Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 220 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 210 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 210 ms)
↳8 AND
    ↳9 IDP
        ↳10 IDependencyGraphProof (⇔, 0 ms)
        ↳11 TRUE
    ↳12 IDP
        ↳13 IDependencyGraphProof (⇔, 0 ms)
        ↳14 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: HanoiR 
public class HanoiR {
    private void solve(int h, int from, int to, int support) {
	if (h < 1) return;
	else if (h == 1) {
	    //System.out.println("from " + from + " to " + to + "\n");
	}
	else {
	    solve(h - 1, from, support, to);
	    //System.out.println("from " + from + " to " + to + "\n");
	    solve(h - 1, support, to, from);
	}
    }

    public static void main(String[] args) {
	Random.args = args;
	new HanoiR().solve(Random.random(),1,2,3);
    }
}


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:
HanoiR.main([Ljava/lang/String;)V: Graph of 110 nodes with 0 SCCs.

HanoiR.solve(IIII)V: Graph of 43 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: HanoiR.solve(IIII)V
SCC calls the following helper methods: HanoiR.solve(IIII)V
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 33 rules for P and 10 rules for R.


P rules:
537_0_solve_ConstantStackPush(EOS(STATIC_537), i98, i98) → 540_0_solve_GE(EOS(STATIC_540), i98, i98, 1)
540_0_solve_GE(EOS(STATIC_540), i105, i105, matching1) → 543_0_solve_GE(EOS(STATIC_543), i105, i105, 1) | =(matching1, 1)
543_0_solve_GE(EOS(STATIC_543), i105, i105, matching1) → 547_0_solve_Load(EOS(STATIC_547), i105) | &&(>=(i105, 1), =(matching1, 1))
547_0_solve_Load(EOS(STATIC_547), i105) → 551_0_solve_ConstantStackPush(EOS(STATIC_551), i105, i105)
551_0_solve_ConstantStackPush(EOS(STATIC_551), i105, i105) → 562_0_solve_NE(EOS(STATIC_562), i105, i105, 1)
562_0_solve_NE(EOS(STATIC_562), i110, i110, matching1) → 573_0_solve_NE(EOS(STATIC_573), i110, i110, 1) | =(matching1, 1)
573_0_solve_NE(EOS(STATIC_573), i110, i110, matching1) → 577_0_solve_Load(EOS(STATIC_577), i110) | &&(!(=(i110, 1)), =(matching1, 1))
577_0_solve_Load(EOS(STATIC_577), i110) → 581_0_solve_Load(EOS(STATIC_581), i110)
581_0_solve_Load(EOS(STATIC_581), i110) → 585_0_solve_ConstantStackPush(EOS(STATIC_585), i110, i110)
585_0_solve_ConstantStackPush(EOS(STATIC_585), i110, i110) → 593_0_solve_IntArithmetic(EOS(STATIC_593), i110, i110, 1)
593_0_solve_IntArithmetic(EOS(STATIC_593), i110, i110, matching1) → 600_0_solve_Load(EOS(STATIC_600), i110, -(i110, 1)) | &&(>(i110, 0), =(matching1, 1))
600_0_solve_Load(EOS(STATIC_600), i110, i115) → 601_0_solve_Load(EOS(STATIC_601), i110, i115)
601_0_solve_Load(EOS(STATIC_601), i110, i115) → 603_0_solve_Load(EOS(STATIC_603), i110, i115)
603_0_solve_Load(EOS(STATIC_603), i110, i115) → 605_0_solve_InvokeMethod(EOS(STATIC_605), i110, i115)
605_0_solve_InvokeMethod(EOS(STATIC_605), i110, i115) → 607_1_solve_InvokeMethod(607_0_solve_Load(EOS(STATIC_607), i115), i110, i115)
607_0_solve_Load(EOS(STATIC_607), i115) → 610_0_solve_Load(EOS(STATIC_610), i115)
607_1_solve_InvokeMethod(579_0_solve_Return(EOS(STATIC_579)), i110, matching1) → 622_0_solve_Return(EOS(STATIC_622), i110, 1) | =(matching1, 1)
607_1_solve_InvokeMethod(793_0_solve_Return(EOS(STATIC_793)), i110, i270) → 822_0_solve_Return(EOS(STATIC_822), i110, i270)
610_0_solve_Load(EOS(STATIC_610), i115) → 535_0_solve_Load(EOS(STATIC_535), i115)
535_0_solve_Load(EOS(STATIC_535), i98) → 537_0_solve_ConstantStackPush(EOS(STATIC_537), i98, i98)
622_0_solve_Return(EOS(STATIC_622), i110, matching1) → 709_0_solve_Return(EOS(STATIC_709), i110, 1) | =(matching1, 1)
709_0_solve_Return(EOS(STATIC_709), i110, i180) → 717_0_solve_Load(EOS(STATIC_717), i110)
717_0_solve_Load(EOS(STATIC_717), i110) → 720_0_solve_Load(EOS(STATIC_720), i110)
720_0_solve_Load(EOS(STATIC_720), i110) → 722_0_solve_ConstantStackPush(EOS(STATIC_722), i110)
722_0_solve_ConstantStackPush(EOS(STATIC_722), i110) → 724_0_solve_IntArithmetic(EOS(STATIC_724), i110, 1)
724_0_solve_IntArithmetic(EOS(STATIC_724), i110, matching1) → 726_0_solve_Load(EOS(STATIC_726), -(i110, 1)) | &&(>(i110, 0), =(matching1, 1))
726_0_solve_Load(EOS(STATIC_726), i192) → 727_0_solve_Load(EOS(STATIC_727), i192)
727_0_solve_Load(EOS(STATIC_727), i192) → 730_0_solve_Load(EOS(STATIC_730), i192)
730_0_solve_Load(EOS(STATIC_730), i192) → 731_0_solve_InvokeMethod(EOS(STATIC_731), i192)
731_0_solve_InvokeMethod(EOS(STATIC_731), i192) → 733_1_solve_InvokeMethod(733_0_solve_Load(EOS(STATIC_733), i192), i192)
733_0_solve_Load(EOS(STATIC_733), i192) → 735_0_solve_Load(EOS(STATIC_735), i192)
735_0_solve_Load(EOS(STATIC_735), i192) → 535_0_solve_Load(EOS(STATIC_535), i192)
822_0_solve_Return(EOS(STATIC_822), i110, i270) → 709_0_solve_Return(EOS(STATIC_709), i110, i270)
R rules:
540_0_solve_GE(EOS(STATIC_540), i103, i103, matching1) → 541_0_solve_GE(EOS(STATIC_541), i103, i103, 1) | =(matching1, 1)
541_0_solve_GE(EOS(STATIC_541), i103, i103, matching1) → 545_0_solve_Return(EOS(STATIC_545), i103) | &&(<(i103, 1), =(matching1, 1))
562_0_solve_NE(EOS(STATIC_562), matching1, matching2, matching3) → 572_0_solve_NE(EOS(STATIC_572), 1, 1, 1) | &&(&&(=(matching1, 1), =(matching2, 1)), =(matching3, 1))
572_0_solve_NE(EOS(STATIC_572), matching1, matching2, matching3) → 575_0_solve_JMP(EOS(STATIC_575)) | &&(&&(=(matching1, 1), =(matching2, 1)), =(matching3, 1))
575_0_solve_JMP(EOS(STATIC_575)) → 579_0_solve_Return(EOS(STATIC_579))
733_1_solve_InvokeMethod(579_0_solve_Return(EOS(STATIC_579)), matching1) → 746_0_solve_Return(EOS(STATIC_746), 1) | =(matching1, 1)
733_1_solve_InvokeMethod(793_0_solve_Return(EOS(STATIC_793)), i277) → 826_0_solve_Return(EOS(STATIC_826), i277)
746_0_solve_Return(EOS(STATIC_746), matching1) → 789_0_solve_Return(EOS(STATIC_789), 1) | =(matching1, 1)
789_0_solve_Return(EOS(STATIC_789), i243) → 793_0_solve_Return(EOS(STATIC_793))
826_0_solve_Return(EOS(STATIC_826), i277) → 789_0_solve_Return(EOS(STATIC_789), i277)

Combined rules. Obtained 3 conditional rules for P and 2 conditional rules for R.


P rules:
537_0_solve_ConstantStackPush(EOS(STATIC_537), x0, x0) → 607_1_solve_InvokeMethod(537_0_solve_ConstantStackPush(EOS(STATIC_537), -(x0, 1), -(x0, 1)), x0, -(x0, 1)) | >(x0, 1)
607_1_solve_InvokeMethod(579_0_solve_Return(EOS(STATIC_579)), x0, 1) → 733_1_solve_InvokeMethod(537_0_solve_ConstantStackPush(EOS(STATIC_537), -(x0, 1), -(x0, 1)), -(x0, 1)) | >(x0, 0)
607_1_solve_InvokeMethod(793_0_solve_Return(EOS(STATIC_793)), x0, x1) → 733_1_solve_InvokeMethod(537_0_solve_ConstantStackPush(EOS(STATIC_537), -(x0, 1), -(x0, 1)), -(x0, 1)) | >(x0, 0)
R rules:
733_1_solve_InvokeMethod(579_0_solve_Return(EOS(STATIC_579)), 1) → 793_0_solve_Return(EOS(STATIC_793))
733_1_solve_InvokeMethod(793_0_solve_Return(EOS(STATIC_793)), x0) → 793_0_solve_Return(EOS(STATIC_793))

Filtered ground terms:



537_0_solve_ConstantStackPush(x1, x2, x3) → 537_0_solve_ConstantStackPush(x2, x3)
Cond_607_1_solve_InvokeMethod1(x1, x2, x3, x4) → Cond_607_1_solve_InvokeMethod1(x1, x3, x4)
793_0_solve_Return(x1) → 793_0_solve_Return
Cond_607_1_solve_InvokeMethod(x1, x2, x3, x4) → Cond_607_1_solve_InvokeMethod(x1, x3)
579_0_solve_Return(x1) → 579_0_solve_Return
Cond_537_0_solve_ConstantStackPush(x1, x2, x3, x4) → Cond_537_0_solve_ConstantStackPush(x1, x3, x4)

Filtered duplicate args:



537_0_solve_ConstantStackPush(x1, x2) → 537_0_solve_ConstantStackPush(x2)
Cond_537_0_solve_ConstantStackPush(x1, x2, x3) → Cond_537_0_solve_ConstantStackPush(x1, x3)

Filtered unneeded arguments:



Cond_607_1_solve_InvokeMethod1(x1, x2, x3) → Cond_607_1_solve_InvokeMethod1(x1, x2)

Combined rules. Obtained 3 conditional rules for P and 2 conditional rules for R.


P rules:
537_0_solve_ConstantStackPush(x0) → 607_1_solve_InvokeMethod(537_0_solve_ConstantStackPush(-(x0, 1)), x0, -(x0, 1)) | >(x0, 1)
607_1_solve_InvokeMethod(579_0_solve_Return, x0, 1) → 733_1_solve_InvokeMethod(537_0_solve_ConstantStackPush(-(x0, 1)), -(x0, 1)) | >(x0, 0)
607_1_solve_InvokeMethod(793_0_solve_Return, x0, x1) → 733_1_solve_InvokeMethod(537_0_solve_ConstantStackPush(-(x0, 1)), -(x0, 1)) | >(x0, 0)
R rules:
733_1_solve_InvokeMethod(579_0_solve_Return, 1) → 793_0_solve_Return
733_1_solve_InvokeMethod(793_0_solve_Return, x0) → 793_0_solve_Return

Performed bisimulation on rules. Used the following equivalence classes: {[579_0_solve_Return, 793_0_solve_Return]=579_0_solve_Return}


Finished conversion. Obtained 7 rules for P and 2 rules for R. System has predefined symbols.


P rules:
537_0_SOLVE_CONSTANTSTACKPUSH(x0) → COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0, 1), x0)
COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0) → 607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0, 1)), x0, -(x0, 1))
COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1))
607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0, 1) → COND_607_1_SOLVE_INVOKEMETHOD(>(x0, 0), 579_0_solve_Return, x0, 1)
COND_607_1_SOLVE_INVOKEMETHOD(TRUE, 579_0_solve_Return, x0, 1) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1))
607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0, x1) → COND_607_1_SOLVE_INVOKEMETHOD1(>(x0, 0), 579_0_solve_Return, x0, x1)
COND_607_1_SOLVE_INVOKEMETHOD1(TRUE, 579_0_solve_Return, x0, x1) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1))
R rules:
733_1_solve_InvokeMethod(579_0_solve_Return, 1) → 579_0_solve_Return
733_1_solve_InvokeMethod(579_0_solve_Return, x0) → 579_0_solve_Return

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Integer


The ITRS R consists of the following rules:
733_1_solve_InvokeMethod(579_0_solve_Return, 1) → 579_0_solve_Return
733_1_solve_InvokeMethod(579_0_solve_Return, x0) → 579_0_solve_Return

The integer pair graph contains the following rules and edges:
(0): 537_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_537_0_SOLVE_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(x0[1] - 1), x0[1], x0[1] - 1)
(2): COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2]) → 537_0_SOLVE_CONSTANTSTACKPUSH(x0[2] - 1)
(3): 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[3], 1) → COND_607_1_SOLVE_INVOKEMETHOD(x0[3] > 0, 579_0_solve_Return, x0[3], 1)
(4): COND_607_1_SOLVE_INVOKEMETHOD(TRUE, 579_0_solve_Return, x0[4], 1) → 537_0_SOLVE_CONSTANTSTACKPUSH(x0[4] - 1)
(5): 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[5], x1[5]) → COND_607_1_SOLVE_INVOKEMETHOD1(x0[5] > 0, 579_0_solve_Return, x0[5], x1[5])
(6): COND_607_1_SOLVE_INVOKEMETHOD1(TRUE, 579_0_solve_Return, x0[6], x1[6]) → 537_0_SOLVE_CONSTANTSTACKPUSH(x0[6] - 1)

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


(0) -> (2), if (x0[0] > 1 ∧x0[0] →* x0[2])


(1) -> (3), if (537_0_solve_ConstantStackPush(x0[1] - 1) →* 579_0_solve_Return∧x0[1] →* x0[3]∧x0[1] - 1 →* 1)


(1) -> (5), if (537_0_solve_ConstantStackPush(x0[1] - 1) →* 579_0_solve_Return∧x0[1] →* x0[5]∧x0[1] - 1 →* x1[5])


(2) -> (0), if (x0[2] - 1 →* x0[0])


(3) -> (4), if (x0[3] > 0 ∧x0[3] →* x0[4])


(4) -> (0), if (x0[4] - 1 →* x0[0])


(5) -> (6), if (x0[5] > 0 ∧x0[5] →* x0[6]∧x1[5] →* x1[6])


(6) -> (0), if (x0[6] - 1 →* x0[0])



The set Q consists of the following terms:
733_1_solve_InvokeMethod(579_0_solve_Return, x0)

(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@6fbdd27a 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 537_0_SOLVE_CONSTANTSTACKPUSH(x0) → COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:
  • We consider the chain 537_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1)) which results in the following constraint:

    (1)    (>(x0[0], 1)=TRUE∧x0[0]=x0[1] ⇒ 537_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧537_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))



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

    (2)    (>(x0[0], 1)=TRUE ⇒ 537_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧537_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))



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

    (3)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (4)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (5)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(5)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



  • We consider the chain 537_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2]) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1)) which results in the following constraint:

    (7)    (>(x0[0], 1)=TRUE∧x0[0]=x0[2] ⇒ 537_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧537_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))



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

    (8)    (>(x0[0], 1)=TRUE ⇒ 537_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧537_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))



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

    (9)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (10)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (11)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(5)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)







For Pair COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0) → 607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0, 1)), x0, -(x0, 1)) the following chains were created:
  • We consider the chain COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1)) which results in the following constraint:

    (13)    (COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1])≥607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))∧(UIncreasing(607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥))



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

    (14)    ((UIncreasing(607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[bni_18] = 0∧[(-1)bso_19] ≥ 0)



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

    (15)    ((UIncreasing(607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[bni_18] = 0∧[(-1)bso_19] ≥ 0)



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

    (16)    ((UIncreasing(607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[bni_18] = 0∧[(-1)bso_19] ≥ 0)



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

    (17)    ((UIncreasing(607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[bni_18] = 0∧0 = 0∧[(-1)bso_19] ≥ 0)







For Pair COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1)) the following chains were created:
  • We consider the chain COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2]) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1)) which results in the following constraint:

    (18)    (COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2])≥NonInfC∧COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2])≥537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))∧(UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥))



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

    (19)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[bni_20] = 0∧[2 + (-1)bso_21] ≥ 0)



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

    (20)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[bni_20] = 0∧[2 + (-1)bso_21] ≥ 0)



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

    (21)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[bni_20] = 0∧[2 + (-1)bso_21] ≥ 0)



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

    (22)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[bni_20] = 0∧0 = 0∧[2 + (-1)bso_21] ≥ 0)







For Pair 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0, 1) → COND_607_1_SOLVE_INVOKEMETHOD(>(x0, 0), 579_0_solve_Return, x0, 1) the following chains were created:
  • We consider the chain 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[3], 1) → COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1), COND_607_1_SOLVE_INVOKEMETHOD(TRUE, 579_0_solve_Return, x0[4], 1) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1)) which results in the following constraint:

    (23)    (>(x0[3], 0)=TRUE∧x0[3]=x0[4] ⇒ 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[3], 1)≥NonInfC∧607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[3], 1)≥COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)∧(UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)), ≥))



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

    (24)    (>(x0[3], 0)=TRUE ⇒ 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[3], 1)≥NonInfC∧607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[3], 1)≥COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)∧(UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)), ≥))



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

    (25)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)), ≥)∧[bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[3] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (26)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)), ≥)∧[bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[3] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (27)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)), ≥)∧[bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[3] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (28)    (x0[3] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)), ≥)∧[(3)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[3] ≥ 0∧[(-1)bso_23] ≥ 0)







For Pair COND_607_1_SOLVE_INVOKEMETHOD(TRUE, 579_0_solve_Return, x0, 1) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1)) the following chains were created:
  • We consider the chain COND_607_1_SOLVE_INVOKEMETHOD(TRUE, 579_0_solve_Return, x0[4], 1) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1)) which results in the following constraint:

    (29)    (COND_607_1_SOLVE_INVOKEMETHOD(TRUE, 579_0_solve_Return, x0[4], 1)≥NonInfC∧COND_607_1_SOLVE_INVOKEMETHOD(TRUE, 579_0_solve_Return, x0[4], 1)≥537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))∧(UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥))



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

    (30)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥)∧[bni_24] = 0∧[2 + (-1)bso_25] ≥ 0)



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

    (31)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥)∧[bni_24] = 0∧[2 + (-1)bso_25] ≥ 0)



    We simplified constraint (31) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (32)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥)∧[bni_24] = 0∧[2 + (-1)bso_25] ≥ 0)



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

    (33)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥)∧[bni_24] = 0∧0 = 0∧[2 + (-1)bso_25] ≥ 0)







For Pair 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0, x1) → COND_607_1_SOLVE_INVOKEMETHOD1(>(x0, 0), 579_0_solve_Return, x0, x1) the following chains were created:
  • We consider the chain 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[5], x1[5]) → COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5]), COND_607_1_SOLVE_INVOKEMETHOD1(TRUE, 579_0_solve_Return, x0[6], x1[6]) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1)) which results in the following constraint:

    (34)    (>(x0[5], 0)=TRUE∧x0[5]=x0[6]∧x1[5]=x1[6] ⇒ 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[5], x1[5])≥NonInfC∧607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[5], x1[5])≥COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])∧(UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])), ≥))



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

    (35)    (>(x0[5], 0)=TRUE ⇒ 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[5], x1[5])≥NonInfC∧607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[5], x1[5])≥COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])∧(UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])), ≥))



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

    (36)    (x0[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])), ≥)∧[bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x0[5] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



    We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (37)    (x0[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])), ≥)∧[bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x0[5] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



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

    (38)    (x0[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])), ≥)∧[bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x0[5] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



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

    (39)    (x0[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])), ≥)∧0 = 0∧[bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x0[5] ≥ 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)



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

    (40)    (x0[5] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])), ≥)∧0 = 0∧[(3)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x0[5] ≥ 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)







For Pair COND_607_1_SOLVE_INVOKEMETHOD1(TRUE, 579_0_solve_Return, x0, x1) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1)) the following chains were created:
  • We consider the chain COND_607_1_SOLVE_INVOKEMETHOD1(TRUE, 579_0_solve_Return, x0[6], x1[6]) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1)) which results in the following constraint:

    (41)    (COND_607_1_SOLVE_INVOKEMETHOD1(TRUE, 579_0_solve_Return, x0[6], x1[6])≥NonInfC∧COND_607_1_SOLVE_INVOKEMETHOD1(TRUE, 579_0_solve_Return, x0[6], x1[6])≥537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))∧(UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥))



    We simplified constraint (41) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (42)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥)∧[bni_28] = 0∧[1 + (-1)bso_29] ≥ 0)



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

    (43)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥)∧[bni_28] = 0∧[1 + (-1)bso_29] ≥ 0)



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

    (44)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥)∧[bni_28] = 0∧[1 + (-1)bso_29] ≥ 0)



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

    (45)    ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥)∧[bni_28] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_29] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 537_0_SOLVE_CONSTANTSTACKPUSH(x0) → COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0, 1), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(5)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(5)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

  • COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0) → 607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0, 1)), x0, -(x0, 1))
    • ((UIncreasing(607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[bni_18] = 0∧0 = 0∧[(-1)bso_19] ≥ 0)

  • COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1))
    • ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[bni_20] = 0∧0 = 0∧[2 + (-1)bso_21] ≥ 0)

  • 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0, 1) → COND_607_1_SOLVE_INVOKEMETHOD(>(x0, 0), 579_0_solve_Return, x0, 1)
    • (x0[3] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)), ≥)∧[(3)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[3] ≥ 0∧[(-1)bso_23] ≥ 0)

  • COND_607_1_SOLVE_INVOKEMETHOD(TRUE, 579_0_solve_Return, x0, 1) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1))
    • ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥)∧[bni_24] = 0∧0 = 0∧[2 + (-1)bso_25] ≥ 0)

  • 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0, x1) → COND_607_1_SOLVE_INVOKEMETHOD1(>(x0, 0), 579_0_solve_Return, x0, x1)
    • (x0[5] ≥ 0 ⇒ (UIncreasing(COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])), ≥)∧0 = 0∧[(3)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x0[5] ≥ 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

  • COND_607_1_SOLVE_INVOKEMETHOD1(TRUE, 579_0_solve_Return, x0, x1) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1))
    • ((UIncreasing(537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥)∧[bni_28] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_29] ≥ 0)




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

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(733_1_solve_InvokeMethod(x1, x2)) = [-1]   
POL(579_0_solve_Return) = 0   
POL(1) = [1]   
POL(537_0_SOLVE_CONSTANTSTACKPUSH(x1)) = [1] + [2]x1   
POL(COND_537_0_SOLVE_CONSTANTSTACKPUSH(x1, x2)) = [1] + [2]x2   
POL(>(x1, x2)) = [-1]   
POL(607_1_SOLVE_INVOKEMETHOD(x1, x2, x3)) = [1] + [2]x2   
POL(537_0_solve_ConstantStackPush(x1)) = x1   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(COND_607_1_SOLVE_INVOKEMETHOD(x1, x2, x3, x4)) = [1] + [2]x3   
POL(0) = 0   
POL(COND_607_1_SOLVE_INVOKEMETHOD1(x1, x2, x3, x4)) = [2]x3   

The following pairs are in P>:

COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2]) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))
COND_607_1_SOLVE_INVOKEMETHOD(TRUE, 579_0_solve_Return, x0[4], 1) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))
607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[5], x1[5]) → COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])
COND_607_1_SOLVE_INVOKEMETHOD1(TRUE, 579_0_solve_Return, x0[6], x1[6]) → 537_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))

The following pairs are in Pbound:

537_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])
607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[3], 1) → COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)
607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[5], x1[5]) → COND_607_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 579_0_solve_Return, x0[5], x1[5])

The following pairs are in P≥:

537_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_537_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])
COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))
607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[3], 1) → COND_607_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 579_0_solve_Return, x0[3], 1)

There are no usable rules.

(8) Complex Obligation (AND)

(9) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Integer


The ITRS R consists of the following rules:
733_1_solve_InvokeMethod(579_0_solve_Return, 1) → 579_0_solve_Return
733_1_solve_InvokeMethod(579_0_solve_Return, x0) → 579_0_solve_Return

The integer pair graph contains the following rules and edges:
(0): 537_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_537_0_SOLVE_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(x0[1] - 1), x0[1], x0[1] - 1)
(3): 607_1_SOLVE_INVOKEMETHOD(579_0_solve_Return, x0[3], 1) → COND_607_1_SOLVE_INVOKEMETHOD(x0[3] > 0, 579_0_solve_Return, x0[3], 1)

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


(1) -> (3), if (537_0_solve_ConstantStackPush(x0[1] - 1) →* 579_0_solve_Return∧x0[1] →* x0[3]∧x0[1] - 1 →* 1)



The set Q consists of the following terms:
733_1_solve_InvokeMethod(579_0_solve_Return, x0)

(10) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(11) TRUE

(12) 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:
733_1_solve_InvokeMethod(579_0_solve_Return, 1) → 579_0_solve_Return
733_1_solve_InvokeMethod(579_0_solve_Return, x0) → 579_0_solve_Return

The integer pair graph contains the following rules and edges:
(1): COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 607_1_SOLVE_INVOKEMETHOD(537_0_solve_ConstantStackPush(x0[1] - 1), x0[1], x0[1] - 1)
(2): COND_537_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2]) → 537_0_SOLVE_CONSTANTSTACKPUSH(x0[2] - 1)
(4): COND_607_1_SOLVE_INVOKEMETHOD(TRUE, 579_0_solve_Return, x0[4], 1) → 537_0_SOLVE_CONSTANTSTACKPUSH(x0[4] - 1)
(6): COND_607_1_SOLVE_INVOKEMETHOD1(TRUE, 579_0_solve_Return, x0[6], x1[6]) → 537_0_SOLVE_CONSTANTSTACKPUSH(x0[6] - 1)


The set Q consists of the following terms:
733_1_solve_InvokeMethod(579_0_solve_Return, x0)

(13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(14) TRUE