(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)))=TRUEjava.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])∧(UIncreasing(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])=TRUE4281_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])∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))∧(UIncreasing(4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))), ≥))



    We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (10)    ((UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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 ⇒ (UIncreasing(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$Node))), +(+(+(+(x3, 1), 1), 1), 1))
    • ((UIncreasing(4281_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), java.lang.Object(LinkedList(java.lang.Object(LinkedList$Node))), +(+(+(+(x3[1], 1), 1), 1), 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[3 + (-1)bso_18] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
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 Pbound:

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.

(27) TRUE