(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] > -1java.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))=TRUEjava.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])))∧(UIncreasing(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)=TRUE521_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])))∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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))))∧(UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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 ⇒ (UIncreasing(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))))
    • ((UIncreasing(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 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(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 Pbound:

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


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])))


The set Q is empty.

(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.

(14) TRUE