Termination proof of /tmp/tmpGRIdw4/CountMetaListRec.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 UsableRulesReductionPairsProof (⇔)

        ↳13 QDP

        ↳14 MRRProof (⇔)

        ↳15 QDP

        ↳16 PisEmptyProof (⇔)

        ↳17 YES

    ↳18 IDP

        ↳19 IDPNonInfProof (⇒)

        ↳20 AND

            ↳21 IDP

                ↳22 IDependencyGraphProof (⇔)

                ↳23 IDP

                ↳24 IDPNonInfProof (⇒)

                ↳25 AND

                    ↳26 IDP

                        ↳27 IDependencyGraphProof (⇔)

                        ↳28 TRUE

                    ↳29 IDP

                        ↳30 IDependencyGraphProof (⇔)

                        ↳31 TRUE

            ↳32 IDP

                ↳33 IDependencyGraphProof (⇔)

                ↳34 IDP

                ↳35 IDPNonInfProof (⇒)

                ↳36 AND

                    ↳37 IDP

                        ↳38 IDependencyGraphProof (⇔)

                        ↳39 TRUE

                    ↳40 IDP

                        ↳41 IDependencyGraphProof (⇔)

                        ↳42 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: CountMetaListRec

public class CountMetaListRec {
	public static void main(String[] args) {
		Random.args = args;
		List l = createMetaList();

		int count = countMetaList(l);
	}

	public static int countMetaList(List cur) {
		if (cur == null) {
			return 0;
		} else {
			if (cur.value instanceof List) {
				List inner = (List) cur.value;
				cur.value = inner.next;
				cur = new List(inner.value, cur);
			}
			return 1 + countMetaList(cur.next);
		}
	}

	public static List createMetaList() {
		int count = Random.random();
		List cur = null;
		for (int i = 0; i < count; i++) {
			int innerCount = Random.random();
			List innerList = null;
			for (int j = innerCount; j > 0; j--) {
				innerList = new List(null, innerList);
			}
			cur = new List(innerList, cur);
		}

		return cur;
	}
}

class List {
	Object value;
	List next;

	public List(Object v, List n) {
		this.value = v;
		this.next = n;
	}
}


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:
CountMetaListRec.main([Ljava/lang/String;)V: Graph of 40 nodes with 0 SCCs.

CountMetaListRec.createMetaList()LList;: Graph of 194 nodes with 1 SCC.

CountMetaListRec.countMetaList(LList;)I: Graph of 89 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 62 rules for P and 25 rules for R.

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

Filtered ground terms:

1150_0_countMetaList_NONNULL(x1, x2, x3) → 1150_0_countMetaList_NONNULL(x2, x3)
List(x1, x2, x3) → List(x2, x3)
2585_0_countMetaList_Return(x1) → 2585_0_countMetaList_Return
1187_0_countMetaList_Return(x1, x2) → 1187_0_countMetaList_Return

Filtered duplicate args:

1150_0_countMetaList_NONNULL(x1, x2) → 1150_0_countMetaList_NONNULL(x2)

Finished conversion. Obtained 3 rules for P and 6 rules for R. System has no predefined symbols.

Log for SCC 1:

Generated 69 rules for P and 49 rules for R.

Combined rules. Obtained 4 rules for P and 0 rules for R.

Filtered ground terms:

3643_0_createMetaList_LE(x1, x2, x3, x4, x5, x6, x7) → 3643_0_createMetaList_LE(x2, x3, x4, x5, x6, x7)
List(x1) → List
Cond_3643_0_createMetaList_LE1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_3643_0_createMetaList_LE1(x1, x3, x4, x5, x6, x7, x8)
3134_0_random_ArrayAccess(x1, x2, x3) → 3134_0_random_ArrayAccess(x2, x3)
Cond_3643_0_createMetaList_LE(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_3643_0_createMetaList_LE(x1, x3, x4, x5, x6)
3166_0_random_IntArithmetic(x1, x2, x3, x4) → 3166_0_random_IntArithmetic(x2, x3)

Filtered duplicate args:

3643_0_createMetaList_LE(x1, x2, x3, x4, x5, x6) → 3643_0_createMetaList_LE(x1, x2, x3, x4, x6)
Cond_3643_0_createMetaList_LE1(x1, x2, x3, x4, x5, x6, x7) → Cond_3643_0_createMetaList_LE1(x1, x2, x3, x4, x5, x7)

Filtered unneeded arguments:

3134_1_createMetaList_InvokeMethod(x1, x2, x3, x4) → 3134_1_createMetaList_InvokeMethod(x1, x2, x4)
Cond_3134_1_createMetaList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_3134_1_createMetaList_InvokeMethod(x1, x2, x3, x5)
3166_1_createMetaList_InvokeMethod(x1, x2, x3, x4) → 3166_1_createMetaList_InvokeMethod(x1, x2, x4)
Cond_3166_1_createMetaList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_3166_1_createMetaList_InvokeMethod(x1, x2, x3, x5)
3643_0_createMetaList_LE(x1, x2, x3, x4, x5) → 3643_0_createMetaList_LE(x1, x3, x5)
Cond_3643_0_createMetaList_LE(x1, x2, x3, x4, x5) → Cond_3643_0_createMetaList_LE(x1, x2, x4)
Cond_3643_0_createMetaList_LE1(x1, x2, x3, x4, x5, x6) → Cond_3643_0_createMetaList_LE1(x1, x2, x4, x6)

Combined rules. Obtained 4 rules for P and 0 rules for R.

Finished conversion. Obtained 4 rules for P and 0 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:
none

The ITRS R consists of the following rules:
1150_0_countMetaList_NONNULL(NULL) → 1187_0_countMetaList_Return
2423_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, java.lang.Object(List(x0, x1))) → 2585_0_countMetaList_Return
1659_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL) → 2585_0_countMetaList_Return
1659_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0) → 2585_0_countMetaList_Return
1630_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL) → 2585_0_countMetaList_Return
1630_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0) → 2585_0_countMetaList_Return

The integer pair graph contains the following rules and edges:
(0): 1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(List(x0[0], x1[0])), x2[0]))) → 1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(x1[0], x2[0])))
(1): 1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(x0[1]), x1[1]))) → 1150_0_COUNTMETALIST_NONNULL(x1[1])
(2): 1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(NULL, x0[2]))) → 1150_0_COUNTMETALIST_NONNULL(x0[2])

(0) -> (0), if ((java.lang.Object(List(x1[0], x2[0])) →^* java.lang.Object(List(java.lang.Object(List(x0[0]', x1[0]')), x2[0]'))))

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

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

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

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

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

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

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

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

The set Q consists of the following terms:
1150_0_countMetaList_NONNULL(NULL)
2423_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, java.lang.Object(List(x0, x1)))
1659_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL)
1659_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0)
1630_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL)
1630_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0)

(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:

1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(List(x0[0], x1[0])), x2[0]))) → 1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(x1[0], x2[0])))
1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(x0[1]), x1[1]))) → 1150_0_COUNTMETALIST_NONNULL(x1[1])
1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(NULL, x0[2]))) → 1150_0_COUNTMETALIST_NONNULL(x0[2])

The TRS R consists of the following rules:

1150_0_countMetaList_NONNULL(NULL) → 1187_0_countMetaList_Return
2423_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, java.lang.Object(List(x0, x1))) → 2585_0_countMetaList_Return
1659_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL) → 2585_0_countMetaList_Return
1659_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0) → 2585_0_countMetaList_Return
1630_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL) → 2585_0_countMetaList_Return
1630_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0) → 2585_0_countMetaList_Return

The set Q consists of the following terms:

1150_0_countMetaList_NONNULL(NULL)
2423_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, java.lang.Object(List(x0, x1)))
1659_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL)
1659_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0)
1630_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL)
1630_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0)

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:

1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(List(x0[0], x1[0])), x2[0]))) → 1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(x1[0], x2[0])))
1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(x0[1]), x1[1]))) → 1150_0_COUNTMETALIST_NONNULL(x1[1])
1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(NULL, x0[2]))) → 1150_0_COUNTMETALIST_NONNULL(x0[2])

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

1150_0_countMetaList_NONNULL(NULL)
2423_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, java.lang.Object(List(x0, x1)))
1659_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL)
1659_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0)
1630_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL)
1630_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0)

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].

1150_0_countMetaList_NONNULL(NULL)
2423_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, java.lang.Object(List(x0, x1)))
1659_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL)
1659_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0)
1630_1_countMetaList_InvokeMethod(1187_0_countMetaList_Return, NULL)
1630_1_countMetaList_InvokeMethod(2585_0_countMetaList_Return, x0)

(11) Obligation:

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

1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(List(x0[0], x1[0])), x2[0]))) → 1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(x1[0], x2[0])))
1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(x0[1]), x1[1]))) → 1150_0_COUNTMETALIST_NONNULL(x1[1])
1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(NULL, x0[2]))) → 1150_0_COUNTMETALIST_NONNULL(x0[2])

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

(12) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(NULL, x0[2]))) → 1150_0_COUNTMETALIST_NONNULL(x0[2])

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(1150_0_COUNTMETALIST_NONNULL(x₁)) = 2·x₁
POL(List(x₁, x₂)) = x₁ + 2·x₂
POL(NULL) = 0
POL(java.lang.Object(x₁)) = 2·x₁

(13) Obligation:

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

1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(List(x0[0], x1[0])), x2[0]))) → 1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(x1[0], x2[0])))
1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(x0[1]), x1[1]))) → 1150_0_COUNTMETALIST_NONNULL(x1[1])

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

(14) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(List(x0[0], x1[0])), x2[0]))) → 1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(x1[0], x2[0])))
1150_0_COUNTMETALIST_NONNULL(java.lang.Object(List(java.lang.Object(x0[1]), x1[1]))) → 1150_0_COUNTMETALIST_NONNULL(x1[1])

Used ordering: Polynomial interpretation [POLO]:

POL(1150_0_COUNTMETALIST_NONNULL(x₁)) = 2·x₁
POL(List(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(java.lang.Object(x₁)) = 2 + 2·x₁

(15) Obligation:

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

(16) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(17) YES

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]) → COND_3134_1_CREATEMETALIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])
(1): COND_3134_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])
(2): 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]) → COND_3166_1_CREATEMETALIST_INVOKEMETHOD(x2[2] > 0, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])
(3): COND_3166_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])
(4): 3643_0_CREATEMETALIST_LE(x0[4], x2[4], 0) → COND_3643_0_CREATEMETALIST_LE(x2[4] >= 0 && x0[4] > x2[4] + 1, x0[4], x2[4], 0)
(5): COND_3643_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0) → 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], x2[5] + 1)
(6): 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])
(7): COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)

(0) -> (1), if ((x2[0] >= 1 && x2[0] < x0[0] →^* TRUE)∧(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →^* 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0] →^* x3[1])∧(x5[0] →^* x5[1]))

(1) -> (2), if ((3166_0_random_IntArithmetic(x6[1], x7[1]) →^* 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1] →^* x4[2])∧(x5[1] →^* x6[2]))

(2) -> (3), if ((x2[2] > 0 →^* TRUE)∧(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]) →^* 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]))∧(x4[2] →^* x4[3])∧(x6[2] →^* x6[3]))

(3) -> (4), if ((x4[3] →^* x0[4])∧(x6[3] →^* x2[4])∧(x1[3] →^* 0))

(3) -> (6), if ((x4[3] →^* x0[6])∧(x6[3] →^* x2[6])∧(x1[3] →^* x4[6]))

(4) -> (5), if ((x2[4] >= 0 && x0[4] > x2[4] + 1 →^* TRUE)∧(x0[4] →^* x0[5])∧(x2[4] →^* x2[5]))

(5) -> (0), if ((3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]) →^* 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x0[5] →^* x3[0])∧(x2[5] + 1 →^* x5[0]))

(6) -> (7), if ((x4[6] > 0 →^* TRUE)∧(x0[6] →^* x0[7])∧(x2[6] →^* x2[7])∧(x4[6] →^* x4[7]))

(7) -> (4), if ((x0[7] →^* x0[4])∧(x2[7] →^* x2[4])∧(x4[7] + -1 →^* 0))

(7) -> (6), if ((x0[7] →^* x0[6])∧(x2[7] →^* x2[6])∧(x4[7] + -1 →^* x4[6]))

The set Q is empty.

(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 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5) → COND_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5) the following chains were created:

We consider the chain 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]) → COND_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]), COND_3134_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1]) which results in the following constraint:
(1)    (&&(>=(x2[0], 1), <(x2[0], x0[0]))=TRUE∧3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])=3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1])∧x3[0]=x3[1]∧x5[0]=x5[1] ⇒ 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])≥NonInfC∧3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])≥COND_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])∧(U^Increasing(COND_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[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 ⇒ 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])≥NonInfC∧3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])≥COND_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])∧(U^Increasing(COND_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[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_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]x5[0] + [bni_31]x3[0] ≥ 0∧[(-1)bso_32] ≥ 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_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]x5[0] + [bni_31]x3[0] ≥ 0∧[(-1)bso_32] ≥ 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_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]x5[0] + [bni_31]x3[0] ≥ 0∧[(-1)bso_32] ≥ 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_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31] = 0∧[bni_31] = 0∧0 = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 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_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31] = 0∧[bni_31] = 0∧0 = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 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_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31] = 0∧[bni_31] = 0∧0 = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 0)

For Pair COND_3134_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5) → 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6, x7), x3, x5) the following chains were created:

We consider the chain COND_3134_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1]) which results in the following constraint:
(9)    (COND_3134_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1])≥NonInfC∧COND_3134_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1])≥3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])∧(U^Increasing(3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    ((U^Increasing(3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥)∧[(-1)bso_34] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    ((U^Increasing(3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥)∧[(-1)bso_34] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    ((U^Increasing(3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥)∧[(-1)bso_34] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    ((U^Increasing(3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_34] ≥ 0)

For Pair 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6) → COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2, 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6) the following chains were created:

We consider the chain 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]) → COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]), COND_3166_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3]) which results in the following constraint:
(14)    (>(x2[2], 0)=TRUE∧3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2])=3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3])∧x4[2]=x4[3]∧x6[2]=x6[3] ⇒ 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])≥NonInfC∧3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])≥COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])∧(U^Increasing(COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥))

We simplified constraint (14) using rules (I), (II), (IV) which results in the following new constraint:
(15)    (>(x2[2], 0)=TRUE ⇒ 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])≥NonInfC∧3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])≥COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])∧(U^Increasing(COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    (x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]x6[2] + [bni_35]x4[2] ≥ 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    (x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]x6[2] + [bni_35]x4[2] ≥ 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    (x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]x6[2] + [bni_35]x4[2] ≥ 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19)    (x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_35] = 0∧[bni_35] = 0∧[(-1)bni_35 + (-1)Bound*bni_35] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(20)    (x2[2] ≥ 0 ⇒ (U^Increasing(COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_35] = 0∧[bni_35] = 0∧[(-1)bni_35 + (-1)Bound*bni_35] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)

For Pair COND_3166_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6) → 3643_0_CREATEMETALIST_LE(x4, x6, x1) the following chains were created:

We consider the chain COND_3166_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3]) which results in the following constraint:
(21)    (COND_3166_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3])≥NonInfC∧COND_3166_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3])≥3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])∧(U^Increasing(3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥))

We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥)∧[(-1)bso_38] ≥ 0)

We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥)∧[(-1)bso_38] ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥)∧[(-1)bso_38] ≥ 0)

We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

For Pair 3643_0_CREATEMETALIST_LE(x0, x2, 0) → COND_3643_0_CREATEMETALIST_LE(&&(>=(x2, 0), >(x0, +(x2, 1))), x0, x2, 0) the following chains were created:

We consider the chain 3643_0_CREATEMETALIST_LE(x0[4], x2[4], 0) → COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0), COND_3643_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0) → 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1)) which results in the following constraint:
(26)    (&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1)))=TRUE∧x0[4]=x0[5]∧x2[4]=x2[5] ⇒ 3643_0_CREATEMETALIST_LE(x0[4], x2[4], 0)≥NonInfC∧3643_0_CREATEMETALIST_LE(x0[4], x2[4], 0)≥COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)∧(U^Increasing(COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥))

We simplified constraint (26) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(27)    (>=(x2[4], 0)=TRUE∧>(x0[4], +(x2[4], 1))=TRUE ⇒ 3643_0_CREATEMETALIST_LE(x0[4], x2[4], 0)≥NonInfC∧3643_0_CREATEMETALIST_LE(x0[4], x2[4], 0)≥COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)∧(U^Increasing(COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥))

We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(28)    (x2[4] ≥ 0∧x0[4] + [-2] + [-1]x2[4] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]x2[4] + [bni_39]x0[4] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(29)    (x2[4] ≥ 0∧x0[4] + [-2] + [-1]x2[4] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]x2[4] + [bni_39]x0[4] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(30)    (x2[4] ≥ 0∧x0[4] + [-2] + [-1]x2[4] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]x2[4] + [bni_39]x0[4] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥)∧[bni_39 + (-1)Bound*bni_39] + [bni_39]x0[4] ≥ 0∧[(-1)bso_40] ≥ 0)

For Pair COND_3643_0_CREATEMETALIST_LE(TRUE, x0, x2, 0) → 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), x0, +(x2, 1)) the following chains were created:

We consider the chain COND_3643_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0) → 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1)) which results in the following constraint:
(32)    (COND_3643_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0)≥NonInfC∧COND_3643_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0)≥3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))∧(U^Increasing(3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥))

We simplified constraint (32) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(33)    ((U^Increasing(3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥)∧[1 + (-1)bso_42] ≥ 0)

We simplified constraint (33) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(34)    ((U^Increasing(3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥)∧[1 + (-1)bso_42] ≥ 0)

We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(35)    ((U^Increasing(3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥)∧[1 + (-1)bso_42] ≥ 0)

We simplified constraint (35) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(36)    ((U^Increasing(3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_42] ≥ 0)

For Pair 3643_0_CREATEMETALIST_LE(x0, x2, x4) → COND_3643_0_CREATEMETALIST_LE1(>(x4, 0), x0, x2, x4) the following chains were created:

We consider the chain 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6]), COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:
(37)    (>(x4[6], 0)=TRUE∧x0[6]=x0[7]∧x2[6]=x2[7]∧x4[6]=x4[7] ⇒ 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

We simplified constraint (37) using rule (IV) which results in the following new constraint:
(38)    (>(x4[6], 0)=TRUE ⇒ 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

We simplified constraint (38) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(39)    (x4[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]x2[6] + [bni_43]x0[6] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(40)    (x4[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]x2[6] + [bni_43]x0[6] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(41)    (x4[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]x2[6] + [bni_43]x0[6] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (41) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(42)    (x4[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43] = 0∧[bni_43] = 0∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(43)    (x4[6] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43] = 0∧[bni_43] = 0∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_44] ≥ 0)

For Pair COND_3643_0_CREATEMETALIST_LE1(TRUE, x0, x2, x4) → 3643_0_CREATEMETALIST_LE(x0, x2, +(x4, -1)) the following chains were created:

We consider the chain COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:
(44)    (COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥NonInfC∧COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))∧(U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥))

We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(45)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[(-1)bso_46] ≥ 0)

We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(46)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[(-1)bso_46] ≥ 0)

We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(47)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[(-1)bso_46] ≥ 0)

We simplified constraint (47) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(48)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_46] ≥ 0)

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

3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5) → COND_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5)
- (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31] = 0∧[bni_31] = 0∧0 = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 0)
COND_3134_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5) → 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6, x7), x3, x5)
- ((U^Increasing(3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_34] ≥ 0)
3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6) → COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2, 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6)
- (x2[2] ≥ 0 ⇒ (U^Increasing(COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_35] = 0∧[bni_35] = 0∧[(-1)bni_35 + (-1)Bound*bni_35] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)
COND_3166_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6) → 3643_0_CREATEMETALIST_LE(x4, x6, x1)
- ((U^Increasing(3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
3643_0_CREATEMETALIST_LE(x0, x2, 0) → COND_3643_0_CREATEMETALIST_LE(&&(>=(x2, 0), >(x0, +(x2, 1))), x0, x2, 0)
- (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥)∧[bni_39 + (-1)Bound*bni_39] + [bni_39]x0[4] ≥ 0∧[(-1)bso_40] ≥ 0)
COND_3643_0_CREATEMETALIST_LE(TRUE, x0, x2, 0) → 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), x0, +(x2, 1))
- ((U^Increasing(3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_42] ≥ 0)
3643_0_CREATEMETALIST_LE(x0, x2, x4) → COND_3643_0_CREATEMETALIST_LE1(>(x4, 0), x0, x2, x4)
- (x4[6] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43] = 0∧[bni_43] = 0∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_44] ≥ 0)
COND_3643_0_CREATEMETALIST_LE1(TRUE, x0, x2, x4) → 3643_0_CREATEMETALIST_LE(x0, x2, +(x4, -1))
- ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_46] ≥ 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(3134_1_CREATEMETALIST_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂ + [-1]x₁
POL(3134_0_random_ArrayAccess(x₁, x₂)) = [-1] + [-1]x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁, x₂)) = [-1]
POL(COND_3134_1_CREATEMETALIST_INVOKEMETHOD(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<(x₁, x₂)) = [-1]
POL(3166_1_CREATEMETALIST_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [-1]x₁ + [-1]x₃ + x₂
POL(3166_0_random_IntArithmetic(x₁, x₂)) = 0
POL(java.lang.String(x₁, x₂)) = [-1] + [-1]x₂ + x₁
POL(COND_3166_1_CREATEMETALIST_INVOKEMETHOD(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + x₃ + [-1]x₂
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(3643_0_CREATEMETALIST_LE(x₁, x₂, x₃)) = [-1] + [-1]x₂ + x₁
POL(COND_3643_0_CREATEMETALIST_LE(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃ + x₂
POL(+(x₁, x₂)) = x₁ + x₂
POL(COND_3643_0_CREATEMETALIST_LE1(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_3643_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0) → 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))

The following pairs are in P_bound:

3643_0_CREATEMETALIST_LE(x0[4], x2[4], 0) → COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)

The following pairs are in P_≥:

3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]) → COND_3134_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])
COND_3134_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])
3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]) → COND_3166_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])
COND_3166_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])
3643_0_CREATEMETALIST_LE(x0[4], x2[4], 0) → COND_3643_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)
3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])
COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))

There are no usable rules.

(20) Complex Obligation (AND)

(21) 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): 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]) → COND_3134_1_CREATEMETALIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])
(1): COND_3134_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])
(2): 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]) → COND_3166_1_CREATEMETALIST_INVOKEMETHOD(x2[2] > 0, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])
(3): COND_3166_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])
(4): 3643_0_CREATEMETALIST_LE(x0[4], x2[4], 0) → COND_3643_0_CREATEMETALIST_LE(x2[4] >= 0 && x0[4] > x2[4] + 1, x0[4], x2[4], 0)
(6): 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])
(7): COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)

(1) -> (2), if ((3166_0_random_IntArithmetic(x6[1], x7[1]) →^* 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1] →^* x4[2])∧(x5[1] →^* x6[2]))

(3) -> (4), if ((x4[3] →^* x0[4])∧(x6[3] →^* x2[4])∧(x1[3] →^* 0))

(7) -> (4), if ((x0[7] →^* x0[4])∧(x2[7] →^* x2[4])∧(x4[7] + -1 →^* 0))

(3) -> (6), if ((x4[3] →^* x0[6])∧(x6[3] →^* x2[6])∧(x1[3] →^* x4[6]))

(7) -> (6), if ((x0[7] →^* x0[6])∧(x2[7] →^* x2[6])∧(x4[7] + -1 →^* x4[6]))

(6) -> (7), if ((x4[6] > 0 →^* TRUE)∧(x0[6] →^* x0[7])∧(x2[6] →^* x2[7])∧(x4[6] →^* x4[7]))

The set Q is empty.

(22) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(7): COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)
(6): 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])

(7) -> (6), if ((x0[7] →^* x0[6])∧(x2[7] →^* x2[6])∧(x4[7] + -1 →^* x4[6]))

(6) -> (7), if ((x4[6] > 0 →^* TRUE)∧(x0[6] →^* x0[7])∧(x2[6] →^* x2[7])∧(x4[6] →^* x4[7]))

The set Q is empty.

(24) 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 COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) the following chains were created:

We consider the chain COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:
(1)    (COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥NonInfC∧COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))∧(U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[2 + (-1)bso_8] ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[2 + (-1)bso_8] ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[2 + (-1)bso_8] ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧[2 + (-1)bso_8] ≥ 0)

For Pair 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6]) the following chains were created:

We consider the chain 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6]), COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:
(6)    (>(x4[6], 0)=TRUE∧x0[6]=x0[7]∧x2[6]=x2[7]∧x4[6]=x4[7] ⇒ 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

We simplified constraint (6) using rule (IV) which results in the following new constraint:
(7)    (>(x4[6], 0)=TRUE ⇒ 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (x4[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x4[6] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (x4[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x4[6] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (x4[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x4[6] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (x4[6] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x4[6] ≥ 0∧[(-1)bso_10] ≥ 0)

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

COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))
- ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧[2 + (-1)bso_8] ≥ 0)
3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])
- (x4[6] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x4[6] ≥ 0∧[(-1)bso_10] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_3643_0_CREATEMETALIST_LE1(x₁, x₂, x₃, x₄)) = [-1] + [2]x₄
POL(3643_0_CREATEMETALIST_LE(x₁, x₂, x₃)) = [-1] + [2]x₃
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0

The following pairs are in P_>:

COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))

The following pairs are in P_bound:

3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])

The following pairs are in P_≥:

3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])

There are no usable rules.

(25) Complex Obligation (AND)

(26) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(6): 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])

The set Q is empty.

(27) IDependencyGraphProof (EQUIVALENT transformation)

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

(28) TRUE

(29) 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

(30) IDependencyGraphProof (EQUIVALENT transformation)

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

(31) TRUE

(32) 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): 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]) → COND_3134_1_CREATEMETALIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])
(1): COND_3134_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])
(2): 3166_1_CREATEMETALIST_INVOKEMETHOD(3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]) → COND_3166_1_CREATEMETALIST_INVOKEMETHOD(x2[2] > 0, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])
(3): COND_3166_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 3643_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])
(5): COND_3643_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0) → 3134_1_CREATEMETALIST_INVOKEMETHOD(3134_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], x2[5] + 1)
(6): 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])
(7): COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)

(1) -> (2), if ((3166_0_random_IntArithmetic(x6[1], x7[1]) →^* 3166_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1] →^* x4[2])∧(x5[1] →^* x6[2]))

(3) -> (6), if ((x4[3] →^* x0[6])∧(x6[3] →^* x2[6])∧(x1[3] →^* x4[6]))

(7) -> (6), if ((x0[7] →^* x0[6])∧(x2[7] →^* x2[6])∧(x4[7] + -1 →^* x4[6]))

(6) -> (7), if ((x4[6] > 0 →^* TRUE)∧(x0[6] →^* x0[7])∧(x2[6] →^* x2[7])∧(x4[6] →^* x4[7]))

The set Q is empty.

(33) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(34) 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

(7) -> (6), if ((x0[7] →^* x0[6])∧(x2[7] →^* x2[6])∧(x4[7] + -1 →^* x4[6]))

(6) -> (7), if ((x4[6] > 0 →^* TRUE)∧(x0[6] →^* x0[7])∧(x2[6] →^* x2[7])∧(x4[6] →^* x4[7]))

The set Q is empty.

(35) 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 COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) the following chains were created:

We consider the chain COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:
(1)    (COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥NonInfC∧COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))∧(U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

For Pair 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6]) the following chains were created:

We consider the chain 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6]), COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:
(6)    (>(x4[6], 0)=TRUE∧x0[6]=x0[7]∧x2[6]=x2[7]∧x4[6]=x4[7] ⇒ 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

We simplified constraint (6) using rule (IV) which results in the following new constraint:
(7)    (>(x4[6], 0)=TRUE ⇒ 3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (x4[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x4[6] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (x4[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x4[6] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (x4[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x4[6] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (x4[6] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x4[6] ≥ 0∧[(-1)bso_9] ≥ 0)

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

COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))
- ((U^Increasing(3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)
3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])
- (x4[6] ≥ 0 ⇒ (U^Increasing(COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x4[6] ≥ 0∧[(-1)bso_9] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_3643_0_CREATEMETALIST_LE1(x₁, x₂, x₃, x₄)) = [1] + x₄
POL(3643_0_CREATEMETALIST_LE(x₁, x₂, x₃)) = [1] + x₃
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0

The following pairs are in P_>:

COND_3643_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 3643_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))

The following pairs are in P_bound:

3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])

The following pairs are in P_≥:

3643_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_3643_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])

There are no usable rules.

(36) Complex Obligation (AND)

(37) 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

(38) IDependencyGraphProof (EQUIVALENT transformation)

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

(39) TRUE

(40) 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

(41) 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) IDPtoQDPProof (SOUND transformation)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) QReductionProof (EQUIVALENT transformation)

(11) Obligation:

(12) UsableRulesReductionPairsProof (EQUIVALENT transformation)

(13) Obligation:

(14) MRRProof (EQUIVALENT transformation)

(15) Obligation:

(16) PisEmptyProof (EQUIVALENT transformation)

(17) YES

(18) Obligation:

(19) IDPNonInfProof (SOUND transformation)

(20) Complex Obligation (AND)

(21) Obligation:

(22) IDependencyGraphProof (EQUIVALENT transformation)

(23) Obligation:

(24) IDPNonInfProof (SOUND transformation)

(25) Complex Obligation (AND)

(26) Obligation:

(27) IDependencyGraphProof (EQUIVALENT transformation)

(28) TRUE

(29) Obligation:

(30) IDependencyGraphProof (EQUIVALENT transformation)

(31) TRUE

(32) Obligation:

(33) IDependencyGraphProof (EQUIVALENT transformation)

(34) Obligation:

(35) IDPNonInfProof (SOUND transformation)

(36) Complex Obligation (AND)

(37) Obligation:

(38) IDependencyGraphProof (EQUIVALENT transformation)

(39) TRUE

(40) Obligation:

(41) IDependencyGraphProof (EQUIVALENT transformation)

(42) TRUE