Termination proof of /tmp/tmpstHGRm/Convert.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPNonInfProof (⇒)

        ↳7 AND

            ↳8 IDP

                ↳9 IDependencyGraphProof (⇔)

                ↳10 TRUE

            ↳11 IDP

                ↳12 IDependencyGraphProof (⇔)

                ↳13 TRUE

    ↳14 IDP

        ↳15 IDPNonInfProof (⇒)

        ↳16 IDP

        ↳17 IDPNonInfProof (⇒)

        ↳18 IDP

        ↳19 IDPNonInfProof (⇒)

        ↳20 IDP

        ↳21 IDependencyGraphProof (⇔)

        ↳22 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) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Convert.main([Ljava/lang/String;)V: Graph of 136 nodes with 1 SCC.

IntList.createIntList()LIntList;: Graph of 162 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 38 rules for P and 49 rules for R.

Combined rules. Obtained 2 rules for P and 0 rules for R.

Filtered ground terms:

2831_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5) → 2831_1_createIntList_InvokeMethod(x1, x2, x3)
IntList(x1) → IntList
2831_0_random_ArrayAccess(x1, x2, x3) → 2831_0_random_ArrayAccess(x2, x3)
Cond_2871_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2871_1_createIntList_InvokeMethod(x1, x2, x3, x4)
2871_0_random_IntArithmetic(x1, x2, x3, x4) → 2871_0_random_IntArithmetic(x2, x3)
2871_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5) → 2871_1_createIntList_InvokeMethod(x1, x2, x3)
Cond_2831_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2831_1_createIntList_InvokeMethod(x1, x2, x3, x4)

Filtered unneeded arguments:

2831_1_createIntList_InvokeMethod(x1, x2, x3) → 2831_1_createIntList_InvokeMethod(x1, x2)
Cond_2831_1_createIntList_InvokeMethod(x1, x2, x3, x4) → Cond_2831_1_createIntList_InvokeMethod(x1, x2, x3)
2871_1_createIntList_InvokeMethod(x1, x2, x3) → 2871_1_createIntList_InvokeMethod(x1, x2)
Cond_2871_1_createIntList_InvokeMethod(x1, x2, x3, x4) → Cond_2871_1_createIntList_InvokeMethod(x1, x2, x3)

Combined rules. Obtained 2 rules for P and 0 rules for R.

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

Log for SCC 1:

Generated 34 rules for P and 2 rules for R.

Combined rules. Obtained 3 rules for P and 1 rules for R.

Filtered ground terms:

3554_0_main_NULL(x1, x2, x3) → 3554_0_main_NULL(x2, x3)
IntList(x1, x2, x3) → IntList(x2, x3)
Cond_3554_0_main_NULL2(x1, x2, x3, x4) → Cond_3554_0_main_NULL2(x1, x3, x4)
Cond_3554_0_main_NULL1(x1, x2, x3, x4) → Cond_3554_0_main_NULL1(x1, x3, x4)
Cond_3554_0_main_NULL(x1, x2, x3, x4) → Cond_3554_0_main_NULL(x1, x3, x4)
3567_0_main_Return(x1) → 3567_0_main_Return

Filtered duplicate args:

3554_0_main_NULL(x1, x2) → 3554_0_main_NULL(x2)
Cond_3554_0_main_NULL2(x1, x2, x3) → Cond_3554_0_main_NULL2(x1, x3)
Cond_3554_0_main_NULL1(x1, x2, x3) → Cond_3554_0_main_NULL1(x1, x3)
Cond_3554_0_main_NULL(x1, x2, x3) → Cond_3554_0_main_NULL(x1, x3)

Combined rules. Obtained 3 rules for P and 1 rules for R.

Finished conversion. Obtained 3 rules for P and 1 rules for R. System has predefined symbols.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2831_1_CREATEINTLIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_2831_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
(2): 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2871_1_CREATEINTLIST_INVOKEMETHOD(x4[2] > 0 && x2[2] > 0 && 0 < x4[2] + -1, 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])
(3): COND_2871_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), x4[3] + -1)

(0) -> (1), if ((x2[0] >= 1 && x2[0] < x0[0] →^* TRUE)∧(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →^* 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0] →^* x3[1]))

(1) -> (2), if ((2871_0_random_IntArithmetic(x5[1], x6[1]) →^* 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1] →^* x4[2]))

(2) -> (3), if ((x4[2] > 0 && x2[2] > 0 && 0 < x4[2] + -1 →^* TRUE)∧(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]) →^* 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]))∧(x4[2] →^* x4[3]))

(3) -> (0), if ((2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]) →^* 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] + -1 →^* x3[0]))

The set Q is empty.

(6) IDPNonInfProof (SOUND transformation)

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 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) the following chains were created:

We consider the chain 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]), COND_2831_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1]) which results in the following constraint:
(1)    (&&(>=(x2[0], 1), <(x2[0], x0[0]))=TRUE∧2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])=2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1])∧x3[0]=x3[1] ⇒ 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(U^Increasing(COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=(x2[0], 1)=TRUE∧<(x2[0], x0[0])=TRUE ⇒ 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(U^Increasing(COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x3[0] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x3[0] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x3[0] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_26] = 0∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x2[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_26] = 0∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(8)    (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_26] = 0∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

For Pair COND_2831_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5, x6), x3) the following chains were created:

We consider the chain COND_2831_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1]) which results in the following constraint:
(9)    (COND_2831_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥NonInfC∧COND_2831_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1])∧(U^Increasing(2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    ((U^Increasing(2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥)∧[(-1)bso_29] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    ((U^Increasing(2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥)∧[(-1)bso_29] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    ((U^Increasing(2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥)∧[(-1)bso_29] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    ((U^Increasing(2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

For Pair 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4, 0), >(x2, 0)), <(0, +(x4, -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) the following chains were created:

We consider the chain 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]), COND_2871_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1)) which results in the following constraint:
(14)    (&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1)))=TRUE∧2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2])=2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3])∧x4[2]=x4[3] ⇒ 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥NonInfC∧2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])∧(U^Increasing(COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥))

We simplified constraint (14) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(15)    (<(0, +(x4[2], -1))=TRUE∧>(x4[2], 0)=TRUE∧>(x2[2], 0)=TRUE ⇒ 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥NonInfC∧2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])∧(U^Increasing(COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    (x4[2] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    (x4[2] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    (x4[2] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19)    (x4[2] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(20)    (x4[2] ≥ 0∧[1] + x4[2] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧0 = 0∧0 = 0∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21)    (x4[2] ≥ 0∧[1] + x4[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧0 = 0∧0 = 0∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

For Pair COND_2871_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), +(x4, -1)) the following chains were created:

We consider the chain COND_2871_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1)) which results in the following constraint:
(22)    (COND_2871_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3])≥NonInfC∧COND_2871_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3])≥2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))∧(U^Increasing(2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23)    ((U^Increasing(2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥)∧[2 + (-1)bso_33] ≥ 0)

We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24)    ((U^Increasing(2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥)∧[2 + (-1)bso_33] ≥ 0)

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25)    ((U^Increasing(2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥)∧[2 + (-1)bso_33] ≥ 0)

We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(26)    ((U^Increasing(2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_33] ≥ 0)

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

2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
- (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_26] = 0∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)
COND_2831_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5, x6), x3)
- ((U^Increasing(2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)
2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4, 0), >(x2, 0)), <(0, +(x4, -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4)
- (x4[2] ≥ 0∧[1] + x4[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧0 = 0∧0 = 0∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)
COND_2871_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), +(x4, -1))
- ((U^Increasing(2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_33] ≥ 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(2831_1_CREATEINTLIST_INVOKEMETHOD(x₁, x₂)) = [-1] + [2]x₂ + [-1]x₁
POL(2831_0_random_ArrayAccess(x₁, x₂)) = [-1] + [-1]x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁, x₂)) = [-1]
POL(COND_2831_1_CREATEINTLIST_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [2]x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<(x₁, x₂)) = [-1]
POL(2871_1_CREATEINTLIST_INVOKEMETHOD(x₁, x₂)) = [-1] + [2]x₂
POL(2871_0_random_IntArithmetic(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(java.lang.String(x₁, x₂)) = [-1]x₂ + [-1]x₁
POL(COND_2871_1_CREATEINTLIST_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [2]x₃
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_2871_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))

The following pairs are in P_bound:

2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])

The following pairs are in P_≥:

2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2831_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
COND_2831_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2871_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])

There are no usable rules.

(7) Complex Obligation (AND)

(8) Obligation:

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

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

The following domains are used:

Boolean, Integer

(1) -> (2), if ((2871_0_random_IntArithmetic(x5[1], x6[1]) →^* 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1] →^* x4[2]))

The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) TRUE

(11) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2831_1_CREATEINTLIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_2831_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2871_1_CREATEINTLIST_INVOKEMETHOD(2871_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
(3): COND_2871_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2871_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2831_1_CREATEINTLIST_INVOKEMETHOD(2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), x4[3] + -1)

(3) -> (0), if ((2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]) →^* 2831_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] + -1 →^* x3[0]))

The set Q is empty.

(12) IDependencyGraphProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
3554_0_main_NULL(NULL) → 3567_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(x0[0] <= 0, java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
(1): COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1]))
(2): 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_3554_0_MAIN_NULL1(x0[2] <= 0, java.lang.Object(IntList(x0[2], NULL)))
(3): COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 3554_0_MAIN_NULL(NULL)
(4): 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(x0[4] > 0, java.lang.Object(IntList(x0[4], x1[4])))
(5): COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[5] - 1, x1[5])))

(0) -> (1), if ((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])))))

(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 →^* TRUE)∧(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 →^* TRUE)∧(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 consists of the following terms:
3554_0_main_NULL(NULL)

(15) IDPNonInfProof (SOUND transformation)

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 3554_0_MAIN_NULL(java.lang.Object(IntList(x0, java.lang.Object(x1)))) → COND_3554_0_MAIN_NULL(<=(x0, 0), java.lang.Object(IntList(x0, java.lang.Object(x1)))) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_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]))) ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_3554_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 ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_3554_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_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_21 + (-1)Bound*bni_21] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_21 + (-1)Bound*bni_21] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_21 + (-1)Bound*bni_21] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_21 + (-1)Bound*bni_21] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_22] ≥ 0)

For Pair COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0, java.lang.Object(x1)))) → 3554_0_MAIN_NULL(java.lang.Object(x1)) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1])), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_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_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥3554_0_MAIN_NULL(java.lang.Object(x1[1]))∧(U^Increasing(3554_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_3554_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_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))))≥3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_24] ≥ 0)
We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1])), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_3554_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_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥3554_0_MAIN_NULL(java.lang.Object(x1[1]))∧(U^Increasing(3554_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_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[2], NULL)))))≥NonInfC∧COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[2], NULL)))))≥3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL)))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (17) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(18)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_24] ≥ 0)
We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1])), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_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_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥3554_0_MAIN_NULL(java.lang.Object(x1[1]))∧(U^Increasing(3554_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_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥NonInfC∧COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_24] ≥ 0)

For Pair 3554_0_MAIN_NULL(java.lang.Object(IntList(x0, NULL))) → COND_3554_0_MAIN_NULL1(<=(x0, 0), java.lang.Object(IntList(x0, NULL))) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 3554_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)) ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL)))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL)))≥COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))∧(U^Increasing(COND_3554_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 ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL)))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL)))≥COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))∧(U^Increasing(COND_3554_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_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥)∧[(5)bni_25 + (-1)Bound*bni_25] ≥ 0∧[8 + (-1)bso_26] ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥)∧[(5)bni_25 + (-1)Bound*bni_25] ≥ 0∧[8 + (-1)bso_26] ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥)∧[(5)bni_25 + (-1)Bound*bni_25] ≥ 0∧[8 + (-1)bso_26] ≥ 0)

We simplified constraint (29) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(30)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥)∧0 ≥ 0∧[(5)bni_25 + (-1)Bound*bni_25] ≥ 0∧0 ≥ 0∧[8 + (-1)bso_26] ≥ 0)

For Pair COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0, NULL))) → 3554_0_MAIN_NULL(NULL) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 3554_0_MAIN_NULL(NULL), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_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_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥NonInfC∧COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥3554_0_MAIN_NULL(NULL)∧(U^Increasing(3554_0_MAIN_NULL(NULL)), ≥))

We solved constraint (31) using rules (I), (II), (DELETE_TRIVIAL_REDUCESTO).
We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 3554_0_MAIN_NULL(NULL), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_3554_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_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥NonInfC∧COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥3554_0_MAIN_NULL(NULL)∧(U^Increasing(3554_0_MAIN_NULL(NULL)), ≥))

We solved constraint (32) using rules (I), (II), (DELETE_TRIVIAL_REDUCESTO).
We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL))), COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 3554_0_MAIN_NULL(NULL), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_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_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥NonInfC∧COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL)))≥3554_0_MAIN_NULL(NULL)∧(U^Increasing(3554_0_MAIN_NULL(NULL)), ≥))

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

For Pair 3554_0_MAIN_NULL(java.lang.Object(IntList(x0, x1))) → COND_3554_0_MAIN_NULL2(>(x0, 0), java.lang.Object(IntList(x0, x1))) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_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])) ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_3554_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 ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_3554_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_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_27 + (-1)Bound*bni_27] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(37)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_27 + (-1)Bound*bni_27] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(38)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_27 + (-1)Bound*bni_27] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (38) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(39)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_27 + (-1)Bound*bni_27] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_28] ≥ 0)

For Pair COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0, x1))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0, 1), x1))) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), java.lang.Object(x1[0]))))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (42) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(43)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(44)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (44) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(45)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_30] ≥ 0)
We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], NULL)))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], NULL)))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), NULL)))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (48) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(49)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(50)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (50) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(51)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧0 ≥ 0∧[(-1)bso_30] ≥ 0)
We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), x1[4])))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (54) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(55)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (55) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(56)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (56) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(57)    (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_30] ≥ 0)

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

3554_0_MAIN_NULL(java.lang.Object(IntList(x0, java.lang.Object(x1)))) → COND_3554_0_MAIN_NULL(<=(x0, 0), java.lang.Object(IntList(x0, java.lang.Object(x1))))
- (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_21 + (-1)Bound*bni_21] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_22] ≥ 0)
COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0, java.lang.Object(x1)))) → 3554_0_MAIN_NULL(java.lang.Object(x1))
- (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_24] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_24] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(5)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_24] ≥ 0)
3554_0_MAIN_NULL(java.lang.Object(IntList(x0, NULL))) → COND_3554_0_MAIN_NULL1(<=(x0, 0), java.lang.Object(IntList(x0, NULL)))
- (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))), ≥)∧0 ≥ 0∧[(5)bni_25 + (-1)Bound*bni_25] ≥ 0∧0 ≥ 0∧[8 + (-1)bso_26] ≥ 0)
COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0, NULL))) → 3554_0_MAIN_NULL(NULL)
3554_0_MAIN_NULL(java.lang.Object(IntList(x0, x1))) → COND_3554_0_MAIN_NULL2(>(x0, 0), java.lang.Object(IntList(x0, x1)))
- (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_27 + (-1)Bound*bni_27] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_28] ≥ 0)
COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0, x1))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0, 1), x1)))
- (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_30] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧0 ≥ 0∧[(-1)bso_30] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(5)bni_29 + (-1)Bound*bni_29] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_30] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(3554_0_main_NULL(x₁)) = 0
POL(NULL) = 0
POL(3567_0_main_Return) = 0
POL(3554_0_MAIN_NULL(x₁)) = [1] + [2]x₁
POL(java.lang.Object(x₁)) = [2]
POL(IntList(x₁, x₂)) = 0
POL(COND_3554_0_MAIN_NULL(x₁, x₂)) = [1] + [2]x₂
POL(<=(x₁, x₂)) = 0
POL(0) = 0
POL(COND_3554_0_MAIN_NULL1(x₁, x₂)) = [-1] + [-1]x₂
POL(COND_3554_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_>:

3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))
COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 3554_0_MAIN_NULL(NULL)

The following pairs are in P_bound:

3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1]))
3554_0_MAIN_NULL(java.lang.Object(IntList(x0[2], NULL))) → COND_3554_0_MAIN_NULL1(<=(x0[2], 0), java.lang.Object(IntList(x0[2], NULL)))
COND_3554_0_MAIN_NULL1(TRUE, java.lang.Object(IntList(x0[3], NULL))) → 3554_0_MAIN_NULL(NULL)
3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_≥:

3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1]))
3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

There are no usable rules.

(16) Obligation:

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

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
3554_0_main_NULL(NULL) → 3567_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(x0[0] <= 0, java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
(1): COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1]))
(4): 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(x0[4] > 0, java.lang.Object(IntList(x0[4], x1[4])))
(5): COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_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 →^* TRUE)∧(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 →^* TRUE)∧(java.lang.Object(IntList(x0[4], x1[4])) →^* java.lang.Object(IntList(x0[5], x1[5]))))

The set Q consists of the following terms:
3554_0_main_NULL(NULL)

(17) IDPNonInfProof (SOUND transformation)

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 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_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 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_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]))) ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_3554_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 ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))≥COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))∧(U^Increasing(COND_3554_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_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_18 + (-1)Bound*bni_18] + [(4)bni_18]x1[0] + [(6)bni_18]x0[0] ≥ 0∧[2 + (-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_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_18 + (-1)Bound*bni_18] + [(4)bni_18]x1[0] + [(6)bni_18]x0[0] ≥ 0∧[2 + (-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_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(5)bni_18 + (-1)Bound*bni_18] + [(4)bni_18]x1[0] + [(6)bni_18]x0[0] ≥ 0∧[2 + (-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_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(4)bni_18] ≥ 0∧[(6)bni_18] ≥ 0∧[(5)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[2 + (-1)bso_19] ≥ 0)

For Pair COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1])) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1])), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_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_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥3554_0_MAIN_NULL(java.lang.Object(x1[1]))∧(U^Increasing(3554_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_3554_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_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))))≥3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0]1, java.lang.Object(x1[0]1))))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(11)bni_20 + (-1)Bound*bni_20] + [(8)bni_20]x1[0]1 + [(12)bni_20]x0[0]1 + [(6)bni_20]x0[0] ≥ 0∧[6 + (-1)bso_21] + [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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(11)bni_20 + (-1)Bound*bni_20] + [(8)bni_20]x1[0]1 + [(12)bni_20]x0[0]1 + [(6)bni_20]x0[0] ≥ 0∧[6 + (-1)bso_21] + [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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(11)bni_20 + (-1)Bound*bni_20] + [(8)bni_20]x1[0]1 + [(12)bni_20]x0[0]1 + [(6)bni_20]x0[0] ≥ 0∧[6 + (-1)bso_21] + [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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(8)bni_20] ≥ 0∧[(12)bni_20] ≥ 0∧[(6)bni_20] ≥ 0∧[(11)bni_20 + (-1)Bound*bni_20] ≥ 0∧[6 + (-1)bso_21] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)
We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))), COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1])), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_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_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1]))))≥3554_0_MAIN_NULL(java.lang.Object(x1[1]))∧(U^Increasing(3554_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_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥NonInfC∧COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[0], java.lang.Object(IntList(x0[4], x1[4])))))≥3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(11)bni_20 + (-1)Bound*bni_20] + [(4)bni_20]x1[4] + [(12)bni_20]x0[4] + [(6)bni_20]x0[0] ≥ 0∧[6 + (-1)bso_21] + [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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(11)bni_20 + (-1)Bound*bni_20] + [(4)bni_20]x1[4] + [(12)bni_20]x0[4] + [(6)bni_20]x0[0] ≥ 0∧[6 + (-1)bso_21] + [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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(11)bni_20 + (-1)Bound*bni_20] + [(4)bni_20]x1[4] + [(12)bni_20]x0[4] + [(6)bni_20]x0[0] ≥ 0∧[6 + (-1)bso_21] + [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(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(4)bni_20] ≥ 0∧[(12)bni_20] ≥ 0∧[(6)bni_20] ≥ 0∧[(11)bni_20 + (-1)Bound*bni_20] ≥ 0∧[6 + (-1)bso_21] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)

For Pair 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_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])) ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_3554_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 ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_3554_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_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x1[4] + [(6)bni_22]x0[4] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x1[4] + [(6)bni_22]x0[4] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(5)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x1[4] + [(6)bni_22]x0[4] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(2)bni_22] ≥ 0∧[(6)bni_22] ≥ 0∧[(5)bni_22 + (-1)Bound*bni_22] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_23] ≥ 0)

For Pair COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], java.lang.Object(x1[0]))))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), java.lang.Object(x1[0]))))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_24 + (-1)Bound*bni_24] + [(4)bni_24]x1[0] + [(6)bni_24]x0[4] ≥ 0∧[(-1)bso_25] + [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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_24 + (-1)Bound*bni_24] + [(4)bni_24]x1[0] + [(6)bni_24]x0[4] ≥ 0∧[(-1)bso_25] + [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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_24 + (-1)Bound*bni_24] + [(4)bni_24]x1[0] + [(6)bni_24]x0[4] ≥ 0∧[(-1)bso_25] + [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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(4)bni_24] ≥ 0∧[(6)bni_24] ≥ 0∧[(5)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0∧[1] ≥ 0)
We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), x1[4])))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x1[4] + [(6)bni_24]x0[4] ≥ 0∧[(-1)bso_25] + [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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x1[4] + [(6)bni_24]x0[4] ≥ 0∧[(-1)bso_25] + [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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(5)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x1[4] + [(6)bni_24]x0[4] ≥ 0∧[(-1)bso_25] + [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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(2)bni_24] ≥ 0∧[(6)bni_24] ≥ 0∧[(5)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0∧[1] ≥ 0)

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

3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
- (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(4)bni_18] ≥ 0∧[(6)bni_18] ≥ 0∧[(5)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[2 + (-1)bso_19] ≥ 0)
COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1]))
- (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(8)bni_20] ≥ 0∧[(12)bni_20] ≥ 0∧[(6)bni_20] ≥ 0∧[(11)bni_20 + (-1)Bound*bni_20] ≥ 0∧[6 + (-1)bso_21] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(x1[1]))), ≥)∧[(4)bni_20] ≥ 0∧[(12)bni_20] ≥ 0∧[(6)bni_20] ≥ 0∧[(11)bni_20 + (-1)Bound*bni_20] ≥ 0∧[6 + (-1)bso_21] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)
3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
- (0 ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(2)bni_22] ≥ 0∧[(6)bni_22] ≥ 0∧[(5)bni_22 + (-1)Bound*bni_22] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_23] ≥ 0)
COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))
- (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(4)bni_24] ≥ 0∧[(6)bni_24] ≥ 0∧[(5)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0∧[1] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(2)bni_24] ≥ 0∧[(6)bni_24] ≥ 0∧[(5)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0∧[1] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(3554_0_main_NULL(x₁)) = 0
POL(NULL) = 0
POL(3567_0_main_Return) = 0
POL(3554_0_MAIN_NULL(x₁)) = [1] + x₁
POL(java.lang.Object(x₁)) = [2]x₁
POL(IntList(x₁, x₂)) = [2] + x₂ + [3]x₁
POL(COND_3554_0_MAIN_NULL(x₁, x₂)) = [-1] + x₂
POL(<=(x₁, x₂)) = 0
POL(0) = 0
POL(COND_3554_0_MAIN_NULL2(x₁, x₂)) = [1] + x₂
POL(>(x₁, x₂)) = 0
POL(-(x₁, x₂)) = 0
POL(1) = 0

The following pairs are in P_>:

3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1]))

The following pairs are in P_bound:

3554_0_MAIN_NULL(java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → COND_3554_0_MAIN_NULL(<=(x0[0], 0), java.lang.Object(IntList(x0[0], java.lang.Object(x1[0]))))
COND_3554_0_MAIN_NULL(TRUE, java.lang.Object(IntList(x0[1], java.lang.Object(x1[1])))) → 3554_0_MAIN_NULL(java.lang.Object(x1[1]))
3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_≥:

3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

There are no usable rules.

(18) Obligation:

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

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
3554_0_main_NULL(NULL) → 3567_0_main_Return

The integer pair graph contains the following rules and edges:
(4): 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(x0[4] > 0, java.lang.Object(IntList(x0[4], x1[4])))
(5): COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_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 →^* TRUE)∧(java.lang.Object(IntList(x0[4], x1[4])) →^* java.lang.Object(IntList(x0[5], x1[5]))))

The set Q consists of the following terms:
3554_0_main_NULL(NULL)

(19) IDPNonInfProof (SOUND transformation)

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 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_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])) ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_3554_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 ⇒ 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4])))≥COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))∧(U^Increasing(COND_3554_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_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(-2)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[4] ≥ 0∧[(-1)bso_19] ≥ 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_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(-2)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[4] ≥ 0∧[(-1)bso_19] ≥ 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_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧[(-2)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[4] ≥ 0∧[(-1)bso_19] ≥ 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_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧0 = 0∧[(-2)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[4] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x0[4] ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧0 = 0∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[4] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)

For Pair COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))) the following chains were created:

We consider the chain 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4]))), COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5]))), 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5])))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))∧(U^Increasing(3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥NonInfC∧COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[4], x1[4])))≥3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[4], 1), x1[4])))∧(U^Increasing(3554_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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(-2)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[4] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(-2)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[4] ≥ 0∧[1 + (-1)bso_21] ≥ 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(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧[(-2)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[4] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    (x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 = 0∧[(-2)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[4] ≥ 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x0[4] ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 = 0∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[4] ≥ 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)

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

3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
- (x0[4] ≥ 0 ⇒ (U^Increasing(COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))), ≥)∧0 = 0∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[4] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)
COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))
- (x0[4] ≥ 0 ⇒ (U^Increasing(3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))), ≥)∧0 = 0∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[4] ≥ 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)

POL(TRUE) = [2]
POL(FALSE) = 0
POL(3554_0_main_NULL(x₁)) = [-1] + [-1]x₁
POL(NULL) = [-1]
POL(3567_0_main_Return) = [-1]
POL(3554_0_MAIN_NULL(x₁)) = [-1] + [-1]x₁
POL(java.lang.Object(x₁)) = [-1]x₁
POL(IntList(x₁, x₂)) = [-1] + x₁
POL(COND_3554_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_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_bound:

3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))
COND_3554_0_MAIN_NULL2(TRUE, java.lang.Object(IntList(x0[5], x1[5]))) → 3554_0_MAIN_NULL(java.lang.Object(IntList(-(x0[5], 1), x1[5])))

The following pairs are in P_≥:

3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(>(x0[4], 0), java.lang.Object(IntList(x0[4], x1[4])))

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

The ITRS R consists of the following rules:
3554_0_main_NULL(NULL) → 3567_0_main_Return

The integer pair graph contains the following rules and edges:
(4): 3554_0_MAIN_NULL(java.lang.Object(IntList(x0[4], x1[4]))) → COND_3554_0_MAIN_NULL2(x0[4] > 0, java.lang.Object(IntList(x0[4], x1[4])))

The set Q consists of the following terms:
3554_0_main_NULL(NULL)

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) IDPNonInfProof (SOUND transformation)

(7) Complex Obligation (AND)

(8) Obligation:

(9) IDependencyGraphProof (EQUIVALENT transformation)

(10) TRUE

(11) Obligation:

(12) IDependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) IDPNonInfProof (SOUND transformation)

(16) Obligation:

(17) IDPNonInfProof (SOUND transformation)

(18) Obligation:

(19) IDPNonInfProof (SOUND transformation)

(20) Obligation:

(21) IDependencyGraphProof (EQUIVALENT transformation)

(22) TRUE