Termination proof of /tmp/tmpzcCd-L/LinkedList.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 UsableRulesProof (⇔)

        ↳18 QDP

        ↳19 QReductionProof (⇔)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔)

        ↳22 YES

    ↳23 IDP

        ↳24 IDPNonInfProof (⇒)

        ↳25 IDP

        ↳26 IDependencyGraphProof (⇔)

        ↳27 TRUE

(0) Obligation:

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

public class Entry
{
    private String item;
    private int count;

    public Entry(String itemData, int countData)
    {
        item = itemData;
        count = countData;
    }

    public String toString( )
    {
        return (item + " " + count);
    }

    public boolean equals(Object otherObject)
    {
        if (otherObject == null)
            return false;
        else if (getClass( ) != otherObject.getClass( ))
            return false;
        else
        {
            Entry otherEntry = (Entry)otherObject;
            return (item.equals(otherEntry.item) 
                      && (count == otherEntry.count));
        }
    }

   // <There should be other constructors and methods, including accessor and 
   //                       mutator methods, but we do not use them in this demonstration.>
}



public class LinkedList<T>

{

    private class Node<T>

    {

        private T data;

        private Node<T> link;



        public Node( )

        {

             data = null;

             link = null;

        }



        public Node(T newData, Node<T> linkValue)

        {

            data = newData;

            link = linkValue;

        }

     }//End of Node<T> inner class



    private Node<T> head;



    public LinkedList( )

    {

        head = null;

    }



    /**

     Adds a node at the start of the list with the specified data.

     The added node will be the first node in the list.

    */

    public void addToStart(T itemData)

    {

        this.head = new Node<T>(itemData, this.head);

    }



    /**

     Removes the head node and returns true if the list contains at least

     one node. Returns false if the list is empty.

    */

    public boolean deleteHeadNode( )

    {

        if (head != null)

        {

            head = head.link;

            return true;

        }

        else

            return false;

    }



    /**

     Returns the number of nodes in the list.

    */

    public int size( )

    {

        int count = 0;

        Node<T> position = head;

        while (position != null)

        {

            count++;

            position = position.link;

        }

        return count;

    }



    public boolean contains(T item)

    {

        return (find(item) != null);

    }



    /**

     Finds the first node containing the target item, and returns a

     reference to that node. If target is not in the list, null is returned.

    */

    private Node<T> find(T target)

    {

        Node<T> position = head;

        T itemAtPosition;

        while (position != null)

        {

            itemAtPosition = position.data;

            if (itemAtPosition.equals(target))

                return position;

            position = position.link;

        }

        return null; //target was not found

    }



    /**

     Finds the first node containing the target and returns a reference

      to the data in that node. If target is not in the list, null is returned.

    */

    public T findData(T target)

    {

        return find(target).data;

    }



    public void outputList( )

    {

        Node<T> position = head;

        while (position != null)

        {

            //System.out.println(position.data);

            position = position.link;

        }

    }



    public boolean isEmpty( )

    {

        return (head == null);

    }



    public void clear( )

    {

        head = null;

    }



   /*

    For two lists to be equal they must contain the same data items in

    the same order. The equals method of T is used to compare data items.

   */

   public boolean equals(Object otherObject)

    {

        if (!(otherObject instanceof LinkedList))

            return false;

        else

        {

            LinkedList<T> otherList = (LinkedList<T>)otherObject;

            if (size( ) != otherList.size( ))

                return false;

            Node<T> position = head;

            Node<T> otherPosition = otherList.head;

            while (position != null)

            {

                if (!(position.data.equals(otherPosition.data)))

                    return false;

                position = position.link;

                otherPosition = otherPosition.link;

            }

            return true; //no mismatch was not found

        }

    }



   public static void main(String[] args)

    {

        LinkedList<Entry> list = new LinkedList<Entry>( );



	for (int i = 1; i < args.length; i++) {

	    Entry entry = new Entry(args[i], i++);

	    list.addToStart(entry);

	    entry = new Entry(args[i], i++);

	    list.addToStart(entry);

	    entry = new Entry(args[i], i++);

	    list.addToStart(entry);

	}



	list.size(); // remove it!

        //System.out.println("List has " + list.size( ) 

	//                  + " nodes.");

        list.outputList( );

        //System.out.println("End of list.");

    }

}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
LinkedList.main([Ljava/lang/String;)V: Graph of 243 nodes with 1 SCC.

LinkedList.size()I: Graph of 44 nodes with 1 SCC.

LinkedList.outputList()V: Graph of 38 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 28 rules for P and 10 rules for R.

Combined rules. Obtained 5 rules for P and 1 rules for R.

Filtered ground terms:

3455_0_outputList_NULL(x1, x2, x3) → 3455_0_outputList_NULL(x2, x3)
LinkedList$Node(x1, x2) → LinkedList$Node(x2)
3807_0_access$000_Return(x1, x2) → 3807_0_access$000_Return(x2)
3775_0_access$000_Return(x1, x2) → 3775_0_access$000_Return(x2)
3546_0_outputList_Return(x1) → 3546_0_outputList_Return

Filtered duplicate args:

3455_0_outputList_NULL(x1, x2) → 3455_0_outputList_NULL(x2)

Finished conversion. Obtained 5 rules for P and 1 rules for R. System has no predefined symbols.

Log for SCC 1:

Generated 29 rules for P and 11 rules for R.

Combined rules. Obtained 5 rules for P and 1 rules for R.

Filtered ground terms:

3573_0_size_NULL(x1, x2, x3) → 3573_0_size_NULL(x2, x3)
LinkedList$Node(x1, x2) → LinkedList$Node(x2)
4030_0_access$000_Return(x1, x2) → 4030_0_access$000_Return(x2)
3998_0_access$000_Return(x1, x2) → 3998_0_access$000_Return(x2)
3608_0_size_Return(x1) → 3608_0_size_Return

Filtered duplicate args:

3573_0_size_NULL(x1, x2) → 3573_0_size_NULL(x2)

Finished conversion. Obtained 5 rules for P and 1 rules for R. System has no predefined symbols.

Log for SCC 2:

Generated 140 rules for P and 170 rules for R.

Combined rules. Obtained 1 rules for P and 0 rules for R.

Filtered ground terms:

4281_0_main_Load(x1, x2, x3, x4, x5) → 4281_0_main_Load(x2, x3, x4, x5)
LinkedList$Node(x1) → LinkedList$Node
LinkedList(x1, x2) → LinkedList(x2)
Cond_4281_0_main_Load(x1, x2, x3, x4, x5, x6) → Cond_4281_0_main_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:

4281_0_main_Load(x1, x2, x3, x4) → 4281_0_main_Load(x1, x2, x4)
Cond_4281_0_main_Load(x1, x2, x3, x4, x5) → Cond_4281_0_main_Load(x1, x2, x3, x5)

Combined rules. Obtained 1 rules for P and 0 rules for R.

Finished conversion. Obtained 1 rules for P and 0 rules for R. System has predefined symbols.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The following domains are used:
none

The ITRS R consists of the following rules:
3455_0_outputList_NULL(NULL) → 3546_0_outputList_Return

The integer pair graph contains the following rules and edges:
(0): 3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[0]))) → 3775_1_OUTPUTLIST_INVOKEMETHOD(3775_0_access$000_Return(x0[0]), java.lang.Object(LinkedList$Node(x0[0])))
(1): 3775_1_OUTPUTLIST_INVOKEMETHOD(3775_0_access$000_Return(x0[1]), java.lang.Object(LinkedList$Node(x0[1]))) → 3455_0_OUTPUTLIST_NULL(x0[1])
(2): 3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[2]))) → 3455_0_OUTPUTLIST_NULL(x0[2])
(3): 3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[3]))) → 3807_1_OUTPUTLIST_INVOKEMETHOD(3807_0_access$000_Return(x0[3]), java.lang.Object(LinkedList$Node(x0[3])))
(4): 3807_1_OUTPUTLIST_INVOKEMETHOD(3807_0_access$000_Return(x0[4]), java.lang.Object(LinkedList$Node(x0[4]))) → 3455_0_OUTPUTLIST_NULL(x0[4])

(0) -> (1), if ((3775_0_access$000_Return(x0[0]) →^* 3775_0_access$000_Return(x0[1]))∧(java.lang.Object(LinkedList$Node(x0[0])) →^* java.lang.Object(LinkedList$Node(x0[1]))))

(1) -> (0), if ((x0[1] →^* java.lang.Object(LinkedList$Node(x0[0]))))

(1) -> (2), if ((x0[1] →^* java.lang.Object(LinkedList$Node(x0[2]))))

(1) -> (3), if ((x0[1] →^* java.lang.Object(LinkedList$Node(x0[3]))))

(2) -> (0), if ((x0[2] →^* java.lang.Object(LinkedList$Node(x0[0]))))

(2) -> (2), if ((x0[2] →^* java.lang.Object(LinkedList$Node(x0[2]'))))

(2) -> (3), if ((x0[2] →^* java.lang.Object(LinkedList$Node(x0[3]))))

(3) -> (4), if ((3807_0_access$000_Return(x0[3]) →^* 3807_0_access$000_Return(x0[4]))∧(java.lang.Object(LinkedList$Node(x0[3])) →^* java.lang.Object(LinkedList$Node(x0[4]))))

(4) -> (0), if ((x0[4] →^* java.lang.Object(LinkedList$Node(x0[0]))))

(4) -> (2), if ((x0[4] →^* java.lang.Object(LinkedList$Node(x0[2]))))

(4) -> (3), if ((x0[4] →^* java.lang.Object(LinkedList$Node(x0[3]))))

The set Q consists of the following terms:
3455_0_outputList_NULL(NULL)

(6) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(7) Obligation:

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

3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[0]))) → 3775_1_OUTPUTLIST_INVOKEMETHOD(3775_0_access$000_Return(x0[0]), java.lang.Object(LinkedList$Node(x0[0])))
3775_1_OUTPUTLIST_INVOKEMETHOD(3775_0_access$000_Return(x0[1]), java.lang.Object(LinkedList$Node(x0[1]))) → 3455_0_OUTPUTLIST_NULL(x0[1])
3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[2]))) → 3455_0_OUTPUTLIST_NULL(x0[2])
3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[3]))) → 3807_1_OUTPUTLIST_INVOKEMETHOD(3807_0_access$000_Return(x0[3]), java.lang.Object(LinkedList$Node(x0[3])))
3807_1_OUTPUTLIST_INVOKEMETHOD(3807_0_access$000_Return(x0[4]), java.lang.Object(LinkedList$Node(x0[4]))) → 3455_0_OUTPUTLIST_NULL(x0[4])

The TRS R consists of the following rules:

3455_0_outputList_NULL(NULL) → 3546_0_outputList_Return

The set Q consists of the following terms:

3455_0_outputList_NULL(NULL)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[0]))) → 3775_1_OUTPUTLIST_INVOKEMETHOD(3775_0_access$000_Return(x0[0]), java.lang.Object(LinkedList$Node(x0[0])))
3775_1_OUTPUTLIST_INVOKEMETHOD(3775_0_access$000_Return(x0[1]), java.lang.Object(LinkedList$Node(x0[1]))) → 3455_0_OUTPUTLIST_NULL(x0[1])
3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[2]))) → 3455_0_OUTPUTLIST_NULL(x0[2])
3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[3]))) → 3807_1_OUTPUTLIST_INVOKEMETHOD(3807_0_access$000_Return(x0[3]), java.lang.Object(LinkedList$Node(x0[3])))
3807_1_OUTPUTLIST_INVOKEMETHOD(3807_0_access$000_Return(x0[4]), java.lang.Object(LinkedList$Node(x0[4]))) → 3455_0_OUTPUTLIST_NULL(x0[4])

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

3455_0_outputList_NULL(NULL)

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

(10) QReductionProof (EQUIVALENT transformation)

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

3455_0_outputList_NULL(NULL)

(11) Obligation:

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

3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[0]))) → 3775_1_OUTPUTLIST_INVOKEMETHOD(3775_0_access$000_Return(x0[0]), java.lang.Object(LinkedList$Node(x0[0])))
3775_1_OUTPUTLIST_INVOKEMETHOD(3775_0_access$000_Return(x0[1]), java.lang.Object(LinkedList$Node(x0[1]))) → 3455_0_OUTPUTLIST_NULL(x0[1])
3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[2]))) → 3455_0_OUTPUTLIST_NULL(x0[2])
3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[3]))) → 3807_1_OUTPUTLIST_INVOKEMETHOD(3807_0_access$000_Return(x0[3]), java.lang.Object(LinkedList$Node(x0[3])))
3807_1_OUTPUTLIST_INVOKEMETHOD(3807_0_access$000_Return(x0[4]), java.lang.Object(LinkedList$Node(x0[4]))) → 3455_0_OUTPUTLIST_NULL(x0[4])

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:

3775_1_OUTPUTLIST_INVOKEMETHOD(3775_0_access$000_Return(x0[1]), java.lang.Object(LinkedList$Node(x0[1]))) → 3455_0_OUTPUTLIST_NULL(x0[1])
The graph contains the following edges 1 > 1, 2 > 1
3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[2]))) → 3455_0_OUTPUTLIST_NULL(x0[2])
The graph contains the following edges 1 > 1
3807_1_OUTPUTLIST_INVOKEMETHOD(3807_0_access$000_Return(x0[4]), java.lang.Object(LinkedList$Node(x0[4]))) → 3455_0_OUTPUTLIST_NULL(x0[4])
The graph contains the following edges 1 > 1, 2 > 1
3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[0]))) → 3775_1_OUTPUTLIST_INVOKEMETHOD(3775_0_access$000_Return(x0[0]), java.lang.Object(LinkedList$Node(x0[0])))
The graph contains the following edges 1 >= 2
3455_0_OUTPUTLIST_NULL(java.lang.Object(LinkedList$Node(x0[3]))) → 3807_1_OUTPUTLIST_INVOKEMETHOD(3807_0_access$000_Return(x0[3]), java.lang.Object(LinkedList$Node(x0[3])))
The graph contains the following edges 1 >= 2

(13) YES

(14) Obligation:

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

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

The following domains are used:
none

The ITRS R consists of the following rules:
3573_0_size_NULL(NULL) → 3608_0_size_Return

The integer pair graph contains the following rules and edges:
(0): 3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[0]))) → 3998_1_SIZE_INVOKEMETHOD(3998_0_access$000_Return(x0[0]), java.lang.Object(LinkedList$Node(x0[0])))
(1): 3998_1_SIZE_INVOKEMETHOD(3998_0_access$000_Return(x0[1]), java.lang.Object(LinkedList$Node(x0[1]))) → 3573_0_SIZE_NULL(x0[1])
(2): 3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[2]))) → 3573_0_SIZE_NULL(x0[2])
(3): 3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[3]))) → 4030_1_SIZE_INVOKEMETHOD(4030_0_access$000_Return(x0[3]), java.lang.Object(LinkedList$Node(x0[3])))
(4): 4030_1_SIZE_INVOKEMETHOD(4030_0_access$000_Return(x0[4]), java.lang.Object(LinkedList$Node(x0[4]))) → 3573_0_SIZE_NULL(x0[4])

(0) -> (1), if ((3998_0_access$000_Return(x0[0]) →^* 3998_0_access$000_Return(x0[1]))∧(java.lang.Object(LinkedList$Node(x0[0])) →^* java.lang.Object(LinkedList$Node(x0[1]))))

(1) -> (0), if ((x0[1] →^* java.lang.Object(LinkedList$Node(x0[0]))))

(1) -> (2), if ((x0[1] →^* java.lang.Object(LinkedList$Node(x0[2]))))

(1) -> (3), if ((x0[1] →^* java.lang.Object(LinkedList$Node(x0[3]))))

(2) -> (0), if ((x0[2] →^* java.lang.Object(LinkedList$Node(x0[0]))))

(2) -> (2), if ((x0[2] →^* java.lang.Object(LinkedList$Node(x0[2]'))))

(2) -> (3), if ((x0[2] →^* java.lang.Object(LinkedList$Node(x0[3]))))

(3) -> (4), if ((4030_0_access$000_Return(x0[3]) →^* 4030_0_access$000_Return(x0[4]))∧(java.lang.Object(LinkedList$Node(x0[3])) →^* java.lang.Object(LinkedList$Node(x0[4]))))

(4) -> (0), if ((x0[4] →^* java.lang.Object(LinkedList$Node(x0[0]))))

(4) -> (2), if ((x0[4] →^* java.lang.Object(LinkedList$Node(x0[2]))))

(4) -> (3), if ((x0[4] →^* java.lang.Object(LinkedList$Node(x0[3]))))

The set Q consists of the following terms:
3573_0_size_NULL(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:

3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[0]))) → 3998_1_SIZE_INVOKEMETHOD(3998_0_access$000_Return(x0[0]), java.lang.Object(LinkedList$Node(x0[0])))
3998_1_SIZE_INVOKEMETHOD(3998_0_access$000_Return(x0[1]), java.lang.Object(LinkedList$Node(x0[1]))) → 3573_0_SIZE_NULL(x0[1])
3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[2]))) → 3573_0_SIZE_NULL(x0[2])
3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[3]))) → 4030_1_SIZE_INVOKEMETHOD(4030_0_access$000_Return(x0[3]), java.lang.Object(LinkedList$Node(x0[3])))
4030_1_SIZE_INVOKEMETHOD(4030_0_access$000_Return(x0[4]), java.lang.Object(LinkedList$Node(x0[4]))) → 3573_0_SIZE_NULL(x0[4])

The TRS R consists of the following rules:

3573_0_size_NULL(NULL) → 3608_0_size_Return

The set Q consists of the following terms:

3573_0_size_NULL(NULL)

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

(17) 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.

(18) Obligation:

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

3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[0]))) → 3998_1_SIZE_INVOKEMETHOD(3998_0_access$000_Return(x0[0]), java.lang.Object(LinkedList$Node(x0[0])))
3998_1_SIZE_INVOKEMETHOD(3998_0_access$000_Return(x0[1]), java.lang.Object(LinkedList$Node(x0[1]))) → 3573_0_SIZE_NULL(x0[1])
3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[2]))) → 3573_0_SIZE_NULL(x0[2])
3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[3]))) → 4030_1_SIZE_INVOKEMETHOD(4030_0_access$000_Return(x0[3]), java.lang.Object(LinkedList$Node(x0[3])))
4030_1_SIZE_INVOKEMETHOD(4030_0_access$000_Return(x0[4]), java.lang.Object(LinkedList$Node(x0[4]))) → 3573_0_SIZE_NULL(x0[4])

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

3573_0_size_NULL(NULL)

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

(19) 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].

3573_0_size_NULL(NULL)

(20) Obligation:

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

3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[0]))) → 3998_1_SIZE_INVOKEMETHOD(3998_0_access$000_Return(x0[0]), java.lang.Object(LinkedList$Node(x0[0])))
3998_1_SIZE_INVOKEMETHOD(3998_0_access$000_Return(x0[1]), java.lang.Object(LinkedList$Node(x0[1]))) → 3573_0_SIZE_NULL(x0[1])
3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[2]))) → 3573_0_SIZE_NULL(x0[2])
3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[3]))) → 4030_1_SIZE_INVOKEMETHOD(4030_0_access$000_Return(x0[3]), java.lang.Object(LinkedList$Node(x0[3])))
4030_1_SIZE_INVOKEMETHOD(4030_0_access$000_Return(x0[4]), java.lang.Object(LinkedList$Node(x0[4]))) → 3573_0_SIZE_NULL(x0[4])

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

(21) 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:

3998_1_SIZE_INVOKEMETHOD(3998_0_access$000_Return(x0[1]), java.lang.Object(LinkedList$Node(x0[1]))) → 3573_0_SIZE_NULL(x0[1])
The graph contains the following edges 1 > 1, 2 > 1
3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[2]))) → 3573_0_SIZE_NULL(x0[2])
The graph contains the following edges 1 > 1
4030_1_SIZE_INVOKEMETHOD(4030_0_access$000_Return(x0[4]), java.lang.Object(LinkedList$Node(x0[4]))) → 3573_0_SIZE_NULL(x0[4])
The graph contains the following edges 1 > 1, 2 > 1
3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[0]))) → 3998_1_SIZE_INVOKEMETHOD(3998_0_access$000_Return(x0[0]), java.lang.Object(LinkedList$Node(x0[0])))
The graph contains the following edges 1 >= 2
3573_0_SIZE_NULL(java.lang.Object(LinkedList$Node(x0[3]))) → 4030_1_SIZE_INVOKEMETHOD(4030_0_access$000_Return(x0[3]), java.lang.Object(LinkedList$Node(x0[3])))
The graph contains the following edges 1 >= 2

(22) YES

(23) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0]) → COND_4281_0_MAIN_LOAD(x3[0] > 1 && x3[0] < x0[0] && x0[0] > x3[0] + 1 && x0[0] > x3[0] + 1 + 1 && 3 < x3[0] + 1 + 1, java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])
(1): COND_4281_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(x2[1])), x3[1]) → 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), x3[1] + 1 + 1 + 1 + 1)

(0) -> (1), if ((x3[0] > 1 && x3[0] < x0[0] && x0[0] > x3[0] + 1 && x0[0] > x3[0] + 1 + 1 && 3 < x3[0] + 1 + 1 →^* TRUE)∧(java.lang.Object(ARRAY(x0[0], x1[0])) →^* java.lang.Object(ARRAY(x0[1], x1[1])))∧(java.lang.Object(LinkedList(x2[0])) →^* java.lang.Object(LinkedList(x2[1])))∧(x3[0] →^* x3[1]))

(1) -> (0), if ((java.lang.Object(ARRAY(x0[1], x1[1])) →^* java.lang.Object(ARRAY(x0[0], x1[0])))∧(java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))) →^* java.lang.Object(LinkedList(x2[0])))∧(x3[1] + 1 + 1 + 1 + 1 →^* x3[0]))

The set Q is empty.

(24) IDPNonInfProof (SOUND transformation)

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

For Pair 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), java.lang.Object(LinkedList(x2)), x3) → COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3, 1), <(x3, x0)), >(x0, +(x3, 1))), >(x0, +(+(x3, 1), 1))), <(3, +(+(x3, 1), 1))), java.lang.Object(ARRAY(x0, x1)), java.lang.Object(LinkedList(x2)), x3) the following chains were created:

We consider the chain 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0]) → COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0]), COND_4281_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(x2[1])), x3[1]) → 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1)) which results in the following constraint:
(1)    (&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1)))=TRUE∧java.lang.Object(ARRAY(x0[0], x1[0]))=java.lang.Object(ARRAY(x0[1], x1[1]))∧java.lang.Object(LinkedList(x2[0]))=java.lang.Object(LinkedList(x2[1]))∧x3[0]=x3[1] ⇒ 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])≥NonInfC∧4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])≥COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])∧(U^Increasing(COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (<(3, +(+(x3[0], 1), 1))=TRUE∧>(x0[0], +(+(x3[0], 1), 1))=TRUE∧>(x0[0], +(x3[0], 1))=TRUE∧>(x3[0], 1)=TRUE∧<(x3[0], x0[0])=TRUE ⇒ 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])≥NonInfC∧4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])≥COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])∧(U^Increasing(COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x3[0] + [-2] ≥ 0∧x0[0] + [-3] + [-1]x3[0] ≥ 0∧x0[0] + [-2] + [-1]x3[0] ≥ 0∧x3[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x3[0] + [bni_15]x0[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x3[0] + [-2] ≥ 0∧x0[0] + [-3] + [-1]x3[0] ≥ 0∧x0[0] + [-2] + [-1]x3[0] ≥ 0∧x3[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x3[0] + [bni_15]x0[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x3[0] + [-2] ≥ 0∧x0[0] + [-3] + [-1]x3[0] ≥ 0∧x0[0] + [-2] + [-1]x3[0] ≥ 0∧x3[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x3[0] + [bni_15]x0[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x3[0] + [-2] ≥ 0∧x0[0] + [-3] + [-1]x3[0] ≥ 0∧x0[0] + [-2] + [-1]x3[0] ≥ 0∧x3[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])), ≥)∧0 = 0∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x3[0] + [bni_15]x0[0] ≥ 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x3[0] ≥ 0∧x0[0] + [-5] + [-1]x3[0] ≥ 0∧x0[0] + [-4] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧x0[0] + [-3] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])), ≥)∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x3[0] + [bni_15]x0[0] ≥ 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(8)    (x3[0] ≥ 0∧x0[0] ≥ 0∧[1] + x0[0] ≥ 0∧x3[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (U^Increasing(COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])), ≥)∧0 = 0∧[(4)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

For Pair COND_4281_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0, x1)), java.lang.Object(LinkedList(x2)), x3) → 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3, 1), 1), 1), 1)) the following chains were created:

We consider the chain COND_4281_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(x2[1])), x3[1]) → 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1)) which results in the following constraint:
(9)    (COND_4281_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(x2[1])), x3[1])≥NonInfC∧COND_4281_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(x2[1])), x3[1])≥4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))∧(U^Increasing(4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    ((U^Increasing(4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))), ≥)∧[3 + (-1)bso_18] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    ((U^Increasing(4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))), ≥)∧[3 + (-1)bso_18] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    ((U^Increasing(4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))), ≥)∧[3 + (-1)bso_18] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    ((U^Increasing(4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[3 + (-1)bso_18] ≥ 0)

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

4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), java.lang.Object(LinkedList(x2)), x3) → COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3, 1), <(x3, x0)), >(x0, +(x3, 1))), >(x0, +(+(x3, 1), 1))), <(3, +(+(x3, 1), 1))), java.lang.Object(ARRAY(x0, x1)), java.lang.Object(LinkedList(x2)), x3)
- (x3[0] ≥ 0∧x0[0] ≥ 0∧[1] + x0[0] ≥ 0∧x3[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (U^Increasing(COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])), ≥)∧0 = 0∧[(4)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)
COND_4281_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0, x1)), java.lang.Object(LinkedList(x2)), x3) → 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3, 1), 1), 1), 1))
- ((U^Increasing(4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[3 + (-1)bso_18] ≥ 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(4281_0_MAIN_LOAD(x₁, x₂, x₃)) = [-1]x₃ + [-1]x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁, x₂)) = [-1] + [-1]x₁
POL(LinkedList(x₁)) = [-1]
POL(COND_4281_0_MAIN_LOAD(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(3) = [3]
POL(LinkedList$Node) = [-1]

The following pairs are in P_>:

4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0]) → COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])
COND_4281_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(x2[1])), x3[1]) → 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))

The following pairs are in P_bound:

4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0]) → COND_4281_0_MAIN_LOAD(&&(&&(&&(&&(>(x3[0], 1), <(x3[0], x0[0])), >(x0[0], +(x3[0], 1))), >(x0[0], +(+(x3[0], 1), 1))), <(3, +(+(x3[0], 1), 1))), java.lang.Object(ARRAY(x0[0], x1[0])), java.lang.Object(LinkedList(x2[0])), x3[0])

The following pairs are in P_≥:
none

There are no usable rules.

(25) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_4281_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(x2[1])), x3[1]) → 4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), x3[1] + 1 + 1 + 1 + 1)

The set Q is empty.

(26) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) IDPtoQDPProof (SOUND transformation)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) QReductionProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) IDPtoQDPProof (SOUND transformation)

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) QReductionProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) IDPNonInfProof (SOUND transformation)

(25) Obligation:

(26) IDependencyGraphProof (EQUIVALENT transformation)

(27) TRUE