Termination proof of /tmp/tmppMh5Yw/IntListSumRec.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 MRRProof (⇔)

        ↳13 QDP

        ↳14 PisEmptyProof (⇔)

        ↳15 YES

    ↳16 IDP

        ↳17 IDPNonInfProof (⇒)

        ↳18 AND

            ↳19 IDP

                ↳20 IDependencyGraphProof (⇔)

                ↳21 TRUE

            ↳22 IDP

                ↳23 IDependencyGraphProof (⇔)

                ↳24 TRUE

(0) Obligation:

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

package IntListSumRec;

public class IntListSumRec {
	public static void main(String[] args) {
		Random.args = args;
		List l = List.createList(Random.random());

		int sum = l.sumList();
	}
}

class List {
	int value;
	List next;

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

	public int sumList() {
		if (this.next == null) {
			return this.value;
		} else {
			int firstTwo = this.value + this.next.value;
			List nextNext = this.next.next;
			List shorter = new List(firstTwo, nextNext);
			return shorter.sumList();
		}
	}

	public static List createList(int l) {
		if (l < 0) {
			return null;
		} else {
			return new List(Random.random(), createList(l - 1));
		}
	}
}

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:
IntListSumRec.IntListSumRec.main([Ljava/lang/String;)V: Graph of 108 nodes with 0 SCCs.

IntListSumRec.List.createList(I)LIntListSumRec/List;: Graph of 117 nodes with 0 SCCs.

IntListSumRec.List.sumList()I: Graph of 48 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 36 rules for P and 11 rules for R.

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

Filtered ground terms:

IntListSumRec.List(x1, x2, x3) → IntListSumRec.List(x2, x3)
1150_0_sumList_FieldAccess(x1, x2, x3) → 1150_0_sumList_FieldAccess(x2, x3)
1268_0_sumList_Return(x1, x2) → 1268_0_sumList_Return(x2)
1215_0_sumList_Return(x1, x2) → 1215_0_sumList_Return(x2)
1162_0_sumList_Return(x1, x2, x3) → 1162_0_sumList_Return(x2, x3)

Filtered duplicate args:

1150_0_sumList_FieldAccess(x1, x2) → 1150_0_sumList_FieldAccess(x2)

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

Log for SCC 1:

Generated 29 rules for P and 85 rules for R.

Combined rules. Obtained 2 rules for P and 6 rules for R.

Filtered ground terms:

786_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → 786_1_createList_InvokeMethod(x1, x4, x5)
IntListSumRec.List(x1) → IntListSumRec.List
704_0_random_ArrayAccess(x1, x2, x3) → 704_0_random_ArrayAccess(x2, x3)
704_1_createList_InvokeMethod(x1, x2, x3, x4) → 704_1_createList_InvokeMethod(x1, x2)
Cond_717_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_717_1_createList_InvokeMethod(x1, x2, x3)
717_0_random_IntArithmetic(x1, x2, x3, x4) → 717_0_random_IntArithmetic(x2, x3)
717_1_createList_InvokeMethod(x1, x2, x3, x4) → 717_1_createList_InvokeMethod(x1, x2)
Cond_704_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_704_1_createList_InvokeMethod(x1, x2, x3)
1090_0_createList_Return(x1, x2) → 1090_0_createList_Return
939_0_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 939_0_createList_InvokeMethod(x2, x5, x6)
java.lang.NullPointerException(x1) → java.lang.NullPointerException
900_0_createList_InvokeMethod(x1, x2, x3, x4, x5) → 900_0_createList_InvokeMethod(x3)
896_0_createList_Return(x1, x2) → 896_0_createList_Return
java.lang.ArrayIndexOutOfBoundsException(x1) → java.lang.ArrayIndexOutOfBoundsException
java.lang.IndexOutOfBoundsException(x1) → java.lang.IndexOutOfBoundsException
838_0_createList_InvokeMethod(x1, x2, x3, x4, x5) → 838_0_createList_InvokeMethod(x3)
692_0_createList_Return(x1, x2, x3) → 692_0_createList_Return

Filtered unneeded arguments:

786_1_createList_InvokeMethod(x1, x2, x3) → 786_1_createList_InvokeMethod(x1, x3)

Filtered all non-integer terms:

717_0_random_IntArithmetic(x1, x2) → 717_0_random_IntArithmetic(x2)
939_0_createList_InvokeMethod(x1, x2, x3) → 939_0_createList_InvokeMethod(x2, x3)

Combined rules. Obtained 2 rules for P and 6 rules for R.

Finished conversion. Obtained 2 rules for P and 6 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:

Integer

The ITRS R consists of the following rules:
1208_1_sumList_InvokeMethod(1162_0_sumList_Return(java.lang.Object(IntListSumRec.List(NULL, x0)), x0), java.lang.Object(IntListSumRec.List(NULL, x0))) → 1215_0_sumList_Return(x0)
1208_1_sumList_InvokeMethod(1215_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(NULL, x1)), x2))) → 1268_0_sumList_Return(x0)
1208_1_sumList_InvokeMethod(1268_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x1, x2)), x3)), x4))) → 1268_0_sumList_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 1150_0_SUMLIST_FIELDACCESS(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x0[0], x1[0])), x2[0]))) → 1150_0_SUMLIST_FIELDACCESS(java.lang.Object(IntListSumRec.List(x0[0], x2[0] + x1[0])))

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

The set Q consists of the following terms:
1208_1_sumList_InvokeMethod(1162_0_sumList_Return(java.lang.Object(IntListSumRec.List(NULL, x0)), x0), java.lang.Object(IntListSumRec.List(NULL, x0)))
1208_1_sumList_InvokeMethod(1215_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(NULL, x1)), x2)))
1208_1_sumList_InvokeMethod(1268_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x1, x2)), x3)), x4)))

(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_SUMLIST_FIELDACCESS(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x0[0], x1[0])), x2[0]))) → 1150_0_SUMLIST_FIELDACCESS(java.lang.Object(IntListSumRec.List(x0[0], plus_int(x2[0], x1[0]))))

The TRS R consists of the following rules:

1208_1_sumList_InvokeMethod(1162_0_sumList_Return(java.lang.Object(IntListSumRec.List(NULL, x0)), x0), java.lang.Object(IntListSumRec.List(NULL, x0))) → 1215_0_sumList_Return(x0)
1208_1_sumList_InvokeMethod(1215_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(NULL, x1)), x2))) → 1268_0_sumList_Return(x0)
1208_1_sumList_InvokeMethod(1268_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x1, x2)), x3)), x4))) → 1268_0_sumList_Return(x0)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

1208_1_sumList_InvokeMethod(1162_0_sumList_Return(java.lang.Object(IntListSumRec.List(NULL, x0)), x0), java.lang.Object(IntListSumRec.List(NULL, x0)))
1208_1_sumList_InvokeMethod(1215_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(NULL, x1)), x2)))
1208_1_sumList_InvokeMethod(1268_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x1, x2)), x3)), x4)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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_SUMLIST_FIELDACCESS(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x0[0], x1[0])), x2[0]))) → 1150_0_SUMLIST_FIELDACCESS(java.lang.Object(IntListSumRec.List(x0[0], plus_int(x2[0], x1[0]))))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

1208_1_sumList_InvokeMethod(1162_0_sumList_Return(java.lang.Object(IntListSumRec.List(NULL, x0)), x0), java.lang.Object(IntListSumRec.List(NULL, x0)))
1208_1_sumList_InvokeMethod(1215_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(NULL, x1)), x2)))
1208_1_sumList_InvokeMethod(1268_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x1, x2)), x3)), x4)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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].

1208_1_sumList_InvokeMethod(1162_0_sumList_Return(java.lang.Object(IntListSumRec.List(NULL, x0)), x0), java.lang.Object(IntListSumRec.List(NULL, x0)))
1208_1_sumList_InvokeMethod(1215_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(NULL, x1)), x2)))
1208_1_sumList_InvokeMethod(1268_0_sumList_Return(x0), java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x1, x2)), x3)), x4)))

(11) Obligation:

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

1150_0_SUMLIST_FIELDACCESS(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x0[0], x1[0])), x2[0]))) → 1150_0_SUMLIST_FIELDACCESS(java.lang.Object(IntListSumRec.List(x0[0], plus_int(x2[0], x1[0]))))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

(12) 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_SUMLIST_FIELDACCESS(java.lang.Object(IntListSumRec.List(java.lang.Object(IntListSumRec.List(x0[0], x1[0])), x2[0]))) → 1150_0_SUMLIST_FIELDACCESS(java.lang.Object(IntListSumRec.List(x0[0], plus_int(x2[0], x1[0]))))

Strictly oriented rules of the TRS R:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(1150_0_SUMLIST_FIELDACCESS(x₁)) = x₁
POL(IntListSumRec.List(x₁, x₂)) = x₁ + x₂
POL(java.lang.Object(x₁)) = 2 + x₁
POL(minus_nat(x₁, x₂)) = x₁ + x₂
POL(neg(x₁)) = x₁
POL(plus_int(x₁, x₂)) = 1 + x₁ + x₂
POL(plus_nat(x₁, x₂)) = x₁ + x₂
POL(pos(x₁)) = 1 + x₁
POL(s(x₁)) = 1 + x₁

(13) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

(14) PisEmptyProof (EQUIVALENT transformation)

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

(15) YES

(16) Obligation:

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

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

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
786_1_createList_InvokeMethod(692_0_createList_Return, -1) → 896_0_createList_Return
786_1_createList_InvokeMethod(838_0_createList_InvokeMethod(x0), x0) → 939_0_createList_InvokeMethod(x1, x0)
786_1_createList_InvokeMethod(896_0_createList_Return, x1) → 1090_0_createList_Return
786_1_createList_InvokeMethod(900_0_createList_InvokeMethod(x0), x0) → 939_0_createList_InvokeMethod(x1, x0)
786_1_createList_InvokeMethod(939_0_createList_InvokeMethod(x1, x2), x4) → 939_0_createList_InvokeMethod(x3, x4)
786_1_createList_InvokeMethod(1090_0_createList_Return, x1) → 1090_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_704_1_CREATELIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_704_1_CREATELIST_INVOKEMETHOD(TRUE, 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1])
(2): 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2[2]), x4[2]) → COND_717_1_CREATELIST_INVOKEMETHOD(x4[2] >= 0 && x2[2] > 0 && 0 <= x4[2] - 1, 717_0_random_IntArithmetic(x2[2]), x4[2])
(3): COND_717_1_CREATELIST_INVOKEMETHOD(TRUE, 717_0_random_IntArithmetic(x2[3]), x4[3]) → 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), x4[3] - 1)

(0) -> (1), if ((x2[0] >= 1 && x2[0] < x0[0] →^* TRUE)∧(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →^* 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0] →^* x3[1]))

(1) -> (2), if ((717_0_random_IntArithmetic(x5[1]) →^* 717_0_random_IntArithmetic(x2[2]))∧(x3[1] →^* x4[2]))

(2) -> (3), if ((x4[2] >= 0 && x2[2] > 0 && 0 <= x4[2] - 1 →^* TRUE)∧(717_0_random_IntArithmetic(x2[2]) →^* 717_0_random_IntArithmetic(x2[3]))∧(x4[2] →^* x4[3]))

(3) -> (0), if ((704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]) →^* 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] - 1 →^* x3[0]))

The set Q consists of the following terms:
786_1_createList_InvokeMethod(692_0_createList_Return, -1)
786_1_createList_InvokeMethod(838_0_createList_InvokeMethod(x0), x0)
786_1_createList_InvokeMethod(896_0_createList_Return, x0)
786_1_createList_InvokeMethod(900_0_createList_InvokeMethod(x0), x0)
786_1_createList_InvokeMethod(939_0_createList_InvokeMethod(x0, x1), x2)
786_1_createList_InvokeMethod(1090_0_createList_Return, x0)

(17) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) the following chains were created:

We consider the chain 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]), COND_704_1_CREATELIST_INVOKEMETHOD(TRUE, 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1]) which results in the following constraint:
(1)    (&&(>=(x2[0], 1), <(x2[0], x0[0]))=TRUE∧704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])=704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1])∧x3[0]=x3[1] ⇒ 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(U^Increasing(COND_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_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 ⇒ 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(U^Increasing(COND_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_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_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)Bound*bni_28] + [bni_28]x3[0] ≥ 0∧[(-1)bso_29] ≥ 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_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)Bound*bni_28] + [bni_28]x3[0] ≥ 0∧[(-1)bso_29] ≥ 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_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)Bound*bni_28] + [bni_28]x3[0] ≥ 0∧[(-1)bso_29] ≥ 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_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[bni_28] = 0∧0 = 0∧[(-1)Bound*bni_28] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 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_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[bni_28] = 0∧0 = 0∧[(-1)Bound*bni_28] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 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_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[bni_28] = 0∧0 = 0∧[(-1)Bound*bni_28] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

For Pair COND_704_1_CREATELIST_INVOKEMETHOD(TRUE, 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5), x3) the following chains were created:

We consider the chain COND_704_1_CREATELIST_INVOKEMETHOD(TRUE, 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1]) which results in the following constraint:
(9)    (COND_704_1_CREATELIST_INVOKEMETHOD(TRUE, 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥NonInfC∧COND_704_1_CREATELIST_INVOKEMETHOD(TRUE, 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1])∧(U^Increasing(717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    ((U^Increasing(717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1])), ≥)∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    ((U^Increasing(717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1])), ≥)∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    ((U^Increasing(717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1])), ≥)∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    ((U^Increasing(717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_31] ≥ 0)

For Pair 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2), x4) → COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4, 0), >(x2, 0)), <=(0, -(x4, 1))), 717_0_random_IntArithmetic(x2), x4) the following chains were created:

We consider the chain 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2[2]), x4[2]) → COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2]), COND_717_1_CREATELIST_INVOKEMETHOD(TRUE, 717_0_random_IntArithmetic(x2[3]), x4[3]) → 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1)) which results in the following constraint:
(14)    (&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1)))=TRUE∧717_0_random_IntArithmetic(x2[2])=717_0_random_IntArithmetic(x2[3])∧x4[2]=x4[3] ⇒ 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2[2]), x4[2])≥NonInfC∧717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2[2]), x4[2])≥COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])∧(U^Increasing(COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(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 ⇒ 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2[2]), x4[2])≥NonInfC∧717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2[2]), x4[2])≥COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])∧(U^Increasing(COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    (x4[2] + [-1] ≥ 0∧x4[2] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x4[2] ≥ 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    (x4[2] + [-1] ≥ 0∧x4[2] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x4[2] ≥ 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    (x4[2] + [-1] ≥ 0∧x4[2] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x4[2] ≥ 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    (x4[2] ≥ 0∧[1] + x4[2] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x4[2] ≥ 0∧[(-1)bso_33] ≥ 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] ≥ 0 ⇒ (U^Increasing(COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x4[2] ≥ 0∧[(-1)bso_33] ≥ 0)

For Pair COND_717_1_CREATELIST_INVOKEMETHOD(TRUE, 717_0_random_IntArithmetic(x2), x4) → 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5, x6)), x7), -(x4, 1)) the following chains were created:

We consider the chain COND_717_1_CREATELIST_INVOKEMETHOD(TRUE, 717_0_random_IntArithmetic(x2[3]), x4[3]) → 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1)) which results in the following constraint:
(21)    (COND_717_1_CREATELIST_INVOKEMETHOD(TRUE, 717_0_random_IntArithmetic(x2[3]), x4[3])≥NonInfC∧COND_717_1_CREATELIST_INVOKEMETHOD(TRUE, 717_0_random_IntArithmetic(x2[3]), x4[3])≥704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))∧(U^Increasing(704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))), ≥))

We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22)    ((U^Increasing(704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))), ≥)∧[(-1)bso_35] ≥ 0)

We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23)    ((U^Increasing(704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))), ≥)∧[(-1)bso_35] ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24)    ((U^Increasing(704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))), ≥)∧[(-1)bso_35] ≥ 0)

We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25)    ((U^Increasing(704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)

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

704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
- (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[bni_28] = 0∧0 = 0∧[(-1)Bound*bni_28] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)
COND_704_1_CREATELIST_INVOKEMETHOD(TRUE, 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5), x3)
- ((U^Increasing(717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_31] ≥ 0)
717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2), x4) → COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4, 0), >(x2, 0)), <=(0, -(x4, 1))), 717_0_random_IntArithmetic(x2), x4)
- (x4[2] ≥ 0∧[1] + x4[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x4[2] ≥ 0∧[(-1)bso_33] ≥ 0)
COND_717_1_CREATELIST_INVOKEMETHOD(TRUE, 717_0_random_IntArithmetic(x2), x4) → 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5, x6)), x7), -(x4, 1))
- ((U^Increasing(704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 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(786_1_createList_InvokeMethod(x₁, x₂)) = [-1]
POL(692_0_createList_Return) = [-1]
POL(-1) = [-1]
POL(896_0_createList_Return) = [-1]
POL(838_0_createList_InvokeMethod(x₁)) = [-1]
POL(939_0_createList_InvokeMethod(x₁, x₂)) = [-1]
POL(1090_0_createList_Return) = [-1]
POL(900_0_createList_InvokeMethod(x₁)) = [-1]
POL(704_1_CREATELIST_INVOKEMETHOD(x₁, x₂)) = [-1] + x₂ + [-1]x₁
POL(704_0_random_ArrayAccess(x₁, x₂)) = [1] + [-1]x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁, x₂)) = [2]
POL(COND_704_1_CREATELIST_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<(x₁, x₂)) = [-1]
POL(717_1_CREATELIST_INVOKEMETHOD(x₁, x₂)) = [-1] + x₂
POL(717_0_random_IntArithmetic(x₁)) = x₁
POL(COND_717_1_CREATELIST_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + x₃
POL(0) = 0
POL(>(x₁, x₂)) = [-1]
POL(<=(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂

The following pairs are in P_>:

COND_704_1_CREATELIST_INVOKEMETHOD(TRUE, 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1])

The following pairs are in P_bound:

717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2[2]), x4[2]) → COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])

The following pairs are in P_≥:

704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_704_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2[2]), x4[2]) → COND_717_1_CREATELIST_INVOKEMETHOD(&&(&&(>=(x4[2], 0), >(x2[2], 0)), <=(0, -(x4[2], 1))), 717_0_random_IntArithmetic(x2[2]), x4[2])
COND_717_1_CREATELIST_INVOKEMETHOD(TRUE, 717_0_random_IntArithmetic(x2[3]), x4[3]) → 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))

There are no usable rules.

(18) Complex Obligation (AND)

(19) 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

The ITRS R consists of the following rules:
786_1_createList_InvokeMethod(692_0_createList_Return, -1) → 896_0_createList_Return
786_1_createList_InvokeMethod(838_0_createList_InvokeMethod(x0), x0) → 939_0_createList_InvokeMethod(x1, x0)
786_1_createList_InvokeMethod(896_0_createList_Return, x1) → 1090_0_createList_Return
786_1_createList_InvokeMethod(900_0_createList_InvokeMethod(x0), x0) → 939_0_createList_InvokeMethod(x1, x0)
786_1_createList_InvokeMethod(939_0_createList_InvokeMethod(x1, x2), x4) → 939_0_createList_InvokeMethod(x3, x4)
786_1_createList_InvokeMethod(1090_0_createList_Return, x1) → 1090_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_704_1_CREATELIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(2): 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x2[2]), x4[2]) → COND_717_1_CREATELIST_INVOKEMETHOD(x4[2] >= 0 && x2[2] > 0 && 0 <= x4[2] - 1, 717_0_random_IntArithmetic(x2[2]), x4[2])
(3): COND_717_1_CREATELIST_INVOKEMETHOD(TRUE, 717_0_random_IntArithmetic(x2[3]), x4[3]) → 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), x4[3] - 1)

(3) -> (0), if ((704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]) →^* 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] - 1 →^* x3[0]))

(2) -> (3), if ((x4[2] >= 0 && x2[2] > 0 && 0 <= x4[2] - 1 →^* TRUE)∧(717_0_random_IntArithmetic(x2[2]) →^* 717_0_random_IntArithmetic(x2[3]))∧(x4[2] →^* x4[3]))

(20) IDependencyGraphProof (EQUIVALENT transformation)

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

(21) TRUE

(22) Obligation:

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

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

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
786_1_createList_InvokeMethod(692_0_createList_Return, -1) → 896_0_createList_Return
786_1_createList_InvokeMethod(838_0_createList_InvokeMethod(x0), x0) → 939_0_createList_InvokeMethod(x1, x0)
786_1_createList_InvokeMethod(896_0_createList_Return, x1) → 1090_0_createList_Return
786_1_createList_InvokeMethod(900_0_createList_InvokeMethod(x0), x0) → 939_0_createList_InvokeMethod(x1, x0)
786_1_createList_InvokeMethod(939_0_createList_InvokeMethod(x1, x2), x4) → 939_0_createList_InvokeMethod(x3, x4)
786_1_createList_InvokeMethod(1090_0_createList_Return, x1) → 1090_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_704_1_CREATELIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_704_1_CREATELIST_INVOKEMETHOD(TRUE, 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 717_1_CREATELIST_INVOKEMETHOD(717_0_random_IntArithmetic(x5[1]), x3[1])
(3): COND_717_1_CREATELIST_INVOKEMETHOD(TRUE, 717_0_random_IntArithmetic(x2[3]), x4[3]) → 704_1_CREATELIST_INVOKEMETHOD(704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), x4[3] - 1)

(3) -> (0), if ((704_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]) →^* 704_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] - 1 →^* x3[0]))

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(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) MRRProof (EQUIVALENT transformation)

(13) Obligation:

(14) PisEmptyProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) IDPNonInfProof (SOUND transformation)

(18) Complex Obligation (AND)

(19) Obligation:

(20) IDependencyGraphProof (EQUIVALENT transformation)

(21) TRUE

(22) Obligation:

(23) IDependencyGraphProof (EQUIVALENT transformation)

(24) TRUE