Termination proof of /tmp/tmpzhdFqb/Ackermann.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 150 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 380 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 370 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:
375_0_ack_GT(EOS(STATIC_375), i40, i38, i40) → 379_0_ack_GT(EOS(STATIC_379), i40, i38, i40)
379_0_ack_GT(EOS(STATIC_379), i40, i38, i40) → 382_0_ack_Load(EOS(STATIC_382), i40, i38) | >(i40, 0)
382_0_ack_Load(EOS(STATIC_382), i40, i38) → 386_0_ack_GT(EOS(STATIC_386), i40, i38, i38)
386_0_ack_GT(EOS(STATIC_386), i40, i43, i43) → 390_0_ack_GT(EOS(STATIC_390), i40, i43, i43)
386_0_ack_GT(EOS(STATIC_386), i40, i44, i44) → 391_0_ack_GT(EOS(STATIC_391), i40, i44, i44)
390_0_ack_GT(EOS(STATIC_390), i40, i43, i43) → 394_0_ack_Load(EOS(STATIC_394), i40, i43) | <=(i43, 0)
394_0_ack_Load(EOS(STATIC_394), i40, i43) → 401_0_ack_ConstantStackPush(EOS(STATIC_401), i40, i43, i40)
401_0_ack_ConstantStackPush(EOS(STATIC_401), i40, i43, i40) → 408_0_ack_IntArithmetic(EOS(STATIC_408), i40, i43, i40, 1)
408_0_ack_IntArithmetic(EOS(STATIC_408), i40, i43, i40, matching1) → 413_0_ack_ConstantStackPush(EOS(STATIC_413), i40, i43, -(i40, 1)) | &&(>(i40, 0), =(matching1, 1))
413_0_ack_ConstantStackPush(EOS(STATIC_413), i40, i43, i46) → 416_0_ack_InvokeMethod(EOS(STATIC_416), i40, i43, i46, 1)
416_0_ack_InvokeMethod(EOS(STATIC_416), i40, i43, i46, matching1) → 419_1_ack_InvokeMethod(419_0_ack_Load(EOS(STATIC_419), i46, 1), i40, i43, i46, 1) | =(matching1, 1)
419_0_ack_Load(EOS(STATIC_419), i46, matching1) → 423_0_ack_Load(EOS(STATIC_423), i46, 1) | =(matching1, 1)
423_0_ack_Load(EOS(STATIC_423), i46, matching1) → 373_0_ack_Load(EOS(STATIC_373), i46, 1) | =(matching1, 1)
373_0_ack_Load(EOS(STATIC_373), i37, i38) → 375_0_ack_GT(EOS(STATIC_375), i37, i38, i37)
391_0_ack_GT(EOS(STATIC_391), i40, i44, i44) → 396_0_ack_Load(EOS(STATIC_396), i40, i44) | >(i44, 0)
396_0_ack_Load(EOS(STATIC_396), i40, i44) → 402_0_ack_ConstantStackPush(EOS(STATIC_402), i40, i44, i40)
402_0_ack_ConstantStackPush(EOS(STATIC_402), i40, i44, i40) → 410_0_ack_IntArithmetic(EOS(STATIC_410), i40, i44, i40, 1)
410_0_ack_IntArithmetic(EOS(STATIC_410), i40, i44, i40, matching1) → 414_0_ack_Load(EOS(STATIC_414), i40, i44, -(i40, 1)) | &&(>(i40, 0), =(matching1, 1))
414_0_ack_Load(EOS(STATIC_414), i40, i44, i47) → 418_0_ack_Load(EOS(STATIC_418), i44, i47, i40)
418_0_ack_Load(EOS(STATIC_418), i44, i47, i40) → 421_0_ack_ConstantStackPush(EOS(STATIC_421), i47, i40, i44)
421_0_ack_ConstantStackPush(EOS(STATIC_421), i47, i40, i44) → 424_0_ack_IntArithmetic(EOS(STATIC_424), i47, i40, i44, 1)
424_0_ack_IntArithmetic(EOS(STATIC_424), i47, i40, i44, matching1) → 426_0_ack_InvokeMethod(EOS(STATIC_426), i47, i40, -(i44, 1)) | &&(>(i44, 0), =(matching1, 1))
426_0_ack_InvokeMethod(EOS(STATIC_426), i47, i40, i48) → 433_1_ack_InvokeMethod(433_0_ack_Load(EOS(STATIC_433), i40, i48), i47, i40, i48)
433_0_ack_Load(EOS(STATIC_433), i40, i48) → 436_0_ack_Load(EOS(STATIC_436), i40, i48)
433_1_ack_InvokeMethod(438_0_ack_Return(EOS(STATIC_438), i54, matching1, i45), i47, i54, matching2) → 450_0_ack_Return(EOS(STATIC_450), i47, i54, 0, i54, 0, i45) | &&(=(matching1, 0), =(matching2, 0))
433_1_ack_InvokeMethod(494_0_ack_Return(EOS(STATIC_494), i81, matching1, i62), i47, i81, matching2) → 513_0_ack_Return(EOS(STATIC_513), i47, i81, 0, i81, 0, i62) | &&(=(matching1, 0), =(matching2, 0))
433_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), i45), i47, i97, i98) → 550_0_ack_Return(EOS(STATIC_550), i47, i97, i98, i45)
433_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), i45), i47, i112, i113) → 572_0_ack_Return(EOS(STATIC_572), i47, i112, i113, i45)
436_0_ack_Load(EOS(STATIC_436), i40, i48) → 373_0_ack_Load(EOS(STATIC_373), i40, i48)
450_0_ack_Return(EOS(STATIC_450), i47, i54, matching1, i54, matching2, i45) → 454_0_ack_InvokeMethod(EOS(STATIC_454), i47, i45) | &&(=(matching1, 0), =(matching2, 0))
454_0_ack_InvokeMethod(EOS(STATIC_454), i47, i45) → 496_0_ack_InvokeMethod(EOS(STATIC_496), i47, i45)
496_0_ack_InvokeMethod(EOS(STATIC_496), i47, i62) → 502_1_ack_InvokeMethod(502_0_ack_Load(EOS(STATIC_502), i47, i62), i47, i62)
502_0_ack_Load(EOS(STATIC_502), i47, i62) → 507_0_ack_Load(EOS(STATIC_507), i47, i62)
507_0_ack_Load(EOS(STATIC_507), i47, i62) → 373_0_ack_Load(EOS(STATIC_373), i47, i62)
513_0_ack_Return(EOS(STATIC_513), i47, i81, matching1, i81, matching2, i62) → 450_0_ack_Return(EOS(STATIC_450), i47, i81, 0, i81, 0, i62) | &&(=(matching1, 0), =(matching2, 0))
550_0_ack_Return(EOS(STATIC_550), i47, i97, i98, i45) → 488_0_ack_Return(EOS(STATIC_488), i47, i97, i98, i45)
488_0_ack_Return(EOS(STATIC_488), i47, i68, i69, i62) → 496_0_ack_InvokeMethod(EOS(STATIC_496), i47, i62)
572_0_ack_Return(EOS(STATIC_572), i47, i112, i113, i45) → 488_0_ack_Return(EOS(STATIC_488), i47, i112, i113, i45)
R rules:
375_0_ack_GT(EOS(STATIC_375), matching1, i38, matching2) → 378_0_ack_GT(EOS(STATIC_378), 0, i38, 0) | &&(=(matching1, 0), =(matching2, 0))
378_0_ack_GT(EOS(STATIC_378), matching1, i38, matching2) → 380_0_ack_Load(EOS(STATIC_380), 0, i38) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
380_0_ack_Load(EOS(STATIC_380), matching1, i38) → 384_0_ack_ConstantStackPush(EOS(STATIC_384), 0, i38, i38) | =(matching1, 0)
384_0_ack_ConstantStackPush(EOS(STATIC_384), matching1, i38, i38) → 388_0_ack_IntArithmetic(EOS(STATIC_388), 0, i38, i38, 1) | =(matching1, 0)
388_0_ack_IntArithmetic(EOS(STATIC_388), matching1, i38, i38, matching2) → 393_0_ack_Return(EOS(STATIC_393), 0, i38, +(i38, 1)) | &&(=(matching1, 0), =(matching2, 1))
419_1_ack_InvokeMethod(393_0_ack_Return(EOS(STATIC_393), matching1, matching2, i45), i40, i43, matching3, matching4) → 434_0_ack_Return(EOS(STATIC_434), i40, i43, 0, 1, 0, 1, i45) | &&(&&(&&(=(matching1, 0), =(matching2, 1)), =(matching3, 0)), =(matching4, 1))
419_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), i45), i40, i43, i95, matching1) → 548_0_ack_Return(EOS(STATIC_548), i40, i43, i95, 1, i45) | =(matching1, 1)
419_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), i45), i40, i43, i110, matching1) → 570_0_ack_Return(EOS(STATIC_570), i40, i43, i110, 1, i45) | =(matching1, 1)
434_0_ack_Return(EOS(STATIC_434), i40, i43, matching1, matching2, matching3, matching4, i45) → 438_0_ack_Return(EOS(STATIC_438), i40, i43, i45) | &&(&&(&&(=(matching1, 0), =(matching2, 1)), =(matching3, 0)), =(matching4, 1))
438_0_ack_Return(EOS(STATIC_438), i40, i43, i45) → 494_0_ack_Return(EOS(STATIC_494), i40, i43, i45)
487_0_ack_Return(EOS(STATIC_487), i40, i43, i66, matching1, i62) → 494_0_ack_Return(EOS(STATIC_494), i40, i43, i62) | =(matching1, 1)
502_1_ack_InvokeMethod(393_0_ack_Return(EOS(STATIC_393), matching1, i86, i45), matching2, i86) → 523_0_ack_Return(EOS(STATIC_523), 0, i86, 0, i86, i45) | &&(=(matching1, 0), =(matching2, 0))
502_1_ack_InvokeMethod(438_0_ack_Return(EOS(STATIC_438), i87, i88, i45), i87, i88) → 524_0_ack_Return(EOS(STATIC_524), i87, i88, i87, i88, i45)
502_1_ack_InvokeMethod(494_0_ack_Return(EOS(STATIC_494), i90, i91, i89), i90, i91) → 527_0_ack_Return(EOS(STATIC_527), i90, i91, i90, i91, i89)
502_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), i45), i101, i102) → 552_0_ack_Return(EOS(STATIC_552), i101, i102, i45)
502_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), i45), i116, i117) → 574_0_ack_Return(EOS(STATIC_574), i116, i117, i45)
523_0_ack_Return(EOS(STATIC_523), matching1, i86, matching2, i86, i45) → 528_0_ack_Return(EOS(STATIC_528), i45) | &&(=(matching1, 0), =(matching2, 0))
524_0_ack_Return(EOS(STATIC_524), i87, i88, i87, i88, i45) → 531_0_ack_Return(EOS(STATIC_531), i45)
527_0_ack_Return(EOS(STATIC_527), i90, i91, i90, i91, i89) → 524_0_ack_Return(EOS(STATIC_524), i90, i91, i90, i91, i89)
528_0_ack_Return(EOS(STATIC_528), i45) → 531_0_ack_Return(EOS(STATIC_531), i45)
531_0_ack_Return(EOS(STATIC_531), i45) → 555_0_ack_Return(EOS(STATIC_555), i45)
548_0_ack_Return(EOS(STATIC_548), i40, i43, i95, matching1, i45) → 487_0_ack_Return(EOS(STATIC_487), i40, i43, i95, 1, i45) | =(matching1, 1)
552_0_ack_Return(EOS(STATIC_552), i101, i102, i45) → 555_0_ack_Return(EOS(STATIC_555), i45)
570_0_ack_Return(EOS(STATIC_570), i40, i43, i110, matching1, i45) → 487_0_ack_Return(EOS(STATIC_487), i40, i43, i110, 1, i45) | =(matching1, 1)
574_0_ack_Return(EOS(STATIC_574), i116, i117, i45) → 552_0_ack_Return(EOS(STATIC_552), i116, i117, i45)

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

P rules:
375_0_ack_GT(EOS(STATIC_375), x0, x1, x0) → 419_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), -(x0, 1), 1, -(x0, 1)), x0, x1, -(x0, 1), 1) | &&(<=(x1, 0), >(x0, 0))
375_0_ack_GT(EOS(STATIC_375), x0, x1, x0) → 433_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), x0, -(x1, 1), x0), -(x0, 1), x0, -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
433_1_ack_InvokeMethod(438_0_ack_Return(EOS(STATIC_438), x0, 0, x2), x3, x0, 0) → 502_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), x3, x2, x3), x3, x2)
433_1_ack_InvokeMethod(494_0_ack_Return(EOS(STATIC_494), x0, 0, x2), x3, x0, 0) → 502_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), x3, x2, x3), x3, x2)
433_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), x0), x1, x2, x3) → 502_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), x1, x0, x1), x1, x0)
433_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), x0), x1, x2, x3) → 502_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), x1, x0, x1), x1, x0)
R rules:
375_0_ack_GT(EOS(STATIC_375), 0, x1, 0) → 393_0_ack_Return(EOS(STATIC_393), 0, x1, +(x1, 1))
419_1_ack_InvokeMethod(393_0_ack_Return(EOS(STATIC_393), 0, 1, x2), x3, x4, 0, 1) → 494_0_ack_Return(EOS(STATIC_494), x3, x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), x0), x1, x2) → 555_0_ack_Return(EOS(STATIC_555), x0)
502_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), x0), x1, x2) → 555_0_ack_Return(EOS(STATIC_555), x0)
502_1_ack_InvokeMethod(438_0_ack_Return(EOS(STATIC_438), x0, x1, x2), x0, x1) → 555_0_ack_Return(EOS(STATIC_555), x2)
502_1_ack_InvokeMethod(494_0_ack_Return(EOS(STATIC_494), x0, x1, x2), x0, x1) → 555_0_ack_Return(EOS(STATIC_555), x2)
502_1_ack_InvokeMethod(393_0_ack_Return(EOS(STATIC_393), 0, x1, x2), 0, x1) → 555_0_ack_Return(EOS(STATIC_555), x2)
419_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), x0), x1, x2, x3, 1) → 494_0_ack_Return(EOS(STATIC_494), x1, x2, x0)
419_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), x0), x1, x2, x3, 1) → 494_0_ack_Return(EOS(STATIC_494), x1, x2, x0)

Filtered ground terms:

375_0_ack_GT(x1, x2, x3, x4) → 375_0_ack_GT(x2, x3, x4)
555_0_ack_Return(x1, x2) → 555_0_ack_Return(x2)
531_0_ack_Return(x1, x2) → 531_0_ack_Return(x2)
494_0_ack_Return(x1, x2, x3, x4) → 494_0_ack_Return(x2, x3, x4)
438_0_ack_Return(x1, x2, x3, x4) → 438_0_ack_Return(x2, x3, x4)
Cond_375_0_ack_GT1(x1, x2, x3, x4, x5) → Cond_375_0_ack_GT1(x1, x3, x4, x5)
419_1_ack_InvokeMethod(x1, x2, x3, x4, x5) → 419_1_ack_InvokeMethod(x1, x2, x3, x4)
Cond_375_0_ack_GT(x1, x2, x3, x4, x5) → Cond_375_0_ack_GT(x1, x3, x4, x5)
393_0_ack_Return(x1, x2, x3, x4) → 393_0_ack_Return(x3, x4)

Filtered duplicate args:

375_0_ack_GT(x1, x2, x3) → 375_0_ack_GT(x2, x3)
Cond_375_0_ack_GT(x1, x2, x3, x4) → Cond_375_0_ack_GT(x1, x3, x4)
Cond_375_0_ack_GT1(x1, x2, x3, x4) → Cond_375_0_ack_GT1(x1, x3, x4)

Filtered unneeded arguments:

419_1_ack_InvokeMethod(x1, x2, x3, x4) → 419_1_ack_InvokeMethod(x1, x3, x4)
433_1_ack_InvokeMethod(x1, x2, x3, x4) → 433_1_ack_InvokeMethod(x1, x2, x4)
438_0_ack_Return(x1, x2, x3) → 438_0_ack_Return(x2, x3)
494_0_ack_Return(x1, x2, x3) → 494_0_ack_Return(x2, x3)

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

P rules:
375_0_ack_GT(x1, x0) → 419_1_ack_InvokeMethod(375_0_ack_GT(1, -(x0, 1)), x1, -(x0, 1)) | &&(<=(x1, 0), >(x0, 0))
375_0_ack_GT(x1, x0) → 433_1_ack_InvokeMethod(375_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
433_1_ack_InvokeMethod(438_0_ack_Return(0, x2), x3, 0) → 502_1_ack_InvokeMethod(375_0_ack_GT(x2, x3), x3, x2)
433_1_ack_InvokeMethod(494_0_ack_Return(0, x2), x3, 0) → 502_1_ack_InvokeMethod(375_0_ack_GT(x2, x3), x3, x2)
433_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x3) → 502_1_ack_InvokeMethod(375_0_ack_GT(x0, x1), x1, x0)
433_1_ack_InvokeMethod(555_0_ack_Return(x0), x1, x3) → 502_1_ack_InvokeMethod(375_0_ack_GT(x0, x1), x1, x0)
R rules:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, +(x1, 1))
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 555_0_ack_Return(x0)
502_1_ack_InvokeMethod(555_0_ack_Return(x0), x1, x2) → 555_0_ack_Return(x0)
502_1_ack_InvokeMethod(438_0_ack_Return(x1, x2), x0, x1) → 555_0_ack_Return(x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 555_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 555_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)
419_1_ack_InvokeMethod(555_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)

Performed bisimulation on rules. Used the following equivalence classes: {[494_0_ack_Return_2, 438_0_ack_Return_2]=494_0_ack_Return_2, [531_0_ack_Return_1, 555_0_ack_Return_1]=531_0_ack_Return_1}

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

P rules:
375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0)
COND_375_0_ACK_GT(TRUE, x1, x0) → 375_0_ACK_GT(1, -(x0, 1))
375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
COND_375_0_ACK_GT1(TRUE, x1, x0) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1))
COND_375_0_ACK_GT1(TRUE, x1, x0) → 375_0_ACK_GT(-(x1, 1), x0)
433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2), x3, 0) → 375_0_ACK_GT(x2, x3)
433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0), x1, x3) → 375_0_ACK_GT(x0, x1)
R rules:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, +(x1, 1))
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 531_0_ack_Return(x0)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 531_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 531_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_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:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, x1 + 1)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 531_0_ack_Return(x0)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 531_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 531_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(0): 375_0_ACK_GT(x1[0], x0[0]) → COND_375_0_ACK_GT(x1[0] <= 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_375_0_ACK_GT(TRUE, x1[1], x0[1]) → 375_0_ACK_GT(1, x0[1] - 1)
(2): 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_375_0_ACK_GT1(TRUE, x1[3], x0[3]) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])
(5): 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0) → 375_0_ACK_GT(x2[5], x3[5])
(6): 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6]) → 375_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 (375_0_ack_GT(x1[3] - 1, x0[3]) →^* 494_0_ack_Return(0, x2[5])∧x0[3] - 1 →^* x3[5]∧x1[3] - 1 →^* 0)

(3) -> (6), if (375_0_ack_GT(x1[3] - 1, x0[3]) →^* 531_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:
375_0_ack_GT(x0, 0)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x0), x1, 0)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x0, x1), x2, x0)
502_1_ack_InvokeMethod(393_0_ack_Return(x0, x1), 0, x0)
419_1_ack_InvokeMethod(531_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@1f65b124 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 375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0) the following chains were created:

We consider the chain 375_0_ACK_GT(x1[0], x0[0]) → COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_375_0_ACK_GT(TRUE, x1[1], x0[1]) → 375_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] ⇒ 375_0_ACK_GT(x1[0], x0[0])≥NonInfC∧375_0_ACK_GT(x1[0], x0[0])≥COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_375_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 ⇒ 375_0_ACK_GT(x1[0], x0[0])≥NonInfC∧375_0_ACK_GT(x1[0], x0[0])≥COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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_375_0_ACK_GT(TRUE, x1, x0) → 375_0_ACK_GT(1, -(x0, 1)) the following chains were created:

We consider the chain COND_375_0_ACK_GT(TRUE, x1[1], x0[1]) → 375_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:
(8)    (COND_375_0_ACK_GT(TRUE, x1[1], x0[1])≥NonInfC∧COND_375_0_ACK_GT(TRUE, x1[1], x0[1])≥375_0_ACK_GT(1, -(x0[1], 1))∧(U^Increasing(375_0_ACK_GT(1, -(x0[1], 1))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(375_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)    ((U^Increasing(375_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)    ((U^Increasing(375_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)    ((U^Increasing(375_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 0)

For Pair 375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:

We consider the chain 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_375_0_ACK_GT1(TRUE, x1[3], x0[3]) → 433_1_ACK_INVOKEMETHOD(375_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] ⇒ 375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_375_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 ⇒ 375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_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] ⇒ 375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_375_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 ⇒ 375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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_375_0_ACK_GT1(TRUE, x1, x0) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1)) the following chains were created:

We consider the chain COND_375_0_ACK_GT1(TRUE, x1[3], x0[3]) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1)) which results in the following constraint:
(27)    (COND_375_0_ACK_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_375_0_ACK_GT1(TRUE, x1[3], x0[3])≥433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))∧(U^Increasing(433_1_ACK_INVOKEMETHOD(375_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)    ((U^Increasing(433_1_ACK_INVOKEMETHOD(375_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)    ((U^Increasing(433_1_ACK_INVOKEMETHOD(375_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)    ((U^Increasing(433_1_ACK_INVOKEMETHOD(375_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)    ((U^Increasing(433_1_ACK_INVOKEMETHOD(375_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_375_0_ACK_GT1(TRUE, x1, x0) → 375_0_ACK_GT(-(x1, 1), x0) the following chains were created:

We consider the chain COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:
(32)    (COND_375_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_375_0_ACK_GT1(TRUE, x1[4], x0[4])≥375_0_ACK_GT(-(x1[4], 1), x0[4])∧(U^Increasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

We simplified constraint (32) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(33)    ((U^Increasing(375_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)    ((U^Increasing(375_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)    ((U^Increasing(375_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)    ((U^Increasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

For Pair 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2), x3, 0) → 375_0_ACK_GT(x2, x3) the following chains were created:

We consider the chain 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0) → 375_0_ACK_GT(x2[5], x3[5]) which results in the following constraint:
(37)    (433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0)≥NonInfC∧433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0)≥375_0_ACK_GT(x2[5], x3[5])∧(U^Increasing(375_0_ACK_GT(x2[5], x3[5])), ≥))

We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(38)    ((U^Increasing(375_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)    ((U^Increasing(375_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)    ((U^Increasing(375_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)    ((U^Increasing(375_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧0 = 0∧0 = 0∧[(-1)bso_33] ≥ 0)

For Pair 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0), x1, x3) → 375_0_ACK_GT(x0, x1) the following chains were created:

We consider the chain 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6]) → 375_0_ACK_GT(x0[6], x1[6]) which results in the following constraint:
(42)    (433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6])≥NonInfC∧433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6])≥375_0_ACK_GT(x0[6], x1[6])∧(U^Increasing(375_0_ACK_GT(x0[6], x1[6])), ≥))

We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(43)    ((U^Increasing(375_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)    ((U^Increasing(375_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)    ((U^Increasing(375_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)    ((U^Increasing(375_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.

375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_375_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_375_0_ACK_GT(TRUE, x1, x0) → 375_0_ACK_GT(1, -(x0, 1))
- ((U^Increasing(375_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 0)
375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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_375_0_ACK_GT1(TRUE, x1, x0) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1))
- ((U^Increasing(433_1_ACK_INVOKEMETHOD(375_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_375_0_ACK_GT1(TRUE, x1, x0) → 375_0_ACK_GT(-(x1, 1), x0)
- ((U^Increasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)
433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2), x3, 0) → 375_0_ACK_GT(x2, x3)
- ((U^Increasing(375_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧0 = 0∧0 = 0∧[(-1)bso_33] ≥ 0)
433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0), x1, x3) → 375_0_ACK_GT(x0, x1)
- ((U^Increasing(375_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 P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(375_0_ack_GT(x₁, x₂)) = [2] + [2]x₂ + x₁
POL(0) = 0
POL(393_0_ack_Return(x₁, x₂)) = [-1] + [-1]x₂ + [2]x₁
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(419_1_ack_InvokeMethod(x₁, x₂, x₃)) = [-1]
POL(494_0_ack_Return(x₁, x₂)) = x₂
POL(502_1_ack_InvokeMethod(x₁, x₂, x₃)) = [-1]
POL(531_0_ack_Return(x₁)) = x₁
POL(375_0_ACK_GT(x₁, x₂)) = [-1] + x₂
POL(COND_375_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_375_0_ACK_GT1(x₁, x₂, x₃)) = [-1] + x₃
POL(433_1_ACK_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + x₂

The following pairs are in P_>:

COND_375_0_ACK_GT(TRUE, x1[1], x0[1]) → 375_0_ACK_GT(1, -(x0[1], 1))
COND_375_0_ACK_GT1(TRUE, x1[3], x0[3]) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))

The following pairs are in P_bound:

375_0_ACK_GT(x1[0], x0[0]) → COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P_≥:

375_0_ACK_GT(x1[0], x0[0]) → COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4])
433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0) → 375_0_ACK_GT(x2[5], x3[5])
433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6]) → 375_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:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, x1 + 1)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 531_0_ack_Return(x0)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 531_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 531_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(0): 375_0_ACK_GT(x1[0], x0[0]) → COND_375_0_ACK_GT(x1[0] <= 0 && x0[0] > 0, x1[0], x0[0])
(2): 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(4): COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])
(5): 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0) → 375_0_ACK_GT(x2[5], x3[5])
(6): 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6]) → 375_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])

(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

(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])

(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_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])
(2): 375_0_ACK_GT(x1[2], x0[2]) → COND_375_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])

(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@1f65b124 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_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4]) the following chains were created:

We consider the chain COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:
(1)    (COND_375_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_375_0_ACK_GT1(TRUE, x1[4], x0[4])≥375_0_ACK_GT(-(x1[4], 1), x0[4])∧(U^Increasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(375_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)    ((U^Increasing(375_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)    ((U^Increasing(375_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)    ((U^Increasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)

For Pair 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) the following chains were created:

We consider the chain 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_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] ⇒ 375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_375_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 ⇒ 375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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 ⇒ (U^Increasing(COND_375_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_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4])
- ((U^Increasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)
375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_375_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)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_375_0_ACK_GT1(x₁, x₂, x₃)) = [-1] + x₃ + x₂
POL(375_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_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4])

The following pairs are in P_bound:

375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P_≥:

375_0_ACK_GT(x1[2], x0[2]) → COND_375_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): 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])

The set Q consists of the following terms:
375_0_ack_GT(x0, 0)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x0), x1, 0)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x0, x1), x2, x0)
502_1_ack_InvokeMethod(393_0_ack_Return(x0, x1), 0, x0)
419_1_ack_InvokeMethod(531_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_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])

The set Q consists of the following terms:
375_0_ack_GT(x0, 0)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x0), x1, 0)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x0, x1), x2, x0)
502_1_ack_InvokeMethod(393_0_ack_Return(x0, x1), 0, x0)
419_1_ack_InvokeMethod(531_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:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, x1 + 1)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 531_0_ack_Return(x0)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 531_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 531_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(1): COND_375_0_ACK_GT(TRUE, x1[1], x0[1]) → 375_0_ACK_GT(1, x0[1] - 1)
(3): COND_375_0_ACK_GT1(TRUE, x1[3], x0[3]) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])
(5): 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0) → 375_0_ACK_GT(x2[5], x3[5])
(6): 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6]) → 375_0_ACK_GT(x0[6], x1[6])

(3) -> (5), if (375_0_ack_GT(x1[3] - 1, x0[3]) →^* 494_0_ack_Return(0, x2[5])∧x0[3] - 1 →^* x3[5]∧x1[3] - 1 →^* 0)

(3) -> (6), if (375_0_ack_GT(x1[3] - 1, x0[3]) →^* 531_0_ack_Return(x0[6])∧x0[3] - 1 →^* x1[6]∧x1[3] - 1 →^* x3[6])

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Obligation:

(5) SCCToIDPv1Proof (SOUND transformation)

(6) Obligation:

(7) IDPNonInfProof (SOUND transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) Obligation:

(12) UsableRulesProof (EQUIVALENT transformation)

(13) Obligation:

(14) IDPNonInfProof (SOUND transformation)

(15) Complex Obligation (AND)

(16) Obligation:

(17) IDependencyGraphProof (EQUIVALENT transformation)

(18) TRUE

(19) Obligation:

(20) IDependencyGraphProof (EQUIVALENT transformation)

(21) TRUE

(22) Obligation:

(23) IDependencyGraphProof (EQUIVALENT transformation)

(24) TRUE