Termination proof of /tmp/tmpcBaTQa/Shuffle.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPtoQDPProof (⇒)

        ↳7 QDP

        ↳8 UsableRulesProof (⇔)

        ↳9 QDP

        ↳10 QReductionProof (⇔)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 IDP

        ↳15 IDPNonInfProof (⇒)

        ↳16 AND

            ↳17 IDP

                ↳18 IDependencyGraphProof (⇔)

                ↳19 TRUE

            ↳20 IDP

                ↳21 IDependencyGraphProof (⇔)

                ↳22 TRUE

    ↳23 IDP

        ↳24 IDPtoQDPProof (⇒)

        ↳25 QDP

        ↳26 DependencyGraphProof (⇔)

        ↳27 QDP

        ↳28 UsableRulesProof (⇔)

        ↳29 QDP

        ↳30 QReductionProof (⇔)

        ↳31 QDP

        ↳32 QDPOrderProof (⇔)

        ↳33 QDP

        ↳34 DependencyGraphProof (⇔)

        ↳35 TRUE

(0) Obligation:

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

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

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


public class Shuffle{

  // adapted from [Walther, 94]
  public static void main(String[] args) {
    Random.args = args;
    IntList l = IntList.createIntList();
    IntList res = null;

    while (l != null) {

      res = new IntList(l.value, res);
      l = l.next;
      if (l != null) l = l.reverse();

    }

  }
}


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();
    int j;

    IntList l = null;

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

    return l;
  }

  public IntList reverse() {

    IntList res = null;
    IntList l = this;

    while (l != null) {
      res = new IntList(l.value, res);
      l = l.next;

    }

    return res;
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Shuffle.main([Ljava/lang/String;)V: Graph of 72 nodes with 1 SCC.

IntList.createIntList()LIntList;: Graph of 164 nodes with 1 SCC.

IntList.reverse()LIntList;: Graph of 52 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 44 rules for P and 3 rules for R.

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

Filtered ground terms:

3120_0_reverse_Store(x1, x2, x3) → 3120_0_reverse_Store(x2, x3)
IntList(x1, x2, x3) → IntList(x2, x3)
2920_0_reverse_NULL(x1, x2, x3, x4) → 2920_0_reverse_NULL(x2, x3, x4)
2939_0_reverse_Return(x1, x2) → 2939_0_reverse_Return(x2)

Filtered duplicate args:

2920_0_reverse_NULL(x1, x2, x3) → 2920_0_reverse_NULL(x1, x3)

Finished conversion. Obtained 3 rules for P and 1 rules for R. System has no predefined symbols.

Log for SCC 1:

Generated 40 rules for P and 49 rules for R.

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

Filtered ground terms:

IntList(x1) → IntList
2386_0_random_ArrayAccess(x1, x2, x3) → 2386_0_random_ArrayAccess(x2, x3)
2409_0_random_IntArithmetic(x1, x2, x3, x4) → 2409_0_random_IntArithmetic(x2, x3)

Filtered unneeded arguments:

2386_1_createIntList_InvokeMethod(x1, x2, x3) → 2386_1_createIntList_InvokeMethod(x1, x2)
Cond_2386_1_createIntList_InvokeMethod(x1, x2, x3, x4) → Cond_2386_1_createIntList_InvokeMethod(x1, x2, x3)
2409_1_createIntList_InvokeMethod(x1, x2, x3) → 2409_1_createIntList_InvokeMethod(x1, x2)
Cond_2409_1_createIntList_InvokeMethod(x1, x2, x3, x4) → Cond_2409_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 2:

Generated 36 rules for P and 56 rules for R.

Combined rules. Obtained 3 rules for P and 6 rules for R.

Filtered ground terms:

2330_0_main_NULL(x1, x2, x3, x4) → 2330_0_main_NULL(x2, x3, x4)
IntList(x1, x2, x3) → IntList(x2, x3)
2939_0_reverse_Return(x1, x2) → 2939_0_reverse_Return(x2)
2587_0_reverse_ConstantStackPush(x1, x2) → 2587_0_reverse_ConstantStackPush(x2)
2920_0_reverse_NULL(x1, x2, x3, x4) → 2920_0_reverse_NULL(x2, x3, x4)
3104_0_reverse_FieldAccess(x1, x2, x3) → 3104_0_reverse_FieldAccess(x2, x3)
2346_0_main_Return(x1) → 2346_0_main_Return

Filtered duplicate args:

2330_0_main_NULL(x1, x2, x3) → 2330_0_main_NULL(x2, x3)
2920_0_reverse_NULL(x1, x2, x3) → 2920_0_reverse_NULL(x1, x3)

Finished conversion. Obtained 3 rules for P and 6 rules for R. System has no 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:
none

The ITRS R consists of the following rules:
2920_0_reverse_NULL(x0, NULL) → 2939_0_reverse_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 3120_0_REVERSE_STORE(java.lang.Object(IntList(x0[0], x1[0])), x2[0]) → 2920_0_REVERSE_NULL(java.lang.Object(IntList(x0[0], x1[0])), x2[0])
(1): 2920_0_REVERSE_NULL(x0[1], java.lang.Object(IntList(x1[1], x2[1]))) → 2920_0_REVERSE_NULL(java.lang.Object(IntList(x1[1], x0[1])), x2[1])
(2): 2920_0_REVERSE_NULL(x0[2], java.lang.Object(IntList(x1[2], x2[2]))) → 3120_0_REVERSE_STORE(java.lang.Object(IntList(x1[2], x0[2])), x2[2])

(0) -> (1), if ((java.lang.Object(IntList(x0[0], x1[0])) →^* x0[1])∧(x2[0] →^* java.lang.Object(IntList(x1[1], x2[1]))))

(0) -> (2), if ((java.lang.Object(IntList(x0[0], x1[0])) →^* x0[2])∧(x2[0] →^* java.lang.Object(IntList(x1[2], x2[2]))))

(1) -> (1), if ((java.lang.Object(IntList(x1[1], x0[1])) →^* x0[1]')∧(x2[1] →^* java.lang.Object(IntList(x1[1]', x2[1]'))))

(1) -> (2), if ((java.lang.Object(IntList(x1[1], x0[1])) →^* x0[2])∧(x2[1] →^* java.lang.Object(IntList(x1[2], x2[2]))))

(2) -> (0), if ((java.lang.Object(IntList(x1[2], x0[2])) →^* java.lang.Object(IntList(x0[0], x1[0])))∧(x2[2] →^* x2[0]))

The set Q consists of the following terms:
2920_0_reverse_NULL(x0, NULL)

(6) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

3120_0_REVERSE_STORE(java.lang.Object(IntList(x0[0], x1[0])), x2[0]) → 2920_0_REVERSE_NULL(java.lang.Object(IntList(x0[0], x1[0])), x2[0])
2920_0_REVERSE_NULL(x0[1], java.lang.Object(IntList(x1[1], x2[1]))) → 2920_0_REVERSE_NULL(java.lang.Object(IntList(x1[1], x0[1])), x2[1])
2920_0_REVERSE_NULL(x0[2], java.lang.Object(IntList(x1[2], x2[2]))) → 3120_0_REVERSE_STORE(java.lang.Object(IntList(x1[2], x0[2])), x2[2])

The TRS R consists of the following rules:

2920_0_reverse_NULL(x0, NULL) → 2939_0_reverse_Return(x0)

The set Q consists of the following terms:

2920_0_reverse_NULL(x0, NULL)

We have to consider all minimal (P,Q,R)-chains.

(8) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(9) Obligation:

Q DP problem:
The TRS P consists of the following rules:

3120_0_REVERSE_STORE(java.lang.Object(IntList(x0[0], x1[0])), x2[0]) → 2920_0_REVERSE_NULL(java.lang.Object(IntList(x0[0], x1[0])), x2[0])
2920_0_REVERSE_NULL(x0[1], java.lang.Object(IntList(x1[1], x2[1]))) → 2920_0_REVERSE_NULL(java.lang.Object(IntList(x1[1], x0[1])), x2[1])
2920_0_REVERSE_NULL(x0[2], java.lang.Object(IntList(x1[2], x2[2]))) → 3120_0_REVERSE_STORE(java.lang.Object(IntList(x1[2], x0[2])), x2[2])

R is empty.
The set Q consists of the following terms:

2920_0_reverse_NULL(x0, NULL)

We have to consider all minimal (P,Q,R)-chains.

(10) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

2920_0_reverse_NULL(x0, NULL)

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

3120_0_REVERSE_STORE(java.lang.Object(IntList(x0[0], x1[0])), x2[0]) → 2920_0_REVERSE_NULL(java.lang.Object(IntList(x0[0], x1[0])), x2[0])
2920_0_REVERSE_NULL(x0[1], java.lang.Object(IntList(x1[1], x2[1]))) → 2920_0_REVERSE_NULL(java.lang.Object(IntList(x1[1], x0[1])), x2[1])
2920_0_REVERSE_NULL(x0[2], java.lang.Object(IntList(x1[2], x2[2]))) → 3120_0_REVERSE_STORE(java.lang.Object(IntList(x1[2], x0[2])), x2[2])

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

2920_0_REVERSE_NULL(x0[2], java.lang.Object(IntList(x1[2], x2[2]))) → 3120_0_REVERSE_STORE(java.lang.Object(IntList(x1[2], x0[2])), x2[2])
The graph contains the following edges 2 > 2
2920_0_REVERSE_NULL(x0[1], java.lang.Object(IntList(x1[1], x2[1]))) → 2920_0_REVERSE_NULL(java.lang.Object(IntList(x1[1], x0[1])), x2[1])
The graph contains the following edges 2 > 2
3120_0_REVERSE_STORE(java.lang.Object(IntList(x0[0], x1[0])), x2[0]) → 2920_0_REVERSE_NULL(java.lang.Object(IntList(x0[0], x1[0])), x2[0])
The graph contains the following edges 1 >= 1, 2 >= 2

(13) YES

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2386_1_CREATEINTLIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_2386_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
(2): 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2409_1_CREATEINTLIST_INVOKEMETHOD(x4[2] > 0 && x2[2] > 0 && 0 < x4[2] + -1, 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])
(3): COND_2409_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_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)∧(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →^* 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0] →^* x3[1]))

(1) -> (2), if ((2409_0_random_IntArithmetic(x5[1], x6[1]) →^* 2409_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)∧(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]) →^* 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]))∧(x4[2] →^* x4[3]))

(3) -> (0), if ((2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]) →^* 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] + -1 →^* x3[0]))

The set Q is empty.

(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 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) the following chains were created:

We consider the chain 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]), COND_2386_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(x5[1], x6[1]), x3[1]) which results in the following constraint:
(1)    (&&(>=(x2[0], 1), <(x2[0], x0[0]))=TRUE∧2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])=2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1])∧x3[0]=x3[1] ⇒ 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(U^Increasing(COND_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_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 ⇒ 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(U^Increasing(COND_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_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_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_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_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_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_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_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_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_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_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_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_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_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_2386_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(x5, x6), x3) the following chains were created:

We consider the chain COND_2386_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(x5[1], x6[1]), x3[1]) which results in the following constraint:
(9)    (COND_2386_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥NonInfC∧COND_2386_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(x5[1], x6[1]), x3[1])∧(U^Increasing(2409_1_CREATEINTLIST_INVOKEMETHOD(2409_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(2409_1_CREATEINTLIST_INVOKEMETHOD(2409_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(2409_1_CREATEINTLIST_INVOKEMETHOD(2409_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(2409_1_CREATEINTLIST_INVOKEMETHOD(2409_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(2409_1_CREATEINTLIST_INVOKEMETHOD(2409_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 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → COND_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4, 0), >(x2, 0)), <(0, +(x4, -1))), 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) the following chains were created:

We consider the chain 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]), COND_2409_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_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∧2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2])=2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3])∧x4[2]=x4[3] ⇒ 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥NonInfC∧2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥COND_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])∧(U^Increasing(COND_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_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 ⇒ 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥NonInfC∧2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥COND_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])∧(U^Increasing(COND_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_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_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_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_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_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_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_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_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_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_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_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_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_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_2409_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), +(x4, -1)) the following chains were created:

We consider the chain COND_2409_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1)) which results in the following constraint:
(22)    (COND_2409_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3])≥NonInfC∧COND_2409_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3])≥2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))∧(U^Increasing(2386_1_CREATEINTLIST_INVOKEMETHOD(2386_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(2386_1_CREATEINTLIST_INVOKEMETHOD(2386_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(2386_1_CREATEINTLIST_INVOKEMETHOD(2386_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(2386_1_CREATEINTLIST_INVOKEMETHOD(2386_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(2386_1_CREATEINTLIST_INVOKEMETHOD(2386_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.

2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
- (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_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_2386_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(x5, x6), x3)
- ((U^Increasing(2409_1_CREATEINTLIST_INVOKEMETHOD(2409_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)
2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → COND_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4, 0), >(x2, 0)), <(0, +(x4, -1))), 2409_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_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_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_2409_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), +(x4, -1))
- ((U^Increasing(2386_1_CREATEINTLIST_INVOKEMETHOD(2386_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(2386_1_CREATEINTLIST_INVOKEMETHOD(x₁, x₂)) = [-1] + [2]x₂ + [-1]x₁
POL(2386_0_random_ArrayAccess(x₁, x₂)) = [-1] + [-1]x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁, x₂)) = [-1]
POL(COND_2386_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(2409_1_CREATEINTLIST_INVOKEMETHOD(x₁, x₂)) = [-1] + [2]x₂
POL(2409_0_random_IntArithmetic(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(java.lang.String(x₁, x₂)) = [-1]x₂ + [-1]x₁
POL(COND_2409_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_2409_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))

The following pairs are in P_bound:

2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])

The following pairs are in P_≥:

2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2386_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
COND_2386_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2409_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])

There are no usable rules.

(16) Complex Obligation (AND)

(17) 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 ((2409_0_random_IntArithmetic(x5[1], x6[1]) →^* 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1] →^* x4[2]))

The set Q is empty.

(18) IDependencyGraphProof (EQUIVALENT transformation)

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

(19) TRUE

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2386_1_CREATEINTLIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_2386_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2409_1_CREATEINTLIST_INVOKEMETHOD(2409_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
(3): COND_2409_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2409_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2386_1_CREATEINTLIST_INVOKEMETHOD(2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), x4[3] + -1)

(3) -> (0), if ((2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]) →^* 2386_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] + -1 →^* x3[0]))

The set Q is empty.

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(22) TRUE

(23) 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:
none

The ITRS R consists of the following rules:
2330_0_main_NULL(x0, NULL) → 2346_0_main_Return
2587_0_reverse_ConstantStackPush(java.lang.Object(x0)) → 2920_0_reverse_NULL(NULL, java.lang.Object(x0))
2920_0_reverse_NULL(x0, NULL) → 2939_0_reverse_Return(x0)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x1, x0)), x2)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 3104_0_reverse_FieldAccess(java.lang.Object(IntList(x1, x0)), java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x0, x1)), x2)

The integer pair graph contains the following rules and edges:
(0): 2330_0_MAIN_NULL(x2[0], java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → 2587_1_MAIN_INVOKEMETHOD(2587_0_reverse_ConstantStackPush(java.lang.Object(x1[0])), java.lang.Object(IntList(x0[0], x2[0])), java.lang.Object(x1[0]))
(1): 2587_1_MAIN_INVOKEMETHOD(2939_0_reverse_Return(x0[1]), java.lang.Object(IntList(x1[1], x2[1])), java.lang.Object(x3[1])) → 2330_0_MAIN_NULL(java.lang.Object(IntList(x1[1], x2[1])), x0[1])
(2): 2330_0_MAIN_NULL(x1[2], java.lang.Object(IntList(x0[2], NULL))) → 2330_0_MAIN_NULL(java.lang.Object(IntList(x0[2], x1[2])), NULL)

(0) -> (1), if ((2587_0_reverse_ConstantStackPush(java.lang.Object(x1[0])) →^* 2939_0_reverse_Return(x0[1]))∧(java.lang.Object(IntList(x0[0], x2[0])) →^* java.lang.Object(IntList(x1[1], x2[1])))∧(java.lang.Object(x1[0]) →^* java.lang.Object(x3[1])))

(1) -> (0), if ((java.lang.Object(IntList(x1[1], x2[1])) →^* x2[0])∧(x0[1] →^* java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))))

(1) -> (2), if ((java.lang.Object(IntList(x1[1], x2[1])) →^* x1[2])∧(x0[1] →^* java.lang.Object(IntList(x0[2], NULL))))

(2) -> (0), if ((java.lang.Object(IntList(x0[2], x1[2])) →^* x2[0])∧(NULL →^* java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))))

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

The set Q consists of the following terms:
2330_0_main_NULL(x0, NULL)
2587_0_reverse_ConstantStackPush(java.lang.Object(x0))
2920_0_reverse_NULL(x0, NULL)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2)))

(24) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

2330_0_MAIN_NULL(x2[0], java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → 2587_1_MAIN_INVOKEMETHOD(2587_0_reverse_ConstantStackPush(java.lang.Object(x1[0])), java.lang.Object(IntList(x0[0], x2[0])), java.lang.Object(x1[0]))
2587_1_MAIN_INVOKEMETHOD(2939_0_reverse_Return(x0[1]), java.lang.Object(IntList(x1[1], x2[1])), java.lang.Object(x3[1])) → 2330_0_MAIN_NULL(java.lang.Object(IntList(x1[1], x2[1])), x0[1])
2330_0_MAIN_NULL(x1[2], java.lang.Object(IntList(x0[2], NULL))) → 2330_0_MAIN_NULL(java.lang.Object(IntList(x0[2], x1[2])), NULL)

The TRS R consists of the following rules:

2330_0_main_NULL(x0, NULL) → 2346_0_main_Return
2587_0_reverse_ConstantStackPush(java.lang.Object(x0)) → 2920_0_reverse_NULL(NULL, java.lang.Object(x0))
2920_0_reverse_NULL(x0, NULL) → 2939_0_reverse_Return(x0)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x1, x0)), x2)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 3104_0_reverse_FieldAccess(java.lang.Object(IntList(x1, x0)), java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x0, x1)), x2)

The set Q consists of the following terms:

2330_0_main_NULL(x0, NULL)
2587_0_reverse_ConstantStackPush(java.lang.Object(x0))
2920_0_reverse_NULL(x0, NULL)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2)))

We have to consider all minimal (P,Q,R)-chains.

(26) DependencyGraphProof (EQUIVALENT transformation)

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

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

2587_1_MAIN_INVOKEMETHOD(2939_0_reverse_Return(x0[1]), java.lang.Object(IntList(x1[1], x2[1])), java.lang.Object(x3[1])) → 2330_0_MAIN_NULL(java.lang.Object(IntList(x1[1], x2[1])), x0[1])
2330_0_MAIN_NULL(x2[0], java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → 2587_1_MAIN_INVOKEMETHOD(2587_0_reverse_ConstantStackPush(java.lang.Object(x1[0])), java.lang.Object(IntList(x0[0], x2[0])), java.lang.Object(x1[0]))

The TRS R consists of the following rules:

2330_0_main_NULL(x0, NULL) → 2346_0_main_Return
2587_0_reverse_ConstantStackPush(java.lang.Object(x0)) → 2920_0_reverse_NULL(NULL, java.lang.Object(x0))
2920_0_reverse_NULL(x0, NULL) → 2939_0_reverse_Return(x0)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x1, x0)), x2)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 3104_0_reverse_FieldAccess(java.lang.Object(IntList(x1, x0)), java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x0, x1)), x2)

The set Q consists of the following terms:

2330_0_main_NULL(x0, NULL)
2587_0_reverse_ConstantStackPush(java.lang.Object(x0))
2920_0_reverse_NULL(x0, NULL)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2)))

We have to consider all minimal (P,Q,R)-chains.

(28) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(29) Obligation:

Q DP problem:
The TRS P consists of the following rules:

2587_1_MAIN_INVOKEMETHOD(2939_0_reverse_Return(x0[1]), java.lang.Object(IntList(x1[1], x2[1])), java.lang.Object(x3[1])) → 2330_0_MAIN_NULL(java.lang.Object(IntList(x1[1], x2[1])), x0[1])
2330_0_MAIN_NULL(x2[0], java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → 2587_1_MAIN_INVOKEMETHOD(2587_0_reverse_ConstantStackPush(java.lang.Object(x1[0])), java.lang.Object(IntList(x0[0], x2[0])), java.lang.Object(x1[0]))

The TRS R consists of the following rules:

2587_0_reverse_ConstantStackPush(java.lang.Object(x0)) → 2920_0_reverse_NULL(NULL, java.lang.Object(x0))
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 3104_0_reverse_FieldAccess(java.lang.Object(IntList(x1, x0)), java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x0, x1)), x2)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x1, x0)), x2)
2920_0_reverse_NULL(x0, NULL) → 2939_0_reverse_Return(x0)

The set Q consists of the following terms:

2330_0_main_NULL(x0, NULL)
2587_0_reverse_ConstantStackPush(java.lang.Object(x0))
2920_0_reverse_NULL(x0, NULL)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2)))

We have to consider all minimal (P,Q,R)-chains.

(30) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

2330_0_main_NULL(x0, NULL)

(31) Obligation:

Q DP problem:
The TRS P consists of the following rules:

2587_1_MAIN_INVOKEMETHOD(2939_0_reverse_Return(x0[1]), java.lang.Object(IntList(x1[1], x2[1])), java.lang.Object(x3[1])) → 2330_0_MAIN_NULL(java.lang.Object(IntList(x1[1], x2[1])), x0[1])
2330_0_MAIN_NULL(x2[0], java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → 2587_1_MAIN_INVOKEMETHOD(2587_0_reverse_ConstantStackPush(java.lang.Object(x1[0])), java.lang.Object(IntList(x0[0], x2[0])), java.lang.Object(x1[0]))

The TRS R consists of the following rules:

2587_0_reverse_ConstantStackPush(java.lang.Object(x0)) → 2920_0_reverse_NULL(NULL, java.lang.Object(x0))
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 3104_0_reverse_FieldAccess(java.lang.Object(IntList(x1, x0)), java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x0, x1)), x2)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x1, x0)), x2)
2920_0_reverse_NULL(x0, NULL) → 2939_0_reverse_Return(x0)

The set Q consists of the following terms:

2587_0_reverse_ConstantStackPush(java.lang.Object(x0))
2920_0_reverse_NULL(x0, NULL)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2)))

We have to consider all minimal (P,Q,R)-chains.

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

2587_1_MAIN_INVOKEMETHOD(2939_0_reverse_Return(x0[1]), java.lang.Object(IntList(x1[1], x2[1])), java.lang.Object(x3[1])) → 2330_0_MAIN_NULL(java.lang.Object(IntList(x1[1], x2[1])), x0[1])

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(2330_0_MAIN_NULL(x₁, x₂)) = x₂
POL(2587_0_reverse_ConstantStackPush(x₁)) = 1 + x₁
POL(2587_1_MAIN_INVOKEMETHOD(x₁, x₂, x₃)) = x₁
POL(2920_0_reverse_NULL(x₁, x₂)) = 1 + x₁ + x₂
POL(2939_0_reverse_Return(x₁)) = 1 + x₁
POL(3104_0_reverse_FieldAccess(x₁, x₂)) = x₁ + x₂
POL(IntList(x₁, x₂)) = 1 + x₂
POL(NULL) = 0
POL(java.lang.Object(x₁)) = x₁

The following usable rules [FROCOS05] were oriented:

2587_0_reverse_ConstantStackPush(java.lang.Object(x0)) → 2920_0_reverse_NULL(NULL, java.lang.Object(x0))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x0, x1)), x2)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 3104_0_reverse_FieldAccess(java.lang.Object(IntList(x1, x0)), java.lang.Object(IntList(x1, x2)))
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x1, x0)), x2)
2920_0_reverse_NULL(x0, NULL) → 2939_0_reverse_Return(x0)

(33) Obligation:

Q DP problem:
The TRS P consists of the following rules:

2330_0_MAIN_NULL(x2[0], java.lang.Object(IntList(x0[0], java.lang.Object(x1[0])))) → 2587_1_MAIN_INVOKEMETHOD(2587_0_reverse_ConstantStackPush(java.lang.Object(x1[0])), java.lang.Object(IntList(x0[0], x2[0])), java.lang.Object(x1[0]))

The TRS R consists of the following rules:

2587_0_reverse_ConstantStackPush(java.lang.Object(x0)) → 2920_0_reverse_NULL(NULL, java.lang.Object(x0))
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 3104_0_reverse_FieldAccess(java.lang.Object(IntList(x1, x0)), java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x0, x1)), x2)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2))) → 2920_0_reverse_NULL(java.lang.Object(IntList(x1, x0)), x2)
2920_0_reverse_NULL(x0, NULL) → 2939_0_reverse_Return(x0)

The set Q consists of the following terms:

2587_0_reverse_ConstantStackPush(java.lang.Object(x0))
2920_0_reverse_NULL(x0, NULL)
2920_0_reverse_NULL(x0, java.lang.Object(IntList(x1, x2)))
3104_0_reverse_FieldAccess(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x2)))

We have to consider all minimal (P,Q,R)-chains.

(34) DependencyGraphProof (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) IDPtoQDPProof (SOUND transformation)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) QReductionProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) IDPNonInfProof (SOUND transformation)

(16) Complex Obligation (AND)

(17) Obligation:

(18) IDependencyGraphProof (EQUIVALENT transformation)

(19) TRUE

(20) Obligation:

(21) IDependencyGraphProof (EQUIVALENT transformation)

(22) TRUE

(23) Obligation:

(24) IDPtoQDPProof (SOUND transformation)

(25) Obligation:

(26) DependencyGraphProof (EQUIVALENT transformation)

(27) Obligation:

(28) UsableRulesProof (EQUIVALENT transformation)

(29) Obligation:

(30) QReductionProof (EQUIVALENT transformation)

(31) Obligation:

(32) QDPOrderProof (EQUIVALENT transformation)

(33) Obligation:

(34) DependencyGraphProof (EQUIVALENT transformation)

(35) TRUE