3), x3, x2)
441_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x3) → 512_1_ack_InvokeMethod(383_0_ack_GT(x0, x1), x1, x0)
441_1_ack_InvokeMethod(572_0_ack_Return(x0), x1, x3) → 512_1_ack_InvokeMethod(383_0_ack_GT(x0, x1), x1, x0)
R rules:
383_0_ack_GT(x1, 0) → 400_0_ack_Return(x1, +(x1, 1))
427_1_ack_InvokeMethod(400_0_ack_Return(1, x2), x4, 0) → 503_0_ack_Return(x4, x2)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2) → 572_0_ack_Return(x0)
512_1_ack_InvokeMethod(572_0_ack_Return(x0), x1, x2) → 572_0_ack_Return(x0)
512_1_ack_InvokeMethod(446_0_ack_Return(x1, x2), x0, x1) → 572_0_ack_Return(x2)
512_1_ack_InvokeMethod(503_0_ack_Return(x1, x2), x0, x1) → 572_0_ack_Return(x2)
512_1_ack_InvokeMethod(400_0_ack_Return(x1, x2), 0, x1) → 572_0_ack_Return(x2)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x2, x3) → 503_0_ack_Return(x2, x0)
427_1_ack_InvokeMethod(572_0_ack_Return(x0), x2, x3) → 503_0_ack_Return(x2, x0)

Performed bisimulation on rules. Used the following equivalence classes: {[503_0_ack_Return_2, 446_0_ack_Return_2]=503_0_ack_Return_2, [544_0_ack_Return_1, 572_0_ack_Return_1]=544_0_ack_Return_1}


Finished conversion. Obtained 7 rules for P and 6 rules for R. System has predefined symbols.


P rules:
383_0_ACK_GT(x1, x0) → COND_383_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0)
COND_383_0_ACK_GT(TRUE, x1, x0) → 383_0_ACK_GT(1, -(x0, 1))
383_0_ACK_GT(x1, x0) → COND_383_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
COND_383_0_ACK_GT1(TRUE, x1, x0) → 441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1))
COND_383_0_ACK_GT1(TRUE, x1, x0) → 383_0_ACK_GT(-(x1, 1), x0)
441_1_ACK_INVOKEMETHOD(503_0_ack_Return(0, x2), x3, 0) → 383_0_ACK_GT(x2, x3)
441_1_ACK_INVOKEMETHOD(544_0_ack_Return(x0), x1, x3) → 383_0_ACK_GT(x0, x1)
R rules:
383_0_ack_GT(x1, 0) → 400_0_ack_Return(x1, +(x1, 1))
427_1_ack_InvokeMethod(400_0_ack_Return(1, x2), x4, 0) → 503_0_ack_Return(x4, x2)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2) → 544_0_ack_Return(x0)
512_1_ack_InvokeMethod(503_0_ack_Return(x1, x2), x0, x1) → 544_0_ack_Return(x2)
512_1_ack_InvokeMethod(400_0_ack_Return(x1, x2), 0, x1) → 544_0_ack_Return(x2)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x2, x3) → 503_0_ack_Return(x2, x0)

(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:

Integer, Boolean


The ITRS R consists of the following rules:
383_0_ack_GT(x1, 0) → 400_0_ack_Return(x1, x1 + 1)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x2), x4, 0) → 503_0_ack_Return(x4, x2)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2) → 544_0_ack_Return(x0)
512_1_ack_InvokeMethod(503_0_ack_Return(x1, x2), x0, x1) → 544_0_ack_Return(x2)
512_1_ack_InvokeMethod(400_0_ack_Return(x1, x2), 0, x1) → 544_0_ack_Return(x2)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x2, x3) → 503_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(0): 383_0_ACK_GT(x1[0], x0[0]) → COND_383_0_ACK_GT(x1[0] <= 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_383_0_ACK_GT(TRUE, x1[1], x0[1]) → 383_0_ACK_GT(1, x0[1] - 1)
(2): 383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_383_0_ACK_GT1(TRUE, x1[3], x0[3]) → 441_1_ACK_INVOKEMETHOD(383_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(x1[4] - 1, x0[4])
(5): 441_1_ACK_INVOKEMETHOD(503_0_ack_Return(0, x2[5]), x3[5], 0) → 383_0_ACK_GT(x2[5], x3[5])
(6): 441_1_ACK_INVOKEMETHOD(544_0_ack_Return(x0[6]), x1[6], x3[6]) → 383_0_ACK_GT(x0[6], x1[6])

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


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


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


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


(2) -> (4), if (x1[2] > 0 && x0[2] > 0 ∧x1[2] →* x1[4]∧x0[2] →* x0[4])


(3) -> (5), if (383_0_ack_GT(x1[3] - 1, x0[3]) →* 503_0_ack_Return(0, x2[5])∧x0[3] - 1 →* x3[5]∧x1[3] - 1 →* 0)


(3) -> (6), if (383_0_ack_GT(x1[3] - 1, x0[3]) →* 544_0_ack_Return(x0[6])∧x0[3] - 1 →* x1[6]∧x1[3] - 1 →* x3[6])


(4) -> (0), if (x1[4] - 1 →* x1[0]∧x0[4] →* x0[0])


(4) -> (2), if (x1[4] - 1 →* x1[2]∧x0[4] →* x0[2])


(5) -> (0), if (x2[5] →* x1[0]∧x3[5] →* x0[0])


(5) -> (2), if (x2[5] →* x1[2]∧x3[5] →* x0[2])


(6) -> (0), if (x0[6] →* x1[0]∧x1[6] →* x0[0])


(6) -> (2), if (x0[6] →* x1[2]∧x1[6] →* x0[2])



The set Q consists of the following terms:
383_0_ack_GT(x0, 0)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x0), x1, 0)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)
512_1_ack_InvokeMethod(503_0_ack_Return(x0, x1), x2, x0)
512_1_ack_InvokeMethod(400_0_ack_Return(x0, x1), 0, x0)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)

(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@57fa9986 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 383_0_ACK_GT(x1, x0) → COND_383_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
  • We consider the chain 383_0_ACK_GT(x1[0], x0[0]) → COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_383_0_ACK_GT(TRUE, x1[1], x0[1]) → 383_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:

    (1)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 383_0_ACK_GT(x1[0], x0[0])≥NonInfC∧383_0_ACK_GT(x1[0], x0[0])≥COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥))



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

    (2)    (<=(x1[0], 0)=TRUE∧>(x0[0], 0)=TRUE ⇒ 383_0_ACK_GT(x1[0], x0[0])≥NonInfC∧383_0_ACK_GT(x1[0], x0[0])≥COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥))



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

    (3)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (4)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (5)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (6)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)







For Pair COND_383_0_ACK_GT(TRUE, x1, x0) → 383_0_ACK_GT(1, -(x0, 1)) the following chains were created:
  • We consider the chain COND_383_0_ACK_GT(TRUE, x1[1], x0[1]) → 383_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:

    (8)    (COND_383_0_ACK_GT(TRUE, x1[1], x0[1])≥NonInfC∧COND_383_0_ACK_GT(TRUE, x1[1], x0[1])≥383_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(383_0_ACK_GT(1, -(x0[1], 1))), ≥))



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

    (9)    ((UIncreasing(383_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)



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

    (10)    ((UIncreasing(383_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)



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

    (11)    ((UIncreasing(383_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)



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

    (12)    ((UIncreasing(383_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 0)







For Pair 383_0_ACK_GT(x1, x0) → COND_383_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
  • We consider the chain 383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_383_0_ACK_GT1(TRUE, x1[3], x0[3]) → 441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1)) which results in the following constraint:

    (13)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3] ⇒ 383_0_ACK_GT(x1[2], x0[2])≥NonInfC∧383_0_ACK_GT(x1[2], x0[2])≥COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



    We simplified constraint (13) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (14)    (>(x1[2], 0)=TRUE∧>(x0[2], 0)=TRUE ⇒ 383_0_ACK_GT(x1[2], x0[2])≥NonInfC∧383_0_ACK_GT(x1[2], x0[2])≥COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (15)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (16)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (17)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)



    We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (18)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (19)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)



  • We consider the chain 383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

    (20)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4] ⇒ 383_0_ACK_GT(x1[2], x0[2])≥NonInfC∧383_0_ACK_GT(x1[2], x0[2])≥COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



    We simplified constraint (20) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (21)    (>(x1[2], 0)=TRUE∧>(x0[2], 0)=TRUE ⇒ 383_0_ACK_GT(x1[2], x0[2])≥NonInfC∧383_0_ACK_GT(x1[2], x0[2])≥COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (22)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)



    We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (23)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (24)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (25)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (26)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)







For Pair COND_383_0_ACK_GT1(TRUE, x1, x0) → 441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1)) the following chains were created:
  • We consider the chain COND_383_0_ACK_GT1(TRUE, x1[3], x0[3]) → 441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1)) which results in the following constraint:

    (27)    (COND_383_0_ACK_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_383_0_ACK_GT1(TRUE, x1[3], x0[3])≥441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))∧(UIncreasing(441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥))



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

    (28)    ((UIncreasing(441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_28] = 0∧[1 + (-1)bso_29] ≥ 0)



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

    (29)    ((UIncreasing(441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_28] = 0∧[1 + (-1)bso_29] ≥ 0)



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

    (30)    ((UIncreasing(441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_28] = 0∧[1 + (-1)bso_29] ≥ 0)



    We simplified constraint (30) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (31)    ((UIncreasing(441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_28] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_29] ≥ 0)







For Pair COND_383_0_ACK_GT1(TRUE, x1, x0) → 383_0_ACK_GT(-(x1, 1), x0) the following chains were created:
  • We consider the chain COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

    (32)    (COND_383_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_383_0_ACK_GT1(TRUE, x1[4], x0[4])≥383_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



    We simplified constraint (32) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (33)    ((UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧[(-1)bso_31] ≥ 0)



    We simplified constraint (33) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (34)    ((UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧[(-1)bso_31] ≥ 0)



    We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (35)    ((UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧[(-1)bso_31] ≥ 0)



    We simplified constraint (35) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (36)    ((UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)







For Pair 441_1_ACK_INVOKEMETHOD(503_0_ack_Return(0, x2), x3, 0) → 383_0_ACK_GT(x2, x3) the following chains were created:
  • We consider the chain 441_1_ACK_INVOKEMETHOD(503_0_ack_Return(0, x2[5]), x3[5], 0) → 383_0_ACK_GT(x2[5], x3[5]) which results in the following constraint:

    (37)    (441_1_ACK_INVOKEMETHOD(503_0_ack_Return(0, x2[5]), x3[5], 0)≥NonInfC∧441_1_ACK_INVOKEMETHOD(503_0_ack_Return(0, x2[5]), x3[5], 0)≥383_0_ACK_GT(x2[5], x3[5])∧(UIncreasing(383_0_ACK_GT(x2[5], x3[5])), ≥))



    We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (38)    ((UIncreasing(383_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (39)    ((UIncreasing(383_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (40)    ((UIncreasing(383_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (40) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (41)    ((UIncreasing(383_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧0 = 0∧0 = 0∧[(-1)bso_33] ≥ 0)







For Pair 441_1_ACK_INVOKEMETHOD(544_0_ack_Return(x0), x1, x3) → 383_0_ACK_GT(x0, x1) the following chains were created:
  • We consider the chain 441_1_ACK_INVOKEMETHOD(544_0_ack_Return(x0[6]), x1[6], x3[6]) → 383_0_ACK_GT(x0[6], x1[6]) which results in the following constraint:

    (42)    (441_1_ACK_INVOKEMETHOD(544_0_ack_Return(x0[6]), x1[6], x3[6])≥NonInfC∧441_1_ACK_INVOKEMETHOD(544_0_ack_Return(x0[6]), x1[6], x3[6])≥383_0_ACK_GT(x0[6], x1[6])∧(UIncreasing(383_0_ACK_GT(x0[6], x1[6])), ≥))



    We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (43)    ((UIncreasing(383_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_34] = 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (43) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (44)    ((UIncreasing(383_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_34] = 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (44) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (45)    ((UIncreasing(383_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_34] = 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (45) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (46)    ((UIncreasing(383_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_34] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 383_0_ACK_GT(x1, x0) → COND_383_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

  • COND_383_0_ACK_GT(TRUE, x1, x0) → 383_0_ACK_GT(1, -(x0, 1))
    • ((UIncreasing(383_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 0)

  • 383_0_ACK_GT(x1, x0) → COND_383_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

  • COND_383_0_ACK_GT1(TRUE, x1, x0) → 441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1))
    • ((UIncreasing(441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_28] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_29] ≥ 0)

  • COND_383_0_ACK_GT1(TRUE, x1, x0) → 383_0_ACK_GT(-(x1, 1), x0)
    • ((UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

  • 441_1_ACK_INVOKEMETHOD(503_0_ack_Return(0, x2), x3, 0) → 383_0_ACK_GT(x2, x3)
    • ((UIncreasing(383_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧0 = 0∧0 = 0∧[(-1)bso_33] ≥ 0)

  • 441_1_ACK_INVOKEMETHOD(544_0_ack_Return(x0), x1, x3) → 383_0_ACK_GT(x0, x1)
    • ((UIncreasing(383_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_34] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 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(383_0_ack_GT(x1, x2)) = [2] + [2]x2 + x1   
POL(0) = 0   
POL(400_0_ack_Return(x1, x2)) = [-1] + [-1]x2 + [2]x1   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(427_1_ack_InvokeMethod(x1, x2, x3)) = [-1]   
POL(503_0_ack_Return(x1, x2)) = x2   
POL(512_1_ack_InvokeMethod(x1, x2, x3)) = [-1]   
POL(544_0_ack_Return(x1)) = x1   
POL(383_0_ACK_GT(x1, x2)) = [-1] + x2   
POL(COND_383_0_ACK_GT(x1, x2, x3)) = [-1] + x3   
POL(&&(x1, x2)) = [-1]   
POL(<=(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(COND_383_0_ACK_GT1(x1, x2, x3)) = [-1] + x3   
POL(441_1_ACK_INVOKEMETHOD(x1, x2, x3)) = [-1] + x2   

The following pairs are in P>:

COND_383_0_ACK_GT(TRUE, x1[1], x0[1]) → 383_0_ACK_GT(1, -(x0[1], 1))
COND_383_0_ACK_GT1(TRUE, x1[3], x0[3]) → 441_1_ACK_INVOKEMETHOD(383_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))

The following pairs are in Pbound:

383_0_ACK_GT(x1[0], x0[0]) → COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P≥:

383_0_ACK_GT(x1[0], x0[0]) → COND_383_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(-(x1[4], 1), x0[4])
441_1_ACK_INVOKEMETHOD(503_0_ack_Return(0, x2[5]), x3[5], 0) → 383_0_ACK_GT(x2[5], x3[5])
441_1_ACK_INVOKEMETHOD(544_0_ack_Return(x0[6]), x1[6], x3[6]) → 383_0_ACK_GT(x0[6], x1[6])

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:

Integer, Boolean


The ITRS R consists of the following rules:
383_0_ack_GT(x1, 0) → 400_0_ack_Return(x1, x1 + 1)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x2), x4, 0) → 503_0_ack_Return(x4, x2)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2) → 544_0_ack_Return(x0)
512_1_ack_InvokeMethod(503_0_ack_Return(x1, x2), x0, x1) → 544_0_ack_Return(x2)
512_1_ack_InvokeMethod(400_0_ack_Return(x1, x2), 0, x1) → 544_0_ack_Return(x2)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x2, x3) → 503_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(0): 383_0_ACK_GT(x1[0], x0[0]) → COND_383_0_ACK_GT(x1[0] <= 0 && x0[0] > 0, x1[0], x0[0])
(2): 383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(4): COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(x1[4] - 1, x0[4])
(5): 441_1_ACK_INVOKEMETHOD(503_0_ack_Return(0, x2[5]), x3[5], 0) → 383_0_ACK_GT(x2[5], x3[5])
(6): 441_1_ACK_INVOKEMETHOD(544_0_ack_Return(x0[6]), x1[6], x3[6]) → 383_0_ACK_GT(x0[6], x1[6])

(4) -> (0), if (x1[4] - 1 →* x1[0]∧x0[4] →* x0[0])


(5) -> (0), if (x2[5] →* x1[0]∧x3[5] →* x0[0])


(6) -> (0), if (x0[6] →* x1[0]∧x1[6] →* x0[0])


(4) -> (2), if (x1[4] - 1 →* x1[2]∧x0[4] →* x0[2])


(5) -> (2), if (x2[5] →* x1[2]∧x3[5] →* x0[2])


(6) -> (2), if (x0[6] →* x1[2]∧x1[6] →* x0[2])


(2) -> (4), if (x1[2] > 0 && x0[2] > 0 ∧x1[2] →* x1[4]∧x0[2] →* x0[4])



The set Q consists of the following terms:
383_0_ack_GT(x0, 0)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x0), x1, 0)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)
512_1_ack_InvokeMethod(503_0_ack_Return(x0, x1), x2, x0)
512_1_ack_InvokeMethod(400_0_ack_Return(x0, x1), 0, x0)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)

(10) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(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:
383_0_ack_GT(x1, 0) → 400_0_ack_Return(x1, x1 + 1)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x2), x4, 0) → 503_0_ack_Return(x4, x2)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2) → 544_0_ack_Return(x0)
512_1_ack_InvokeMethod(503_0_ack_Return(x1, x2), x0, x1) → 544_0_ack_Return(x2)
512_1_ack_InvokeMethod(400_0_ack_Return(x1, x2), 0, x1) → 544_0_ack_Return(x2)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x2, x3) → 503_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(4): COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(x1[4] - 1, x0[4])
(2): 383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])

(4) -> (2), if (x1[4] - 1 →* x1[2]∧x0[4] →* x0[2])


(2) -> (4), if (x1[2] > 0 && x0[2] > 0 ∧x1[2] →* x1[4]∧x0[2] →* x0[4])



The set Q consists of the following terms:
383_0_ack_GT(x0, 0)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x0), x1, 0)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)
512_1_ack_InvokeMethod(503_0_ack_Return(x0, x1), x2, x0)
512_1_ack_InvokeMethod(400_0_ack_Return(x0, x1), 0, x0)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)

(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:
(4): COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(x1[4] - 1, x0[4])
(2): 383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])

(4) -> (2), if (x1[4] - 1 →* x1[2]∧x0[4] →* x0[2])


(2) -> (4), if (x1[2] > 0 && x0[2] > 0 ∧x1[2] →* x1[4]∧x0[2] →* x0[4])



The set Q consists of the following terms:
383_0_ack_GT(x0, 0)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x0), x1, 0)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)
512_1_ack_InvokeMethod(503_0_ack_Return(x0, x1), x2, x0)
512_1_ack_InvokeMethod(400_0_ack_Return(x0, x1), 0, x0)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)

(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@57fa9986 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_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(-(x1[4], 1), x0[4]) the following chains were created:
  • We consider the chain COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

    (1)    (COND_383_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_383_0_ACK_GT1(TRUE, x1[4], x0[4])≥383_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



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

    (2)    ((UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧[1 + (-1)bso_10] ≥ 0)



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

    (3)    ((UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧[1 + (-1)bso_10] ≥ 0)



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

    (4)    ((UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧[1 + (-1)bso_10] ≥ 0)



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

    (5)    ((UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)







For Pair 383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) the following chains were created:
  • We consider the chain 383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

    (6)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4] ⇒ 383_0_ACK_GT(x1[2], x0[2])≥NonInfC∧383_0_ACK_GT(x1[2], x0[2])≥COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (7)    (>(x1[2], 0)=TRUE∧>(x0[2], 0)=TRUE ⇒ 383_0_ACK_GT(x1[2], x0[2])≥NonInfC∧383_0_ACK_GT(x1[2], x0[2])≥COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (8)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (9)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (10)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (11)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (12)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_11 + bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(-(x1[4], 1), x0[4])
    • ((UIncreasing(383_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)

  • 383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_11 + bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 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_383_0_ACK_GT1(x1, x2, x3)) = [-1] + x3 + x2   
POL(383_0_ACK_GT(x1, x2)) = [-1] + x1 + x2   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(-(x1[4], 1), x0[4])

The following pairs are in Pbound:

383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P≥:

383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

There are no usable rules.

(15) Complex Obligation (AND)

(16) 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:
(2): 383_0_ACK_GT(x1[2], x0[2]) → COND_383_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])


The set Q consists of the following terms:
383_0_ack_GT(x0, 0)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x0), x1, 0)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)
512_1_ack_InvokeMethod(503_0_ack_Return(x0, x1), x2, x0)
512_1_ack_InvokeMethod(400_0_ack_Return(x0, x1), 0, x0)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE

(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:
(4): COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(x1[4] - 1, x0[4])


The set Q consists of the following terms:
383_0_ack_GT(x0, 0)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x0), x1, 0)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)
512_1_ack_InvokeMethod(503_0_ack_Return(x0, x1), x2, x0)
512_1_ack_InvokeMethod(400_0_ack_Return(x0, x1), 0, x0)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)

(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


The ITRS R consists of the following rules:
383_0_ack_GT(x1, 0) → 400_0_ack_Return(x1, x1 + 1)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x2), x4, 0) → 503_0_ack_Return(x4, x2)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2) → 544_0_ack_Return(x0)
512_1_ack_InvokeMethod(503_0_ack_Return(x1, x2), x0, x1) → 544_0_ack_Return(x2)
512_1_ack_InvokeMethod(400_0_ack_Return(x1, x2), 0, x1) → 544_0_ack_Return(x2)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x2, x3) → 503_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(1): COND_383_0_ACK_GT(TRUE, x1[1], x0[1]) → 383_0_ACK_GT(1, x0[1] - 1)
(3): COND_383_0_ACK_GT1(TRUE, x1[3], x0[3]) → 441_1_ACK_INVOKEMETHOD(383_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_383_0_ACK_GT1(TRUE, x1[4], x0[4]) → 383_0_ACK_GT(x1[4] - 1, x0[4])
(5): 441_1_ACK_INVOKEMETHOD(503_0_ack_Return(0, x2[5]), x3[5], 0) → 383_0_ACK_GT(x2[5], x3[5])
(6): 441_1_ACK_INVOKEMETHOD(544_0_ack_Return(x0[6]), x1[6], x3[6]) → 383_0_ACK_GT(x0[6], x1[6])

(3) -> (5), if (383_0_ack_GT(x1[3] - 1, x0[3]) →* 503_0_ack_Return(0, x2[5])∧x0[3] - 1 →* x3[5]∧x1[3] - 1 →* 0)


(3) -> (6), if (383_0_ack_GT(x1[3] - 1, x0[3]) →* 544_0_ack_Return(x0[6])∧x0[3] - 1 →* x1[6]∧x1[3] - 1 →* x3[6])



The set Q consists of the following terms:
383_0_ack_GT(x0, 0)
427_1_ack_InvokeMethod(400_0_ack_Return(1, x0), x1, 0)
512_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)
512_1_ack_InvokeMethod(503_0_ack_Return(x0, x1), x2, x0)
512_1_ack_InvokeMethod(400_0_ack_Return(x0, x1), 0, x0)
427_1_ack_InvokeMethod(544_0_ack_Return(x0), x1, x2)

(23) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.

(24) TRUE