Termination proof of /tmp/tmpjPmcxm/Convert.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 630 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 10 ms)

↳4 AND

    ↳5 JBCTerminationSCC

        ↳6 SCCToIDPv1Proof (⇒, 170 ms)

        ↳7 IDP

        ↳8 IDPNonInfProof (⇒, 160 ms)

        ↳9 AND

            ↳10 IDP

                ↳11 IDependencyGraphProof (⇔, 0 ms)

                ↳12 TRUE

            ↳13 IDP

                ↳14 IDependencyGraphProof (⇔, 0 ms)

                ↳15 TRUE

    ↳16 JBCTerminationSCC

        ↳17 SCCToIDPv1Proof (⇒, 180 ms)

        ↳18 IDP

        ↳19 IDPNonInfProof (⇒, 1010 ms)

        ↳20 IDP

        ↳21 IDPNonInfProof (⇒, 360 ms)

        ↳22 IDP

        ↳23 IDPNonInfProof (⇒, 220 ms)

        ↳24 IDP

        ↳25 IDependencyGraphProof (⇔, 0 ms)

        ↳26 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Convert

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: UsedFieldsAnalysis

(6) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 38 rules for P and 0 rules for R.

P rules:
1704_0_createIntList_LE(EOS(STATIC_1704), i439, i439) → 1713_0_createIntList_LE(EOS(STATIC_1713), i439, i439)
1713_0_createIntList_LE(EOS(STATIC_1713), i439, i439) → 1723_0_createIntList_New(EOS(STATIC_1723), i439) | >(i439, 0)
1723_0_createIntList_New(EOS(STATIC_1723), i439) → 1732_0_createIntList_Duplicate(EOS(STATIC_1732), i439)
1732_0_createIntList_Duplicate(EOS(STATIC_1732), i439) → 1740_0_createIntList_InvokeMethod(EOS(STATIC_1740), i439)
1740_0_createIntList_InvokeMethod(EOS(STATIC_1740), i439) → 1751_0_random_FieldAccess(EOS(STATIC_1751), i439)
1751_0_random_FieldAccess(EOS(STATIC_1751), i439) → 1770_0_random_FieldAccess(EOS(STATIC_1770), i439)
1770_0_random_FieldAccess(EOS(STATIC_1770), i439) → 1775_0_random_ArrayAccess(EOS(STATIC_1775), i439)
1775_0_random_ArrayAccess(EOS(STATIC_1775), i439) → 1779_0_random_ArrayAccess(EOS(STATIC_1779), i439)
1779_0_random_ArrayAccess(EOS(STATIC_1779), i439) → 1785_0_random_Store(EOS(STATIC_1785), i439, o676)
1785_0_random_Store(EOS(STATIC_1785), i439, o676) → 1792_0_random_FieldAccess(EOS(STATIC_1792), i439, o676)
1792_0_random_FieldAccess(EOS(STATIC_1792), i439, o676) → 1796_0_random_ConstantStackPush(EOS(STATIC_1796), i439, o676)
1796_0_random_ConstantStackPush(EOS(STATIC_1796), i439, o676) → 1801_0_random_IntArithmetic(EOS(STATIC_1801), i439, o676)
1801_0_random_IntArithmetic(EOS(STATIC_1801), i439, o676) → 1807_0_random_FieldAccess(EOS(STATIC_1807), i439, o676)
1807_0_random_FieldAccess(EOS(STATIC_1807), i439, o676) → 1813_0_random_Load(EOS(STATIC_1813), i439, o676)
1813_0_random_Load(EOS(STATIC_1813), i439, o676) → 1821_0_random_InvokeMethod(EOS(STATIC_1821), i439, o676)
1821_0_random_InvokeMethod(EOS(STATIC_1821), i439, java.lang.Object(o697sub)) → 1827_0_random_InvokeMethod(EOS(STATIC_1827), i439, java.lang.Object(o697sub))
1827_0_random_InvokeMethod(EOS(STATIC_1827), i439, java.lang.Object(o697sub)) → 1833_0_length_Load(EOS(STATIC_1833), i439, java.lang.Object(o697sub), java.lang.Object(o697sub))
1833_0_length_Load(EOS(STATIC_1833), i439, java.lang.Object(o697sub), java.lang.Object(o697sub)) → 1848_0_length_FieldAccess(EOS(STATIC_1848), i439, java.lang.Object(o697sub), java.lang.Object(o697sub))
1848_0_length_FieldAccess(EOS(STATIC_1848), i439, java.lang.Object(java.lang.String(o709sub, i483)), java.lang.Object(java.lang.String(o709sub, i483))) → 1856_0_length_FieldAccess(EOS(STATIC_1856), i439, java.lang.Object(java.lang.String(o709sub, i483)), java.lang.Object(java.lang.String(o709sub, i483))) | &&(>=(i483, 0), >=(i484, 0))
1856_0_length_FieldAccess(EOS(STATIC_1856), i439, java.lang.Object(java.lang.String(o709sub, i483)), java.lang.Object(java.lang.String(o709sub, i483))) → 1864_0_length_Return(EOS(STATIC_1864), i439, java.lang.Object(java.lang.String(o709sub, i483)))
1864_0_length_Return(EOS(STATIC_1864), i439, java.lang.Object(java.lang.String(o709sub, i483))) → 1875_0_random_Return(EOS(STATIC_1875), i439)
1875_0_random_Return(EOS(STATIC_1875), i439) → 1879_0_createIntList_Load(EOS(STATIC_1879), i439)
1879_0_createIntList_Load(EOS(STATIC_1879), i439) → 1888_0_createIntList_InvokeMethod(EOS(STATIC_1888), i439)
1888_0_createIntList_InvokeMethod(EOS(STATIC_1888), i439) → 1896_0_<init>_Load(EOS(STATIC_1896), i439)
1896_0_<init>_Load(EOS(STATIC_1896), i439) → 1909_0_<init>_InvokeMethod(EOS(STATIC_1909), i439)
1909_0_<init>_InvokeMethod(EOS(STATIC_1909), i439) → 1917_0_<init>_Load(EOS(STATIC_1917), i439)
1917_0_<init>_Load(EOS(STATIC_1917), i439) → 1923_0_<init>_Load(EOS(STATIC_1923), i439)
1923_0_<init>_Load(EOS(STATIC_1923), i439) → 1930_0_<init>_FieldAccess(EOS(STATIC_1930), i439)
1930_0_<init>_FieldAccess(EOS(STATIC_1930), i439) → 1938_0_<init>_Load(EOS(STATIC_1938), i439)
1938_0_<init>_Load(EOS(STATIC_1938), i439) → 1946_0_<init>_Load(EOS(STATIC_1946), i439)
1946_0_<init>_Load(EOS(STATIC_1946), i439) → 1954_0_<init>_FieldAccess(EOS(STATIC_1954), i439)
1954_0_<init>_FieldAccess(EOS(STATIC_1954), i439) → 1963_0_<init>_Return(EOS(STATIC_1963), i439)
1963_0_<init>_Return(EOS(STATIC_1963), i439) → 1972_0_createIntList_Store(EOS(STATIC_1972), i439)
1972_0_createIntList_Store(EOS(STATIC_1972), i439) → 1984_0_createIntList_Inc(EOS(STATIC_1984), i439)
1984_0_createIntList_Inc(EOS(STATIC_1984), i439) → 1993_0_createIntList_JMP(EOS(STATIC_1993), +(i439, -1)) | >(i439, 0)
1993_0_createIntList_JMP(EOS(STATIC_1993), i535) → 2005_0_createIntList_Load(EOS(STATIC_2005), i535)
2005_0_createIntList_Load(EOS(STATIC_2005), i535) → 1692_0_createIntList_Load(EOS(STATIC_1692), i535)
1692_0_createIntList_Load(EOS(STATIC_1692), i435) → 1704_0_createIntList_LE(EOS(STATIC_1704), i435, i435)
R rules:

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

P rules:
1704_0_createIntList_LE(EOS(STATIC_1704), x0, x0) → 1704_0_createIntList_LE(EOS(STATIC_1704), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

1704_0_createIntList_LE(x1, x2, x3) → 1704_0_createIntList_LE(x2, x3)
EOS(x1) → EOS
Cond_1704_0_createIntList_LE(x1, x2, x3, x4) → Cond_1704_0_createIntList_LE(x1, x3, x4)

Filtered duplicate args:

1704_0_createIntList_LE(x1, x2) → 1704_0_createIntList_LE(x2)
Cond_1704_0_createIntList_LE(x1, x2, x3) → Cond_1704_0_createIntList_LE(x1, x3)

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

P rules:
1704_0_createIntList_LE(x0) → 1704_0_createIntList_LE(+(x0, -1)) | >(x0, 0)
R rules:

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

P rules:
1704_0_CREATEINTLIST_LE(x0) → COND_1704_0_CREATEINTLIST_LE(>(x0, 0), x0)
COND_1704_0_CREATEINTLIST_LE(TRUE, x0) → 1704_0_CREATEINTLIST_LE(+(x0, -1))
R rules:

(7) 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): 1704_0_CREATEINTLIST_LE(x0[0]) → COND_1704_0_CREATEINTLIST_LE(x0[0] > 0, x0[0])
(1): COND_1704_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1704_0_CREATEINTLIST_LE(x0[1] + -1)

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

(1) -> (0), if (x0[1] + -1 →^* x0[0])

The set Q is empty.

(8) 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@5c2c91f4 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 1704_0_CREATEINTLIST_LE(x0) → COND_1704_0_CREATEINTLIST_LE(>(x0, 0), x0) the following chains were created:

We consider the chain 1704_0_CREATEINTLIST_LE(x0[0]) → COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0]), COND_1704_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1704_0_CREATEINTLIST_LE(+(x0[1], -1)) which results in the following constraint:
(1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 1704_0_CREATEINTLIST_LE(x0[0])≥NonInfC∧1704_0_CREATEINTLIST_LE(x0[0])≥COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE ⇒ 1704_0_CREATEINTLIST_LE(x0[0])≥NonInfC∧1704_0_CREATEINTLIST_LE(x0[0])≥COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_1704_0_CREATEINTLIST_LE(TRUE, x0) → 1704_0_CREATEINTLIST_LE(+(x0, -1)) the following chains were created:

We consider the chain COND_1704_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1704_0_CREATEINTLIST_LE(+(x0[1], -1)) which results in the following constraint:
(7)    (COND_1704_0_CREATEINTLIST_LE(TRUE, x0[1])≥NonInfC∧COND_1704_0_CREATEINTLIST_LE(TRUE, x0[1])≥1704_0_CREATEINTLIST_LE(+(x0[1], -1))∧(U^Increasing(1704_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(1704_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(1704_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(1704_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(1704_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

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

1704_0_CREATEINTLIST_LE(x0) → COND_1704_0_CREATEINTLIST_LE(>(x0, 0), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)
COND_1704_0_CREATEINTLIST_LE(TRUE, x0) → 1704_0_CREATEINTLIST_LE(+(x0, -1))
- ((U^Increasing(1704_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(1704_0_CREATEINTLIST_LE(x₁)) = [2]x₁
POL(COND_1704_0_CREATEINTLIST_LE(x₁, x₂)) = [2]x₂
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_1704_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1704_0_CREATEINTLIST_LE(+(x0[1], -1))

The following pairs are in P_bound:

1704_0_CREATEINTLIST_LE(x0[0]) → COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

1704_0_CREATEINTLIST_LE(x0[0]) → COND_1704_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])

There are no usable rules.

(9) Complex Obligation (AND)

(10) 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): 1704_0_CREATEINTLIST_LE(x0[0]) → COND_1704_0_CREATEINTLIST_LE(x0[0] > 0, x0[0])

The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE

(13) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1704_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1704_0_CREATEINTLIST_LE(x0[1] + -1)

The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) 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: UsedFieldsAnalysis

(17) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 34 rules for P and 0 rules for R.

P rules:
2082_0_main_NULL(EOS(STATIC_2082), java.lang.Object(o848sub), java.lang.Object(o848sub)) → 2083_0_main_NULL(EOS(STATIC_2083), java.lang.Object(o848sub), java.lang.Object(o848sub))
2083_0_main_NULL(EOS(STATIC_2083), java.lang.Object(o848sub), java.lang.Object(o848sub)) → 2086_0_main_Load(EOS(STATIC_2086), java.lang.Object(o848sub))
2086_0_main_Load(EOS(STATIC_2086), java.lang.Object(o848sub)) → 2089_0_main_FieldAccess(EOS(STATIC_2089), java.lang.Object(o848sub), java.lang.Object(o848sub))
2089_0_main_FieldAccess(EOS(STATIC_2089), java.lang.Object(IntList(EOC, i572, o852)), java.lang.Object(IntList(EOC, i572, o852))) → 2091_0_main_FieldAccess(EOS(STATIC_2091), java.lang.Object(IntList(EOC, i572, o852)), java.lang.Object(IntList(EOC, i572, o852)))
2091_0_main_FieldAccess(EOS(STATIC_2091), java.lang.Object(IntList(EOC, i572, o852)), java.lang.Object(IntList(EOC, i572, o852))) → 2093_0_main_GT(EOS(STATIC_2093), java.lang.Object(IntList(EOC, i572, o852)), i572)
2093_0_main_GT(EOS(STATIC_2093), java.lang.Object(IntList(EOC, i575, o852)), i575) → 2094_0_main_GT(EOS(STATIC_2094), java.lang.Object(IntList(EOC, i575, o852)), i575)
2093_0_main_GT(EOS(STATIC_2093), java.lang.Object(IntList(EOC, i576, o852)), i576) → 2095_0_main_GT(EOS(STATIC_2095), java.lang.Object(IntList(EOC, i576, o852)), i576)
2094_0_main_GT(EOS(STATIC_2094), java.lang.Object(IntList(EOC, i575, o852)), i575) → 2097_0_main_Load(EOS(STATIC_2097), java.lang.Object(IntList(EOC, i575, o852))) | <=(i575, 0)
2097_0_main_Load(EOS(STATIC_2097), java.lang.Object(IntList(EOC, i575, o852))) → 2100_0_main_FieldAccess(EOS(STATIC_2100), java.lang.Object(IntList(EOC, i575, o852)))
2100_0_main_FieldAccess(EOS(STATIC_2100), java.lang.Object(IntList(EOC, i575, o852))) → 2103_0_main_Store(EOS(STATIC_2103), o852)
2103_0_main_Store(EOS(STATIC_2103), o852) → 2105_0_main_Load(EOS(STATIC_2105), o852)
2105_0_main_Load(EOS(STATIC_2105), o852) → 2109_0_main_NULL(EOS(STATIC_2109), o852, o852)
2109_0_main_NULL(EOS(STATIC_2109), java.lang.Object(o854sub), java.lang.Object(o854sub)) → 2111_0_main_NULL(EOS(STATIC_2111), java.lang.Object(o854sub), java.lang.Object(o854sub))
2109_0_main_NULL(EOS(STATIC_2109), NULL, NULL) → 2113_0_main_NULL(EOS(STATIC_2113), NULL, NULL)
2111_0_main_NULL(EOS(STATIC_2111), java.lang.Object(o854sub), java.lang.Object(o854sub)) → 2116_0_main_Load(EOS(STATIC_2116), java.lang.Object(o854sub))
2116_0_main_Load(EOS(STATIC_2116), java.lang.Object(o854sub)) → 2121_0_main_Load(EOS(STATIC_2121), java.lang.Object(o854sub))
2121_0_main_Load(EOS(STATIC_2121), java.lang.Object(o854sub)) → 2124_0_main_IntArithmetic(EOS(STATIC_2124), java.lang.Object(o854sub))
2124_0_main_IntArithmetic(EOS(STATIC_2124), java.lang.Object(o854sub)) → 2127_0_main_Store(EOS(STATIC_2127), java.lang.Object(o854sub))
2127_0_main_Store(EOS(STATIC_2127), java.lang.Object(o854sub)) → 2129_0_main_JMP(EOS(STATIC_2129), java.lang.Object(o854sub))
2129_0_main_JMP(EOS(STATIC_2129), java.lang.Object(o854sub)) → 2131_0_main_Load(EOS(STATIC_2131), java.lang.Object(o854sub))
2131_0_main_Load(EOS(STATIC_2131), java.lang.Object(o854sub)) → 2080_0_main_Load(EOS(STATIC_2080), java.lang.Object(o854sub))
2080_0_main_Load(EOS(STATIC_2080), o842) → 2082_0_main_NULL(EOS(STATIC_2082), o842, o842)
2113_0_main_NULL(EOS(STATIC_2113), NULL, NULL) → 2118_0_main_Load(EOS(STATIC_2118), NULL)
2118_0_main_Load(EOS(STATIC_2118), NULL) → 2080_0_main_Load(EOS(STATIC_2080), NULL)
2095_0_main_GT(EOS(STATIC_2095), java.lang.Object(IntList(EOC, i576, o852)), i576) → 2098_0_main_Load(EOS(STATIC_2098), java.lang.Object(IntList(EOC, i576, o852))) | >(i576, 0)
2098_0_main_Load(EOS(STATIC_2098), java.lang.Object(IntList(EOC, i576, o852))) → 2101_0_main_Duplicate(EOS(STATIC_2101), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852)))
2101_0_main_Duplicate(EOS(STATIC_2101), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852))) → 2104_0_main_FieldAccess(EOS(STATIC_2104), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852)))
2104_0_main_FieldAccess(EOS(STATIC_2104), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852))) → 2107_0_main_ConstantStackPush(EOS(STATIC_2107), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852)), i576)
2107_0_main_ConstantStackPush(EOS(STATIC_2107), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852)), i576) → 2110_0_main_IntArithmetic(EOS(STATIC_2110), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852)), i576, 1)
2110_0_main_IntArithmetic(EOS(STATIC_2110), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852)), i576, matching1) → 2114_0_main_FieldAccess(EOS(STATIC_2114), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852)), -(i576, 1)) | &&(>(i576, 0), =(matching1, 1))
2114_0_main_FieldAccess(EOS(STATIC_2114), java.lang.Object(IntList(EOC, i576, o852)), java.lang.Object(IntList(EOC, i576, o852)), i580) → 2120_0_main_Inc(EOS(STATIC_2120), java.lang.Object(IntList(EOC, i580, o852)))
2120_0_main_Inc(EOS(STATIC_2120), java.lang.Object(IntList(EOC, i580, o852))) → 2122_0_main_JMP(EOS(STATIC_2122), java.lang.Object(IntList(EOC, i580, o852)))
2122_0_main_JMP(EOS(STATIC_2122), java.lang.Object(IntList(EOC, i580, o852))) → 2126_0_main_Load(EOS(STATIC_2126), java.lang.Object(IntList(EOC, i580, o852)))
2126_0_main_Load(EOS(STATIC_2126), java.lang.Object(IntList(EOC, i580, o852))) → 2080_0_main_Load(EOS(STATIC_2080), java.lang.Object(IntList(EOC, i580, o852)))
R rules:

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

P rules:
2082_0_main_NULL(EOS(STATIC_2082), java.lang.Object(IntList(EOC, x0, java.lang.Object(x1))), java.lang.Object(IntList(EOC, x0, java.lang.Object(x1)))) → 2082_0_main_NULL(EOS(STATIC_2082), java.lang.Object(x1), java.lang.Object(x1)) | <=(x0, 0)
2082_0_main_NULL(EOS(STATIC_2082), java.lang.Object(IntList(EOC, x0, NULL)), java.lang.Object(IntList(EOC, x0, NULL))) → 2082_0_main_NULL(EOS(STATIC_2082), NULL, NULL) | <=(x0, 0)
2082_0_main_NULL(EOS(STATIC_2082), java.lang.Object(IntList(EOC, x0, x1)), java.lang.Object(IntList(EOC, x0, x1))) → 2082_0_main_NULL(EOS(STATIC_2082), java.lang.Object(IntList(EOC, -(x0, 1), x1)), java.lang.Object(IntList(EOC, -(x0, 1), x1))) | >(x0, 0)
R rules:

Filtered ground terms:

2082_0_main_NULL(x1, x2, x3) → 2082_0_main_NULL(x2, x3)
IntList(x1, x2, x3) → IntList(x2, x3)
EOS(x1) → EOS
Cond_2082_0_main_NULL2(x1, x2, x3, x4) → Cond_2082_0_main_NULL2(x1, x3, x4)
Cond_2082_0_main_NULL1(x1, x2, x3, x4) → Cond_2082_0_main_NULL1(x1, x3, x4)
Cond_2082_0_main_NULL(x1, x2, x3, x4) → Cond_2082_0_main_NULL(x1, x3, x4)

Filtered duplicate args:

2082_0_main_NULL(x1, x2) → 2082_0_main_NULL(x2)
Cond_2082_0_main_NULL(x1, x2, x3) → Cond_2082_0_main_NULL(x1, x3)
Cond_2082_0_main_NULL1(x1, x2, x3) → Cond_2082_0_main_NULL1(x1, x3)
Cond_2082_0_main_NULL2(x1, x2, x3) → Cond_2082_0_main_NULL2(x1, x3)

Filtered unneeded arguments:

Cond_2082_0_main_NULL1(x1, x2) → Cond_2082_0_main_NULL1(x1)

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

P rules:
2082_0_main_NULL(java.lang.Object(IntList(x0, java.lang.Object(x1)))) → 2082_0_main_NULL(java.lang.Object(x1)) | <=(x0, 0)
2082_0_main_NULL(java.lang.Object(IntList(x0, NULL))) → 2082_0_main_NULL(NULL) | <=(x0, 0)
2082_0_main_NULL(java.lang.Object(IntList(x0, x1))) → 2082_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:
2082_0_MAIN_NULL(java.lang.Object(IntList(x0, java.lang.Object(x1)))) → COND_2082_0_MAIN_NULL(<=(x0, 0), java.lang.Object(IntList(x0, java.lang.Object(x1))))
COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0, java.lang.Object(x1)))) → 2082_0_MAIN_NULL(java.lang.Object(x1))
2082_0_MAIN_NULL(java.lang.Object(IntList(x0, NULL))) → COND_2082_0_MAIN_NULL1(<=(x0, 0), java.lang.Object(IntList(x0, NULL)))
COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0, NULL))) → 2082_0_MAIN_NULL(NULL)
2082_0_MAIN_NULL(java.lang.Object(IntList(x0, x1))) → COND_2082_0_MAIN_NULL2(>(x0, 0), java.lang.Object(IntList(x0, x1)))
COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0, x1))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0, 1), x1)))
R 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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(x0[0] <= 0, java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
(1): COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1]))
(2): 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_2082_0_MAIN_NULL1(x0[2] <= 0, java.lang.Object(IntList(x0[2], NULL)))
(3): COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 2082_0_MAIN_NULL(NULL)
(4): 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(x0[4] > 0, java.lang.Object(IntList(x0[4], x1[4])))
(5): COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_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.

(19) 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@72225972 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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0, java.lang.Object(x1)))) → COND_2082_0_MAIN_NULL(<=(x0, 0), java.lang.Object(IntList(x0, java.lang.Object(x1)))) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_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]))) ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_2082_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 ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_2082_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_2082_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_2082_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_2082_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_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0, java.lang.Object(x1)))) → 2082_0_MAIN_NULL(java.lang.Object(x1)) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1])), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥2082_0_MAIN_NULL(java.lang.Object(x1[1]))∧(U^Increasing(2082_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_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))))≥2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1])), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥2082_0_MAIN_NULL(java.lang.Object(x1[1]))∧(U^Increasing(2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[2], NULL)))))≥NonInfC∧COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[2], NULL)))))≥2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL)))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1])), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥2082_0_MAIN_NULL(java.lang.Object(x1[1]))∧(U^Increasing(2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥NonInfC∧COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0, NULL))) → COND_2082_0_MAIN_NULL1(<=(x0, 0), java.lang.Object(IntList(x0, NULL))) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_2082_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 2082_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)) ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL)))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL)))≥COND_2082_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))∧(U^Increasing(COND_2082_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 ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL)))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL)))≥COND_2082_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))∧(U^Increasing(COND_2082_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_2082_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_2082_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_2082_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_2082_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_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0, NULL))) → 2082_0_MAIN_NULL(NULL) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_2082_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 2082_0_MAIN_NULL(NULL), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_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_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥NonInfC∧COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥2082_0_MAIN_NULL(NULL)∧(U^Increasing(2082_0_MAIN_NULL(NULL)), ≥))

We solved constraint (31) using rules (I), (II), (DELETE_TRIVIAL_REDUCESTO).
We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_2082_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 2082_0_MAIN_NULL(NULL), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_2082_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_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥NonInfC∧COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥2082_0_MAIN_NULL(NULL)∧(U^Increasing(2082_0_MAIN_NULL(NULL)), ≥))

We solved constraint (32) using rules (I), (II), (DELETE_TRIVIAL_REDUCESTO).
We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_2082_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 2082_0_MAIN_NULL(NULL), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_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_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥NonInfC∧COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥2082_0_MAIN_NULL(NULL)∧(U^Increasing(2082_0_MAIN_NULL(NULL)), ≥))

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

For Pair 2082_0_MAIN_NULL(java.lang.Object(IntList(x0, x1))) → COND_2082_0_MAIN_NULL2(>(x0, 0), java.lang.Object(IntList(x0, x1))) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_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])) ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_2082_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 ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_2082_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_2082_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_2082_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_2082_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_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0, x1))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0, 1), x1))) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), java.lang.Object(x1[0]))))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], NULL)))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], NULL)))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), NULL)))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), x1[4])))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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.

2082_0_MAIN_NULL(java.lang.Object(IntList(x0, java.lang.Object(x1)))) → COND_2082_0_MAIN_NULL(<=(x0, 0), java.lang.Object(IntList(x0, java.lang.Object(x1))))
- (0 ≥ 0 ⇒ (U^Increasing(COND_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0, java.lang.Object(x1)))) → 2082_0_MAIN_NULL(java.lang.Object(x1))
- (0 ≥ 0 ⇒ (U^Increasing(2082_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(2082_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(2082_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)
2082_0_MAIN_NULL(java.lang.Object(IntList(x0, NULL))) → COND_2082_0_MAIN_NULL1(<=(x0, 0), java.lang.Object(IntList(x0, NULL)))
- (0 ≥ 0 ⇒ (U^Increasing(COND_2082_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_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0, NULL))) → 2082_0_MAIN_NULL(NULL)
2082_0_MAIN_NULL(java.lang.Object(IntList(x0, x1))) → COND_2082_0_MAIN_NULL2(>(x0, 0), java.lang.Object(IntList(x0, x1)))
- (0 ≥ 0 ⇒ (U^Increasing(COND_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0, x1))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0, 1), x1)))
- (0 ≥ 0 ⇒ (U^Increasing(2082_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(2082_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(2082_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)

POL(TRUE) = 0
POL(FALSE) = 0
POL(2082_0_MAIN_NULL(x₁)) = [1] + [2]x₁
POL(java.lang.Object(x₁)) = [2]
POL(IntList(x₁, x₂)) = 0
POL(COND_2082_0_MAIN_NULL(x₁, x₂)) = [1] + [2]x₂
POL(<=(x₁, x₂)) = 0
POL(0) = 0
POL(NULL) = 0
POL(COND_2082_0_MAIN_NULL1(x₁, x₂)) = [-1] + [-1]x₂
POL(COND_2082_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_>:

2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_2082_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))
COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 2082_0_MAIN_NULL(NULL)

The following pairs are in P_bound:

2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1]))
2082_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_2082_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))
COND_2082_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 2082_0_MAIN_NULL(NULL)
2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_≥:

2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1]))
2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

There are no usable rules.

(20) 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): 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(x0[0] <= 0, java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
(1): COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1]))
(4): 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(x0[4] > 0, java.lang.Object(IntList(x0[4], x1[4])))
(5): COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_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.

(21) 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@72225972 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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_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]))) ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_2082_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 ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_2082_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_2082_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_2082_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_2082_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_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1])) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1])), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥2082_0_MAIN_NULL(java.lang.Object(x1[1]))∧(U^Increasing(2082_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_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))))≥2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1])), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥2082_0_MAIN_NULL(java.lang.Object(x1[1]))∧(U^Increasing(2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥NonInfC∧COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_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])) ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_2082_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 ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_2082_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_2082_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_2082_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_2082_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_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), java.lang.Object(x1[0]))))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), x1[4])))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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.

2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
- (0 ≥ 0 ⇒ (U^Increasing(COND_2082_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_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1]))
- (0 ≥ 0 ⇒ (U^Increasing(2082_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(2082_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)
2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
- (0 ≥ 0 ⇒ (U^Increasing(COND_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))
- (0 ≥ 0 ⇒ (U^Increasing(2082_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(2082_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(2082_0_MAIN_NULL(x₁)) = [1] + x₁
POL(java.lang.Object(x₁)) = [2]x₁
POL(IntList(x₁, x₂)) = [2] + x₂ + [3]x₁
POL(COND_2082_0_MAIN_NULL(x₁, x₂)) = [-1] + x₂
POL(<=(x₁, x₂)) = 0
POL(0) = 0
POL(COND_2082_0_MAIN_NULL2(x₁, x₂)) = [1] + x₂
POL(>(x₁, x₂)) = 0
POL(-(x₁, x₂)) = 0
POL(1) = 0

The following pairs are in P_>:

2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1]))

The following pairs are in P_bound:

2082_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_2082_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_2082_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 2082_0_MAIN_NULL(java.lang.Object(x1[1]))
2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_≥:

2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

There are no usable rules.

(22) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(4): 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(x0[4] > 0, java.lang.Object(IntList(x0[4], x1[4])))
(5): COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_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.

(23) 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@be72834 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 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_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])) ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_2082_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 ⇒ 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_2082_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_2082_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_2082_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_2082_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_2082_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_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))) the following chains were created:

We consider the chain 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), x1[4])))∧(U^Increasing(2082_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(2082_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(2082_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(2082_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(2082_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(2082_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.

2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
- (x0[4] ≥ 0 ⇒ (U^Increasing(COND_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))
- (x0[4] ≥ 0 ⇒ (U^Increasing(2082_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(2082_0_MAIN_NULL(x₁)) = [-1] + [-1]x₁
POL(java.lang.Object(x₁)) = [-1]x₁
POL(IntList(x₁, x₂)) = [-1] + x₁
POL(COND_2082_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_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_bound:

2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_2082_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 2082_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_≥:

2082_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_2082_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))

There are no usable rules.

(24) 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

(25) 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) SCCToIDPv1Proof (SOUND transformation)

(7) Obligation:

(8) IDPNonInfProof (SOUND transformation)

(9) Complex Obligation (AND)

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE

(13) Obligation:

(14) IDependencyGraphProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) SCCToIDPv1Proof (SOUND transformation)

(18) Obligation:

(19) IDPNonInfProof (SOUND transformation)

(20) Obligation:

(21) IDPNonInfProof (SOUND transformation)

(22) Obligation:

(23) IDPNonInfProof (SOUND transformation)

(24) Obligation:

(25) IDependencyGraphProof (EQUIVALENT transformation)

(26) TRUE