Termination proof of /tmp/tmpYWE9aM/Shuffle.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPtoQDPProof (⇒)

        ↳7 QDP

        ↳8 UsableRulesProof (⇔)

        ↳9 QDP

        ↳10 QReductionProof (⇔)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 IDP

        ↳15 IDPtoQDPProof (⇒)

        ↳16 QDP

        ↳17 DependencyGraphProof (⇔)

        ↳18 QDP

        ↳19 UsableRulesProof (⇔)

        ↳20 QDP

        ↳21 QReductionProof (⇔)

        ↳22 QDP

        ↳23 QDPOrderProof (⇔)

        ↳24 QDP

        ↳25 DependencyGraphProof (⇔)

        ↳26 TRUE

    ↳27 IDP

        ↳28 IDPNonInfProof (⇒)

        ↳29 AND

            ↳30 IDP

                ↳31 IDependencyGraphProof (⇔)

                ↳32 TRUE

            ↳33 IDP

                ↳34 IDependencyGraphProof (⇔)

                ↳35 TRUE

(0) Obligation:

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

package Shuffle;

public class List {

  int value;
  List next;

  public List() {}

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

  public static List copy(List x) {
    if (x == null) return null;
    else return new List(x.value,copy(x.next));
  }
}


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

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


package Shuffle; 
public class Shuffle {

	// from Walther (1994)
	public static List reverse(final List l) {
		if (l == null || l.next==null) return l;
		final List nextItem = l.next;
		final List reverseRest = reverse(nextItem);
		l.next = null;
		nextItem.next = l;
		return reverseRest;
	}

	public static List shuffle(List xs) {
		if (xs == null) {
			return null;
		} else {
			List next = xs.next;
//			xs.next = null;
			return new List(xs.value, shuffle(reverse(next)));
		}
	} 

  public static void main(String[] args) {
	Random.args = args;

	List xs = createList(Random.random());
	List ys = shuffle(xs);
  }

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

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Shuffle.Shuffle.main([Ljava/lang/String;)V: Graph of 121 nodes with 0 SCCs.

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

Shuffle.Shuffle.shuffle(LShuffle/List;)LShuffle/List;: Graph of 79 nodes with 0 SCCs.

Shuffle.Shuffle.reverse(LShuffle/List;)LShuffle/List;: Graph of 51 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 16 rules for P and 34 rules for R.

Combined rules. Obtained 1 rules for P and 6 rules for R.

Filtered ground terms:

Shuffle.List(x1, x2) → Shuffle.List(x2)
1248_0_reverse_NULL(x1, x2, x3) → 1248_0_reverse_NULL(x2, x3)
1820_0_reverse_Return(x1, x2) → 1820_0_reverse_Return(x2)
1751_0_reverse_FieldAccess(x1, x2, x3, x4) → 1751_0_reverse_FieldAccess(x2)
1387_0_reverse_Return(x1, x2) → 1387_0_reverse_Return
1321_0_reverse_Return(x1, x2, x3) → 1321_0_reverse_Return
1270_0_reverse_Return(x1, x2, x3) → 1270_0_reverse_Return

Filtered duplicate args:

1346_1_reverse_InvokeMethod(x1, x2, x3, x4) → 1346_1_reverse_InvokeMethod(x1, x2, x4)
1248_0_reverse_NULL(x1, x2) → 1248_0_reverse_NULL(x2)

Finished conversion. Obtained 1 rules for P and 6 rules for R. System has no predefined symbols.

Log for SCC 1:

Generated 31 rules for P and 103 rules for R.

Combined rules. Obtained 5 rules for P and 18 rules for R.

Filtered ground terms:

1599_1_shuffle_InvokeMethod(x1, x2, x3, x4, x5) → 1599_1_shuffle_InvokeMethod(x1, x4, x5)
Shuffle.List(x1, x2, x3) → Shuffle.List(x2, x3)
1110_0_shuffle_NONNULL(x1, x2, x3) → 1110_0_shuffle_NONNULL(x2, x3)
1235_1_shuffle_InvokeMethod(x1, x2, x3, x4, x5) → 1235_1_shuffle_InvokeMethod(x1, x4, x5)
1820_0_reverse_Return(x1, x2) → 1820_0_reverse_Return(x2)
1387_0_reverse_Return(x1, x2) → 1387_0_reverse_Return(x2)
1338_1_shuffle_InvokeMethod(x1, x2, x3, x4, x5) → 1338_1_shuffle_InvokeMethod(x1, x4, x5)
1321_0_reverse_Return(x1, x2, x3) → 1321_0_reverse_Return(x2, x3)
1270_0_reverse_Return(x1, x2, x3) → 1270_0_reverse_Return
1235_0_reverse_Load(x1, x2) → 1235_0_reverse_Load(x2)
1353_0_reverse_Return(x1, x2, x3) → 1353_0_reverse_Return
1346_0_reverse_Load(x1, x2) → 1346_0_reverse_Load(x2)
1751_0_reverse_FieldAccess(x1, x2, x3, x4) → 1751_0_reverse_FieldAccess(x2, x3, x4)
1248_0_reverse_NULL(x1, x2, x3) → 1248_0_reverse_NULL(x2, x3)
2159_0_shuffle_Return(x1, x2) → 2159_0_shuffle_Return(x2)
2148_0_shuffle_Return(x1, x2) → 2148_0_shuffle_Return(x2)
1398_0_shuffle_Return(x1, x2) → 1398_0_shuffle_Return(x2)
1134_0_shuffle_Return(x1, x2, x3) → 1134_0_shuffle_Return

Filtered duplicate args:

1110_0_shuffle_NONNULL(x1, x2) → 1110_0_shuffle_NONNULL(x2)
1321_0_reverse_Return(x1, x2) → 1321_0_reverse_Return(x2)
1346_1_reverse_InvokeMethod(x1, x2, x3, x4) → 1346_1_reverse_InvokeMethod(x1, x2, x4)
1248_0_reverse_NULL(x1, x2) → 1248_0_reverse_NULL(x2)

Filtered unneeded arguments:

1235_1_shuffle_InvokeMethod(x1, x2, x3) → 1235_1_shuffle_InvokeMethod(x1, x3)
1338_1_shuffle_InvokeMethod(x1, x2, x3) → 1338_1_shuffle_InvokeMethod(x1, x3)
1599_1_shuffle_InvokeMethod(x1, x2, x3) → 1599_1_shuffle_InvokeMethod(x1, x3)

Filtered all free variables:

1398_0_shuffle_Return(x1) → 1398_0_shuffle_Return
2148_0_shuffle_Return(x1) → 2148_0_shuffle_Return
2159_0_shuffle_Return(x1) → 2159_0_shuffle_Return
1751_0_reverse_FieldAccess(x1, x2, x3) → 1751_0_reverse_FieldAccess(x1, x3)

Finished conversion. Obtained 5 rules for P and 18 rules for R. System has no predefined symbols.

Log for SCC 2:

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:

851_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → 851_1_createList_InvokeMethod(x1, x4, x5)
Shuffle.List(x1) → Shuffle.List
745_0_random_ArrayAccess(x1, x2, x3) → 745_0_random_ArrayAccess(x2, x3)
745_1_createList_InvokeMethod(x1, x2, x3, x4) → 745_1_createList_InvokeMethod(x1, x2)
Cond_760_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_760_1_createList_InvokeMethod(x1, x2, x3)
760_0_random_IntArithmetic(x1, x2, x3, x4) → 760_0_random_IntArithmetic(x2, x3)
760_1_createList_InvokeMethod(x1, x2, x3, x4) → 760_1_createList_InvokeMethod(x1, x2)
Cond_745_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_745_1_createList_InvokeMethod(x1, x2, x3)
1292_0_createList_Return(x1, x2) → 1292_0_createList_Return
1051_0_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 1051_0_createList_InvokeMethod(x2, x5, x6)
java.lang.NullPointerException(x1) → java.lang.NullPointerException
1005_0_createList_InvokeMethod(x1, x2, x3, x4, x5) → 1005_0_createList_InvokeMethod(x3)
999_0_createList_Return(x1, x2) → 999_0_createList_Return
java.lang.ArrayIndexOutOfBoundsException(x1) → java.lang.ArrayIndexOutOfBoundsException
java.lang.IndexOutOfBoundsException(x1) → java.lang.IndexOutOfBoundsException
923_0_createList_InvokeMethod(x1, x2, x3, x4, x5) → 923_0_createList_InvokeMethod(x3)
727_0_createList_Return(x1, x2, x3) → 727_0_createList_Return

Filtered unneeded arguments:

851_1_createList_InvokeMethod(x1, x2, x3) → 851_1_createList_InvokeMethod(x1, x3)

Filtered all non-integer terms:

760_0_random_IntArithmetic(x1, x2) → 760_0_random_IntArithmetic(x2)
1051_0_createList_InvokeMethod(x1, x2, x3) → 1051_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:
none

The ITRS R consists of the following rules:
1248_0_reverse_NULL(NULL) → 1270_0_reverse_Return
1346_1_reverse_InvokeMethod(1321_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL)))), java.lang.Object(Shuffle.List(NULL))) → 1387_0_reverse_Return
1346_1_reverse_InvokeMethod(1387_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0))), java.lang.Object(x0)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL)))))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0)))), java.lang.Object(Shuffle.List(java.lang.Object(x1))), java.lang.Object(x1)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0))))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0)))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0))))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL))))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL)))))))

The integer pair graph contains the following rules and edges:
(0): 1248_0_REVERSE_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[0])))) → 1248_0_REVERSE_NULL(java.lang.Object(x0[0]))

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

The set Q consists of the following terms:
1248_0_reverse_NULL(NULL)
1346_1_reverse_InvokeMethod(1321_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL)))), java.lang.Object(Shuffle.List(NULL)))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0))), java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0)))), java.lang.Object(Shuffle.List(java.lang.Object(x1))), java.lang.Object(x1))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(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:

1248_0_REVERSE_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[0])))) → 1248_0_REVERSE_NULL(java.lang.Object(x0[0]))

The TRS R consists of the following rules:

1248_0_reverse_NULL(NULL) → 1270_0_reverse_Return
1346_1_reverse_InvokeMethod(1321_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL)))), java.lang.Object(Shuffle.List(NULL))) → 1387_0_reverse_Return
1346_1_reverse_InvokeMethod(1387_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0))), java.lang.Object(x0)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL)))))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0)))), java.lang.Object(Shuffle.List(java.lang.Object(x1))), java.lang.Object(x1)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0))))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0)))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0))))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL))))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL)))))))

The set Q consists of the following terms:

1248_0_reverse_NULL(NULL)
1346_1_reverse_InvokeMethod(1321_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL)))), java.lang.Object(Shuffle.List(NULL)))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0))), java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0)))), java.lang.Object(Shuffle.List(java.lang.Object(x1))), java.lang.Object(x1))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(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:

1248_0_REVERSE_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[0])))) → 1248_0_REVERSE_NULL(java.lang.Object(x0[0]))

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

1248_0_reverse_NULL(NULL)
1346_1_reverse_InvokeMethod(1321_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL)))), java.lang.Object(Shuffle.List(NULL)))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0))), java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0)))), java.lang.Object(Shuffle.List(java.lang.Object(x1))), java.lang.Object(x1))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(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].

1248_0_reverse_NULL(NULL)
1346_1_reverse_InvokeMethod(1321_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL)))), java.lang.Object(Shuffle.List(NULL)))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0))), java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0)))), java.lang.Object(Shuffle.List(java.lang.Object(x1))), java.lang.Object(x1))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0))))

(11) Obligation:

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

1248_0_REVERSE_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[0])))) → 1248_0_REVERSE_NULL(java.lang.Object(x0[0]))

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

1248_0_REVERSE_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[0])))) → 1248_0_REVERSE_NULL(java.lang.Object(x0[0]))
The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

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

The following domains are used:
none

The ITRS R consists of the following rules:
1110_0_shuffle_NONNULL(NULL) → 1134_0_shuffle_Return
1235_0_reverse_Load(x0) → 1248_0_reverse_NULL(x0)
1338_1_shuffle_InvokeMethod(1134_0_shuffle_Return, NULL) → 1398_0_shuffle_Return
1338_1_shuffle_InvokeMethod(1398_0_shuffle_Return, java.lang.Object(Shuffle.List(NULL, x0))) → 2148_0_shuffle_Return
1338_1_shuffle_InvokeMethod(2148_0_shuffle_Return, x6) → 2148_0_shuffle_Return
1338_1_shuffle_InvokeMethod(2159_0_shuffle_Return, x7) → 2148_0_shuffle_Return
1599_1_shuffle_InvokeMethod(2148_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x6), x7))) → 2159_0_shuffle_Return
1599_1_shuffle_InvokeMethod(2159_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x7), x8))) → 2159_0_shuffle_Return
1248_0_reverse_NULL(NULL) → 1270_0_reverse_Return
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0))) → 1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))) → 1346_1_reverse_InvokeMethod(1346_0_reverse_Load(java.lang.Object(x0)), java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(x0))
1346_0_reverse_Load(java.lang.Object(x0)) → 1248_0_reverse_NULL(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x2))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x2)), x0)), x1)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0))) → 1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x1)), x0)))
1346_0_reverse_Load(NULL) → 1353_0_reverse_Return

The integer pair graph contains the following rules and edges:
(0): 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(x0[0], x1[0]))) → 1235_1_SHUFFLE_INVOKEMETHOD(1235_0_reverse_Load(x0[0]), x0[0])
(1): 1235_1_SHUFFLE_INVOKEMETHOD(1270_0_reverse_Return, NULL) → 1110_0_SHUFFLE_NONNULL(NULL)
(2): 1235_1_SHUFFLE_INVOKEMETHOD(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0[2]))), java.lang.Object(Shuffle.List(NULL, x0[2]))) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(NULL, x0[2])))
(3): 1235_1_SHUFFLE_INVOKEMETHOD(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3]))), x5[3]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3])))
(4): 1235_1_SHUFFLE_INVOKEMETHOD(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4]))), x5[4]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4])))

(0) -> (1), if ((1235_0_reverse_Load(x0[0]) →^* 1270_0_reverse_Return)∧(x0[0] →^* NULL))

(0) -> (2), if ((1235_0_reverse_Load(x0[0]) →^* 1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0[2]))))∧(x0[0] →^* java.lang.Object(Shuffle.List(NULL, x0[2]))))

(0) -> (3), if ((1235_0_reverse_Load(x0[0]) →^* 1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3]))))∧(x0[0] →^* x5[3]))

(0) -> (4), if ((1235_0_reverse_Load(x0[0]) →^* 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4]))))∧(x0[0] →^* x5[4]))

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

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

(3) -> (0), if ((java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3])) →^* java.lang.Object(Shuffle.List(x0[0], x1[0]))))

(4) -> (0), if ((java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4])) →^* java.lang.Object(Shuffle.List(x0[0], x1[0]))))

The set Q consists of the following terms:
1110_0_shuffle_NONNULL(NULL)
1235_0_reverse_Load(x0)
1338_1_shuffle_InvokeMethod(1134_0_shuffle_Return, NULL)
1338_1_shuffle_InvokeMethod(1398_0_shuffle_Return, java.lang.Object(Shuffle.List(NULL, x0)))
1338_1_shuffle_InvokeMethod(2148_0_shuffle_Return, x0)
1338_1_shuffle_InvokeMethod(2159_0_shuffle_Return, x0)
1599_1_shuffle_InvokeMethod(2148_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1599_1_shuffle_InvokeMethod(2159_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1248_0_reverse_NULL(NULL)
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1346_0_reverse_Load(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x2)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0)))
1346_0_reverse_Load(NULL)

(15) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(16) Obligation:

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

1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(x0[0], x1[0]))) → 1235_1_SHUFFLE_INVOKEMETHOD(1235_0_reverse_Load(x0[0]), x0[0])
1235_1_SHUFFLE_INVOKEMETHOD(1270_0_reverse_Return, NULL) → 1110_0_SHUFFLE_NONNULL(NULL)
1235_1_SHUFFLE_INVOKEMETHOD(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0[2]))), java.lang.Object(Shuffle.List(NULL, x0[2]))) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(NULL, x0[2])))
1235_1_SHUFFLE_INVOKEMETHOD(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3]))), x5[3]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3])))
1235_1_SHUFFLE_INVOKEMETHOD(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4]))), x5[4]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4])))

The TRS R consists of the following rules:

1110_0_shuffle_NONNULL(NULL) → 1134_0_shuffle_Return
1235_0_reverse_Load(x0) → 1248_0_reverse_NULL(x0)
1338_1_shuffle_InvokeMethod(1134_0_shuffle_Return, NULL) → 1398_0_shuffle_Return
1338_1_shuffle_InvokeMethod(1398_0_shuffle_Return, java.lang.Object(Shuffle.List(NULL, x0))) → 2148_0_shuffle_Return
1338_1_shuffle_InvokeMethod(2148_0_shuffle_Return, x6) → 2148_0_shuffle_Return
1338_1_shuffle_InvokeMethod(2159_0_shuffle_Return, x7) → 2148_0_shuffle_Return
1599_1_shuffle_InvokeMethod(2148_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x6), x7))) → 2159_0_shuffle_Return
1599_1_shuffle_InvokeMethod(2159_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x7), x8))) → 2159_0_shuffle_Return
1248_0_reverse_NULL(NULL) → 1270_0_reverse_Return
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0))) → 1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))) → 1346_1_reverse_InvokeMethod(1346_0_reverse_Load(java.lang.Object(x0)), java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(x0))
1346_0_reverse_Load(java.lang.Object(x0)) → 1248_0_reverse_NULL(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x2))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x2)), x0)), x1)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0))) → 1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x1)), x0)))
1346_0_reverse_Load(NULL) → 1353_0_reverse_Return

The set Q consists of the following terms:

1110_0_shuffle_NONNULL(NULL)
1235_0_reverse_Load(x0)
1338_1_shuffle_InvokeMethod(1134_0_shuffle_Return, NULL)
1338_1_shuffle_InvokeMethod(1398_0_shuffle_Return, java.lang.Object(Shuffle.List(NULL, x0)))
1338_1_shuffle_InvokeMethod(2148_0_shuffle_Return, x0)
1338_1_shuffle_InvokeMethod(2159_0_shuffle_Return, x0)
1599_1_shuffle_InvokeMethod(2148_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1599_1_shuffle_InvokeMethod(2159_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1248_0_reverse_NULL(NULL)
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1346_0_reverse_Load(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x2)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0)))
1346_0_reverse_Load(NULL)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

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

(18) Obligation:

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

1235_1_SHUFFLE_INVOKEMETHOD(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0[2]))), java.lang.Object(Shuffle.List(NULL, x0[2]))) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(NULL, x0[2])))
1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(x0[0], x1[0]))) → 1235_1_SHUFFLE_INVOKEMETHOD(1235_0_reverse_Load(x0[0]), x0[0])
1235_1_SHUFFLE_INVOKEMETHOD(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3]))), x5[3]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3])))
1235_1_SHUFFLE_INVOKEMETHOD(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4]))), x5[4]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4])))

The TRS R consists of the following rules:

1110_0_shuffle_NONNULL(NULL) → 1134_0_shuffle_Return
1235_0_reverse_Load(x0) → 1248_0_reverse_NULL(x0)
1338_1_shuffle_InvokeMethod(1134_0_shuffle_Return, NULL) → 1398_0_shuffle_Return
1338_1_shuffle_InvokeMethod(1398_0_shuffle_Return, java.lang.Object(Shuffle.List(NULL, x0))) → 2148_0_shuffle_Return
1338_1_shuffle_InvokeMethod(2148_0_shuffle_Return, x6) → 2148_0_shuffle_Return
1338_1_shuffle_InvokeMethod(2159_0_shuffle_Return, x7) → 2148_0_shuffle_Return
1599_1_shuffle_InvokeMethod(2148_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x6), x7))) → 2159_0_shuffle_Return
1599_1_shuffle_InvokeMethod(2159_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x7), x8))) → 2159_0_shuffle_Return
1248_0_reverse_NULL(NULL) → 1270_0_reverse_Return
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0))) → 1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))) → 1346_1_reverse_InvokeMethod(1346_0_reverse_Load(java.lang.Object(x0)), java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(x0))
1346_0_reverse_Load(java.lang.Object(x0)) → 1248_0_reverse_NULL(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x2))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x2)), x0)), x1)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0))) → 1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x1)), x0)))
1346_0_reverse_Load(NULL) → 1353_0_reverse_Return

The set Q consists of the following terms:

1110_0_shuffle_NONNULL(NULL)
1235_0_reverse_Load(x0)
1338_1_shuffle_InvokeMethod(1134_0_shuffle_Return, NULL)
1338_1_shuffle_InvokeMethod(1398_0_shuffle_Return, java.lang.Object(Shuffle.List(NULL, x0)))
1338_1_shuffle_InvokeMethod(2148_0_shuffle_Return, x0)
1338_1_shuffle_InvokeMethod(2159_0_shuffle_Return, x0)
1599_1_shuffle_InvokeMethod(2148_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1599_1_shuffle_InvokeMethod(2159_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1248_0_reverse_NULL(NULL)
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1346_0_reverse_Load(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x2)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0)))
1346_0_reverse_Load(NULL)

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

(19) 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.

(20) Obligation:

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

1235_1_SHUFFLE_INVOKEMETHOD(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0[2]))), java.lang.Object(Shuffle.List(NULL, x0[2]))) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(NULL, x0[2])))
1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(x0[0], x1[0]))) → 1235_1_SHUFFLE_INVOKEMETHOD(1235_0_reverse_Load(x0[0]), x0[0])
1235_1_SHUFFLE_INVOKEMETHOD(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3]))), x5[3]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3])))
1235_1_SHUFFLE_INVOKEMETHOD(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4]))), x5[4]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4])))

The TRS R consists of the following rules:

1235_0_reverse_Load(x0) → 1248_0_reverse_NULL(x0)
1248_0_reverse_NULL(NULL) → 1270_0_reverse_Return
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0))) → 1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))) → 1346_1_reverse_InvokeMethod(1346_0_reverse_Load(java.lang.Object(x0)), java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(x0))
1346_0_reverse_Load(java.lang.Object(x0)) → 1248_0_reverse_NULL(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0))) → 1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x1)), x0)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x2))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x2)), x0)), x1)))

The set Q consists of the following terms:

1110_0_shuffle_NONNULL(NULL)
1235_0_reverse_Load(x0)
1338_1_shuffle_InvokeMethod(1134_0_shuffle_Return, NULL)
1338_1_shuffle_InvokeMethod(1398_0_shuffle_Return, java.lang.Object(Shuffle.List(NULL, x0)))
1338_1_shuffle_InvokeMethod(2148_0_shuffle_Return, x0)
1338_1_shuffle_InvokeMethod(2159_0_shuffle_Return, x0)
1599_1_shuffle_InvokeMethod(2148_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1599_1_shuffle_InvokeMethod(2159_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1248_0_reverse_NULL(NULL)
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1346_0_reverse_Load(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x2)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0)))
1346_0_reverse_Load(NULL)

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

(21) 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].

1110_0_shuffle_NONNULL(NULL)
1338_1_shuffle_InvokeMethod(1134_0_shuffle_Return, NULL)
1338_1_shuffle_InvokeMethod(1398_0_shuffle_Return, java.lang.Object(Shuffle.List(NULL, x0)))
1338_1_shuffle_InvokeMethod(2148_0_shuffle_Return, x0)
1338_1_shuffle_InvokeMethod(2159_0_shuffle_Return, x0)
1599_1_shuffle_InvokeMethod(2148_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1599_1_shuffle_InvokeMethod(2159_0_shuffle_Return, java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))

(22) Obligation:

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

1235_1_SHUFFLE_INVOKEMETHOD(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0[2]))), java.lang.Object(Shuffle.List(NULL, x0[2]))) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(NULL, x0[2])))
1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(x0[0], x1[0]))) → 1235_1_SHUFFLE_INVOKEMETHOD(1235_0_reverse_Load(x0[0]), x0[0])
1235_1_SHUFFLE_INVOKEMETHOD(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3]))), x5[3]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3])))
1235_1_SHUFFLE_INVOKEMETHOD(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4]))), x5[4]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4])))

The TRS R consists of the following rules:

1235_0_reverse_Load(x0) → 1248_0_reverse_NULL(x0)
1248_0_reverse_NULL(NULL) → 1270_0_reverse_Return
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0))) → 1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))) → 1346_1_reverse_InvokeMethod(1346_0_reverse_Load(java.lang.Object(x0)), java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(x0))
1346_0_reverse_Load(java.lang.Object(x0)) → 1248_0_reverse_NULL(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0))) → 1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x1)), x0)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x2))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x2)), x0)), x1)))

The set Q consists of the following terms:

1235_0_reverse_Load(x0)
1248_0_reverse_NULL(NULL)
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1346_0_reverse_Load(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x2)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0)))
1346_0_reverse_Load(NULL)

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(x0[0], x1[0]))) → 1235_1_SHUFFLE_INVOKEMETHOD(1235_0_reverse_Load(x0[0]), x0[0])

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

POL(1110_0_SHUFFLE_NONNULL(x₁)) = x₁
POL(1235_0_reverse_Load(x₁)) = x₁
POL(1235_1_SHUFFLE_INVOKEMETHOD(x₁, x₂)) = x₁
POL(1248_0_reverse_NULL(x₁)) = x₁
POL(1270_0_reverse_Return) = 0
POL(1321_0_reverse_Return(x₁)) = 1
POL(1346_0_reverse_Load(x₁)) = x₁
POL(1346_1_reverse_InvokeMethod(x₁, x₂, x₃)) = 1 + x₁
POL(1387_0_reverse_Return(x₁)) = x₁
POL(1751_0_reverse_FieldAccess(x₁, x₂)) = 1 + x₁
POL(1820_0_reverse_Return(x₁)) = x₁
POL(NULL) = 0
POL(Shuffle.List(x₁, x₂)) = 1 + x₁
POL(java.lang.Object(x₁)) = x₁

The following usable rules [FROCOS05] were oriented:

1235_0_reverse_Load(x0) → 1248_0_reverse_NULL(x0)
1248_0_reverse_NULL(NULL) → 1270_0_reverse_Return
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0))) → 1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))) → 1346_1_reverse_InvokeMethod(1346_0_reverse_Load(java.lang.Object(x0)), java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0))) → 1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x1)), x0)))
1346_0_reverse_Load(java.lang.Object(x0)) → 1248_0_reverse_NULL(java.lang.Object(x0))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x2))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x2)), x0)), x1)))

(24) Obligation:

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

1235_1_SHUFFLE_INVOKEMETHOD(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0[2]))), java.lang.Object(Shuffle.List(NULL, x0[2]))) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(NULL, x0[2])))
1235_1_SHUFFLE_INVOKEMETHOD(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3]))), x5[3]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0[3])), x1[3])))
1235_1_SHUFFLE_INVOKEMETHOD(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4]))), x5[4]) → 1110_0_SHUFFLE_NONNULL(java.lang.Object(Shuffle.List(java.lang.Object(x0[4]), x1[4])))

The TRS R consists of the following rules:

1235_0_reverse_Load(x0) → 1248_0_reverse_NULL(x0)
1248_0_reverse_NULL(NULL) → 1270_0_reverse_Return
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0))) → 1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))) → 1346_1_reverse_InvokeMethod(1346_0_reverse_Load(java.lang.Object(x0)), java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(x0))
1346_0_reverse_Load(java.lang.Object(x0)) → 1248_0_reverse_NULL(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2)) → 1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0))) → 1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x1)), x0)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x3))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x2))) → 1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x2)), x0)), x1)))

The set Q consists of the following terms:

1235_0_reverse_Load(x0)
1248_0_reverse_NULL(NULL)
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(NULL, x0)))
1248_0_reverse_NULL(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)))
1346_0_reverse_Load(java.lang.Object(x0))
1346_1_reverse_InvokeMethod(1387_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1346_1_reverse_InvokeMethod(1820_0_reverse_Return(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1))), java.lang.Object(Shuffle.List(java.lang.Object(x2), x3)), java.lang.Object(x2))
1751_0_reverse_FieldAccess(java.lang.Object(Shuffle.List(java.lang.Object(x0), x1)), java.lang.Object(Shuffle.List(NULL, x2)))
1346_1_reverse_InvokeMethod(1321_0_reverse_Return(java.lang.Object(Shuffle.List(NULL, x0))), java.lang.Object(Shuffle.List(java.lang.Object(Shuffle.List(NULL, x0)), x1)), java.lang.Object(Shuffle.List(NULL, x0)))
1346_0_reverse_Load(NULL)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE

(27) 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:
851_1_createList_InvokeMethod(727_0_createList_Return, 0) → 999_0_createList_Return
851_1_createList_InvokeMethod(923_0_createList_InvokeMethod(x0), x0) → 1051_0_createList_InvokeMethod(x1, x0)
851_1_createList_InvokeMethod(999_0_createList_Return, x1) → 1292_0_createList_Return
851_1_createList_InvokeMethod(1005_0_createList_InvokeMethod(x0), x0) → 1051_0_createList_InvokeMethod(x1, x0)
851_1_createList_InvokeMethod(1051_0_createList_InvokeMethod(x1, x2), x4) → 1051_0_createList_InvokeMethod(x3, x4)
851_1_createList_InvokeMethod(1292_0_createList_Return, x1) → 1292_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_745_1_CREATELIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_745_1_CREATELIST_INVOKEMETHOD(TRUE, 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1])
(2): 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2[2]), x4[2]) → COND_760_1_CREATELIST_INVOKEMETHOD(x4[2] > 0 && x2[2] > 0 && 0 < x4[2] - 1, 760_0_random_IntArithmetic(x2[2]), x4[2])
(3): COND_760_1_CREATELIST_INVOKEMETHOD(TRUE, 760_0_random_IntArithmetic(x2[3]), x4[3]) → 745_1_CREATELIST_INVOKEMETHOD(745_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)∧(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →^* 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0] →^* x3[1]))

(1) -> (2), if ((760_0_random_IntArithmetic(x5[1]) →^* 760_0_random_IntArithmetic(x2[2]))∧(x3[1] →^* x4[2]))

(2) -> (3), if ((x4[2] > 0 && x2[2] > 0 && 0 < x4[2] - 1 →^* TRUE)∧(760_0_random_IntArithmetic(x2[2]) →^* 760_0_random_IntArithmetic(x2[3]))∧(x4[2] →^* x4[3]))

(3) -> (0), if ((745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]) →^* 745_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:
851_1_createList_InvokeMethod(727_0_createList_Return, 0)
851_1_createList_InvokeMethod(923_0_createList_InvokeMethod(x0), x0)
851_1_createList_InvokeMethod(999_0_createList_Return, x0)
851_1_createList_InvokeMethod(1005_0_createList_InvokeMethod(x0), x0)
851_1_createList_InvokeMethod(1051_0_createList_InvokeMethod(x0, x1), x2)
851_1_createList_InvokeMethod(1292_0_createList_Return, x0)

(28) 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 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) the following chains were created:

We consider the chain 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]), COND_745_1_CREATELIST_INVOKEMETHOD(TRUE, 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1]) which results in the following constraint:
(1)    (&&(>=(x2[0], 1), <(x2[0], x0[0]))=TRUE∧745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])=745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1])∧x3[0]=x3[1] ⇒ 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(U^Increasing(COND_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_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 ⇒ 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(U^Increasing(COND_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_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_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]x3[0] ≥ 0∧[1 + (-1)bso_28] ≥ 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_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]x3[0] ≥ 0∧[1 + (-1)bso_28] ≥ 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_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]x3[0] ≥ 0∧[1 + (-1)bso_28] ≥ 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_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_27] = 0∧0 = 0∧[(-1)Bound*bni_27] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_28] ≥ 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_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_27] = 0∧0 = 0∧[(-1)Bound*bni_27] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_28] ≥ 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_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_27] = 0∧0 = 0∧[(-1)Bound*bni_27] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)

For Pair COND_745_1_CREATELIST_INVOKEMETHOD(TRUE, 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5), x3) the following chains were created:

We consider the chain COND_745_1_CREATELIST_INVOKEMETHOD(TRUE, 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1]) which results in the following constraint:
(9)    (COND_745_1_CREATELIST_INVOKEMETHOD(TRUE, 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥NonInfC∧COND_745_1_CREATELIST_INVOKEMETHOD(TRUE, 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1])∧(U^Increasing(760_1_CREATELIST_INVOKEMETHOD(760_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(760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1])), ≥)∧[(-1)bso_30] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    ((U^Increasing(760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1])), ≥)∧[(-1)bso_30] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    ((U^Increasing(760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1])), ≥)∧[(-1)bso_30] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    ((U^Increasing(760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

For Pair 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2), x4) → COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4, 0), >(x2, 0)), <(0, -(x4, 1))), 760_0_random_IntArithmetic(x2), x4) the following chains were created:

We consider the chain 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2[2]), x4[2]) → COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2]), COND_760_1_CREATELIST_INVOKEMETHOD(TRUE, 760_0_random_IntArithmetic(x2[3]), x4[3]) → 745_1_CREATELIST_INVOKEMETHOD(745_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∧760_0_random_IntArithmetic(x2[2])=760_0_random_IntArithmetic(x2[3])∧x4[2]=x4[3] ⇒ 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2[2]), x4[2])≥NonInfC∧760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2[2]), x4[2])≥COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2])∧(U^Increasing(COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_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 ⇒ 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2[2]), x4[2])≥NonInfC∧760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2[2]), x4[2])≥COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2])∧(U^Increasing(COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_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] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x4[2] ≥ 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    (x4[2] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x4[2] ≥ 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    (x4[2] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x4[2] ≥ 0∧[(-1)bso_32] ≥ 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_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(3)bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x4[2] ≥ 0∧[(-1)bso_32] ≥ 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_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(3)bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x4[2] ≥ 0∧[(-1)bso_32] ≥ 0)

For Pair COND_760_1_CREATELIST_INVOKEMETHOD(TRUE, 760_0_random_IntArithmetic(x2), x4) → 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5, x6)), x7), -(x4, 1)) the following chains were created:

We consider the chain COND_760_1_CREATELIST_INVOKEMETHOD(TRUE, 760_0_random_IntArithmetic(x2[3]), x4[3]) → 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1)) which results in the following constraint:
(21)    (COND_760_1_CREATELIST_INVOKEMETHOD(TRUE, 760_0_random_IntArithmetic(x2[3]), x4[3])≥NonInfC∧COND_760_1_CREATELIST_INVOKEMETHOD(TRUE, 760_0_random_IntArithmetic(x2[3]), x4[3])≥745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))∧(U^Increasing(745_1_CREATELIST_INVOKEMETHOD(745_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(745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))), ≥)∧[1 + (-1)bso_34] ≥ 0)

We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23)    ((U^Increasing(745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))), ≥)∧[1 + (-1)bso_34] ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24)    ((U^Increasing(745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))), ≥)∧[1 + (-1)bso_34] ≥ 0)

We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25)    ((U^Increasing(745_1_CREATELIST_INVOKEMETHOD(745_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 + (-1)bso_34] ≥ 0)

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

745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
- (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_27] = 0∧0 = 0∧[(-1)Bound*bni_27] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)
COND_745_1_CREATELIST_INVOKEMETHOD(TRUE, 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5), x3)
- ((U^Increasing(760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 0)
760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2), x4) → COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4, 0), >(x2, 0)), <(0, -(x4, 1))), 760_0_random_IntArithmetic(x2), x4)
- (x4[2] ≥ 0∧[1] + x4[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2])), ≥)∧[(3)bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x4[2] ≥ 0∧[(-1)bso_32] ≥ 0)
COND_760_1_CREATELIST_INVOKEMETHOD(TRUE, 760_0_random_IntArithmetic(x2), x4) → 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5, x6)), x7), -(x4, 1))
- ((U^Increasing(745_1_CREATELIST_INVOKEMETHOD(745_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 + (-1)bso_34] ≥ 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(851_1_createList_InvokeMethod(x₁, x₂)) = [-1]
POL(727_0_createList_Return) = [-1]
POL(0) = 0
POL(999_0_createList_Return) = [-1]
POL(923_0_createList_InvokeMethod(x₁)) = [-1]
POL(1051_0_createList_InvokeMethod(x₁, x₂)) = [-1]
POL(1292_0_createList_Return) = [-1]
POL(1005_0_createList_InvokeMethod(x₁)) = [-1]
POL(745_1_CREATELIST_INVOKEMETHOD(x₁, x₂)) = [2]x₂ + [-1]x₁
POL(745_0_random_ArrayAccess(x₁, x₂)) = [-1] + [-1]x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁, x₂)) = [-1]
POL(COND_745_1_CREATELIST_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [2]x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<(x₁, x₂)) = [-1]
POL(760_1_CREATELIST_INVOKEMETHOD(x₁, x₂)) = [-1] + [2]x₂
POL(760_0_random_IntArithmetic(x₁)) = x₁
POL(COND_760_1_CREATELIST_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [2]x₃
POL(>(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂

The following pairs are in P_>:

745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_745_1_CREATELIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
COND_760_1_CREATELIST_INVOKEMETHOD(TRUE, 760_0_random_IntArithmetic(x2[3]), x4[3]) → 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), -(x4[3], 1))

The following pairs are in P_bound:

760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2[2]), x4[2]) → COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2])

The following pairs are in P_≥:

COND_745_1_CREATELIST_INVOKEMETHOD(TRUE, 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1])
760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2[2]), x4[2]) → COND_760_1_CREATELIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, -(x4[2], 1))), 760_0_random_IntArithmetic(x2[2]), x4[2])

There are no usable rules.

(29) Complex Obligation (AND)

(30) 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:
851_1_createList_InvokeMethod(727_0_createList_Return, 0) → 999_0_createList_Return
851_1_createList_InvokeMethod(923_0_createList_InvokeMethod(x0), x0) → 1051_0_createList_InvokeMethod(x1, x0)
851_1_createList_InvokeMethod(999_0_createList_Return, x1) → 1292_0_createList_Return
851_1_createList_InvokeMethod(1005_0_createList_InvokeMethod(x0), x0) → 1051_0_createList_InvokeMethod(x1, x0)
851_1_createList_InvokeMethod(1051_0_createList_InvokeMethod(x1, x2), x4) → 1051_0_createList_InvokeMethod(x3, x4)
851_1_createList_InvokeMethod(1292_0_createList_Return, x1) → 1292_0_createList_Return

The integer pair graph contains the following rules and edges:
(1): COND_745_1_CREATELIST_INVOKEMETHOD(TRUE, 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1])
(2): 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x2[2]), x4[2]) → COND_760_1_CREATELIST_INVOKEMETHOD(x4[2] > 0 && x2[2] > 0 && 0 < x4[2] - 1, 760_0_random_IntArithmetic(x2[2]), x4[2])

(1) -> (2), if ((760_0_random_IntArithmetic(x5[1]) →^* 760_0_random_IntArithmetic(x2[2]))∧(x3[1] →^* x4[2]))

(31) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(32) TRUE

(33) 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:
851_1_createList_InvokeMethod(727_0_createList_Return, 0) → 999_0_createList_Return
851_1_createList_InvokeMethod(923_0_createList_InvokeMethod(x0), x0) → 1051_0_createList_InvokeMethod(x1, x0)
851_1_createList_InvokeMethod(999_0_createList_Return, x1) → 1292_0_createList_Return
851_1_createList_InvokeMethod(1005_0_createList_InvokeMethod(x0), x0) → 1051_0_createList_InvokeMethod(x1, x0)
851_1_createList_InvokeMethod(1051_0_createList_InvokeMethod(x1, x2), x4) → 1051_0_createList_InvokeMethod(x3, x4)
851_1_createList_InvokeMethod(1292_0_createList_Return, x1) → 1292_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_745_1_CREATELIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_745_1_CREATELIST_INVOKEMETHOD(TRUE, 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 760_1_CREATELIST_INVOKEMETHOD(760_0_random_IntArithmetic(x5[1]), x3[1])
(3): COND_760_1_CREATELIST_INVOKEMETHOD(TRUE, 760_0_random_IntArithmetic(x2[3]), x4[3]) → 745_1_CREATELIST_INVOKEMETHOD(745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]), x4[3] - 1)

(3) -> (0), if ((745_0_random_ArrayAccess(java.lang.Object(ARRAY(x5[3], x6[3])), x7[3]) →^* 745_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] - 1 →^* x3[0]))

(34) 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) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) IDPtoQDPProof (SOUND transformation)

(16) Obligation:

(17) DependencyGraphProof (EQUIVALENT transformation)

(18) Obligation:

(19) UsableRulesProof (EQUIVALENT transformation)

(20) Obligation:

(21) QReductionProof (EQUIVALENT transformation)

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) DependencyGraphProof (EQUIVALENT transformation)

(26) TRUE

(27) Obligation:

(28) IDPNonInfProof (SOUND transformation)

(29) Complex Obligation (AND)

(30) Obligation:

(31) IDependencyGraphProof (EQUIVALENT transformation)

(32) TRUE

(33) Obligation:

(34) IDependencyGraphProof (EQUIVALENT transformation)

(35) TRUE