Termination proof of /tmp/tmpcFmZmd/AckR.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 200 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 300 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 310 ms)

↳8 AND

    ↳9 IDP

        ↳10 IDependencyGraphProof (⇔, 0 ms)

        ↳11 IDP

        ↳12 UsableRulesProof (⇔, 0 ms)

        ↳13 IDP

        ↳14 IDPNonInfProof (⇒, 30 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_20 (Apple Inc.) Main-Class: AckermannR

public class AckermannR {

    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) {
	Random.args = args;
	ack(Random.random(),Random.random());
    }
}

public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
      if (index >= args.length)
	  return 0;

      String string = args[index];
      index++;
      return string.length();
  }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
AckermannR.main([Ljava/lang/String;)V: Graph of 167 nodes with 0 SCCs.

AckermannR.ack(II)I: Graph of 52 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: AckermannR.ack(II)I
SCC calls the following helper methods: AckermannR.ack(II)I
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 38 rules for P and 20 rules for R.

P rules:
429_0_ack_GT(EOS(STATIC_429), i78, i73, i78) → 432_0_ack_GT(EOS(STATIC_432), i78, i73, i78)
432_0_ack_GT(EOS(STATIC_432), i78, i73, i78) → 438_0_ack_Load(EOS(STATIC_438), i78, i73) | >(i78, 0)
438_0_ack_Load(EOS(STATIC_438), i78, i73) → 445_0_ack_GT(EOS(STATIC_445), i78, i73, i73)
445_0_ack_GT(EOS(STATIC_445), i78, matching1, matching2) → 452_0_ack_GT(EOS(STATIC_452), i78, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
445_0_ack_GT(EOS(STATIC_445), i78, i81, i81) → 453_0_ack_GT(EOS(STATIC_453), i78, i81, i81)
452_0_ack_GT(EOS(STATIC_452), i78, matching1, matching2) → 460_0_ack_Load(EOS(STATIC_460), i78, 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
460_0_ack_Load(EOS(STATIC_460), i78, matching1) → 473_0_ack_ConstantStackPush(EOS(STATIC_473), i78, 0, i78) | =(matching1, 0)
473_0_ack_ConstantStackPush(EOS(STATIC_473), i78, matching1, i78) → 496_0_ack_IntArithmetic(EOS(STATIC_496), i78, 0, i78, 1) | =(matching1, 0)
496_0_ack_IntArithmetic(EOS(STATIC_496), i78, matching1, i78, matching2) → 526_0_ack_ConstantStackPush(EOS(STATIC_526), i78, 0, -(i78, 1)) | &&(&&(>(i78, 0), =(matching1, 0)), =(matching2, 1))
526_0_ack_ConstantStackPush(EOS(STATIC_526), i78, matching1, i95) → 539_0_ack_InvokeMethod(EOS(STATIC_539), i78, 0, i95, 1) | =(matching1, 0)
539_0_ack_InvokeMethod(EOS(STATIC_539), i78, matching1, i95, matching2) → 552_1_ack_InvokeMethod(552_0_ack_Load(EOS(STATIC_552), i95, 1), i78, 0, i95, 1) | &&(=(matching1, 0), =(matching2, 1))
552_0_ack_Load(EOS(STATIC_552), i95, matching1) → 564_0_ack_Load(EOS(STATIC_564), i95, 1) | =(matching1, 1)
564_0_ack_Load(EOS(STATIC_564), i95, matching1) → 424_0_ack_Load(EOS(STATIC_424), i95, 1) | =(matching1, 1)
424_0_ack_Load(EOS(STATIC_424), i72, i73) → 429_0_ack_GT(EOS(STATIC_429), i72, i73, i72)
453_0_ack_GT(EOS(STATIC_453), i78, i81, i81) → 461_0_ack_Load(EOS(STATIC_461), i78, i81) | >(i81, 0)
461_0_ack_Load(EOS(STATIC_461), i78, i81) → 476_0_ack_ConstantStackPush(EOS(STATIC_476), i78, i81, i78)
476_0_ack_ConstantStackPush(EOS(STATIC_476), i78, i81, i78) → 498_0_ack_IntArithmetic(EOS(STATIC_498), i78, i81, i78, 1)
498_0_ack_IntArithmetic(EOS(STATIC_498), i78, i81, i78, matching1) → 528_0_ack_Load(EOS(STATIC_528), i78, i81, -(i78, 1)) | &&(>(i78, 0), =(matching1, 1))
528_0_ack_Load(EOS(STATIC_528), i78, i81, i96) → 541_0_ack_Load(EOS(STATIC_541), i81, i96, i78)
541_0_ack_Load(EOS(STATIC_541), i81, i96, i78) → 554_0_ack_ConstantStackPush(EOS(STATIC_554), i96, i78, i81)
554_0_ack_ConstantStackPush(EOS(STATIC_554), i96, i78, i81) → 566_0_ack_IntArithmetic(EOS(STATIC_566), i96, i78, i81, 1)
566_0_ack_IntArithmetic(EOS(STATIC_566), i96, i78, i81, matching1) → 571_0_ack_InvokeMethod(EOS(STATIC_571), i96, i78, -(i81, 1)) | &&(>(i81, 0), =(matching1, 1))
571_0_ack_InvokeMethod(EOS(STATIC_571), i96, i78, i101) → 584_1_ack_InvokeMethod(584_0_ack_Load(EOS(STATIC_584), i78, i101), i96, i78, i101)
584_0_ack_Load(EOS(STATIC_584), i78, i101) → 590_0_ack_Load(EOS(STATIC_590), i78, i101)
584_1_ack_InvokeMethod(594_0_ack_Return(EOS(STATIC_594), i114, matching1, i82), i96, i114, matching2) → 621_0_ack_Return(EOS(STATIC_621), i96, i114, 0, i114, 0, i82) | &&(=(matching1, 0), =(matching2, 0))
584_1_ack_InvokeMethod(709_0_ack_Return(EOS(STATIC_709), i178, matching1, i133), i96, i178, matching2) → 755_0_ack_Return(EOS(STATIC_755), i96, i178, 0, i178, 0, i133) | &&(=(matching1, 0), =(matching2, 0))
584_1_ack_InvokeMethod(772_0_ack_Return(EOS(STATIC_772), i82), i96, i213, i214) → 807_0_ack_Return(EOS(STATIC_807), i96, i213, i214, i82)
584_1_ack_InvokeMethod(813_0_ack_Return(EOS(STATIC_813), i82), i96, i252, i253) → 850_0_ack_Return(EOS(STATIC_850), i96, i252, i253, i82)
590_0_ack_Load(EOS(STATIC_590), i78, i101) → 424_0_ack_Load(EOS(STATIC_424), i78, i101)
621_0_ack_Return(EOS(STATIC_621), i96, i114, matching1, i114, matching2, i82) → 629_0_ack_InvokeMethod(EOS(STATIC_629), i96, i82) | &&(=(matching1, 0), =(matching2, 0))
629_0_ack_InvokeMethod(EOS(STATIC_629), i96, i82) → 713_0_ack_InvokeMethod(EOS(STATIC_713), i96, i82)
713_0_ack_InvokeMethod(EOS(STATIC_713), i96, i133) → 719_1_ack_InvokeMethod(719_0_ack_Load(EOS(STATIC_719), i96, i133), i96, i133)
719_0_ack_Load(EOS(STATIC_719), i96, i133) → 741_0_ack_Load(EOS(STATIC_741), i96, i133)
741_0_ack_Load(EOS(STATIC_741), i96, i133) → 424_0_ack_Load(EOS(STATIC_424), i96, i133)
755_0_ack_Return(EOS(STATIC_755), i96, i178, matching1, i178, matching2, i133) → 621_0_ack_Return(EOS(STATIC_621), i96, i178, 0, i178, 0, i133) | &&(=(matching1, 0), =(matching2, 0))
807_0_ack_Return(EOS(STATIC_807), i96, i213, i214, i82) → 699_0_ack_Return(EOS(STATIC_699), i96, i213, i214, i82)
699_0_ack_Return(EOS(STATIC_699), i96, i148, i149, i133) → 713_0_ack_InvokeMethod(EOS(STATIC_713), i96, i133)
850_0_ack_Return(EOS(STATIC_850), i96, i252, i253, i82) → 699_0_ack_Return(EOS(STATIC_699), i96, i252, i253, i82)
R rules:
429_0_ack_GT(EOS(STATIC_429), matching1, i73, matching2) → 431_0_ack_GT(EOS(STATIC_431), 0, i73, 0) | &&(=(matching1, 0), =(matching2, 0))
431_0_ack_GT(EOS(STATIC_431), matching1, i73, matching2) → 436_0_ack_Load(EOS(STATIC_436), 0, i73) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
436_0_ack_Load(EOS(STATIC_436), matching1, i73) → 443_0_ack_ConstantStackPush(EOS(STATIC_443), 0, i73, i73) | =(matching1, 0)
443_0_ack_ConstantStackPush(EOS(STATIC_443), matching1, i73, i73) → 450_0_ack_IntArithmetic(EOS(STATIC_450), 0, i73, i73, 1) | =(matching1, 0)
450_0_ack_IntArithmetic(EOS(STATIC_450), matching1, i73, i73, matching2) → 457_0_ack_Return(EOS(STATIC_457), 0, i73, +(i73, 1)) | &&(&&(>=(i73, 0), =(matching1, 0)), =(matching2, 1))
552_1_ack_InvokeMethod(457_0_ack_Return(EOS(STATIC_457), matching1, matching2, i82), i78, matching3, matching4, matching5) → 588_0_ack_Return(EOS(STATIC_588), i78, 0, 0, 1, 0, 1, i82) | &&(&&(&&(&&(=(matching1, 0), =(matching2, 1)), =(matching3, 0)), =(matching4, 0)), =(matching5, 1))
552_1_ack_InvokeMethod(772_0_ack_Return(EOS(STATIC_772), i82), i78, matching1, i209, matching2) → 804_0_ack_Return(EOS(STATIC_804), i78, 0, i209, 1, i82) | &&(=(matching1, 0), =(matching2, 1))
552_1_ack_InvokeMethod(813_0_ack_Return(EOS(STATIC_813), i82), i78, matching1, i248, matching2) → 848_0_ack_Return(EOS(STATIC_848), i78, 0, i248, 1, i82) | &&(=(matching1, 0), =(matching2, 1))
588_0_ack_Return(EOS(STATIC_588), i78, matching1, matching2, matching3, matching4, matching5, i82) → 594_0_ack_Return(EOS(STATIC_594), i78, 0, i82) | &&(&&(&&(&&(=(matching1, 0), =(matching2, 0)), =(matching3, 1)), =(matching4, 0)), =(matching5, 1))
594_0_ack_Return(EOS(STATIC_594), i78, matching1, i82) → 709_0_ack_Return(EOS(STATIC_709), i78, 0, i82) | =(matching1, 0)
696_0_ack_Return(EOS(STATIC_696), i78, matching1, i144, matching2, i133) → 709_0_ack_Return(EOS(STATIC_709), i78, 0, i133) | &&(=(matching1, 0), =(matching2, 1))
719_1_ack_InvokeMethod(457_0_ack_Return(EOS(STATIC_457), matching1, i197, i82), matching2, i197) → 767_0_ack_Return(EOS(STATIC_767), 0, i197, 0, i197, i82) | &&(=(matching1, 0), =(matching2, 0))
719_1_ack_InvokeMethod(772_0_ack_Return(EOS(STATIC_772), i82), i220, i221) → 809_0_ack_Return(EOS(STATIC_809), i220, i221, i82)
719_1_ack_InvokeMethod(813_0_ack_Return(EOS(STATIC_813), i82), i259, i260) → 853_0_ack_Return(EOS(STATIC_853), i259, i260, i82)
767_0_ack_Return(EOS(STATIC_767), matching1, i197, matching2, i197, i82) → 772_0_ack_Return(EOS(STATIC_772), i82) | &&(=(matching1, 0), =(matching2, 0))
772_0_ack_Return(EOS(STATIC_772), i82) → 813_0_ack_Return(EOS(STATIC_813), i82)
804_0_ack_Return(EOS(STATIC_804), i78, matching1, i209, matching2, i82) → 696_0_ack_Return(EOS(STATIC_696), i78, 0, i209, 1, i82) | &&(=(matching1, 0), =(matching2, 1))
809_0_ack_Return(EOS(STATIC_809), i220, i221, i82) → 813_0_ack_Return(EOS(STATIC_813), i82)
848_0_ack_Return(EOS(STATIC_848), i78, matching1, i248, matching2, i82) → 696_0_ack_Return(EOS(STATIC_696), i78, 0, i248, 1, i82) | &&(=(matching1, 0), =(matching2, 1))
853_0_ack_Return(EOS(STATIC_853), i259, i260, i82) → 809_0_ack_Return(EOS(STATIC_809), i259, i260, i82)

Combined rules. Obtained 6 conditional rules for P and 7 conditional rules for R.

P rules:
429_0_ack_GT(EOS(STATIC_429), x0, 0, x0) → 552_1_ack_InvokeMethod(429_0_ack_GT(EOS(STATIC_429), -(x0, 1), 1, -(x0, 1)), x0, 0, -(x0, 1), 1) | >(x0, 0)
429_0_ack_GT(EOS(STATIC_429), x0, x1, x0) → 584_1_ack_InvokeMethod(429_0_ack_GT(EOS(STATIC_429), x0, -(x1, 1), x0), -(x0, 1), x0, -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
584_1_ack_InvokeMethod(594_0_ack_Return(EOS(STATIC_594), x0, 0, x2), x3, x0, 0) → 719_1_ack_InvokeMethod(429_0_ack_GT(EOS(STATIC_429), x3, x2, x3), x3, x2)
584_1_ack_InvokeMethod(709_0_ack_Return(EOS(STATIC_709), x0, 0, x2), x3, x0, 0) → 719_1_ack_InvokeMethod(429_0_ack_GT(EOS(STATIC_429), x3, x2, x3), x3, x2)
584_1_ack_InvokeMethod(772_0_ack_Return(EOS(STATIC_772), x0), x1, x2, x3) → 719_1_ack_InvokeMethod(429_0_ack_GT(EOS(STATIC_429), x1, x0, x1), x1, x0)
584_1_ack_InvokeMethod(813_0_ack_Return(EOS(STATIC_813), x0), x1, x2, x3) → 719_1_ack_InvokeMethod(429_0_ack_GT(EOS(STATIC_429), x1, x0, x1), x1, x0)
R rules:
429_0_ack_GT(EOS(STATIC_429), 0, x1, 0) → 457_0_ack_Return(EOS(STATIC_457), 0, x1, +(x1, 1)) | >(+(x1, 1), 0)
552_1_ack_InvokeMethod(457_0_ack_Return(EOS(STATIC_457), 0, 1, x2), x3, 0, 0, 1) → 709_0_ack_Return(EOS(STATIC_709), x3, 0, x2)
719_1_ack_InvokeMethod(772_0_ack_Return(EOS(STATIC_772), x0), x1, x2) → 813_0_ack_Return(EOS(STATIC_813), x0)
719_1_ack_InvokeMethod(813_0_ack_Return(EOS(STATIC_813), x0), x1, x2) → 813_0_ack_Return(EOS(STATIC_813), x0)
719_1_ack_InvokeMethod(457_0_ack_Return(EOS(STATIC_457), 0, x1, x2), 0, x1) → 813_0_ack_Return(EOS(STATIC_813), x2)
552_1_ack_InvokeMethod(772_0_ack_Return(EOS(STATIC_772), x0), x1, 0, x3, 1) → 709_0_ack_Return(EOS(STATIC_709), x1, 0, x0)
552_1_ack_InvokeMethod(813_0_ack_Return(EOS(STATIC_813), x0), x1, 0, x3, 1) → 709_0_ack_Return(EOS(STATIC_709), x1, 0, x0)

Filtered ground terms:

429_0_ack_GT(x1, x2, x3, x4) → 429_0_ack_GT(x2, x3, x4)
813_0_ack_Return(x1, x2) → 813_0_ack_Return(x2)
772_0_ack_Return(x1, x2) → 772_0_ack_Return(x2)
709_0_ack_Return(x1, x2, x3, x4) → 709_0_ack_Return(x2, x4)
594_0_ack_Return(x1, x2, x3, x4) → 594_0_ack_Return(x2, x4)
Cond_429_0_ack_GT1(x1, x2, x3, x4, x5) → Cond_429_0_ack_GT1(x1, x3, x4, x5)
552_1_ack_InvokeMethod(x1, x2, x3, x4, x5) → 552_1_ack_InvokeMethod(x1, x2, x4)
Cond_429_0_ack_GT(x1, x2, x3, x4, x5) → Cond_429_0_ack_GT(x1, x3, x4, x5)
457_0_ack_Return(x1, x2, x3, x4) → 457_0_ack_Return(x3, x4)

Filtered duplicate args:

429_0_ack_GT(x1, x2, x3) → 429_0_ack_GT(x2, x3)
Cond_429_0_ack_GT(x1, x2, x3, x4) → Cond_429_0_ack_GT(x1, x3, x4)
Cond_429_0_ack_GT1(x1, x2, x3, x4) → Cond_429_0_ack_GT1(x1, x3, x4)

Filtered unneeded arguments:

552_1_ack_InvokeMethod(x1, x2, x3) → 552_1_ack_InvokeMethod(x1, x3)
584_1_ack_InvokeMethod(x1, x2, x3, x4) → 584_1_ack_InvokeMethod(x1, x2, x4)
709_0_ack_Return(x1, x2) → 709_0_ack_Return(x2)
594_0_ack_Return(x1, x2) → 594_0_ack_Return(x2)

Combined rules. Obtained 6 conditional rules for P and 7 conditional rules for R.

P rules:
429_0_ack_GT(0, x0) → 552_1_ack_InvokeMethod(429_0_ack_GT(1, -(x0, 1)), -(x0, 1)) | >(x0, 0)
429_0_ack_GT(x1, x0) → 584_1_ack_InvokeMethod(429_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
584_1_ack_InvokeMethod(594_0_ack_Return(x2), x3, 0) → 719_1_ack_InvokeMethod(429_0_ack_GT(x2, x3), x3, x2)
584_1_ack_InvokeMethod(709_0_ack_Return(x2), x3, 0) → 719_1_ack_InvokeMethod(429_0_ack_GT(x2, x3), x3, x2)
584_1_ack_InvokeMethod(772_0_ack_Return(x0), x1, x3) → 719_1_ack_InvokeMethod(429_0_ack_GT(x0, x1), x1, x0)
584_1_ack_InvokeMethod(813_0_ack_Return(x0), x1, x3) → 719_1_ack_InvokeMethod(429_0_ack_GT(x0, x1), x1, x0)
R rules:
429_0_ack_GT(x1, 0) → 457_0_ack_Return(x1, +(x1, 1)) | >(x1, -1)
552_1_ack_InvokeMethod(457_0_ack_Return(1, x2), 0) → 709_0_ack_Return(x2)
719_1_ack_InvokeMethod(772_0_ack_Return(x0), x1, x2) → 813_0_ack_Return(x0)
719_1_ack_InvokeMethod(813_0_ack_Return(x0), x1, x2) → 813_0_ack_Return(x0)
719_1_ack_InvokeMethod(457_0_ack_Return(x1, x2), 0, x1) → 813_0_ack_Return(x2)
552_1_ack_InvokeMethod(772_0_ack_Return(x0), x3) → 709_0_ack_Return(x0)
552_1_ack_InvokeMethod(813_0_ack_Return(x0), x3) → 709_0_ack_Return(x0)

Performed bisimulation on rules. Used the following equivalence classes: {[709_0_ack_Return_1, 772_0_ack_Return_1, 813_0_ack_Return_1, 594_0_ack_Return_1]=709_0_ack_Return_1}

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

P rules:
429_0_ACK_GT(0, x0) → COND_429_0_ACK_GT(>(x0, 0), 0, x0)
COND_429_0_ACK_GT(TRUE, 0, x0) → 429_0_ACK_GT(1, -(x0, 1))
429_0_ACK_GT(x1, x0) → COND_429_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
COND_429_0_ACK_GT1(TRUE, x1, x0) → 584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1))
COND_429_0_ACK_GT1(TRUE, x1, x0) → 429_0_ACK_GT(-(x1, 1), x0)
584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x2), x3, 0) → 429_0_ACK_GT(x2, x3)
584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x0), x1, x3) → 429_0_ACK_GT(x0, x1)
R rules:
429_0_ack_GT(x1, 0) → Cond_429_0_ack_GT(>(x1, -1), x1, 0)
Cond_429_0_ack_GT(TRUE, x1, 0) → 457_0_ack_Return(x1, +(x1, 1))
552_1_ack_InvokeMethod(457_0_ack_Return(1, x2), 0) → 709_0_ack_Return(x2)
719_1_ack_InvokeMethod(709_0_ack_Return(x0), x1, x2) → 709_0_ack_Return(x0)
719_1_ack_InvokeMethod(457_0_ack_Return(x1, x2), 0, x1) → 709_0_ack_Return(x2)
552_1_ack_InvokeMethod(709_0_ack_Return(x0), x3) → 709_0_ack_Return(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:
429_0_ack_GT(x1, 0) → Cond_429_0_ack_GT(x1 > -1, x1, 0)
Cond_429_0_ack_GT(TRUE, x1, 0) → 457_0_ack_Return(x1, x1 + 1)
552_1_ack_InvokeMethod(457_0_ack_Return(1, x2), 0) → 709_0_ack_Return(x2)
719_1_ack_InvokeMethod(709_0_ack_Return(x0), x1, x2) → 709_0_ack_Return(x0)
719_1_ack_InvokeMethod(457_0_ack_Return(x1, x2), 0, x1) → 709_0_ack_Return(x2)
552_1_ack_InvokeMethod(709_0_ack_Return(x0), x3) → 709_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 429_0_ACK_GT(0, x0[0]) → COND_429_0_ACK_GT(x0[0] > 0, 0, x0[0])
(1): COND_429_0_ACK_GT(TRUE, 0, x0[1]) → 429_0_ACK_GT(1, x0[1] - 1)
(2): 429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_429_0_ACK_GT1(TRUE, x1[3], x0[3]) → 584_1_ACK_INVOKEMETHOD(429_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(x1[4] - 1, x0[4])
(5): 584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x2[5]), x3[5], 0) → 429_0_ACK_GT(x2[5], x3[5])
(6): 584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x0[6]), x1[6], x3[6]) → 429_0_ACK_GT(x0[6], x1[6])

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

(1) -> (0), if (1 →^* 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 (429_0_ack_GT(x1[3] - 1, x0[3]) →^* 709_0_ack_Return(x2[5])∧x0[3] - 1 →^* x3[5]∧x1[3] - 1 →^* 0)

(3) -> (6), if (429_0_ack_GT(x1[3] - 1, x0[3]) →^* 709_0_ack_Return(x0[6])∧x0[3] - 1 →^* x1[6]∧x1[3] - 1 →^* x3[6])

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

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

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

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

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

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

The set Q consists of the following terms:
429_0_ack_GT(x0, 0)
Cond_429_0_ack_GT(TRUE, x0, 0)
552_1_ack_InvokeMethod(457_0_ack_Return(1, x0), 0)
719_1_ack_InvokeMethod(709_0_ack_Return(x0), x1, x2)
719_1_ack_InvokeMethod(457_0_ack_Return(x0, x1), 0, x0)
552_1_ack_InvokeMethod(709_0_ack_Return(x0), x1)

(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@17cdc4a5 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 429_0_ACK_GT(0, x0) → COND_429_0_ACK_GT(>(x0, 0), 0, x0) the following chains were created:

We consider the chain 429_0_ACK_GT(0, x0[0]) → COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0]), COND_429_0_ACK_GT(TRUE, 0, x0[1]) → 429_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:
(1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 429_0_ACK_GT(0, x0[0])≥NonInfC∧429_0_ACK_GT(0, x0[0])≥COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])∧(U^Increasing(COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE ⇒ 429_0_ACK_GT(0, x0[0])≥NonInfC∧429_0_ACK_GT(0, x0[0])≥COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])∧(U^Increasing(COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)

For Pair COND_429_0_ACK_GT(TRUE, 0, x0) → 429_0_ACK_GT(1, -(x0, 1)) the following chains were created:

We consider the chain COND_429_0_ACK_GT(TRUE, 0, x0[1]) → 429_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:
(7)    (COND_429_0_ACK_GT(TRUE, 0, x0[1])≥NonInfC∧COND_429_0_ACK_GT(TRUE, 0, x0[1])≥429_0_ACK_GT(1, -(x0[1], 1))∧(U^Increasing(429_0_ACK_GT(1, -(x0[1], 1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(429_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_25] = 0∧[1 + (-1)bso_26] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(429_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_25] = 0∧[1 + (-1)bso_26] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(429_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_25] = 0∧[1 + (-1)bso_26] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(429_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_25] = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

For Pair 429_0_ACK_GT(x1, x0) → COND_429_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:

We consider the chain 429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_429_0_ACK_GT1(TRUE, x1[3], x0[3]) → 584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1)) which results in the following constraint:
(12)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3] ⇒ 429_0_ACK_GT(x1[2], x0[2])≥NonInfC∧429_0_ACK_GT(x1[2], x0[2])≥COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

We simplified constraint (12) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(13)    (>(x1[2], 0)=TRUE∧>(x0[2], 0)=TRUE ⇒ 429_0_ACK_GT(x1[2], x0[2])≥NonInfC∧429_0_ACK_GT(x1[2], x0[2])≥COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(17)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)
We consider the chain 429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:
(19)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4] ⇒ 429_0_ACK_GT(x1[2], x0[2])≥NonInfC∧429_0_ACK_GT(x1[2], x0[2])≥COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

We simplified constraint (19) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(20)    (>(x1[2], 0)=TRUE∧>(x0[2], 0)=TRUE ⇒ 429_0_ACK_GT(x1[2], x0[2])≥NonInfC∧429_0_ACK_GT(x1[2], x0[2])≥COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (23) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(24)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (24) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(25)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

For Pair COND_429_0_ACK_GT1(TRUE, x1, x0) → 584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1)) the following chains were created:

We consider the chain COND_429_0_ACK_GT1(TRUE, x1[3], x0[3]) → 584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1)) which results in the following constraint:
(26)    (COND_429_0_ACK_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_429_0_ACK_GT1(TRUE, x1[3], x0[3])≥584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))∧(U^Increasing(584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥))

We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27)    ((U^Increasing(584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28)    ((U^Increasing(584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29)    ((U^Increasing(584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (29) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(30)    ((U^Increasing(584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_29] = 0∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

For Pair COND_429_0_ACK_GT1(TRUE, x1, x0) → 429_0_ACK_GT(-(x1, 1), x0) the following chains were created:

We consider the chain COND_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:
(31)    (COND_429_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_429_0_ACK_GT1(TRUE, x1[4], x0[4])≥429_0_ACK_GT(-(x1[4], 1), x0[4])∧(U^Increasing(429_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

We simplified constraint (31) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(32)    ((U^Increasing(429_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_31] = 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (32) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(33)    ((U^Increasing(429_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_31] = 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (33) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(34)    ((U^Increasing(429_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_31] = 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (34) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(35)    ((U^Increasing(429_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_31] = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 0)

For Pair 584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x2), x3, 0) → 429_0_ACK_GT(x2, x3) the following chains were created:

We consider the chain 584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x2[5]), x3[5], 0) → 429_0_ACK_GT(x2[5], x3[5]) which results in the following constraint:
(36)    (584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x2[5]), x3[5], 0)≥NonInfC∧584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x2[5]), x3[5], 0)≥429_0_ACK_GT(x2[5], x3[5])∧(U^Increasing(429_0_ACK_GT(x2[5], x3[5])), ≥))

We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(37)    ((U^Increasing(429_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_33] = 0∧[1 + (-1)bso_34] ≥ 0)

We simplified constraint (37) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(38)    ((U^Increasing(429_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_33] = 0∧[1 + (-1)bso_34] ≥ 0)

We simplified constraint (38) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(39)    ((U^Increasing(429_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_33] = 0∧[1 + (-1)bso_34] ≥ 0)

We simplified constraint (39) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(40)    ((U^Increasing(429_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_33] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_34] ≥ 0)

For Pair 584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x0), x1, x3) → 429_0_ACK_GT(x0, x1) the following chains were created:

We consider the chain 584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x0[6]), x1[6], x3[6]) → 429_0_ACK_GT(x0[6], x1[6]) which results in the following constraint:
(41)    (584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x0[6]), x1[6], x3[6])≥NonInfC∧584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x0[6]), x1[6], x3[6])≥429_0_ACK_GT(x0[6], x1[6])∧(U^Increasing(429_0_ACK_GT(x0[6], x1[6])), ≥))

We simplified constraint (41) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(42)    ((U^Increasing(429_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_35] = 0∧[1 + (-1)bso_36] ≥ 0)

We simplified constraint (42) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(43)    ((U^Increasing(429_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_35] = 0∧[1 + (-1)bso_36] ≥ 0)

We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(44)    ((U^Increasing(429_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_35] = 0∧[1 + (-1)bso_36] ≥ 0)

We simplified constraint (44) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(45)    ((U^Increasing(429_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_36] ≥ 0)

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

429_0_ACK_GT(0, x0) → COND_429_0_ACK_GT(>(x0, 0), 0, x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)
COND_429_0_ACK_GT(TRUE, 0, x0) → 429_0_ACK_GT(1, -(x0, 1))
- ((U^Increasing(429_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_25] = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)
429_0_ACK_GT(x1, x0) → COND_429_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)
COND_429_0_ACK_GT1(TRUE, x1, x0) → 584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1))
- ((U^Increasing(584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_29] = 0∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 0)
COND_429_0_ACK_GT1(TRUE, x1, x0) → 429_0_ACK_GT(-(x1, 1), x0)
- ((U^Increasing(429_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_31] = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 0)
584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x2), x3, 0) → 429_0_ACK_GT(x2, x3)
- ((U^Increasing(429_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_33] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_34] ≥ 0)
584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x0), x1, x3) → 429_0_ACK_GT(x0, x1)
- ((U^Increasing(429_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_36] ≥ 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(429_0_ack_GT(x₁, x₂)) = [-1]x₂ + [-1]x₁
POL(0) = 0
POL(Cond_429_0_ack_GT(x₁, x₂, x₃)) = [-1] + x₂
POL(>(x₁, x₂)) = [-1]
POL(-1) = [-1]
POL(457_0_ack_Return(x₁, x₂)) = [-1] + [-1]x₂
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(552_1_ack_InvokeMethod(x₁, x₂)) = [-1]
POL(709_0_ack_Return(x₁)) = x₁
POL(719_1_ack_InvokeMethod(x₁, x₂, x₃)) = [-1]
POL(429_0_ACK_GT(x₁, x₂)) = [-1] + x₂
POL(COND_429_0_ACK_GT(x₁, x₂, x₃)) = [-1] + x₃
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(COND_429_0_ACK_GT1(x₁, x₂, x₃)) = [-1] + x₃
POL(&&(x₁, x₂)) = [-1]
POL(584_1_ACK_INVOKEMETHOD(x₁, x₂, x₃)) = x₂

The following pairs are in P_>:

COND_429_0_ACK_GT(TRUE, 0, x0[1]) → 429_0_ACK_GT(1, -(x0[1], 1))
584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x2[5]), x3[5], 0) → 429_0_ACK_GT(x2[5], x3[5])
584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x0[6]), x1[6], x3[6]) → 429_0_ACK_GT(x0[6], x1[6])

The following pairs are in P_bound:

429_0_ACK_GT(0, x0[0]) → COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])
429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P_≥:

429_0_ACK_GT(0, x0[0]) → COND_429_0_ACK_GT(>(x0[0], 0), 0, x0[0])
429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_429_0_ACK_GT1(TRUE, x1[3], x0[3]) → 584_1_ACK_INVOKEMETHOD(429_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))
COND_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(-(x1[4], 1), x0[4])

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:
429_0_ack_GT(x1, 0) → Cond_429_0_ack_GT(x1 > -1, x1, 0)
Cond_429_0_ack_GT(TRUE, x1, 0) → 457_0_ack_Return(x1, x1 + 1)
552_1_ack_InvokeMethod(457_0_ack_Return(1, x2), 0) → 709_0_ack_Return(x2)
719_1_ack_InvokeMethod(709_0_ack_Return(x0), x1, x2) → 709_0_ack_Return(x0)
719_1_ack_InvokeMethod(457_0_ack_Return(x1, x2), 0, x1) → 709_0_ack_Return(x2)
552_1_ack_InvokeMethod(709_0_ack_Return(x0), x3) → 709_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 429_0_ACK_GT(0, x0[0]) → COND_429_0_ACK_GT(x0[0] > 0, 0, x0[0])
(2): 429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_429_0_ACK_GT1(TRUE, x1[3], x0[3]) → 584_1_ACK_INVOKEMETHOD(429_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(x1[4] - 1, x0[4])

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

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

(10) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 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_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(x1[4] - 1, x0[4])
(2): 429_0_ACK_GT(x1[2], x0[2]) → COND_429_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@17cdc4a5 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_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(-(x1[4], 1), x0[4]) the following chains were created:

We consider the chain COND_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:
(1)    (COND_429_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_429_0_ACK_GT1(TRUE, x1[4], x0[4])≥429_0_ACK_GT(-(x1[4], 1), x0[4])∧(U^Increasing(429_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(429_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(429_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(429_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(429_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)

For Pair 429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) the following chains were created:

We consider the chain 429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_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] ⇒ 429_0_ACK_GT(x1[2], x0[2])≥NonInfC∧429_0_ACK_GT(x1[2], x0[2])≥COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_429_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 ⇒ 429_0_ACK_GT(x1[2], x0[2])≥NonInfC∧429_0_ACK_GT(x1[2], x0[2])≥COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(U^Increasing(COND_429_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_429_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_429_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_429_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_429_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_429_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_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(-(x1[4], 1), x0[4])
- ((U^Increasing(429_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)
429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_429_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_429_0_ACK_GT1(x₁, x₂, x₃)) = [-1] + x₃ + x₂
POL(429_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_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(-(x1[4], 1), x0[4])

The following pairs are in P_bound:

429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P_≥:

429_0_ACK_GT(x1[2], x0[2]) → COND_429_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): 429_0_ACK_GT(x1[2], x0[2]) → COND_429_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])

The set Q consists of the following terms:
429_0_ack_GT(x0, 0)
Cond_429_0_ack_GT(TRUE, x0, 0)
552_1_ack_InvokeMethod(457_0_ack_Return(1, x0), 0)
719_1_ack_InvokeMethod(709_0_ack_Return(x0), x1, x2)
719_1_ack_InvokeMethod(457_0_ack_Return(x0, x1), 0, x0)
552_1_ack_InvokeMethod(709_0_ack_Return(x0), x1)

(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_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(x1[4] - 1, x0[4])

The set Q consists of the following terms:
429_0_ack_GT(x0, 0)
Cond_429_0_ack_GT(TRUE, x0, 0)
552_1_ack_InvokeMethod(457_0_ack_Return(1, x0), 0)
719_1_ack_InvokeMethod(709_0_ack_Return(x0), x1, x2)
719_1_ack_InvokeMethod(457_0_ack_Return(x0, x1), 0, x0)
552_1_ack_InvokeMethod(709_0_ack_Return(x0), x1)

(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:
429_0_ack_GT(x1, 0) → Cond_429_0_ack_GT(x1 > -1, x1, 0)
Cond_429_0_ack_GT(TRUE, x1, 0) → 457_0_ack_Return(x1, x1 + 1)
552_1_ack_InvokeMethod(457_0_ack_Return(1, x2), 0) → 709_0_ack_Return(x2)
719_1_ack_InvokeMethod(709_0_ack_Return(x0), x1, x2) → 709_0_ack_Return(x0)
719_1_ack_InvokeMethod(457_0_ack_Return(x1, x2), 0, x1) → 709_0_ack_Return(x2)
552_1_ack_InvokeMethod(709_0_ack_Return(x0), x3) → 709_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(1): COND_429_0_ACK_GT(TRUE, 0, x0[1]) → 429_0_ACK_GT(1, x0[1] - 1)
(3): COND_429_0_ACK_GT1(TRUE, x1[3], x0[3]) → 584_1_ACK_INVOKEMETHOD(429_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_429_0_ACK_GT1(TRUE, x1[4], x0[4]) → 429_0_ACK_GT(x1[4] - 1, x0[4])
(5): 584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x2[5]), x3[5], 0) → 429_0_ACK_GT(x2[5], x3[5])
(6): 584_1_ACK_INVOKEMETHOD(709_0_ack_Return(x0[6]), x1[6], x3[6]) → 429_0_ACK_GT(x0[6], x1[6])

(3) -> (5), if (429_0_ack_GT(x1[3] - 1, x0[3]) →^* 709_0_ack_Return(x2[5])∧x0[3] - 1 →^* x3[5]∧x1[3] - 1 →^* 0)

(3) -> (6), if (429_0_ack_GT(x1[3] - 1, x0[3]) →^* 709_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