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 DependencyGraphProof (⇔)
        ↳9 QDP
        ↳10 UsableRulesProof (⇔)
        ↳11 QDP
        ↳12 QReductionProof (⇔)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔)
        ↳15 YES
    ↳16 IDP
        ↳17 IDPtoQDPProof (⇒)
        ↳18 QDP
        ↳19 UsableRulesProof (⇔)
        ↳20 QDP
        ↳21 QReductionProof (⇔)
        ↳22 QDP
        ↳23 QDPSizeChangeProof (⇔)
        ↳24 YES
    ↳25 IDP
        ↳26 IDPNonInfProof (⇒)
        ↳27 AND
            ↳28 IDP
                ↳29 IDependencyGraphProof (⇔)
                ↳30 TRUE
            ↳31 IDP
                ↳32 IDependencyGraphProof (⇔)
                ↳33 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_14 (Sun Microsystems Inc.) Main-Class: DoublyLinkedList/MainCopy 
package DoublyLinkedList;

/**
 * A linked list with pointers to the previous and next elements
 * @author cotto
 */
public class DoublyLinkedList {
    public int value;
    public DoublyLinkedList prev;
    public DoublyLinkedList next;

    public DoublyLinkedList(final int v) {
        this.value = v;
    }

    public DoublyLinkedList getFirst() {
        if (this.prev == null) {
            return this;
        }
        return this.prev.getFirst();
    }

    public void move(final int relativePosition) {
        if (relativePosition == 0) {
            return;
        }
        if (relativePosition > 0 && this.next != null) {
            final DoublyLinkedList temp = this.next;
            if (this.prev != null) {
                this.prev.next = temp;
            }
            temp.prev = this.prev;
            this.next = temp.next;
            temp.next = this;
            this.prev = temp;
            move(relativePosition - 1);
        }
        if (relativePosition < 0 && this.prev != null) {
            final DoublyLinkedList temp = this.prev;
            if (this.next != null) {
                this.next.prev = temp;
            }
            temp.next = this.next;
            this.prev = temp.prev;
            temp.prev = this;
            this.next = temp;
            move(relativePosition - 1);
        }
    }

    public DoublyLinkedList get(final int index) {
        DoublyLinkedList current = this.getFirst();
        while (index > 0 && current != null) {
            current = current.next;
        }
        return current;
    }

    public DoublyLinkedList find(final int v) {
        final DoublyLinkedList first = this.getFirst();
        return first.findR(v);
    }

    private DoublyLinkedList findR(final int v) {
        if (this.value == v) {
            return this;
        }
        if (this.next != null) {
            return this.next.findR(v);
        }
        return null;
    }

    public void delete(final int v) {
        final DoublyLinkedList elem = find(v);
        if (elem != null) {
            if (elem.prev != null) {
                elem.prev.next = elem.next;
            }
            if (elem.next != null) {
                elem.next.prev = elem.prev;
            }
        }
    }

    public DoublyLinkedList copy() {
        final DoublyLinkedList first = this.getFirst();
        return first.copyR(null);
    }

    private DoublyLinkedList copyR(final DoublyLinkedList p) {
        final DoublyLinkedList copy = new DoublyLinkedList(this.value);
        copy.prev = p;
        if (p != null) {
            p.next = copy;
        }
        if (this.next != null) {
            this.next.copyR(copy);
        }
        return copy;
    }

    static DoublyLinkedList createList() {
        final int count = Random.random();
        DoublyLinkedList cur = null;
        for (int i = 0; i < count; i++) {
            final DoublyLinkedList old = cur;
            cur = new DoublyLinkedList(Random.random());
            cur.prev = old;
            if (old != null) {
              old.next = cur;
            }
        }

        return cur;
    }
}


package DoublyLinkedList;

/**
 *
 * @author cotto
 */
public class MainCopy {
    public static void main(final String[] args) {
        Random.args = args;
        final DoublyLinkedList list = DoublyLinkedList.createList();
        list.copy();
    }
}


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

    public static int random() {
        if (args.length <= index) {
            return 0;
        }
        final String string = args[index];
        index++;
        if (string == null) {
            return 0;
        }
        return string.length();
    }
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
DoublyLinkedList.MainCopy.main([Ljava/lang/String;)V: Graph of 105 nodes with 0 SCCs.

DoublyLinkedList.DoublyLinkedList.createList()LDoublyLinkedList/DoublyLinkedList;: Graph of 205 nodes with 1 SCC.

DoublyLinkedList.DoublyLinkedList.getFirst()LDoublyLinkedList/DoublyLinkedList;: Graph of 29 nodes with 0 SCCs.

DoublyLinkedList.DoublyLinkedList.copyR(LDoublyLinkedList/DoublyLinkedList;)LDoublyLinkedList/DoublyLinkedList;: Graph of 190 nodes with 0 SCCs.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 113 rules for P and 96 rules for R.


Combined rules. Obtained 8 rules for P and 16 rules for R.


Filtered ground terms:


DoublyLinkedList.DoublyLinkedList(x1, x2, x3) → DoublyLinkedList.DoublyLinkedList(x2, x3)
6402_0_copyR_Duplicate(x1, x2, x3, x4) → 6402_0_copyR_Duplicate(x2, x3)
8314_0_copyR_Load(x1, x2, x3) → 8314_0_copyR_Load(x2, x3)
8185_0_copyR_Load(x1, x2, x3) → 8185_0_copyR_Load(x2, x3)
10878_0_copyR_Return(x1, x2) → 10878_0_copyR_Return(x2)
11086_0_copyR_Return(x1, x2) → 11086_0_copyR_Return(x2)
10972_0_copyR_Return(x1, x2) → 10972_0_copyR_Return(x2)
8969_0_copyR_Return(x1, x2) → 8969_0_copyR_Return(x2)
10830_0_copyR_Return(x1, x2) → 10830_0_copyR_Return(x2)
8895_0_copyR_Return(x1, x2) → 8895_0_copyR_Return(x2)

Filtered duplicate args:


8677_1_copyR_InvokeMethod(x1, x2, x3, x4) → 8677_1_copyR_InvokeMethod(x1, x3, x4)
9434_1_copyR_InvokeMethod(x1, x2, x3, x4) → 9434_1_copyR_InvokeMethod(x1, x3, x4)
8580_1_copyR_InvokeMethod(x1, x2, x3, x4) → 8580_1_copyR_InvokeMethod(x1, x3, x4)
9279_1_copyR_InvokeMethod(x1, x2, x3, x4) → 9279_1_copyR_InvokeMethod(x1, x3, x4)

Filtered all free variables:


10878_0_copyR_Return(x1) → 10878_0_copyR_Return
11086_0_copyR_Return(x1) → 11086_0_copyR_Return
10830_0_copyR_Return(x1) → 10830_0_copyR_Return
10972_0_copyR_Return(x1) → 10972_0_copyR_Return

Finished conversion. Obtained 8 rules for P and 16 rules for R. System has no predefined symbols.



Log for SCC 1:

Generated 16 rules for P and 14 rules for R.


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


Filtered ground terms:


3076_0_getFirst_FieldAccess(x1, x2, x3) → 3076_0_getFirst_FieldAccess(x2, x3)
DoublyLinkedList.DoublyLinkedList(x1, x2) → DoublyLinkedList.DoublyLinkedList(x2)
3210_0_getFirst_NONNULL(x1, x2, x3) → 3210_0_getFirst_NONNULL(x2, x3)
3573_0_getFirst_Return(x1) → 3573_0_getFirst_Return
3347_0_getFirst_Return(x1, x2) → 3347_0_getFirst_Return
3257_0_getFirst_Return(x1, x2, x3) → 3257_0_getFirst_Return

Filtered duplicate args:


3076_0_getFirst_FieldAccess(x1, x2) → 3076_0_getFirst_FieldAccess(x2)

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



Log for SCC 2:

Generated 105 rules for P and 60 rules for R.


Combined rules. Obtained 14 rules for P and 0 rules for R.


Filtered ground terms:


6857_0_createList_NULL(x1, x2, x3, x4, x5, x6) → 6857_0_createList_NULL(x2, x4, x5, x6)
DoublyLinkedList.DoublyLinkedList(x1) → DoublyLinkedList.DoublyLinkedList
Cond_5958_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_5958_1_createList_InvokeMethod(x1, x2, x3, x4, x5)
5958_0_random_GT(x1, x2, x3) → 5958_0_random_GT(x2, x3)
5958_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 5958_1_createList_InvokeMethod(x1, x2, x3, x4)
4536_0_createList_Load(x1, x2, x3, x4, x5) → 4536_0_createList_Load(x2, x3, x4, x5)
7317_0_createList_Inc(x1, x2, x3, x4) → 7317_0_createList_Inc(x2, x4)
Cond_6524_1_createList_InvokeMethod3(x1, x2, x3, x4, x5, x6, x7) → Cond_6524_1_createList_InvokeMethod3(x1, x2, x3, x4, x5)
6524_0_random_IntArithmetic(x1, x2, x3, x4) → 6524_0_random_IntArithmetic(x2, x3)
6524_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 6524_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_6524_1_createList_InvokeMethod2(x1, x2, x3, x4, x5, x6, x7) → Cond_6524_1_createList_InvokeMethod2(x1, x2, x3, x4)
Cond_6524_1_createList_InvokeMethod1(x1, x2, x3, x4, x5, x6, x7) → Cond_6524_1_createList_InvokeMethod1(x1, x2, x3, x4)
10511_0_createList_Inc(x1, x2, x3, x4) → 10511_0_createList_Inc(x2, x4)
Cond_6524_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_6524_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_6418_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_6418_1_createList_InvokeMethod(x1, x2, x3, x4, x5)
6418_0_random_ArrayAccess(x1, x2, x3) → 6418_0_random_ArrayAccess(x2, x3)
6418_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 6418_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_5956_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_5956_1_createList_InvokeMethod(x1, x2, x3, x4, x5)
5956_0_random_GT(x1, x2, x3) → 5956_0_random_GT(x2, x3)
5956_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 5956_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_4536_0_createList_Load1(x1, x2, x3, x4, x5, x6) → Cond_4536_0_createList_Load1(x1, x3, x4, x5, x6)
Cond_4536_0_createList_Load(x1, x2, x3, x4, x5, x6) → Cond_4536_0_createList_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:


6857_0_createList_NULL(x1, x2, x3, x4) → 6857_0_createList_NULL(x1, x2, x4)
4536_0_createList_Load(x1, x2, x3, x4) → 4536_0_createList_Load(x1, x2, x4)
Cond_4536_0_createList_Load1(x1, x2, x3, x4, x5) → Cond_4536_0_createList_Load1(x1, x2, x3, x5)
Cond_4536_0_createList_Load(x1, x2, x3, x4, x5) → Cond_4536_0_createList_Load(x1, x2, x3, x5)

Filtered all non-integer terms:


6524_1_createList_InvokeMethod(x1, x2, x3, x4) → 6524_1_createList_InvokeMethod(x1, x2, x3)
6524_0_random_IntArithmetic(x1, x2) → 6524_0_random_IntArithmetic(x2)
4536_0_createList_Load(x1, x2, x3) → 4536_0_createList_Load(x1, x3)
6857_0_createList_NULL(x1, x2, x3) → 6857_0_createList_NULL(x1, x2)

Filtered all free variables:


5956_1_createList_InvokeMethod(x1, x2, x3, x4) → 5956_1_createList_InvokeMethod(x2, x3, x4)
5958_1_createList_InvokeMethod(x1, x2, x3, x4) → 5958_1_createList_InvokeMethod(x2, x3, x4)
Cond_5956_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_5956_1_createList_InvokeMethod(x1, x3, x4, x5)
6418_1_createList_InvokeMethod(x1, x2, x3, x4) → 6418_1_createList_InvokeMethod(x2, x3, x4)
Cond_6418_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_6418_1_createList_InvokeMethod(x1, x3, x4, x5)
6524_1_createList_InvokeMethod(x1, x2, x3) → 6524_1_createList_InvokeMethod(x2, x3)
Cond_6524_1_createList_InvokeMethod(x1, x2, x3, x4) → Cond_6524_1_createList_InvokeMethod(x1, x3, x4)
Cond_6524_1_createList_InvokeMethod1(x1, x2, x3, x4) → Cond_6524_1_createList_InvokeMethod1(x1, x3, x4)
Cond_6524_1_createList_InvokeMethod2(x1, x2, x3, x4) → Cond_6524_1_createList_InvokeMethod2(x1, x3, x4)
Cond_6524_1_createList_InvokeMethod3(x1, x2, x3, x4, x5) → Cond_6524_1_createList_InvokeMethod3(x1, x3, x4, x5)
Cond_5958_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_5958_1_createList_InvokeMethod(x1, x3, x4, x5)

Combined rules. Obtained 5 rules for P and 0 rules for R.


Finished conversion. Obtained 5 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:
8677_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10878_0_copyR_Return
9434_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 11086_0_copyR_Return
8580_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10830_0_copyR_Return
9279_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10972_0_copyR_Return
9279_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10972_0_copyR_Return
9279_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10972_0_copyR_Return
9279_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10972_0_copyR_Return
8580_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10830_0_copyR_Return
8580_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10830_0_copyR_Return
8580_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10830_0_copyR_Return
9434_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 11086_0_copyR_Return
9434_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 11086_0_copyR_Return
9434_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 11086_0_copyR_Return
8677_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10878_0_copyR_Return
8677_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10878_0_copyR_Return
8677_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10878_0_copyR_Return

The integer pair graph contains the following rules and edges:
(0): 8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[0]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL)))
(1): 6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[1], x3[1]))) → 8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], NULL)))
(2): 6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], java.lang.Object(x1[2]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[2], x3[2]))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[2]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], NULL)))
(3): 6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[3], java.lang.Object(x1[3]))), NULL) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[3]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[3], NULL)))

(0) -> (1), if ((java.lang.Object(x1[0]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])))∧(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL)) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[1], x3[1]))))


(0) -> (2), if ((java.lang.Object(x1[0]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], java.lang.Object(x1[2]))))∧(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL)) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[2], x3[2]))))


(0) -> (3), if ((java.lang.Object(x1[0]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[3], java.lang.Object(x1[3]))))∧(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL)) →* NULL))


(1) -> (0), if ((java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))∧(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], NULL)) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL))))


(2) -> (1), if ((java.lang.Object(x1[2]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])))∧(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], NULL)) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[1], x3[1]))))


(2) -> (2), if ((java.lang.Object(x1[2]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2]', java.lang.Object(x1[2]'))))∧(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], NULL)) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[2]', x3[2]'))))


(2) -> (3), if ((java.lang.Object(x1[2]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[3], java.lang.Object(x1[3]))))∧(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], NULL)) →* NULL))


(3) -> (1), if ((java.lang.Object(x1[3]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])))∧(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[3], NULL)) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[1], x3[1]))))


(3) -> (2), if ((java.lang.Object(x1[3]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], java.lang.Object(x1[2]))))∧(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[3], NULL)) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[2], x3[2]))))


(3) -> (3), if ((java.lang.Object(x1[3]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[3]', java.lang.Object(x1[3]'))))∧(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[3], NULL)) →* NULL))



The set Q consists of the following terms:
8677_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9434_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8580_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9279_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8580_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8580_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8580_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9434_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9434_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9434_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8677_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8677_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8677_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, 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:

8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[0]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL)))
6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[1], x3[1]))) → 8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], NULL)))
6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], java.lang.Object(x1[2]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[2], x3[2]))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[2]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], NULL)))
6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[3], java.lang.Object(x1[3]))), NULL) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[3]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[3], NULL)))

The TRS R consists of the following rules:

8677_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10878_0_copyR_Return
9434_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 11086_0_copyR_Return
8580_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10830_0_copyR_Return
9279_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10972_0_copyR_Return
9279_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10972_0_copyR_Return
9279_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10972_0_copyR_Return
9279_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10972_0_copyR_Return
8580_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10830_0_copyR_Return
8580_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10830_0_copyR_Return
8580_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10830_0_copyR_Return
9434_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 11086_0_copyR_Return
9434_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 11086_0_copyR_Return
9434_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 11086_0_copyR_Return
8677_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10878_0_copyR_Return
8677_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10878_0_copyR_Return
8677_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10878_0_copyR_Return

The set Q consists of the following terms:

8677_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9434_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8580_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9279_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8580_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8580_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8580_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9434_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9434_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9434_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8677_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8677_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8677_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))

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

(8) DependencyGraphProof (EQUIVALENT transformation)

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

(9) Obligation:

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

6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[1], x3[1]))) → 8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], NULL)))
8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[0]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL)))
6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], java.lang.Object(x1[2]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[2], x3[2]))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[2]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], NULL)))

The TRS R consists of the following rules:

8677_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10878_0_copyR_Return
9434_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 11086_0_copyR_Return
8580_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10830_0_copyR_Return
9279_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10972_0_copyR_Return
9279_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10972_0_copyR_Return
9279_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10972_0_copyR_Return
9279_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10972_0_copyR_Return
8580_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10830_0_copyR_Return
8580_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10830_0_copyR_Return
8580_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10830_0_copyR_Return
9434_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 11086_0_copyR_Return
9434_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 11086_0_copyR_Return
9434_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 11086_0_copyR_Return
8677_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL))) → 10878_0_copyR_Return
8677_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10878_0_copyR_Return
8677_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x4, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x5, x6)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL))) → 10878_0_copyR_Return

The set Q consists of the following terms:

8677_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9434_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8580_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9279_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8580_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8580_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8580_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9434_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9434_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9434_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8677_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8677_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8677_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))

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

(10) 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.

(11) Obligation:

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

6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[1], x3[1]))) → 8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], NULL)))
8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[0]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL)))
6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], java.lang.Object(x1[2]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[2], x3[2]))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[2]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], NULL)))

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

8677_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9434_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8580_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9279_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8580_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8580_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8580_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9434_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9434_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9434_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8677_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8677_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8677_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))

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

(12) 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].

8677_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9434_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8580_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(8895_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9279_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9279_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8580_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8580_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8580_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9434_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
9434_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
9434_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8677_1_copyR_InvokeMethod(8969_0_copyR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))
8677_1_copyR_InvokeMethod(10972_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))
8677_1_copyR_InvokeMethod(11086_0_copyR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, x2)))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x3, NULL)))

(13) Obligation:

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

6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[1], x3[1]))) → 8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], NULL)))
8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[0]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL)))
6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], java.lang.Object(x1[2]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[2], x3[2]))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[2]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], NULL)))

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

(14) 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:

  • 8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[0]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], NULL)))
    The graph contains the following edges 1 > 1, 2 >= 2

  • 6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], java.lang.Object(x1[2]))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[2], x3[2]))) → 6402_0_COPYR_DUPLICATE(java.lang.Object(x1[2]), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[2], NULL)))
    The graph contains the following edges 1 > 1

  • 6402_0_COPYR_DUPLICATE(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2[1], x3[1]))) → 8185_0_COPYR_LOAD(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], x1[1])), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], NULL)))
    The graph contains the following edges 1 >= 1

(15) YES

(16) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:
none


The ITRS R consists of the following rules:
3319_1_getFirst_InvokeMethod(3257_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))) → 3347_0_getFirst_Return
3319_1_getFirst_InvokeMethod(3347_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))))) → 3573_0_getFirst_Return
3319_1_getFirst_InvokeMethod(3573_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0))))))) → 3573_0_getFirst_Return

The integer pair graph contains the following rules and edges:
(0): 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 3210_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
(1): 3210_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
(2): 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))

(0) -> (1), if ((java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))))∧(x0[0] →* java.lang.Object(x0[1])))


(1) -> (0), if ((java.lang.Object(x0[1]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))))


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


(2) -> (0), if ((java.lang.Object(x0[2]) →* java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))))


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



The set Q consists of the following terms:
3319_1_getFirst_InvokeMethod(3257_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
3319_1_getFirst_InvokeMethod(3347_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
3319_1_getFirst_InvokeMethod(3573_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0)))))))

(17) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(18) Obligation:

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

3076_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 3210_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
3210_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
3076_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))

The TRS R consists of the following rules:

3319_1_getFirst_InvokeMethod(3257_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))) → 3347_0_getFirst_Return
3319_1_getFirst_InvokeMethod(3347_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))))) → 3573_0_getFirst_Return
3319_1_getFirst_InvokeMethod(3573_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0))))))) → 3573_0_getFirst_Return

The set Q consists of the following terms:

3319_1_getFirst_InvokeMethod(3257_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
3319_1_getFirst_InvokeMethod(3347_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
3319_1_getFirst_InvokeMethod(3573_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0)))))))

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:

3076_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 3210_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
3210_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
3076_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))

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

3319_1_getFirst_InvokeMethod(3257_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
3319_1_getFirst_InvokeMethod(3347_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
3319_1_getFirst_InvokeMethod(3573_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0)))))))

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].

3319_1_getFirst_InvokeMethod(3257_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
3319_1_getFirst_InvokeMethod(3347_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
3319_1_getFirst_InvokeMethod(3573_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0)))))))

(22) Obligation:

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

3076_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 3210_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
3210_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
3076_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))

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

(23) 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:

  • 3210_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
    The graph contains the following edges 1 > 1, 2 >= 1

  • 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))
    The graph contains the following edges 1 > 1

  • 3076_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 3210_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
    The graph contains the following edges 1 >= 1, 1 > 2

(24) YES

(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:
(0): 10511_0_CREATELIST_INC(x0[0], x1[0]) → 4536_0_CREATELIST_LOAD(x0[0], x1[0] + 1)
(1): 4536_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_4536_0_CREATELIST_LOAD(x1[1] < x0[1], x0[1], x1[1])
(2): COND_4536_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10511_0_CREATELIST_INC(x0[2], x1[2])
(3): 4536_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_4536_0_CREATELIST_LOAD1(x1[3] < x0[3], x0[3], x1[3])
(4): COND_4536_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 4536_0_CREATELIST_LOAD(x0[4], x1[4] + 1)

(0) -> (1), if ((x0[0] →* x0[1])∧(x1[0] + 1 →* x1[1]))


(0) -> (3), if ((x0[0] →* x0[3])∧(x1[0] + 1 →* x1[3]))


(1) -> (2), if ((x1[1] < x0[1] →* TRUE)∧(x0[1] →* x0[2])∧(x1[1] →* x1[2]))


(2) -> (0), if ((x0[2] →* x0[0])∧(x1[2] →* x1[0]))


(3) -> (4), if ((x1[3] < x0[3] →* TRUE)∧(x0[3] →* x0[4])∧(x1[3] →* x1[4]))


(4) -> (1), if ((x0[4] →* x0[1])∧(x1[4] + 1 →* x1[1]))


(4) -> (3), if ((x0[4] →* x0[3])∧(x1[4] + 1 →* x1[3]))



The set Q is empty.

(26) 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 10511_0_CREATELIST_INC(x0, x1) → 4536_0_CREATELIST_LOAD(x0, +(x1, 1)) the following chains were created:
  • We consider the chain 10511_0_CREATELIST_INC(x0[0], x1[0]) → 4536_0_CREATELIST_LOAD(x0[0], +(x1[0], 1)) which results in the following constraint:

    (1)    (10511_0_CREATELIST_INC(x0[0], x1[0])≥NonInfC∧10511_0_CREATELIST_INC(x0[0], x1[0])≥4536_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))∧(UIncreasing(4536_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥))



    We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (2)    ((UIncreasing(4536_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥)∧[1 + (-1)bso_12] ≥ 0)



    We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (3)    ((UIncreasing(4536_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥)∧[1 + (-1)bso_12] ≥ 0)



    We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (4)    ((UIncreasing(4536_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥)∧[1 + (-1)bso_12] ≥ 0)



    We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (5)    ((UIncreasing(4536_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)







For Pair 4536_0_CREATELIST_LOAD(x0, x1) → COND_4536_0_CREATELIST_LOAD(<(x1, x0), x0, x1) the following chains were created:
  • We consider the chain 4536_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1]), COND_4536_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10511_0_CREATELIST_INC(x0[2], x1[2]) which results in the following constraint:

    (6)    (<(x1[1], x0[1])=TRUE∧x0[1]=x0[2]∧x1[1]=x1[2] ⇒ 4536_0_CREATELIST_LOAD(x0[1], x1[1])≥NonInfC∧4536_0_CREATELIST_LOAD(x0[1], x1[1])≥COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])∧(UIncreasing(COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])), ≥))



    We simplified constraint (6) using rule (IV) which results in the following new constraint:

    (7)    (<(x1[1], x0[1])=TRUE ⇒ 4536_0_CREATELIST_LOAD(x0[1], x1[1])≥NonInfC∧4536_0_CREATELIST_LOAD(x0[1], x1[1])≥COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])∧(UIncreasing(COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])), ≥))



    We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (8)    (x0[1] + [-1] + [-1]x1[1] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[1] + [bni_13]x0[1] ≥ 0∧[(-1)bso_14] ≥ 0)



    We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (9)    (x0[1] + [-1] + [-1]x1[1] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[1] + [bni_13]x0[1] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (10)    (x0[1] + [-1] + [-1]x1[1] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[1] + [bni_13]x0[1] ≥ 0∧[(-1)bso_14] ≥ 0)



    We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (11)    (x0[1] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[1] ≥ 0∧[(-1)bso_14] ≥ 0)



    We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (12)    (x0[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[1] ≥ 0∧[(-1)bso_14] ≥ 0)


    (13)    (x0[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[1] ≥ 0∧[(-1)bso_14] ≥ 0)







For Pair COND_4536_0_CREATELIST_LOAD(TRUE, x0, x1) → 10511_0_CREATELIST_INC(x0, x1) the following chains were created:
  • We consider the chain COND_4536_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10511_0_CREATELIST_INC(x0[2], x1[2]), 10511_0_CREATELIST_INC(x0[0], x1[0]) → 4536_0_CREATELIST_LOAD(x0[0], +(x1[0], 1)) which results in the following constraint:

    (14)    (x0[2]=x0[0]∧x1[2]=x1[0] ⇒ COND_4536_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥NonInfC∧COND_4536_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥10511_0_CREATELIST_INC(x0[2], x1[2])∧(UIncreasing(10511_0_CREATELIST_INC(x0[2], x1[2])), ≥))



    We simplified constraint (14) using rule (IV) which results in the following new constraint:

    (15)    (COND_4536_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥NonInfC∧COND_4536_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥10511_0_CREATELIST_INC(x0[2], x1[2])∧(UIncreasing(10511_0_CREATELIST_INC(x0[2], x1[2])), ≥))



    We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (16)    ((UIncreasing(10511_0_CREATELIST_INC(x0[2], x1[2])), ≥)∧[(-1)bso_16] ≥ 0)



    We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (17)    ((UIncreasing(10511_0_CREATELIST_INC(x0[2], x1[2])), ≥)∧[(-1)bso_16] ≥ 0)



    We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (18)    ((UIncreasing(10511_0_CREATELIST_INC(x0[2], x1[2])), ≥)∧[(-1)bso_16] ≥ 0)



    We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (19)    ((UIncreasing(10511_0_CREATELIST_INC(x0[2], x1[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)







For Pair 4536_0_CREATELIST_LOAD(x0, x1) → COND_4536_0_CREATELIST_LOAD1(<(x1, x0), x0, x1) the following chains were created:
  • We consider the chain 4536_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3]), COND_4536_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 4536_0_CREATELIST_LOAD(x0[4], +(x1[4], 1)) which results in the following constraint:

    (20)    (<(x1[3], x0[3])=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4] ⇒ 4536_0_CREATELIST_LOAD(x0[3], x1[3])≥NonInfC∧4536_0_CREATELIST_LOAD(x0[3], x1[3])≥COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])∧(UIncreasing(COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])), ≥))



    We simplified constraint (20) using rule (IV) which results in the following new constraint:

    (21)    (<(x1[3], x0[3])=TRUE ⇒ 4536_0_CREATELIST_LOAD(x0[3], x1[3])≥NonInfC∧4536_0_CREATELIST_LOAD(x0[3], x1[3])≥COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])∧(UIncreasing(COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])), ≥))



    We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (22)    (x0[3] + [-1] + [-1]x1[3] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x1[3] + [bni_17]x0[3] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (23)    (x0[3] + [-1] + [-1]x1[3] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x1[3] + [bni_17]x0[3] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (24)    (x0[3] + [-1] + [-1]x1[3] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x1[3] + [bni_17]x0[3] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (24) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (25)    (x0[3] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[3] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (26)    (x0[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[3] ≥ 0∧[(-1)bso_18] ≥ 0)


    (27)    (x0[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[3] ≥ 0∧[(-1)bso_18] ≥ 0)







For Pair COND_4536_0_CREATELIST_LOAD1(TRUE, x0, x1) → 4536_0_CREATELIST_LOAD(x0, +(x1, 1)) the following chains were created:
  • We consider the chain COND_4536_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 4536_0_CREATELIST_LOAD(x0[4], +(x1[4], 1)) which results in the following constraint:

    (28)    (COND_4536_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4])≥NonInfC∧COND_4536_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4])≥4536_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))∧(UIncreasing(4536_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))), ≥))



    We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (29)    ((UIncreasing(4536_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))), ≥)∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (30)    ((UIncreasing(4536_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))), ≥)∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (31)    ((UIncreasing(4536_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))), ≥)∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (31) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (32)    ((UIncreasing(4536_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 10511_0_CREATELIST_INC(x0, x1) → 4536_0_CREATELIST_LOAD(x0, +(x1, 1))
    • ((UIncreasing(4536_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  • 4536_0_CREATELIST_LOAD(x0, x1) → COND_4536_0_CREATELIST_LOAD(<(x1, x0), x0, x1)
    • (x0[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[1] ≥ 0∧[(-1)bso_14] ≥ 0)
    • (x0[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[1] ≥ 0∧[(-1)bso_14] ≥ 0)

  • COND_4536_0_CREATELIST_LOAD(TRUE, x0, x1) → 10511_0_CREATELIST_INC(x0, x1)
    • ((UIncreasing(10511_0_CREATELIST_INC(x0[2], x1[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

  • 4536_0_CREATELIST_LOAD(x0, x1) → COND_4536_0_CREATELIST_LOAD1(<(x1, x0), x0, x1)
    • (x0[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[3] ≥ 0∧[(-1)bso_18] ≥ 0)
    • (x0[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[3] ≥ 0∧[(-1)bso_18] ≥ 0)

  • COND_4536_0_CREATELIST_LOAD1(TRUE, x0, x1) → 4536_0_CREATELIST_LOAD(x0, +(x1, 1))
    • ((UIncreasing(4536_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 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(10511_0_CREATELIST_INC(x1, x2)) = [-1] + x1 + [-1]x2   
POL(4536_0_CREATELIST_LOAD(x1, x2)) = [-1] + [-1]x2 + x1   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(COND_4536_0_CREATELIST_LOAD(x1, x2, x3)) = [-1] + [-1]x3 + x2   
POL(<(x1, x2)) = [-1]   
POL(COND_4536_0_CREATELIST_LOAD1(x1, x2, x3)) = [-1] + [-1]x3 + x2   

The following pairs are in P>:

10511_0_CREATELIST_INC(x0[0], x1[0]) → 4536_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))
COND_4536_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 4536_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))

The following pairs are in Pbound:

4536_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])
4536_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])

The following pairs are in P≥:

4536_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_4536_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])
COND_4536_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10511_0_CREATELIST_INC(x0[2], x1[2])
4536_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_4536_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])

There are no usable rules.

(27) Complex Obligation (AND)

(28) 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): 4536_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_4536_0_CREATELIST_LOAD(x1[1] < x0[1], x0[1], x1[1])
(2): COND_4536_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10511_0_CREATELIST_INC(x0[2], x1[2])
(3): 4536_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_4536_0_CREATELIST_LOAD1(x1[3] < x0[3], x0[3], x1[3])

(1) -> (2), if ((x1[1] < x0[1] →* TRUE)∧(x0[1] →* x0[2])∧(x1[1] →* x1[2]))



The set Q is empty.

(29) IDependencyGraphProof (EQUIVALENT transformation)

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

(30) TRUE

(31) 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:
(0): 10511_0_CREATELIST_INC(x0[0], x1[0]) → 4536_0_CREATELIST_LOAD(x0[0], x1[0] + 1)
(2): COND_4536_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10511_0_CREATELIST_INC(x0[2], x1[2])
(4): COND_4536_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 4536_0_CREATELIST_LOAD(x0[4], x1[4] + 1)

(2) -> (0), if ((x0[2] →* x0[0])∧(x1[2] →* x1[0]))



The set Q is empty.

(32) IDependencyGraphProof (EQUIVALENT transformation)

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

(33) TRUE