Termination proof of /tmp/tmpHmiKJp/Convert.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 653 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 AND

    ↳5 JBCTerminationSCC

        ↳6 SCCToIntTRSProof (⇒, 80 ms)

        ↳7 intTRS

        ↳8 PolynomialOrderProcessor (⇔, 0 ms)

        ↳9 YES

    ↳10 JBCTerminationSCC

        ↳11 SCCToIDPv1Proof (⇒, 45 ms)

        ↳12 IDP

        ↳13 IDPNonInfProof (⇒, 772 ms)

        ↳14 IDP

        ↳15 IDPNonInfProof (⇒, 333 ms)

        ↳16 IDP

        ↳17 IDPNonInfProof (⇒, 83 ms)

        ↳18 IDP

        ↳19 IDependencyGraphProof (⇔, 0 ms)

        ↳20 TRUE

(0) Obligation:

JBC Problem based on JBC Program:

public class Convert{

  // adapted from [Giesl, 95]
  // converts a number to decimal notation

  public static void main(String[] args) {
    Random.args = args;
    IntList l = IntList.createIntList();

    int b = Random.random();
    int res = 0;

    while (l != null) {

      if (l.value <= 0) {
        l = l.next;
        if (l != null) res = res * b;
      }
      else {
        l.value--;
        res++;
      }
    }
  }
}

class IntList {
  int value;
  IntList next;

  public IntList(int value, IntList next) {
    this.value = value;
    this.next = next;
  }

  public static IntList createIntList() {

    int i = Random.random();
    IntList l = null;

    while (i > 0) {
      l = new IntList(Random.random(), l);
      i--;
    }

    return l;
  }
}


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:
Convert.main([Ljava/lang/String;)V: Graph of 158 nodes with 1 SCC.

IntList.createIntList()LIntList;: Graph of 163 nodes with 1 SCC.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 2 SCCss.

(4) Complex Obligation (AND)

(5) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: IntList.createIntList()LIntList;
SCC calls the following helper methods:
Performed SCC analyses:

Used field analysis yielded the following read fields:
- java.lang.String: [count]
Marker field analysis yielded the following relations that could be markers:

(6) SCCToIntTRSProof (SOUND transformation)

Transformed FIGraph SCCs to intTRSs. Log:

Generated rules. Obtained 38 IRules

P rules:
f2161_0_createIntList_LE(EOS, i480, i480) → f2166_0_createIntList_LE(EOS, i480, i480)
f2166_0_createIntList_LE(EOS, i480, i480) → f2173_0_createIntList_New(EOS, i480) | >(i480, 0)
f2173_0_createIntList_New(EOS, i480) → f2180_0_createIntList_Duplicate(EOS, i480)
f2180_0_createIntList_Duplicate(EOS, i480) → f2187_0_createIntList_InvokeMethod(EOS, i480)
f2187_0_createIntList_InvokeMethod(EOS, i480) → f2200_0_random_FieldAccess(EOS, i480)
f2200_0_random_FieldAccess(EOS, i480) → f2206_0_random_FieldAccess(EOS, i480)
f2206_0_random_FieldAccess(EOS, i480) → f2210_0_random_ArrayAccess(EOS, i480)
f2210_0_random_ArrayAccess(EOS, i480) → f2216_0_random_ArrayAccess(EOS, i480)
f2216_0_random_ArrayAccess(EOS, i480) → f2221_0_random_Store(EOS, i480, o633)
f2221_0_random_Store(EOS, i480, o633) → f2229_0_random_FieldAccess(EOS, i480, o633)
f2229_0_random_FieldAccess(EOS, i480, o633) → f2235_0_random_ConstantStackPush(EOS, i480, o633)
f2235_0_random_ConstantStackPush(EOS, i480, o633) → f2243_0_random_IntArithmetic(EOS, i480, o633)
f2243_0_random_IntArithmetic(EOS, i480, o633) → f2255_0_random_FieldAccess(EOS, i480, o633)
f2255_0_random_FieldAccess(EOS, i480, o633) → f2262_0_random_Load(EOS, i480, o633)
f2262_0_random_Load(EOS, i480, o633) → f2273_0_random_InvokeMethod(EOS, i480, o633)
f2273_0_random_InvokeMethod(EOS, i480, java.lang.Object(o663sub)) → f2291_0_random_InvokeMethod(EOS, i480, java.lang.Object(o663sub))
f2291_0_random_InvokeMethod(EOS, i480, java.lang.Object(o663sub)) → f2296_0_length_Load(EOS, i480, java.lang.Object(o663sub), java.lang.Object(o663sub))
f2296_0_length_Load(EOS, i480, java.lang.Object(o663sub), java.lang.Object(o663sub)) → f2318_0_length_FieldAccess(EOS, i480, java.lang.Object(o663sub), java.lang.Object(o663sub))
f2318_0_length_FieldAccess(EOS, i480, java.lang.Object(java.lang.String(o678sub, i529)), java.lang.Object(java.lang.String(o678sub, i529))) → f2326_0_length_FieldAccess(EOS, i480, java.lang.Object(java.lang.String(o678sub, i529)), java.lang.Object(java.lang.String(o678sub, i529)))
f2326_0_length_FieldAccess(EOS, i480, java.lang.Object(java.lang.String(o678sub, i529)), java.lang.Object(java.lang.String(o678sub, i529))) → f2339_0_length_Return(EOS, i480, java.lang.Object(java.lang.String(o678sub, i529)))
f2339_0_length_Return(EOS, i480, java.lang.Object(java.lang.String(o678sub, i529))) → f2349_0_random_Return(EOS, i480)
f2349_0_random_Return(EOS, i480) → f2356_0_createIntList_Load(EOS, i480)
f2356_0_createIntList_Load(EOS, i480) → f2370_0_createIntList_InvokeMethod(EOS, i480)
f2370_0_createIntList_InvokeMethod(EOS, i480) → f2386_0__init__Load(EOS, i480)
f2386_0__init__Load(EOS, i480) → f2418_0__init__InvokeMethod(EOS, i480)
f2418_0__init__InvokeMethod(EOS, i480) → f2432_0__init__Load(EOS, i480)
f2432_0__init__Load(EOS, i480) → f2442_0__init__Load(EOS, i480)
f2442_0__init__Load(EOS, i480) → f2464_0__init__FieldAccess(EOS, i480)
f2464_0__init__FieldAccess(EOS, i480) → f2479_0__init__Load(EOS, i480)
f2479_0__init__Load(EOS, i480) → f2493_0__init__Load(EOS, i480)
f2493_0__init__Load(EOS, i480) → f2506_0__init__FieldAccess(EOS, i480)
f2506_0__init__FieldAccess(EOS, i480) → f2523_0__init__Return(EOS, i480)
f2523_0__init__Return(EOS, i480) → f2539_0_createIntList_Store(EOS, i480)
f2539_0_createIntList_Store(EOS, i480) → f2562_0_createIntList_Inc(EOS, i480)
f2562_0_createIntList_Inc(EOS, i480) → f2575_0_createIntList_JMP(EOS, +(i480, -1)) | >(i480, 0)
f2575_0_createIntList_JMP(EOS, i603) → f2589_0_createIntList_Load(EOS, i603)
f2589_0_createIntList_Load(EOS, i603) → f2149_0_createIntList_Load(EOS, i603)
f2149_0_createIntList_Load(EOS, i475) → f2161_0_createIntList_LE(EOS, i475, i475)

Combined rules. Obtained 1 IRules

P rules:
f2161_0_createIntList_LE(EOS, x0, x0) → f2161_0_createIntList_LE(EOS, -(x0, 1), -(x0, 1)) | >(x0, 0)

Filtered ground terms:

f2161_0_createIntList_LE(x1, x2, x3) → f2161_0_createIntList_LE(x2, x3)
Cond_f2161_0_createIntList_LE(x1, x2, x3, x4) → Cond_f2161_0_createIntList_LE(x1, x3, x4)

Filtered duplicate terms:

f2161_0_createIntList_LE(x1, x2) → f2161_0_createIntList_LE(x2)
Cond_f2161_0_createIntList_LE(x1, x2, x3) → Cond_f2161_0_createIntList_LE(x1, x3)

Prepared 1 rules for path length conversion:

P rules:
f2161_0_createIntList_LE(x0) → f2161_0_createIntList_LE(-(x0, 1)) | >(x0, 0)

Finished conversion. Obtained 1 rules.

P rules:
f2161_0_createIntList_LE(x0) → f2161_0_createIntList_LE(-(x0, 1)) | >(x0, 0)

(7) Obligation:

Rules:
f2161_0_createIntList_LE(x0) → f2161_0_createIntList_LE(-(x0, 1)) | >(x0, 0)

(8) PolynomialOrderProcessor (EQUIVALENT transformation)

Found the following polynomial interpretation:

[f2161_0_createIntList_LE(x2)] = x2

Therefore the following rule(s) have been dropped:

f2161_0_createIntList_LE(x0) → f2161_0_createIntList_LE(-(x0, 1)) | >(x0, 0)

(9) YES

(10) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: Convert.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses:

Used field analysis yielded the following read fields:
- IntList: [value, next]
Marker field analysis yielded the following relations that could be markers:

(11) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 34 rules for P and 0 rules for R.

P rules:
f2700_0_main_NULL(EOS(STATIC_2700), java.lang.Object(o824sub), java.lang.Object(o824sub)) → f2701_0_main_NULL(EOS(STATIC_2701), java.lang.Object(o824sub), java.lang.Object(o824sub))
f2701_0_main_NULL(EOS(STATIC_2701), java.lang.Object(o824sub), java.lang.Object(o824sub)) → f2704_0_main_Load(EOS(STATIC_2704), java.lang.Object(o824sub))
f2704_0_main_Load(EOS(STATIC_2704), java.lang.Object(o824sub)) → f2707_0_main_FieldAccess(EOS(STATIC_2707), java.lang.Object(o824sub), java.lang.Object(o824sub))
f2707_0_main_FieldAccess(EOS(STATIC_2707), java.lang.Object(IntList(EOC, i645, o828)), java.lang.Object(IntList(EOC, i645, o828))) → f2710_0_main_FieldAccess(EOS(STATIC_2710), java.lang.Object(IntList(EOC, i645, o828)), java.lang.Object(IntList(EOC, i645, o828)))
f2710_0_main_FieldAccess(EOS(STATIC_2710), java.lang.Object(IntList(EOC, i645, o828)), java.lang.Object(IntList(EOC, i645, o828))) → f2712_0_main_GT(EOS(STATIC_2712), java.lang.Object(IntList(EOC, i645, o828)), i645)
f2712_0_main_GT(EOS(STATIC_2712), java.lang.Object(IntList(EOC, i648, o828)), i648) → f2713_0_main_GT(EOS(STATIC_2713), java.lang.Object(IntList(EOC, i648, o828)), i648)
f2712_0_main_GT(EOS(STATIC_2712), java.lang.Object(IntList(EOC, i649, o828)), i649) → f2714_0_main_GT(EOS(STATIC_2714), java.lang.Object(IntList(EOC, i649, o828)), i649)
f2713_0_main_GT(EOS(STATIC_2713), java.lang.Object(IntList(EOC, i648, o828)), i648) → f2715_0_main_Load(EOS(STATIC_2715), java.lang.Object(IntList(EOC, i648, o828))) | <=(i648, 0)
f2715_0_main_Load(EOS(STATIC_2715), java.lang.Object(IntList(EOC, i648, o828))) → f2718_0_main_FieldAccess(EOS(STATIC_2718), java.lang.Object(IntList(EOC, i648, o828)))
f2718_0_main_FieldAccess(EOS(STATIC_2718), java.lang.Object(IntList(EOC, i648, o828))) → f2721_0_main_Store(EOS(STATIC_2721), o828)
f2721_0_main_Store(EOS(STATIC_2721), o828) → f2724_0_main_Load(EOS(STATIC_2724), o828)
f2724_0_main_Load(EOS(STATIC_2724), o828) → f2727_0_main_NULL(EOS(STATIC_2727), o828, o828)
f2727_0_main_NULL(EOS(STATIC_2727), java.lang.Object(o830sub), java.lang.Object(o830sub)) → f2730_0_main_NULL(EOS(STATIC_2730), java.lang.Object(o830sub), java.lang.Object(o830sub))
f2727_0_main_NULL(EOS(STATIC_2727), NULL, NULL) → f2731_0_main_NULL(EOS(STATIC_2731), NULL, NULL)
f2730_0_main_NULL(EOS(STATIC_2730), java.lang.Object(o830sub), java.lang.Object(o830sub)) → f2734_0_main_Load(EOS(STATIC_2734), java.lang.Object(o830sub))
f2734_0_main_Load(EOS(STATIC_2734), java.lang.Object(o830sub)) → f2740_0_main_Load(EOS(STATIC_2740), java.lang.Object(o830sub))
f2740_0_main_Load(EOS(STATIC_2740), java.lang.Object(o830sub)) → f2743_0_main_IntArithmetic(EOS(STATIC_2743), java.lang.Object(o830sub))
f2743_0_main_IntArithmetic(EOS(STATIC_2743), java.lang.Object(o830sub)) → f2749_0_main_Store(EOS(STATIC_2749), java.lang.Object(o830sub))
f2749_0_main_Store(EOS(STATIC_2749), java.lang.Object(o830sub)) → f2750_0_main_JMP(EOS(STATIC_2750), java.lang.Object(o830sub))
f2750_0_main_JMP(EOS(STATIC_2750), java.lang.Object(o830sub)) → f2754_0_main_Load(EOS(STATIC_2754), java.lang.Object(o830sub))
f2754_0_main_Load(EOS(STATIC_2754), java.lang.Object(o830sub)) → f2697_0_main_Load(EOS(STATIC_2697), java.lang.Object(o830sub))
f2697_0_main_Load(EOS(STATIC_2697), o818) → f2700_0_main_NULL(EOS(STATIC_2700), o818, o818)
f2731_0_main_NULL(EOS(STATIC_2731), NULL, NULL) → f2737_0_main_Load(EOS(STATIC_2737), NULL)
f2737_0_main_Load(EOS(STATIC_2737), NULL) → f2697_0_main_Load(EOS(STATIC_2697), NULL)
f2714_0_main_GT(EOS(STATIC_2714), java.lang.Object(IntList(EOC, i649, o828)), i649) → f2717_0_main_Load(EOS(STATIC_2717), java.lang.Object(IntList(EOC, i649, o828))) | >(i649, 0)
f2717_0_main_Load(EOS(STATIC_2717), java.lang.Object(IntList(EOC, i649, o828))) → f2720_0_main_Duplicate(EOS(STATIC_2720), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828)))
f2720_0_main_Duplicate(EOS(STATIC_2720), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828))) → f2722_0_main_FieldAccess(EOS(STATIC_2722), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828)))
f2722_0_main_FieldAccess(EOS(STATIC_2722), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828))) → f2725_0_main_ConstantStackPush(EOS(STATIC_2725), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828)), i649)
f2725_0_main_ConstantStackPush(EOS(STATIC_2725), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828)), i649) → f2728_0_main_IntArithmetic(EOS(STATIC_2728), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828)), i649, 1)
f2728_0_main_IntArithmetic(EOS(STATIC_2728), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828)), i649, matching1) → f2732_0_main_FieldAccess(EOS(STATIC_2732), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828)), -(i649, 1)) | &&(>(i649, 0), =(matching1, 1))
f2732_0_main_FieldAccess(EOS(STATIC_2732), java.lang.Object(IntList(EOC, i649, o828)), java.lang.Object(IntList(EOC, i649, o828)), i653) → f2739_0_main_Inc(EOS(STATIC_2739), java.lang.Object(IntList(EOC, i653, o828)))
f2739_0_main_Inc(EOS(STATIC_2739), java.lang.Object(IntList(EOC, i653, o828))) → f2741_0_main_JMP(EOS(STATIC_2741), java.lang.Object(IntList(EOC, i653, o828)))
f2741_0_main_JMP(EOS(STATIC_2741), java.lang.Object(IntList(EOC, i653, o828))) → f2747_0_main_Load(EOS(STATIC_2747), java.lang.Object(IntList(EOC, i653, o828)))
f2747_0_main_Load(EOS(STATIC_2747), java.lang.Object(IntList(EOC, i653, o828))) → f2697_0_main_Load(EOS(STATIC_2697), java.lang.Object(IntList(EOC, i653, o828)))
R rules:

Combined rules. Obtained 3 conditional rules for P and 0 conditional rules for R.

P rules:
f2700_0_main_NULL(EOS(STATIC_2700), java.lang.Object(IntList(EOC, x0, java.lang.Object(x1))), java.lang.Object(IntList(EOC, x0, java.lang.Object(x1)))) → f2700_0_main_NULL(EOS(STATIC_2700), java.lang.Object(x1), java.lang.Object(x1)) | <=(x0, 0)
f2700_0_main_NULL(EOS(STATIC_2700), java.lang.Object(IntList(EOC, x0, NULL)), java.lang.Object(IntList(EOC, x0, NULL))) → f2700_0_main_NULL(EOS(STATIC_2700), NULL, NULL) | <=(x0, 0)
f2700_0_main_NULL(EOS(STATIC_2700), java.lang.Object(IntList(EOC, x0, x1)), java.lang.Object(IntList(EOC, x0, x1))) → f2700_0_main_NULL(EOS(STATIC_2700), java.lang.Object(IntList(EOC, -(x0, 1), x1)), java.lang.Object(IntList(EOC, -(x0, 1), x1))) | >(x0, 0)
R rules:

Filtered ground terms:

f2700_0_main_NULL(x1, x2, x3) → f2700_0_main_NULL(x2, x3)
Cond_f2700_0_main_NULL(x1, x2, x3, x4) → Cond_f2700_0_main_NULL(x1, x3, x4)
Cond_f2700_0_main_NULL1(x1, x2, x3, x4) → Cond_f2700_0_main_NULL1(x1, x3, x4)
Cond_f2700_0_main_NULL2(x1, x2, x3, x4) → Cond_f2700_0_main_NULL2(x1, x3, x4)
EOS(x1) → EOS
IntList(x1, x2, x3) → IntList(x2, x3)

Filtered unneeded arguments:

Cond_f2700_0_main_NULL1(x1, x2, x3) → Cond_f2700_0_main_NULL1(x1)

Filtered duplicate args:

f2700_0_main_NULL(x1, x2) → f2700_0_main_NULL(x2)
Cond_f2700_0_main_NULL(x1, x2, x3) → Cond_f2700_0_main_NULL(x1, x3)
Cond_f2700_0_main_NULL2(x1, x2, x3) → Cond_f2700_0_main_NULL2(x1, x3)

Combined rules. Obtained 3 conditional rules for P and 0 conditional rules for R.

P rules:
F2700_0_MAIN_NULL(java.lang.Object(IntList(x0, java.lang.Object(x1)))) → F2700_0_MAIN_NULL(java.lang.Object(x1)) | <=(x0, 0)
F2700_0_MAIN_NULL(java.lang.Object(IntList(x0, NULL))) → F2700_0_MAIN_NULL(NULL) | <=(x0, 0)
F2700_0_MAIN_NULL(java.lang.Object(IntList(x0, x1))) → F2700_0_MAIN_NULL(java.lang.Object(IntList(-(x0, 1), x1))) | >(x0, 0)
R rules:

Finished conversion. Obtained 6 rules for P and 0 rules for R. System has predefined symbols.

P rules:
F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0, java.lang.Object(x1)))) → COND_F2700_0_MAIN_NULL(<=(x0, 0), java.lang.Object(IntList(x0, java.lang.Object(x1))))
COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0, java.lang.Object(x1)))) → F2700_0_MAIN_NULL'(java.lang.Object(x1))
F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0, NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0, 0), java.lang.Object(IntList(x0, NULL)))
COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0, NULL))) → F2700_0_MAIN_NULL'(NULL)
F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0, x1))) → COND_F2700_0_MAIN_NULL2(>(x0, 0), java.lang.Object(IntList(x0, x1)))
COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0, x1))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0, 1), x1)))
R rules:

(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:
(0): F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(x0[0] <= 0, java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
(1): COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))
(2): F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL))) → COND_F2700_0_MAIN_NULL1(x0[2] <= 0, java.lang.Object(IntList(x0[2], NULL)))
(3): COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → F2700_0_MAIN_NULL'(NULL)
(4): F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(x0[4] > 0, java.lang.Object(IntList(x0[4], x1[4])))
(5): COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[5] - 1, x1[5])))

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

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

(1) -> (2), if (java.lang.Object(x1[1]) →^* java.lang.Object(IntList(x0[2], NULL)))

(1) -> (4), if (java.lang.Object(x1[1]) →^* java.lang.Object(IntList(x0[4], x1[4])))

(2) -> (3), if (x0[2] <= 0 ∧java.lang.Object(IntList(x0[2], NULL)) →^* java.lang.Object(IntList(x0[3], NULL)))

(3) -> (0), if (NULL →^* java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))

(3) -> (2), if (NULL →^* java.lang.Object(IntList(x0[2], NULL)))

(3) -> (4), if (NULL →^* java.lang.Object(IntList(x0[4], x1[4])))

(4) -> (5), if (x0[4] > 0 ∧java.lang.Object(IntList(x0[4], x1[4])) →^* java.lang.Object(IntList(x0[5], x1[5])))

(5) -> (0), if (java.lang.Object(IntList(x0[5] - 1, x1[5])) →^* java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))

(5) -> (2), if (java.lang.Object(IntList(x0[5] - 1, x1[5])) →^* java.lang.Object(IntList(x0[2], NULL)))

(5) -> (4), if (java.lang.Object(IntList(x0[5] - 1, x1[5])) →^* java.lang.Object(IntList(x0[4], x1[4])))

The set Q is empty.

(13) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: true Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@3002ff1e Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0, java.lang.Object(x1)))) → COND_F2700_0_MAIN_NULL(<=(x0, 0), java.lang.Object(IntList(x0, java.lang.Object(x1)))) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1])) which results in the following constraint:
(1)    (<=(x0[0], 0)=TRUE∧java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))=java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))) ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥))

We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
(2)    (<=(x0[0], 0)=TRUE ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_18 + (-1)Bound*bni_18] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_18 + (-1)Bound*bni_18] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_18 + (-1)Bound*bni_18] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0, java.lang.Object(x1)))) → F2700_0_MAIN_NULL'(java.lang.Object(x1)) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1])), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) which results in the following constraint:
(7)    (<=(x0[0], 0)=TRUE∧java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))=java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))∧java.lang.Object(x1[1])=java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))) ⇒ COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥))

We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:
(8)    (<=(x0[0], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))))≥NonInfC∧COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)
We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1])), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))) which results in the following constraint:
(13)    (<=(x0[0], 0)=TRUE∧java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))=java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))∧java.lang.Object(x1[1])=java.lang.Object(IntList(x0[2], NULL)) ⇒ COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥))

We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint:
(14)    (<=(x0[0], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[2], NULL)))))≥NonInfC∧COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[2], NULL)))))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL)))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (17) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(18)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)
We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1])), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) which results in the following constraint:
(19)    (<=(x0[0], 0)=TRUE∧java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))=java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))∧java.lang.Object(x1[1])=java.lang.Object(IntList(x0[4], x1[4])) ⇒ COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥))

We simplified constraint (19) using rules (I), (II), (III) which results in the following new constraint:
(20)    (<=(x0[0], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥NonInfC∧COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0, NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0, 0), java.lang.Object(IntList(x0, NULL))) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → F2700_0_MAIN_NULL'(NULL) which results in the following constraint:
(25)    (<=(x0[2], 0)=TRUE∧java.lang.Object(IntList(x0[2], NULL))=java.lang.Object(IntList(x0[3], NULL)) ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL)))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL)))≥COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))∧(U^Increasing(COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥))

We simplified constraint (25) using rules (I), (II), (IV), (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint:
(26)    (<=(x0[2], 0)=TRUE ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL)))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL)))≥COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))∧(U^Increasing(COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥))

We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥)∧[(5)bni_22 + (-1)Bound*bni_22] ≥ 0∧[8 + (-1)bso_23] ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥)∧[(5)bni_22 + (-1)Bound*bni_22] ≥ 0∧[8 + (-1)bso_23] ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥)∧[(5)bni_22 + (-1)Bound*bni_22] ≥ 0∧[8 + (-1)bso_23] ≥ 0)

We simplified constraint (29) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(30)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥)∧0 ≥ 0∧[(5)bni_22 + (-1)Bound*bni_22] ≥ 0∧0 ≥ 0∧[8 + (-1)bso_23] ≥ 0)

For Pair COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0, NULL))) → F2700_0_MAIN_NULL'(NULL) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → F2700_0_MAIN_NULL'(NULL), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) which results in the following constraint:
(31) (<=(x0[2], 0)=TRUE∧java.lang.Object(IntList(x0[2], NULL))=java.lang.Object(IntList(x0[3], NULL))∧NULL=java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))) ⇒ COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥NonInfC∧COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥F2700_0_MAIN_NULL'(NULL)∧(U^Increasing(F2700_0_MAIN_NULL'(NULL)), ≥))

We solved constraint (31) using rules (I), (II), (DELETE_TRIVIAL_REDUCESTO).
We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → F2700_0_MAIN_NULL'(NULL), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))) which results in the following constraint:
(32) (<=(x0[2], 0)=TRUE∧java.lang.Object(IntList(x0[2], NULL))=java.lang.Object(IntList(x0[3], NULL))∧NULL=java.lang.Object(IntList(x0[2]1, NULL)) ⇒ COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥NonInfC∧COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥F2700_0_MAIN_NULL'(NULL)∧(U^Increasing(F2700_0_MAIN_NULL'(NULL)), ≥))

We solved constraint (32) using rules (I), (II), (DELETE_TRIVIAL_REDUCESTO).
We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → F2700_0_MAIN_NULL'(NULL), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) which results in the following constraint:
(33) (<=(x0[2], 0)=TRUE∧java.lang.Object(IntList(x0[2], NULL))=java.lang.Object(IntList(x0[3], NULL))∧NULL=java.lang.Object(IntList(x0[4], x1[4])) ⇒ COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥NonInfC∧COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥F2700_0_MAIN_NULL'(NULL)∧(U^Increasing(F2700_0_MAIN_NULL'(NULL)), ≥))

We solved constraint (33) using rules (I), (II), (DELETE_TRIVIAL_REDUCESTO).

For Pair F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0, x1))) → COND_F2700_0_MAIN_NULL2(>(x0, 0), java.lang.Object(IntList(x0, x1))) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))) which results in the following constraint:
(34)    (>(x0[4], 0)=TRUE∧java.lang.Object(IntList(x0[4], x1[4]))=java.lang.Object(IntList(x0[5], x1[5])) ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥))

We simplified constraint (34) using rules (I), (II), (IV) which results in the following new constraint:
(35)    (>(x0[4], 0)=TRUE ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥))

We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(36)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(37)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(38)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (38) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(39)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0)

For Pair COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0, x1))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0, 1), x1))) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) which results in the following constraint:
(40)    (>(x0[4], 0)=TRUE∧java.lang.Object(IntList(x0[4], x1[4]))=java.lang.Object(IntList(x0[5], x1[5]))∧java.lang.Object(IntList(-(x0[5], 1), x1[5]))=java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))) ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (40) using rules (I), (II), (III), (IV) which results in the following new constraint:
(41)    (>(x0[4], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[4], 1), java.lang.Object(x1[0]))))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (41) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(42)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (42) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(43)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(44)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (44) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(45)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_27] ≥ 0)
We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))) which results in the following constraint:
(46)    (>(x0[4], 0)=TRUE∧java.lang.Object(IntList(x0[4], x1[4]))=java.lang.Object(IntList(x0[5], x1[5]))∧java.lang.Object(IntList(-(x0[5], 1), x1[5]))=java.lang.Object(IntList(x0[2], NULL)) ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (46) using rules (I), (II), (III), (IV) which results in the following new constraint:
(47)    (>(x0[4], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], NULL)))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], NULL)))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[4], 1), NULL)))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (47) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(48)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (48) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(49)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(50)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (50) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(51)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 ≥ 0∧[(-1)bso_27] ≥ 0)
We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) which results in the following constraint:
(52)    (>(x0[4], 0)=TRUE∧java.lang.Object(IntList(x0[4], x1[4]))=java.lang.Object(IntList(x0[5], x1[5]))∧java.lang.Object(IntList(-(x0[5], 1), x1[5]))=java.lang.Object(IntList(x0[4]1, x1[4]1)) ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (52) using rules (I), (II), (III), (IV) which results in the following new constraint:
(53)    (>(x0[4], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[4], 1), x1[4])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (53) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(54)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (54) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(55)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (55) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(56)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (56) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(57)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_27] ≥ 0)

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

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0, java.lang.Object(x1)))) → COND_F2700_0_MAIN_NULL(<=(x0, 0), java.lang.Object(IntList(x0, java.lang.Object(x1))))
- (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_19] ≥ 0)
COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0, java.lang.Object(x1)))) → F2700_0_MAIN_NULL'(java.lang.Object(x1))
- (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)
F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0, NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0, 0), java.lang.Object(IntList(x0, NULL)))
- (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥)∧0 ≥ 0∧[(5)bni_22 + (-1)Bound*bni_22] ≥ 0∧0 ≥ 0∧[8 + (-1)bso_23] ≥ 0)
COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0, NULL))) → F2700_0_MAIN_NULL'(NULL)
F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0, x1))) → COND_F2700_0_MAIN_NULL2(>(x0, 0), java.lang.Object(IntList(x0, x1)))
- (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0)
COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0, x1))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0, 1), x1)))
- (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_27] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 ≥ 0∧[(-1)bso_27] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_27] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(F2700_0_MAIN_NULL'(x₁)) = [1] + [2]x₁
POL(java.lang.Object(x₁)) = [2]
POL(IntList(x₁, x₂)) = 0
POL(COND_F2700_0_MAIN_NULL(x₁, x₂)) = [1] + [2]x₂
POL(<=(x₁, x₂)) = 0
POL(0) = 0
POL(NULL) = 0
POL(COND_F2700_0_MAIN_NULL1(x₁, x₂)) = [-1] + [-1]x₂
POL(COND_F2700_0_MAIN_NULL2(x₁, x₂)) = [1] + [2]x₂
POL(>(x₁, x₂)) = 0
POL(-(x₁, x₂)) = 0
POL(1) = 0

The following pairs are in P_>:

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))
COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → F2700_0_MAIN_NULL'(NULL)

The following pairs are in P_bound:

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))
F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[2], NULL))) → COND_F2700_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))
COND_F2700_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → F2700_0_MAIN_NULL'(NULL)
F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_≥:

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))
F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

There are no usable rules.

(14) 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:
(0): F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(x0[0] <= 0, java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
(1): COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))
(4): F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(x0[4] > 0, java.lang.Object(IntList(x0[4], x1[4])))
(5): COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[5] - 1, x1[5])))

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

(5) -> (0), if (java.lang.Object(IntList(x0[5] - 1, x1[5])) →^* java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))

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

(1) -> (4), if (java.lang.Object(x1[1]) →^* java.lang.Object(IntList(x0[4], x1[4])))

(5) -> (4), if (java.lang.Object(IntList(x0[5] - 1, x1[5])) →^* java.lang.Object(IntList(x0[4], x1[4])))

(4) -> (5), if (x0[4] > 0 ∧java.lang.Object(IntList(x0[4], x1[4])) →^* java.lang.Object(IntList(x0[5], x1[5])))

The set Q is empty.

(15) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: true Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@3002ff1e Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1])) which results in the following constraint:
(1)    (<=(x0[0], 0)=TRUE∧java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))=java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))) ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥))

We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
(2)    (<=(x0[0], 0)=TRUE ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_14 + (-1)Bound*bni_14] + [(4)bni_14]x1[0] + [(6)bni_14]x0[0] ≥ 0∧[2 + (-1)bso_15] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_14 + (-1)Bound*bni_14] + [(4)bni_14]x1[0] + [(6)bni_14]x0[0] ≥ 0∧[2 + (-1)bso_15] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_14 + (-1)Bound*bni_14] + [(4)bni_14]x1[0] + [(6)bni_14]x0[0] ≥ 0∧[2 + (-1)bso_15] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(4)bni_14] ≥ 0∧[(6)bni_14] ≥ 0∧[(5)bni_14 + (-1)Bound*bni_14] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[2 + (-1)bso_15] ≥ 0)

For Pair COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1])) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1])), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) which results in the following constraint:
(7)    (<=(x0[0], 0)=TRUE∧java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))=java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))∧java.lang.Object(x1[1])=java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))) ⇒ COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥))

We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:
(8)    (<=(x0[0], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))))≥NonInfC∧COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(11)bni_16 + (-1)Bound*bni_16] + [(8)bni_16]x1[0]1 + [(12)bni_16]x0[0]1 + [(6)bni_16]x0[0] ≥ 0∧[6 + (-1)bso_17] + [4]x1[0]1 + [6]x0[0]1 + [6]x0[0] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(11)bni_16 + (-1)Bound*bni_16] + [(8)bni_16]x1[0]1 + [(12)bni_16]x0[0]1 + [(6)bni_16]x0[0] ≥ 0∧[6 + (-1)bso_17] + [4]x1[0]1 + [6]x0[0]1 + [6]x0[0] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(11)bni_16 + (-1)Bound*bni_16] + [(8)bni_16]x1[0]1 + [(12)bni_16]x0[0]1 + [(6)bni_16]x0[0] ≥ 0∧[6 + (-1)bso_17] + [4]x1[0]1 + [6]x0[0]1 + [6]x0[0] ≥ 0)

We simplified constraint (11) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(12)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(8)bni_16] ≥ 0∧[(12)bni_16] ≥ 0∧[(6)bni_16] ≥ 0∧[(11)bni_16 + (-1)Bound*bni_16] ≥ 0∧[6 + (-1)bso_17] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)
We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1])), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) which results in the following constraint:
(13)    (<=(x0[0], 0)=TRUE∧java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))=java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))∧java.lang.Object(x1[1])=java.lang.Object(IntList(x0[4], x1[4])) ⇒ COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥))

We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint:
(14)    (<=(x0[0], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥NonInfC∧COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(11)bni_16 + (-1)Bound*bni_16] + [(4)bni_16]x1[4] + [(12)bni_16]x0[4] + [(6)bni_16]x0[0] ≥ 0∧[6 + (-1)bso_17] + [2]x1[4] + [6]x0[4] + [6]x0[0] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(11)bni_16 + (-1)Bound*bni_16] + [(4)bni_16]x1[4] + [(12)bni_16]x0[4] + [(6)bni_16]x0[0] ≥ 0∧[6 + (-1)bso_17] + [2]x1[4] + [6]x0[4] + [6]x0[0] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(11)bni_16 + (-1)Bound*bni_16] + [(4)bni_16]x1[4] + [(12)bni_16]x0[4] + [(6)bni_16]x0[0] ≥ 0∧[6 + (-1)bso_17] + [2]x1[4] + [6]x0[4] + [6]x0[0] ≥ 0)

We simplified constraint (17) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(18)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(4)bni_16] ≥ 0∧[(12)bni_16] ≥ 0∧[(6)bni_16] ≥ 0∧[(11)bni_16 + (-1)Bound*bni_16] ≥ 0∧[6 + (-1)bso_17] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)

For Pair F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))) which results in the following constraint:
(19)    (>(x0[4], 0)=TRUE∧java.lang.Object(IntList(x0[4], x1[4]))=java.lang.Object(IntList(x0[5], x1[5])) ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥))

We simplified constraint (19) using rules (I), (II), (IV) which results in the following new constraint:
(20)    (>(x0[4], 0)=TRUE ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x1[4] + [(6)bni_18]x0[4] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x1[4] + [(6)bni_18]x0[4] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x1[4] + [(6)bni_18]x0[4] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(2)bni_18] ≥ 0∧[(6)bni_18] ≥ 0∧[(5)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) which results in the following constraint:
(25)    (>(x0[4], 0)=TRUE∧java.lang.Object(IntList(x0[4], x1[4]))=java.lang.Object(IntList(x0[5], x1[5]))∧java.lang.Object(IntList(-(x0[5], 1), x1[5]))=java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))) ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (25) using rules (I), (II), (III), (IV) which results in the following new constraint:
(26)    (>(x0[4], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[4], 1), java.lang.Object(x1[0]))))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] + [(4)bni_20]x1[0] + [(6)bni_20]x0[4] ≥ 0∧[(-1)bso_21] + [6]x0[4] ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] + [(4)bni_20]x1[0] + [(6)bni_20]x0[4] ≥ 0∧[(-1)bso_21] + [6]x0[4] ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] + [(4)bni_20]x1[0] + [(6)bni_20]x0[4] ≥ 0∧[(-1)bso_21] + [6]x0[4] ≥ 0)

We simplified constraint (29) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(30)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(4)bni_20] ≥ 0∧[(6)bni_20] ≥ 0∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0∧[1] ≥ 0)
We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) which results in the following constraint:
(31)    (>(x0[4], 0)=TRUE∧java.lang.Object(IntList(x0[4], x1[4]))=java.lang.Object(IntList(x0[5], x1[5]))∧java.lang.Object(IntList(-(x0[5], 1), x1[5]))=java.lang.Object(IntList(x0[4]1, x1[4]1)) ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (31) using rules (I), (II), (III), (IV) which results in the following new constraint:
(32)    (>(x0[4], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[4], 1), x1[4])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (32) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(33)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] + [(2)bni_20]x1[4] + [(6)bni_20]x0[4] ≥ 0∧[(-1)bso_21] + [6]x0[4] ≥ 0)

We simplified constraint (33) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(34)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] + [(2)bni_20]x1[4] + [(6)bni_20]x0[4] ≥ 0∧[(-1)bso_21] + [6]x0[4] ≥ 0)

We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(35)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_20 + (-1)Bound*bni_20] + [(2)bni_20]x1[4] + [(6)bni_20]x0[4] ≥ 0∧[(-1)bso_21] + [6]x0[4] ≥ 0)

We simplified constraint (35) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(36)    (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(2)bni_20] ≥ 0∧[(6)bni_20] ≥ 0∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0∧[1] ≥ 0)

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

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
- (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(4)bni_14] ≥ 0∧[(6)bni_14] ≥ 0∧[(5)bni_14 + (-1)Bound*bni_14] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[2 + (-1)bso_15] ≥ 0)
COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))
- (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(8)bni_16] ≥ 0∧[(12)bni_16] ≥ 0∧[(6)bni_16] ≥ 0∧[(11)bni_16 + (-1)Bound*bni_16] ≥ 0∧[6 + (-1)bso_17] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))), ≥)∧[(4)bni_16] ≥ 0∧[(12)bni_16] ≥ 0∧[(6)bni_16] ≥ 0∧[(11)bni_16 + (-1)Bound*bni_16] ≥ 0∧[6 + (-1)bso_17] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)
F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
- (0 ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(2)bni_18] ≥ 0∧[(6)bni_18] ≥ 0∧[(5)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_19] ≥ 0)
COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))
- (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(4)bni_20] ≥ 0∧[(6)bni_20] ≥ 0∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0∧[1] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(2)bni_20] ≥ 0∧[(6)bni_20] ≥ 0∧[(5)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0∧[1] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(F2700_0_MAIN_NULL'(x₁)) = [1] + x₁
POL(java.lang.Object(x₁)) = [2]x₁
POL(IntList(x₁, x₂)) = [2] + x₂ + [3]x₁
POL(COND_F2700_0_MAIN_NULL(x₁, x₂)) = [-1] + x₂
POL(<=(x₁, x₂)) = 0
POL(0) = 0
POL(COND_F2700_0_MAIN_NULL2(x₁, x₂)) = [1] + x₂
POL(>(x₁, x₂)) = 0
POL(-(x₁, x₂)) = 0
POL(1) = 0

The following pairs are in P_>:

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))

The following pairs are in P_bound:

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_F2700_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_F2700_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → F2700_0_MAIN_NULL'(java.lang.Object(x1[1]))
F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_≥:

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

There are no usable rules.

(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(4): F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(x0[4] > 0, java.lang.Object(IntList(x0[4], x1[4])))
(5): COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[5] - 1, x1[5])))

(5) -> (4), if (java.lang.Object(IntList(x0[5] - 1, x1[5])) →^* java.lang.Object(IntList(x0[4], x1[4])))

(4) -> (5), if (x0[4] > 0 ∧java.lang.Object(IntList(x0[4], x1[4])) →^* java.lang.Object(IntList(x0[5], x1[5])))

The set Q is empty.

(17) 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.IdpDefaultShapeHeuristic@49cbb862 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))) which results in the following constraint:
(1)    (>(x0[4], 0)=TRUE∧java.lang.Object(IntList(x0[4], x1[4]))=java.lang.Object(IntList(x0[5], x1[5])) ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥))

We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
(2)    (>(x0[4], 0)=TRUE ⇒ F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4])))≥COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(-2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(-2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(-2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧0 = 0∧[(-2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x0[4] ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧0 = 0∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))) the following chains were created:

We consider the chain F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) which results in the following constraint:
(8)    (>(x0[4], 0)=TRUE∧java.lang.Object(IntList(x0[4], x1[4]))=java.lang.Object(IntList(x0[5], x1[5]))∧java.lang.Object(IntList(-(x0[5], 1), x1[5]))=java.lang.Object(IntList(x0[4]1, x1[4]1)) ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (8) using rules (I), (II), (III), (IV) which results in the following new constraint:
(9)    (>(x0[4], 0)=TRUE ⇒ COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[4], 1), x1[4])))∧(U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[4] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[4] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[4] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    (x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 = 0∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[4] ≥ 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x0[4] ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 = 0∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[4] ≥ 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

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

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
- (x0[4] ≥ 0 ⇒ (U^Increasing(COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧0 = 0∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)
COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))
- (x0[4] ≥ 0 ⇒ (U^Increasing(F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 = 0∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[4] ≥ 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

POL(TRUE) = [2]
POL(FALSE) = 0
POL(F2700_0_MAIN_NULL'(x₁)) = [-1] + [-1]x₁
POL(java.lang.Object(x₁)) = [-1]x₁
POL(IntList(x₁, x₂)) = [-1] + x₁
POL(COND_F2700_0_MAIN_NULL2(x₁, x₂)) = [-1] + [-1]x₂
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]

The following pairs are in P_>:

COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_bound:

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_F2700_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → F2700_0_MAIN_NULL'(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_≥:

F2700_0_MAIN_NULL'(java.lang.Object(IntList(x0[4], x1[4]))) → COND_F2700_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))

There are no usable rules.

(18) 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

(19) 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) Complex Obligation (AND)

(5) Obligation:

(6) SCCToIntTRSProof (SOUND transformation)

(7) Obligation:

(8) PolynomialOrderProcessor (EQUIVALENT transformation)

(9) YES

(10) Obligation:

(11) SCCToIDPv1Proof (SOUND transformation)

(12) Obligation:

(13) IDPNonInfProof (SOUND transformation)

(14) Obligation:

(15) IDPNonInfProof (SOUND transformation)

(16) Obligation:

(17) IDPNonInfProof (SOUND transformation)

(18) Obligation:

(19) IDependencyGraphProof (EQUIVALENT transformation)

(20) TRUE