Termination proof of /tmp/tmplLonMe/MainGet.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPNonInfProof (⇒)

        ↳7 IDP

        ↳8 IDependencyGraphProof (⇔)

        ↳9 TRUE

    ↳10 IDP

        ↳11 IDPtoQDPProof (⇒)

        ↳12 QDP

        ↳13 UsableRulesProof (⇔)

        ↳14 QDP

        ↳15 QReductionProof (⇔)

        ↳16 QDP

        ↳17 QDPSizeChangeProof (⇔)

        ↳18 YES

    ↳19 IDP

        ↳20 IDPNonInfProof (⇒)

        ↳21 AND

            ↳22 IDP

                ↳23 IDependencyGraphProof (⇔)

                ↳24 TRUE

            ↳25 IDP

                ↳26 IDependencyGraphProof (⇔)

                ↳27 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Sun Microsystems Inc.) Main-Class: DoublyLinkedList/MainGet

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) {
        return this.getFirst().getR(index);
    }

    private DoublyLinkedList getR(final int index) {
      if (index == 0 || this.next == null) {
        return this;
      }
      return this.next.getR(index - 1);
    }

    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 MainGet {
    public static void main(final String[] args) {
        Random.args = args;
        final DoublyLinkedList list = DoublyLinkedList.createList();
        list.get(Random.random());
    }
}


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.MainGet.main([Ljava/lang/String;)V: Graph of 250 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.getR(I)LDoublyLinkedList/DoublyLinkedList;: Graph of 127 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 62 rules for P and 90 rules for R.

Combined rules. Obtained 3 rules for P and 31 rules for R.

Filtered ground terms:

6218_0_getR_EQ(x1, x2, x3, x4) → 6218_0_getR_EQ(x2, x3, x4)
Cond_6218_0_getR_EQ2(x1, x2, x3, x4, x5) → Cond_6218_0_getR_EQ2(x1, x3, x4, x5)
DoublyLinkedList.DoublyLinkedList(x1, x2) → DoublyLinkedList.DoublyLinkedList(x2)
Cond_6218_0_getR_EQ1(x1, x2, x3, x4, x5) → Cond_6218_0_getR_EQ1(x1, x3, x4, x5)
Cond_6218_0_getR_EQ(x1, x2, x3, x4, x5) → Cond_6218_0_getR_EQ(x1, x3, x4, x5)
9096_0_getR_Return(x1) → 9096_0_getR_Return
9251_0_getR_Return(x1) → 9251_0_getR_Return
9162_0_getR_Return(x1) → 9162_0_getR_Return
8302_0_getR_Return(x1, x2) → 8302_0_getR_Return(x2)
8280_0_getR_Return(x1, x2) → 8280_0_getR_Return(x2)
8265_0_getR_Return(x1, x2) → 8265_0_getR_Return(x2)
7348_0_getR_Return(x1, x2, x3, x4) → 7348_0_getR_Return(x3)
7273_0_getR_Return(x1, x2, x3, x4) → 7273_0_getR_Return(x3)
7202_0_getR_Return(x1, x2, x3, x4) → 7202_0_getR_Return(x3)
6286_0_getR_Return(x1, x2, x3, x4) → 6286_0_getR_Return(x2, x4)

Filtered duplicate args:

6218_0_getR_EQ(x1, x2, x3) → 6218_0_getR_EQ(x1, x3)
Cond_6218_0_getR_EQ2(x1, x2, x3, x4) → Cond_6218_0_getR_EQ2(x1, x2, x4)
Cond_6218_0_getR_EQ1(x1, x2, x3, x4) → Cond_6218_0_getR_EQ1(x1, x2, x4)
Cond_6218_0_getR_EQ(x1, x2, x3, x4) → Cond_6218_0_getR_EQ(x1, x2, x4)
6286_0_getR_Return(x1, x2) → 6286_0_getR_Return(x2)

Combined rules. Obtained 3 rules for P and 31 rules for R.

Finished conversion. Obtained 3 rules for P and 31 rules for R. System has 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:

3788_0_getFirst_FieldAccess(x1, x2, x3) → 3788_0_getFirst_FieldAccess(x2, x3)
DoublyLinkedList.DoublyLinkedList(x1, x2) → DoublyLinkedList.DoublyLinkedList(x2)
4849_0_getFirst_NONNULL(x1, x2, x3) → 4849_0_getFirst_NONNULL(x2, x3)
5394_0_getFirst_Return(x1) → 5394_0_getFirst_Return
5050_0_getFirst_Return(x1, x2) → 5050_0_getFirst_Return
5001_0_getFirst_Return(x1, x2, x3) → 5001_0_getFirst_Return

Filtered duplicate args:

3788_0_getFirst_FieldAccess(x1, x2) → 3788_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:

9874_0_createList_NULL(x1, x2, x3, x4, x5, x6) → 9874_0_createList_NULL(x2, x4, x5, x6)
DoublyLinkedList.DoublyLinkedList(x1) → DoublyLinkedList.DoublyLinkedList
Cond_9255_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_9255_1_createList_InvokeMethod(x1, x2, x3, x4, x5)
9255_0_random_GT(x1, x2, x3) → 9255_0_random_GT(x2, x3)
9255_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 9255_1_createList_InvokeMethod(x1, x2, x3, x4)
6830_0_createList_Load(x1, x2, x3, x4, x5) → 6830_0_createList_Load(x2, x3, x4, x5)
10029_0_createList_Inc(x1, x2, x3, x4) → 10029_0_createList_Inc(x2, x4)
Cond_9803_1_createList_InvokeMethod3(x1, x2, x3, x4, x5, x6, x7) → Cond_9803_1_createList_InvokeMethod3(x1, x2, x3, x4, x5)
9803_0_random_IntArithmetic(x1, x2, x3, x4) → 9803_0_random_IntArithmetic(x2, x3)
9803_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 9803_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_9803_1_createList_InvokeMethod2(x1, x2, x3, x4, x5, x6, x7) → Cond_9803_1_createList_InvokeMethod2(x1, x2, x3, x4)
Cond_9803_1_createList_InvokeMethod1(x1, x2, x3, x4, x5, x6, x7) → Cond_9803_1_createList_InvokeMethod1(x1, x2, x3, x4)
10240_0_createList_Inc(x1, x2, x3, x4) → 10240_0_createList_Inc(x2, x4)
Cond_9803_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_9803_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_9767_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_9767_1_createList_InvokeMethod(x1, x2, x3, x4, x5)
9767_0_random_ArrayAccess(x1, x2, x3) → 9767_0_random_ArrayAccess(x2, x3)
9767_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 9767_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_9253_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_9253_1_createList_InvokeMethod(x1, x2, x3, x4, x5)
9253_0_random_GT(x1, x2, x3) → 9253_0_random_GT(x2, x3)
9253_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 9253_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_6830_0_createList_Load1(x1, x2, x3, x4, x5, x6) → Cond_6830_0_createList_Load1(x1, x3, x4, x5, x6)
Cond_6830_0_createList_Load(x1, x2, x3, x4, x5, x6) → Cond_6830_0_createList_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:

9874_0_createList_NULL(x1, x2, x3, x4) → 9874_0_createList_NULL(x1, x2, x4)
6830_0_createList_Load(x1, x2, x3, x4) → 6830_0_createList_Load(x1, x2, x4)
Cond_6830_0_createList_Load1(x1, x2, x3, x4, x5) → Cond_6830_0_createList_Load1(x1, x2, x3, x5)
Cond_6830_0_createList_Load(x1, x2, x3, x4, x5) → Cond_6830_0_createList_Load(x1, x2, x3, x5)

Filtered all non-integer terms:

9803_1_createList_InvokeMethod(x1, x2, x3, x4) → 9803_1_createList_InvokeMethod(x1, x2, x3)
9803_0_random_IntArithmetic(x1, x2) → 9803_0_random_IntArithmetic(x2)
6830_0_createList_Load(x1, x2, x3) → 6830_0_createList_Load(x1, x3)
9874_0_createList_NULL(x1, x2, x3) → 9874_0_createList_NULL(x1, x2)

Filtered all free variables:

9253_1_createList_InvokeMethod(x1, x2, x3, x4) → 9253_1_createList_InvokeMethod(x2, x3, x4)
9255_1_createList_InvokeMethod(x1, x2, x3, x4) → 9255_1_createList_InvokeMethod(x2, x3, x4)
Cond_9253_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_9253_1_createList_InvokeMethod(x1, x3, x4, x5)
9767_1_createList_InvokeMethod(x1, x2, x3, x4) → 9767_1_createList_InvokeMethod(x2, x3, x4)
Cond_9767_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_9767_1_createList_InvokeMethod(x1, x3, x4, x5)
9803_1_createList_InvokeMethod(x1, x2, x3) → 9803_1_createList_InvokeMethod(x2, x3)
Cond_9803_1_createList_InvokeMethod(x1, x2, x3, x4) → Cond_9803_1_createList_InvokeMethod(x1, x3, x4)
Cond_9803_1_createList_InvokeMethod1(x1, x2, x3, x4) → Cond_9803_1_createList_InvokeMethod1(x1, x3, x4)
Cond_9803_1_createList_InvokeMethod2(x1, x2, x3, x4) → Cond_9803_1_createList_InvokeMethod2(x1, x3, x4)
Cond_9803_1_createList_InvokeMethod3(x1, x2, x3, x4, x5) → Cond_9803_1_createList_InvokeMethod3(x1, x3, x4, x5)
Cond_9255_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_9255_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:

Boolean, Integer

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

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

The set Q consists of the following terms:
8041_1_getR_InvokeMethod(8265_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
8169_1_getR_InvokeMethod(8265_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
7891_1_getR_InvokeMethod(8265_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
6218_0_getR_EQ(java.lang.Object(x0), 0)
8041_1_getR_InvokeMethod(6286_0_getR_Return(java.lang.Object(x0)), java.lang.Object(x0), 0)
8041_1_getR_InvokeMethod(7202_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8041_1_getR_InvokeMethod(7273_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8041_1_getR_InvokeMethod(7348_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8041_1_getR_InvokeMethod(8280_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
8041_1_getR_InvokeMethod(8302_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
8041_1_getR_InvokeMethod(9096_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
8041_1_getR_InvokeMethod(9162_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
8041_1_getR_InvokeMethod(9251_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
8169_1_getR_InvokeMethod(6286_0_getR_Return(java.lang.Object(x0)), java.lang.Object(x0), 0)
8169_1_getR_InvokeMethod(7202_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8169_1_getR_InvokeMethod(7273_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8169_1_getR_InvokeMethod(7348_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8169_1_getR_InvokeMethod(8280_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
8169_1_getR_InvokeMethod(8302_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
8169_1_getR_InvokeMethod(9096_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
8169_1_getR_InvokeMethod(9162_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
8169_1_getR_InvokeMethod(9251_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
7891_1_getR_InvokeMethod(6286_0_getR_Return(java.lang.Object(x0)), java.lang.Object(x0), 0)
7891_1_getR_InvokeMethod(7202_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
7891_1_getR_InvokeMethod(7273_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
7891_1_getR_InvokeMethod(7348_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
7891_1_getR_InvokeMethod(8280_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
7891_1_getR_InvokeMethod(8302_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
7891_1_getR_InvokeMethod(9096_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
7891_1_getR_InvokeMethod(9162_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
7891_1_getR_InvokeMethod(9251_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)

(6) 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 6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → COND_6218_0_GETR_EQ(!(=(x1, 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) the following chains were created:

We consider the chain 6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0]) → COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0]), COND_6218_0_GETR_EQ(TRUE, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), x1[1]) → 6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1)) which results in the following constraint:
(1)    (!(=(x1[0], 0))=TRUE∧java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0])))=java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1])))∧x1[0]=x1[1] ⇒ 6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])≥NonInfC∧6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])≥COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])∧(U^Increasing(COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
(2)    (!(=(x1[0], 0))=TRUE ⇒ 6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])≥NonInfC∧6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])≥COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])∧(U^Increasing(COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (0 ≥ 0 ⇒ (U^Increasing(COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])), ≥)∧[(80)bni_45 + (-1)Bound*bni_45] + [bni_45]x1[0] + [(54)bni_45]x0[0] ≥ 0∧[(-1)bso_46] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (0 ≥ 0 ⇒ (U^Increasing(COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])), ≥)∧[(80)bni_45 + (-1)Bound*bni_45] + [bni_45]x1[0] + [(54)bni_45]x0[0] ≥ 0∧[(-1)bso_46] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (0 ≥ 0 ⇒ (U^Increasing(COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])), ≥)∧[(80)bni_45 + (-1)Bound*bni_45] + [bni_45]x1[0] + [(54)bni_45]x0[0] ≥ 0∧[(-1)bso_46] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (0 ≥ 0 ⇒ (U^Increasing(COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])), ≥)∧[bni_45] ≥ 0∧[(54)bni_45] ≥ 0∧[(80)bni_45 + (-1)Bound*bni_45] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_46] ≥ 0)

For Pair COND_6218_0_GETR_EQ(TRUE, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 6218_0_GETR_EQ(java.lang.Object(x0), -(x1, 1)) the following chains were created:

We consider the chain 6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0]) → COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0]), COND_6218_0_GETR_EQ(TRUE, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), x1[1]) → 6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1)), 6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0]) → COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0]) which results in the following constraint:
(7)    (!(=(x1[0], 0))=TRUE∧java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0])))=java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1])))∧x1[0]=x1[1]∧java.lang.Object(x0[1])=java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]1)))∧-(x1[1], 1)=x1[0]1 ⇒ COND_6218_0_GETR_EQ(TRUE, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), x1[1])≥NonInfC∧COND_6218_0_GETR_EQ(TRUE, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), x1[1])≥6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1))∧(U^Increasing(6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1))), ≥))

We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:
(8)    (!(=(x1[0], 0))=TRUE ⇒ COND_6218_0_GETR_EQ(TRUE, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]1))))), x1[0])≥NonInfC∧COND_6218_0_GETR_EQ(TRUE, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]1))))), x1[0])≥6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]1))), -(x1[0], 1))∧(U^Increasing(6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    (0 ≥ 0 ⇒ (U^Increasing(6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1))), ≥)∧[(728)bni_47 + (-1)Bound*bni_47] + [bni_47]x1[0] + [(486)bni_47]x0[0]1 ≥ 0∧[648 + (-1)bso_48] + x1[0] + [432]x0[0]1 ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    (0 ≥ 0 ⇒ (U^Increasing(6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1))), ≥)∧[(728)bni_47 + (-1)Bound*bni_47] + [bni_47]x1[0] + [(486)bni_47]x0[0]1 ≥ 0∧[648 + (-1)bso_48] + x1[0] + [432]x0[0]1 ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (0 ≥ 0 ⇒ (U^Increasing(6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1))), ≥)∧[(728)bni_47 + (-1)Bound*bni_47] + [bni_47]x1[0] + [(486)bni_47]x0[0]1 ≥ 0∧[648 + (-1)bso_48] + x1[0] + [432]x0[0]1 ≥ 0)

We simplified constraint (11) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(12)    (0 ≥ 0 ⇒ (U^Increasing(6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1))), ≥)∧[bni_47] ≥ 0∧[(486)bni_47] ≥ 0∧[(728)bni_47 + (-1)Bound*bni_47] ≥ 0∧[1] ≥ 0∧[648 + (-1)bso_48] ≥ 0∧[1] ≥ 0)

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

6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → COND_6218_0_GETR_EQ(!(=(x1, 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
- (0 ≥ 0 ⇒ (U^Increasing(COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])), ≥)∧[bni_45] ≥ 0∧[(54)bni_45] ≥ 0∧[(80)bni_45 + (-1)Bound*bni_45] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_46] ≥ 0)
COND_6218_0_GETR_EQ(TRUE, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 6218_0_GETR_EQ(java.lang.Object(x0), -(x1, 1))
- (0 ≥ 0 ⇒ (U^Increasing(6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1))), ≥)∧[bni_47] ≥ 0∧[(486)bni_47] ≥ 0∧[(728)bni_47 + (-1)Bound*bni_47] ≥ 0∧[1] ≥ 0∧[648 + (-1)bso_48] ≥ 0∧[1] ≥ 0)

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(8041_1_getR_InvokeMethod(x₁, x₂, x₃)) = 0
POL(8265_0_getR_Return(x₁)) = 0
POL(java.lang.Object(x₁)) = [3] + [3]x₁
POL(DoublyLinkedList.DoublyLinkedList(x₁)) = [3] + [3]x₁
POL(9162_0_getR_Return) = 0
POL(8169_1_getR_InvokeMethod(x₁, x₂, x₃)) = 0
POL(9251_0_getR_Return) = 0
POL(7891_1_getR_InvokeMethod(x₁, x₂, x₃)) = 0
POL(9096_0_getR_Return) = 0
POL(6218_0_getR_EQ(x₁, x₂)) = 0
POL(0) = 0
POL(6286_0_getR_Return(x₁)) = 0
POL(8280_0_getR_Return(x₁)) = 0
POL(7202_0_getR_Return(x₁)) = 0
POL(NULL) = 0
POL(7273_0_getR_Return(x₁)) = 0
POL(7348_0_getR_Return(x₁)) = 0
POL(8302_0_getR_Return(x₁)) = 0
POL(6218_0_GETR_EQ(x₁, x₂)) = [2] + x₂ + [2]x₁
POL(COND_6218_0_GETR_EQ(x₁, x₂, x₃)) = [2] + x₃ + [2]x₂
POL(!(x₁)) = 0
POL(=(x₁, x₂)) = 0
POL(-(x₁, x₂)) = 0
POL(1) = 0

The following pairs are in P_>:

COND_6218_0_GETR_EQ(TRUE, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), x1[1]) → 6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1))

The following pairs are in P_bound:

6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0]) → COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])
COND_6218_0_GETR_EQ(TRUE, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), x1[1]) → 6218_0_GETR_EQ(java.lang.Object(x0[1]), -(x1[1], 1))

The following pairs are in P_≥:

6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0]) → COND_6218_0_GETR_EQ(!(=(x1[0], 0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])

At least the following rules have been oriented under context sensitive arithmetic replacement:

!(TRUE)¹ → FALSE¹
!(FALSE)¹ → TRUE¹

(7) Obligation:

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

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

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
8041_1_getR_InvokeMethod(8265_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 9162_0_getR_Return
8169_1_getR_InvokeMethod(8265_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 9251_0_getR_Return
7891_1_getR_InvokeMethod(8265_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 9096_0_getR_Return
6218_0_getR_EQ(java.lang.Object(x0), 0) → 6286_0_getR_Return(java.lang.Object(x0))
8041_1_getR_InvokeMethod(6286_0_getR_Return(java.lang.Object(x0)), java.lang.Object(x0), 0) → 8280_0_getR_Return(java.lang.Object(x0))
8041_1_getR_InvokeMethod(7202_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0) → 8280_0_getR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
8041_1_getR_InvokeMethod(7273_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0) → 8280_0_getR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
8041_1_getR_InvokeMethod(7348_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0) → 8280_0_getR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
8041_1_getR_InvokeMethod(8280_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 9162_0_getR_Return
8041_1_getR_InvokeMethod(8302_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 9162_0_getR_Return
8041_1_getR_InvokeMethod(9096_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1) → 9162_0_getR_Return
8041_1_getR_InvokeMethod(9162_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1) → 9162_0_getR_Return
8041_1_getR_InvokeMethod(9251_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1) → 9162_0_getR_Return
8169_1_getR_InvokeMethod(6286_0_getR_Return(java.lang.Object(x0)), java.lang.Object(x0), 0) → 8302_0_getR_Return(java.lang.Object(x0))
8169_1_getR_InvokeMethod(7202_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0) → 8302_0_getR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
8169_1_getR_InvokeMethod(7273_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0) → 8302_0_getR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
8169_1_getR_InvokeMethod(7348_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0) → 8302_0_getR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
8169_1_getR_InvokeMethod(8280_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 9251_0_getR_Return
8169_1_getR_InvokeMethod(8302_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 9251_0_getR_Return
8169_1_getR_InvokeMethod(9096_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1) → 9251_0_getR_Return
8169_1_getR_InvokeMethod(9162_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1) → 9251_0_getR_Return
8169_1_getR_InvokeMethod(9251_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1) → 9251_0_getR_Return
7891_1_getR_InvokeMethod(6286_0_getR_Return(java.lang.Object(x0)), java.lang.Object(x0), 0) → 8265_0_getR_Return(java.lang.Object(x0))
7891_1_getR_InvokeMethod(7202_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0) → 8265_0_getR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
7891_1_getR_InvokeMethod(7273_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0) → 8265_0_getR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
7891_1_getR_InvokeMethod(7348_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0) → 8265_0_getR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
7891_1_getR_InvokeMethod(8280_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 9096_0_getR_Return
7891_1_getR_InvokeMethod(8302_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1) → 9096_0_getR_Return
7891_1_getR_InvokeMethod(9096_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1) → 9096_0_getR_Return
7891_1_getR_InvokeMethod(9162_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1) → 9096_0_getR_Return
7891_1_getR_InvokeMethod(9251_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1) → 9096_0_getR_Return

The integer pair graph contains the following rules and edges:
(0): 6218_0_GETR_EQ(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0]) → COND_6218_0_GETR_EQ(!(x1[0] = 0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[0]))), x1[0])

The set Q consists of the following terms:
8041_1_getR_InvokeMethod(8265_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
8169_1_getR_InvokeMethod(8265_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
7891_1_getR_InvokeMethod(8265_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
6218_0_getR_EQ(java.lang.Object(x0), 0)
8041_1_getR_InvokeMethod(6286_0_getR_Return(java.lang.Object(x0)), java.lang.Object(x0), 0)
8041_1_getR_InvokeMethod(7202_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8041_1_getR_InvokeMethod(7273_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8041_1_getR_InvokeMethod(7348_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8041_1_getR_InvokeMethod(8280_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
8041_1_getR_InvokeMethod(8302_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
8041_1_getR_InvokeMethod(9096_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
8041_1_getR_InvokeMethod(9162_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
8041_1_getR_InvokeMethod(9251_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
8169_1_getR_InvokeMethod(6286_0_getR_Return(java.lang.Object(x0)), java.lang.Object(x0), 0)
8169_1_getR_InvokeMethod(7202_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8169_1_getR_InvokeMethod(7273_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8169_1_getR_InvokeMethod(7348_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
8169_1_getR_InvokeMethod(8280_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
8169_1_getR_InvokeMethod(8302_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
8169_1_getR_InvokeMethod(9096_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
8169_1_getR_InvokeMethod(9162_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
8169_1_getR_InvokeMethod(9251_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
7891_1_getR_InvokeMethod(6286_0_getR_Return(java.lang.Object(x0)), java.lang.Object(x0), 0)
7891_1_getR_InvokeMethod(7202_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
7891_1_getR_InvokeMethod(7273_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
7891_1_getR_InvokeMethod(7348_0_getR_Return(x0), java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)), x0)
7891_1_getR_InvokeMethod(8280_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
7891_1_getR_InvokeMethod(8302_0_getR_Return(java.lang.Object(x0)), java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))), x1)
7891_1_getR_InvokeMethod(9096_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
7891_1_getR_InvokeMethod(9162_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)
7891_1_getR_InvokeMethod(9251_0_getR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0))))), x1)

(8) IDependencyGraphProof (EQUIVALENT transformation)

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

(9) TRUE

(10) 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:
5030_1_getFirst_InvokeMethod(5001_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))) → 5050_0_getFirst_Return
5030_1_getFirst_InvokeMethod(5050_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))))) → 5394_0_getFirst_Return
5030_1_getFirst_InvokeMethod(5394_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0))))))) → 5394_0_getFirst_Return

The integer pair graph contains the following rules and edges:
(0): 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 4849_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
(1): 4849_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
(2): 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 3788_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:
5030_1_getFirst_InvokeMethod(5001_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
5030_1_getFirst_InvokeMethod(5050_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
5030_1_getFirst_InvokeMethod(5394_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0)))))))

(11) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(12) Obligation:

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

3788_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 4849_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
4849_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
3788_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))

The TRS R consists of the following rules:

5030_1_getFirst_InvokeMethod(5001_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))) → 5050_0_getFirst_Return
5030_1_getFirst_InvokeMethod(5050_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))))) → 5394_0_getFirst_Return
5030_1_getFirst_InvokeMethod(5394_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0))))))) → 5394_0_getFirst_Return

The set Q consists of the following terms:

5030_1_getFirst_InvokeMethod(5001_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
5030_1_getFirst_InvokeMethod(5050_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
5030_1_getFirst_InvokeMethod(5394_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.

(13) 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.

(14) Obligation:

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

3788_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 4849_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
4849_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
3788_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))

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

5030_1_getFirst_InvokeMethod(5001_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
5030_1_getFirst_InvokeMethod(5050_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
5030_1_getFirst_InvokeMethod(5394_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.

(15) 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].

5030_1_getFirst_InvokeMethod(5001_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
5030_1_getFirst_InvokeMethod(5050_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
5030_1_getFirst_InvokeMethod(5394_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0)))))))

(16) Obligation:

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

3788_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 4849_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
4849_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
3788_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))

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

(17) 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:

4849_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
The graph contains the following edges 1 > 1, 2 >= 1
3788_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 3788_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))
The graph contains the following edges 1 > 1
3788_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 4849_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
The graph contains the following edges 1 >= 1, 1 > 2

(18) YES

(19) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 10240_0_CREATELIST_INC(x0[0], x1[0]) → 6830_0_CREATELIST_LOAD(x0[0], x1[0] + 1)
(1): 6830_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_6830_0_CREATELIST_LOAD(x1[1] < x0[1], x0[1], x1[1])
(2): COND_6830_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10240_0_CREATELIST_INC(x0[2], x1[2])
(3): 6830_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_6830_0_CREATELIST_LOAD1(x1[3] < x0[3], x0[3], x1[3])
(4): COND_6830_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 6830_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.

(20) 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 10240_0_CREATELIST_INC(x0, x1) → 6830_0_CREATELIST_LOAD(x0, +(x1, 1)) the following chains were created:

We consider the chain 10240_0_CREATELIST_INC(x0[0], x1[0]) → 6830_0_CREATELIST_LOAD(x0[0], +(x1[0], 1)) which results in the following constraint:
(1)    (10240_0_CREATELIST_INC(x0[0], x1[0])≥NonInfC∧10240_0_CREATELIST_INC(x0[0], x1[0])≥6830_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))∧(U^Increasing(6830_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(6830_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)    ((U^Increasing(6830_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)    ((U^Increasing(6830_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)    ((U^Increasing(6830_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

For Pair 6830_0_CREATELIST_LOAD(x0, x1) → COND_6830_0_CREATELIST_LOAD(<(x1, x0), x0, x1) the following chains were created:

We consider the chain 6830_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_6830_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1]), COND_6830_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10240_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] ⇒ 6830_0_CREATELIST_LOAD(x0[1], x1[1])≥NonInfC∧6830_0_CREATELIST_LOAD(x0[1], x1[1])≥COND_6830_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])∧(U^Increasing(COND_6830_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 ⇒ 6830_0_CREATELIST_LOAD(x0[1], x1[1])≥NonInfC∧6830_0_CREATELIST_LOAD(x0[1], x1[1])≥COND_6830_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])∧(U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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_6830_0_CREATELIST_LOAD(TRUE, x0, x1) → 10240_0_CREATELIST_INC(x0, x1) the following chains were created:

We consider the chain COND_6830_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10240_0_CREATELIST_INC(x0[2], x1[2]), 10240_0_CREATELIST_INC(x0[0], x1[0]) → 6830_0_CREATELIST_LOAD(x0[0], +(x1[0], 1)) which results in the following constraint:
(14)    (x0[2]=x0[0]∧x1[2]=x1[0] ⇒ COND_6830_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥NonInfC∧COND_6830_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥10240_0_CREATELIST_INC(x0[2], x1[2])∧(U^Increasing(10240_0_CREATELIST_INC(x0[2], x1[2])), ≥))

We simplified constraint (14) using rule (IV) which results in the following new constraint:
(15)    (COND_6830_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥NonInfC∧COND_6830_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥10240_0_CREATELIST_INC(x0[2], x1[2])∧(U^Increasing(10240_0_CREATELIST_INC(x0[2], x1[2])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    ((U^Increasing(10240_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)    ((U^Increasing(10240_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)    ((U^Increasing(10240_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)    ((U^Increasing(10240_0_CREATELIST_INC(x0[2], x1[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

For Pair 6830_0_CREATELIST_LOAD(x0, x1) → COND_6830_0_CREATELIST_LOAD1(<(x1, x0), x0, x1) the following chains were created:

We consider the chain 6830_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_6830_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3]), COND_6830_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 6830_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] ⇒ 6830_0_CREATELIST_LOAD(x0[3], x1[3])≥NonInfC∧6830_0_CREATELIST_LOAD(x0[3], x1[3])≥COND_6830_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])∧(U^Increasing(COND_6830_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 ⇒ 6830_0_CREATELIST_LOAD(x0[3], x1[3])≥NonInfC∧6830_0_CREATELIST_LOAD(x0[3], x1[3])≥COND_6830_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])∧(U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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_6830_0_CREATELIST_LOAD1(TRUE, x0, x1) → 6830_0_CREATELIST_LOAD(x0, +(x1, 1)) the following chains were created:

We consider the chain COND_6830_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 6830_0_CREATELIST_LOAD(x0[4], +(x1[4], 1)) which results in the following constraint:
(28)    (COND_6830_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4])≥NonInfC∧COND_6830_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4])≥6830_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))∧(U^Increasing(6830_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29)    ((U^Increasing(6830_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)    ((U^Increasing(6830_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)    ((U^Increasing(6830_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)    ((U^Increasing(6830_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.

10240_0_CREATELIST_INC(x0, x1) → 6830_0_CREATELIST_LOAD(x0, +(x1, 1))
- ((U^Increasing(6830_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)
6830_0_CREATELIST_LOAD(x0, x1) → COND_6830_0_CREATELIST_LOAD(<(x1, x0), x0, x1)
- (x0[1] ≥ 0∧x1[1] ≥ 0 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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_6830_0_CREATELIST_LOAD(TRUE, x0, x1) → 10240_0_CREATELIST_INC(x0, x1)
- ((U^Increasing(10240_0_CREATELIST_INC(x0[2], x1[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)
6830_0_CREATELIST_LOAD(x0, x1) → COND_6830_0_CREATELIST_LOAD1(<(x1, x0), x0, x1)
- (x0[3] ≥ 0∧x1[3] ≥ 0 ⇒ (U^Increasing(COND_6830_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 ⇒ (U^Increasing(COND_6830_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_6830_0_CREATELIST_LOAD1(TRUE, x0, x1) → 6830_0_CREATELIST_LOAD(x0, +(x1, 1))
- ((U^Increasing(6830_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(10240_0_CREATELIST_INC(x₁, x₂)) = [-1] + x₁ + [-1]x₂
POL(6830_0_CREATELIST_LOAD(x₁, x₂)) = [-1] + [-1]x₂ + x₁
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(COND_6830_0_CREATELIST_LOAD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂
POL(<(x₁, x₂)) = [-1]
POL(COND_6830_0_CREATELIST_LOAD1(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂

The following pairs are in P_>:

10240_0_CREATELIST_INC(x0[0], x1[0]) → 6830_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))
COND_6830_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 6830_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))

The following pairs are in P_bound:

6830_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_6830_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])
6830_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_6830_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])

The following pairs are in P_≥:

6830_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_6830_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])
COND_6830_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10240_0_CREATELIST_INC(x0[2], x1[2])
6830_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_6830_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])

There are no usable rules.

(21) Complex Obligation (AND)

(22) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): 6830_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_6830_0_CREATELIST_LOAD(x1[1] < x0[1], x0[1], x1[1])
(2): COND_6830_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10240_0_CREATELIST_INC(x0[2], x1[2])
(3): 6830_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_6830_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.

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE

(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): 10240_0_CREATELIST_INC(x0[0], x1[0]) → 6830_0_CREATELIST_LOAD(x0[0], x1[0] + 1)
(2): COND_6830_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 10240_0_CREATELIST_INC(x0[2], x1[2])
(4): COND_6830_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 6830_0_CREATELIST_LOAD(x0[4], x1[4] + 1)

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

The set Q is empty.

(26) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) IDPNonInfProof (SOUND transformation)

(7) Obligation:

(8) IDependencyGraphProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) IDPtoQDPProof (SOUND transformation)

(12) Obligation:

(13) UsableRulesProof (EQUIVALENT transformation)

(14) Obligation:

(15) QReductionProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPSizeChangeProof (EQUIVALENT transformation)

(18) YES

(19) Obligation:

(20) IDPNonInfProof (SOUND transformation)

(21) Complex Obligation (AND)

(22) Obligation:

(23) IDependencyGraphProof (EQUIVALENT transformation)

(24) TRUE

(25) Obligation:

(26) IDependencyGraphProof (EQUIVALENT transformation)

(27) TRUE