Termination proof of /tmp/tmpOivo1Y/IntRTA.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 120 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 80 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 190 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 (Sun Microsystems Inc.) Main-Class: IntRTA

public class IntRTA {
  // only wrap a primitive int
  private int val;

  // count up to the value
  // in "limit"
  public static void count(
      IntRTA orig, IntRTA limit) {

    if (orig == null
        || limit == null) {
      return;
    }

    // introduce sharing
    IntRTA copy = orig;

    while (orig.val < limit.val) {
      copy.val++;
    }
  }

  public static void main(String[] args) {
    Random.args = args;
    IntRTA x = new IntRTA();
    x.val = Random.random();
    IntRTA y = new IntRTA();
    y.val = Random.random();
    count(x, y);
  }
}


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

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
IntRTA.main([Ljava/lang/String;)V: Graph of 189 nodes with 0 SCCs.

IntRTA.count(LIntRTA;LIntRTA;)V: Graph of 24 nodes with 1 SCC.

(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: IntRTA.count(LIntRTA;LIntRTA;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 14 rules for P and 0 rules for R.

P rules:
521_0_count_FieldAccess(EOS(STATIC_521), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62))) → 523_0_count_Load(EOS(STATIC_523), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), i62)
523_0_count_Load(EOS(STATIC_523), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), i62) → 525_0_count_FieldAccess(EOS(STATIC_525), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), i62, java.lang.Object(IntRTA(EOC, i47)))
525_0_count_FieldAccess(EOS(STATIC_525), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), i62, java.lang.Object(IntRTA(EOC, i47))) → 527_0_count_GE(EOS(STATIC_527), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), i62, i47)
527_0_count_GE(EOS(STATIC_527), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), i62, i47) → 530_0_count_GE(EOS(STATIC_530), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), i62, i47)
530_0_count_GE(EOS(STATIC_530), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), i62, i47) → 535_0_count_Load(EOS(STATIC_535), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62))) | <(i62, i47)
535_0_count_Load(EOS(STATIC_535), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62))) → 539_0_count_Duplicate(EOS(STATIC_539), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)))
539_0_count_Duplicate(EOS(STATIC_539), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62))) → 547_0_count_FieldAccess(EOS(STATIC_547), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)))
547_0_count_FieldAccess(EOS(STATIC_547), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62))) → 553_0_count_ConstantStackPush(EOS(STATIC_553), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)), i62)
553_0_count_ConstantStackPush(EOS(STATIC_553), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)), i62) → 555_0_count_IntArithmetic(EOS(STATIC_555), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)), i62, 1)
555_0_count_IntArithmetic(EOS(STATIC_555), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)), i62, matching1) → 557_0_count_FieldAccess(EOS(STATIC_557), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)), +(i62, 1)) | &&(>=(i62, 0), =(matching1, 1))
557_0_count_FieldAccess(EOS(STATIC_557), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)), i73) → 560_0_count_JMP(EOS(STATIC_560), java.lang.Object(IntRTA(EOC, i73)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i73)))
560_0_count_JMP(EOS(STATIC_560), java.lang.Object(IntRTA(EOC, i73)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i73))) → 565_0_count_Load(EOS(STATIC_565), java.lang.Object(IntRTA(EOC, i73)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i73)))
565_0_count_Load(EOS(STATIC_565), java.lang.Object(IntRTA(EOC, i73)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i73))) → 517_0_count_Load(EOS(STATIC_517), java.lang.Object(IntRTA(EOC, i73)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i73)))
517_0_count_Load(EOS(STATIC_517), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62))) → 521_0_count_FieldAccess(EOS(STATIC_521), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i47)), java.lang.Object(IntRTA(EOC, i62)), java.lang.Object(IntRTA(EOC, i62)))
R rules:

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
521_0_count_FieldAccess(EOS(STATIC_521), java.lang.Object(IntRTA(EOC, x0)), java.lang.Object(IntRTA(EOC, x1)), java.lang.Object(IntRTA(EOC, x0)), java.lang.Object(IntRTA(EOC, x0))) → 521_0_count_FieldAccess(EOS(STATIC_521), java.lang.Object(IntRTA(EOC, +(x0, 1))), java.lang.Object(IntRTA(EOC, x1)), java.lang.Object(IntRTA(EOC, +(x0, 1))), java.lang.Object(IntRTA(EOC, +(x0, 1)))) | &&(>(x1, x0), >(+(x0, 1), 0))
R rules:

Filtered ground terms:

521_0_count_FieldAccess(x1, x2, x3, x4, x5) → 521_0_count_FieldAccess(x2, x3, x4, x5)
IntRTA(x1, x2) → IntRTA(x2)
EOS(x1) → EOS
Cond_521_0_count_FieldAccess(x1, x2, x3, x4, x5, x6) → Cond_521_0_count_FieldAccess(x1, x3, x4, x5, x6)

Filtered duplicate args:

521_0_count_FieldAccess(x1, x2, x3, x4) → 521_0_count_FieldAccess(x2, x4)
Cond_521_0_count_FieldAccess(x1, x2, x3, x4, x5) → Cond_521_0_count_FieldAccess(x1, x3, x5)

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
521_0_count_FieldAccess(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → 521_0_count_FieldAccess(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(+(x0, 1)))) | &&(>(x1, x0), >(x0, -1))
R rules:

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

P rules:
521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → COND_521_0_COUNT_FIELDACCESS(&&(>(x1, x0), >(x0, -1)), java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0)))
COND_521_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(+(x0, 1))))
R rules:

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0]))) → COND_521_0_COUNT_FIELDACCESS(x1[0] > x0[0] && x0[0] > -1, java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))
(1): COND_521_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1]))) → 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1] + 1)))

(0) -> (1), if (x1[0] > x0[0] && x0[0] > -1 ∧java.lang.Object(IntRTA(x1[0])) →^* java.lang.Object(IntRTA(x1[1]))∧java.lang.Object(IntRTA(x0[0])) →^* java.lang.Object(IntRTA(x0[1])))

(1) -> (0), if (java.lang.Object(IntRTA(x1[1])) →^* java.lang.Object(IntRTA(x1[0]))∧java.lang.Object(IntRTA(x0[1] + 1)) →^* java.lang.Object(IntRTA(x0[0])))

The set Q is empty.

(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@4afbf04a 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 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → COND_521_0_COUNT_FIELDACCESS(&&(>(x1, x0), >(x0, -1)), java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) the following chains were created:

We consider the chain 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0]))) → COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0]))), COND_521_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1]))) → 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1)))) which results in the following constraint:
(1)    (&&(>(x1[0], x0[0]), >(x0[0], -1))=TRUE∧java.lang.Object(IntRTA(x1[0]))=java.lang.Object(IntRTA(x1[1]))∧java.lang.Object(IntRTA(x0[0]))=java.lang.Object(IntRTA(x0[1])) ⇒ 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))≥NonInfC∧521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))≥COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))∧(U^Increasing(COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], x0[0])=TRUE∧>(x0[0], -1)=TRUE ⇒ 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))≥NonInfC∧521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))≥COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))∧(U^Increasing(COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [(-1)bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [(-1)bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [(-1)bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 0)

For Pair COND_521_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(+(x0, 1)))) the following chains were created:

We consider the chain COND_521_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1]))) → 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1)))) which results in the following constraint:
(7)    (COND_521_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1])))≥NonInfC∧COND_521_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1])))≥521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))∧(U^Increasing(521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥)∧[bni_11] = 0∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥)∧[bni_11] = 0∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥)∧[bni_11] = 0∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥)∧[bni_11] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

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

521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → COND_521_0_COUNT_FIELDACCESS(&&(>(x1, x0), >(x0, -1)), java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0)))
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 0)
COND_521_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(+(x0, 1))))
- ((U^Increasing(521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥)∧[bni_11] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 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(521_0_COUNT_FIELDACCESS(x₁, x₂)) = [-1] + [-1]x₂ + [2]x₁
POL(java.lang.Object(x₁)) = x₁
POL(IntRTA(x₁)) = x₁
POL(COND_521_0_COUNT_FIELDACCESS(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(-1) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]

The following pairs are in P_>:

COND_521_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1]))) → 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))

The following pairs are in P_bound:

521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0]))) → COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))

The following pairs are in P_≥:

521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0]))) → COND_521_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >(x0[0], -1)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))

There are no usable rules.

(8) Complex Obligation (AND)

(9) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(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

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_521_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1]))) → 521_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1] + 1)))

The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(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) TRUE

(12) Obligation:

(13) IDependencyGraphProof (EQUIVALENT transformation)

(14) TRUE