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_14 (Sun Microsystems Inc.) Main-Class: DoublyLinkedList/MainFind 
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 MainFind {
    public static void main(final String[] args) {
        Random.args = args;
        final DoublyLinkedList list = DoublyLinkedList.createList();
        list.find(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.MainFind.main([Ljava/lang/String;)V: Graph of 294 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.findR(I)LDoublyLinkedList/DoublyLinkedList;: Graph of 159 nodes with 0 SCCs.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 68 rules for P and 123 rules for R.


Combined rules. Obtained 9 rules for P and 45 rules for R.


Filtered ground terms:


6491_0_findR_FieldAccess(x1, x2, x3, x4) → 6491_0_findR_FieldAccess(x2, x3, x4)
Cond_6491_0_findR_FieldAccess2(x1, x2, x3, x4, x5) → Cond_6491_0_findR_FieldAccess2(x1, x3, x4, x5)
DoublyLinkedList.DoublyLinkedList(x1, x2, x3) → DoublyLinkedList.DoublyLinkedList(x2, x3)
Cond_6834_0_findR_Load(x1, x2, x3, x4, x5) → Cond_6834_0_findR_Load(x1, x3, x4, x5)
6834_0_findR_Load(x1, x2, x3, x4) → 6834_0_findR_Load(x2, x3, x4)
Cond_6491_0_findR_FieldAccess1(x1, x2, x3, x4, x5) → Cond_6491_0_findR_FieldAccess1(x1, x3, x4, x5)
Cond_6991_0_findR_Load(x1, x2, x3, x4, x5) → Cond_6991_0_findR_Load(x1, x3, x4, x5)
6991_0_findR_Load(x1, x2, x3, x4) → 6991_0_findR_Load(x2, x3, x4)
Cond_6491_0_findR_FieldAccess(x1, x2, x3, x4, x5) → Cond_6491_0_findR_FieldAccess(x1, x3, x4, x5)
Cond_6916_0_findR_Load(x1, x2, x3, x4, x5) → Cond_6916_0_findR_Load(x1, x3, x4, x5)
6916_0_findR_Load(x1, x2, x3, x4) → 6916_0_findR_Load(x2, x3, x4)
11292_0_findR_Return(x1) → 11292_0_findR_Return
11507_0_findR_Return(x1) → 11507_0_findR_Return
11387_0_findR_Return(x1) → 11387_0_findR_Return
9657_0_findR_Return(x1, x2) → 9657_0_findR_Return
9630_0_findR_Return(x1, x2) → 9630_0_findR_Return(x2)
9583_0_findR_Return(x1, x2) → 9583_0_findR_Return
9569_0_findR_Return(x1, x2) → 9569_0_findR_Return(x2)
9551_0_findR_Return(x1, x2) → 9551_0_findR_Return
9546_0_findR_Return(x1, x2) → 9546_0_findR_Return(x2)
8362_0_findR_Return(x1, x2) → 8362_0_findR_Return
8298_0_findR_Return(x1, x2) → 8298_0_findR_Return
7470_0_findR_Return(x1, x2, x3, x4) → 7470_0_findR_Return(x2, x3, x4)
7392_0_findR_Return(x1, x2, x3, x4) → 7392_0_findR_Return(x2, x3, x4)
8250_0_findR_Return(x1, x2) → 8250_0_findR_Return
7332_0_findR_Return(x1, x2, x3, x4) → 7332_0_findR_Return(x2, x3, x4)

Filtered duplicate args:


6491_0_findR_FieldAccess(x1, x2, x3) → 6491_0_findR_FieldAccess(x2, x3)
Cond_6491_0_findR_FieldAccess2(x1, x2, x3, x4) → Cond_6491_0_findR_FieldAccess2(x1, x3, x4)
Cond_6491_0_findR_FieldAccess1(x1, x2, x3, x4) → Cond_6491_0_findR_FieldAccess1(x1, x3, x4)
Cond_6491_0_findR_FieldAccess(x1, x2, x3, x4) → Cond_6491_0_findR_FieldAccess(x1, x3, x4)
7470_0_findR_Return(x1, x2, x3) → 7470_0_findR_Return(x2, x3)
7392_0_findR_Return(x1, x2, x3) → 7392_0_findR_Return(x2, x3)
7332_0_findR_Return(x1, x2, x3) → 7332_0_findR_Return(x2, x3)

Combined rules. Obtained 6 rules for P and 45 rules for R.


Finished conversion. Obtained 6 rules for P and 45 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:


4641_0_getFirst_FieldAccess(x1, x2, x3) → 4641_0_getFirst_FieldAccess(x2, x3)
DoublyLinkedList.DoublyLinkedList(x1, x2) → DoublyLinkedList.DoublyLinkedList(x2)
4942_0_getFirst_NONNULL(x1, x2, x3) → 4942_0_getFirst_NONNULL(x2, x3)
5429_0_getFirst_Return(x1) → 5429_0_getFirst_Return
5132_0_getFirst_Return(x1, x2) → 5132_0_getFirst_Return
4999_0_getFirst_Return(x1, x2, x3) → 4999_0_getFirst_Return

Filtered duplicate args:


4641_0_getFirst_FieldAccess(x1, x2) → 4641_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:


12388_0_createList_NULL(x1, x2, x3, x4, x5, x6) → 12388_0_createList_NULL(x2, x4, x5, x6)
DoublyLinkedList.DoublyLinkedList(x1) → DoublyLinkedList.DoublyLinkedList
Cond_11511_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_11511_1_createList_InvokeMethod(x1, x2, x3, x4, x5)
11511_0_random_GT(x1, x2, x3) → 11511_0_random_GT(x2, x3)
11511_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 11511_1_createList_InvokeMethod(x1, x2, x3, x4)
7090_0_createList_Load(x1, x2, x3, x4, x5) → 7090_0_createList_Load(x2, x3, x4, x5)
12546_0_createList_Inc(x1, x2, x3, x4) → 12546_0_createList_Inc(x2, x4)
Cond_12329_1_createList_InvokeMethod3(x1, x2, x3, x4, x5, x6, x7) → Cond_12329_1_createList_InvokeMethod3(x1, x2, x3, x4, x5)
12329_0_random_IntArithmetic(x1, x2, x3, x4) → 12329_0_random_IntArithmetic(x2, x3)
12329_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 12329_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_12329_1_createList_InvokeMethod2(x1, x2, x3, x4, x5, x6, x7) → Cond_12329_1_createList_InvokeMethod2(x1, x2, x3, x4)
Cond_12329_1_createList_InvokeMethod1(x1, x2, x3, x4, x5, x6, x7) → Cond_12329_1_createList_InvokeMethod1(x1, x2, x3, x4)
12751_0_createList_Inc(x1, x2, x3, x4) → 12751_0_createList_Inc(x2, x4)
Cond_12329_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_12329_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_12294_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_12294_1_createList_InvokeMethod(x1, x2, x3, x4, x5)
12294_0_random_ArrayAccess(x1, x2, x3) → 12294_0_random_ArrayAccess(x2, x3)
12294_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 12294_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_11509_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_11509_1_createList_InvokeMethod(x1, x2, x3, x4, x5)
11509_0_random_GT(x1, x2, x3) → 11509_0_random_GT(x2, x3)
11509_1_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 11509_1_createList_InvokeMethod(x1, x2, x3, x4)
Cond_7090_0_createList_Load1(x1, x2, x3, x4, x5, x6) → Cond_7090_0_createList_Load1(x1, x3, x4, x5, x6)
Cond_7090_0_createList_Load(x1, x2, x3, x4, x5, x6) → Cond_7090_0_createList_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:


12388_0_createList_NULL(x1, x2, x3, x4) → 12388_0_createList_NULL(x1, x2, x4)
7090_0_createList_Load(x1, x2, x3, x4) → 7090_0_createList_Load(x1, x2, x4)
Cond_7090_0_createList_Load1(x1, x2, x3, x4, x5) → Cond_7090_0_createList_Load1(x1, x2, x3, x5)
Cond_7090_0_createList_Load(x1, x2, x3, x4, x5) → Cond_7090_0_createList_Load(x1, x2, x3, x5)

Filtered all non-integer terms:


12329_1_createList_InvokeMethod(x1, x2, x3, x4) → 12329_1_createList_InvokeMethod(x1, x2, x3)
12329_0_random_IntArithmetic(x1, x2) → 12329_0_random_IntArithmetic(x2)
7090_0_createList_Load(x1, x2, x3) → 7090_0_createList_Load(x1, x3)
12388_0_createList_NULL(x1, x2, x3) → 12388_0_createList_NULL(x1, x2)

Filtered all free variables:


11509_1_createList_InvokeMethod(x1, x2, x3, x4) → 11509_1_createList_InvokeMethod(x2, x3, x4)
11511_1_createList_InvokeMethod(x1, x2, x3, x4) → 11511_1_createList_InvokeMethod(x2, x3, x4)
Cond_11509_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_11509_1_createList_InvokeMethod(x1, x3, x4, x5)
12294_1_createList_InvokeMethod(x1, x2, x3, x4) → 12294_1_createList_InvokeMethod(x2, x3, x4)
Cond_12294_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_12294_1_createList_InvokeMethod(x1, x3, x4, x5)
12329_1_createList_InvokeMethod(x1, x2, x3) → 12329_1_createList_InvokeMethod(x2, x3)
Cond_12329_1_createList_InvokeMethod(x1, x2, x3, x4) → Cond_12329_1_createList_InvokeMethod(x1, x3, x4)
Cond_12329_1_createList_InvokeMethod1(x1, x2, x3, x4) → Cond_12329_1_createList_InvokeMethod1(x1, x3, x4)
Cond_12329_1_createList_InvokeMethod2(x1, x2, x3, x4) → Cond_12329_1_createList_InvokeMethod2(x1, x3, x4)
Cond_12329_1_createList_InvokeMethod3(x1, x2, x3, x4, x5) → Cond_12329_1_createList_InvokeMethod3(x1, x3, x4, x5)
Cond_11511_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_11511_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


The ITRS R consists of the following rules:
9154_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9154_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9583_0_findR_Return
9353_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9353_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9657_0_findR_Return
8848_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
8848_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9551_0_findR_Return
9154_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9154_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9154_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9583_0_findR_Return
9154_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9583_0_findR_Return
9154_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11387_0_findR_Return
9353_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9353_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9353_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9657_0_findR_Return
9353_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9657_0_findR_Return
9353_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11507_0_findR_Return
8848_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
8848_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
8848_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9551_0_findR_Return
8848_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9551_0_findR_Return
8848_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11292_0_findR_Return

The integer pair graph contains the following rules and edges:
(0): 6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))) → COND_6491_0_FINDR_FIELDACCESS(!(x0[0] = x2[0]), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))
(1): COND_6491_0_FINDR_FIELDACCESS(TRUE, x2[1], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], java.lang.Object(x1[1])))) → 6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))

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


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



The set Q consists of the following terms:
9154_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9154_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9353_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9353_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
8848_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
8848_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9154_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9154_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9154_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9154_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9154_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9154_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9154_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9154_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9154_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9154_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9154_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
9154_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
9154_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
9353_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9353_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9353_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9353_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9353_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9353_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9353_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9353_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9353_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9353_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9353_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
9353_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
9353_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
8848_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
8848_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
8848_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
8848_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
8848_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
8848_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
8848_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
8848_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
8848_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
8848_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
8848_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
8848_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
8848_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)

(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 6491_0_FINDR_FIELDACCESS(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(x1)))) → COND_6491_0_FINDR_FIELDACCESS(!(=(x0, x2)), x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(x1)))) the following chains were created:
  • We consider the chain 6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))) → COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))), COND_6491_0_FINDR_FIELDACCESS(TRUE, x2[1], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], java.lang.Object(x1[1])))) → 6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1])) which results in the following constraint:

    (1)    (!(=(x0[0], x2[0]))=TRUE∧x2[0]=x2[1]∧java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))=java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], java.lang.Object(x1[1]))) ⇒ 6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))≥COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))∧(UIncreasing(COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))), ≥))



    We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

    (2)    (!(=(x0[0], x2[0]))=TRUE ⇒ 6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))≥NonInfC∧6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))≥COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))∧(UIncreasing(COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))), ≥))



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

    (3)    (0 ≥ 0 ⇒ (UIncreasing(COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(6)bni_50 + (-1)Bound*bni_50] + [(8)bni_50]x1[0] + [(2)bni_50]x2[0] ≥ 0∧[2 + (-1)bso_51] ≥ 0)



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

    (4)    (0 ≥ 0 ⇒ (UIncreasing(COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(6)bni_50 + (-1)Bound*bni_50] + [(8)bni_50]x1[0] + [(2)bni_50]x2[0] ≥ 0∧[2 + (-1)bso_51] ≥ 0)



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

    (5)    (0 ≥ 0 ⇒ (UIncreasing(COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(6)bni_50 + (-1)Bound*bni_50] + [(8)bni_50]x1[0] + [(2)bni_50]x2[0] ≥ 0∧[2 + (-1)bso_51] ≥ 0)



    We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (6)    (0 ≥ 0 ⇒ (UIncreasing(COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(8)bni_50] ≥ 0∧0 ≥ 0∧[(2)bni_50] ≥ 0∧[(6)bni_50 + (-1)Bound*bni_50] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[2 + (-1)bso_51] ≥ 0)







For Pair COND_6491_0_FINDR_FIELDACCESS(TRUE, x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(x1)))) → 6491_0_FINDR_FIELDACCESS(x2, java.lang.Object(x1)) the following chains were created:
  • We consider the chain 6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))) → COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))), COND_6491_0_FINDR_FIELDACCESS(TRUE, x2[1], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], java.lang.Object(x1[1])))) → 6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1])), 6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))) → COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))) which results in the following constraint:

    (7)    (!(=(x0[0], x2[0]))=TRUE∧x2[0]=x2[1]∧java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))=java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], java.lang.Object(x1[1])))∧x2[1]=x2[0]1∧java.lang.Object(x1[1])=java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]1, java.lang.Object(x1[0]1))) ⇒ COND_6491_0_FINDR_FIELDACCESS(TRUE, x2[1], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], java.lang.Object(x1[1]))))≥NonInfC∧COND_6491_0_FINDR_FIELDACCESS(TRUE, x2[1], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], java.lang.Object(x1[1]))))≥6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))∧(UIncreasing(6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))), ≥))



    We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:

    (8)    (!(=(x0[0], x2[0]))=TRUE ⇒ COND_6491_0_FINDR_FIELDACCESS(TRUE, x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]1, java.lang.Object(x1[0]1))))))≥NonInfC∧COND_6491_0_FINDR_FIELDACCESS(TRUE, x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]1, java.lang.Object(x1[0]1))))))≥6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]1, java.lang.Object(x1[0]1))))∧(UIncreasing(6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))), ≥))



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

    (9)    (0 ≥ 0 ⇒ (UIncreasing(6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))), ≥)∧[(20)bni_52 + (-1)Bound*bni_52] + [(32)bni_52]x1[0]1 + [(2)bni_52]x2[0] ≥ 0∧[14 + (-1)bso_53] + [24]x1[0]1 ≥ 0)



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

    (10)    (0 ≥ 0 ⇒ (UIncreasing(6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))), ≥)∧[(20)bni_52 + (-1)Bound*bni_52] + [(32)bni_52]x1[0]1 + [(2)bni_52]x2[0] ≥ 0∧[14 + (-1)bso_53] + [24]x1[0]1 ≥ 0)



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

    (11)    (0 ≥ 0 ⇒ (UIncreasing(6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))), ≥)∧[(20)bni_52 + (-1)Bound*bni_52] + [(32)bni_52]x1[0]1 + [(2)bni_52]x2[0] ≥ 0∧[14 + (-1)bso_53] + [24]x1[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 ⇒ (UIncreasing(6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))), ≥)∧[(32)bni_52] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(2)bni_52] ≥ 0∧[(20)bni_52 + (-1)Bound*bni_52] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[14 + (-1)bso_53] ≥ 0∧[1] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 6491_0_FINDR_FIELDACCESS(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(x1)))) → COND_6491_0_FINDR_FIELDACCESS(!(=(x0, x2)), x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(x1))))
    • (0 ≥ 0 ⇒ (UIncreasing(COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))), ≥)∧[(8)bni_50] ≥ 0∧0 ≥ 0∧[(2)bni_50] ≥ 0∧[(6)bni_50 + (-1)Bound*bni_50] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[2 + (-1)bso_51] ≥ 0)

  • COND_6491_0_FINDR_FIELDACCESS(TRUE, x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(x1)))) → 6491_0_FINDR_FIELDACCESS(x2, java.lang.Object(x1))
    • (0 ≥ 0 ⇒ (UIncreasing(6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))), ≥)∧[(32)bni_52] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(2)bni_52] ≥ 0∧[(20)bni_52 + (-1)Bound*bni_52] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[14 + (-1)bso_53] ≥ 0∧[1] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(9154_1_findR_InvokeMethod(x1, x2, x3)) = 0   
POL(7332_0_findR_Return(x1, x2)) = 0   
POL(java.lang.Object(x1)) = [1] + [2]x1   
POL(DoublyLinkedList.DoublyLinkedList(x1, x2)) = [2]x2   
POL(9569_0_findR_Return(x1)) = 0   
POL(8250_0_findR_Return) = 0   
POL(NULL) = 0   
POL(9583_0_findR_Return) = 0   
POL(9353_1_findR_InvokeMethod(x1, x2, x3)) = 0   
POL(9630_0_findR_Return(x1)) = 0   
POL(9657_0_findR_Return) = 0   
POL(8848_1_findR_InvokeMethod(x1, x2, x3)) = 0   
POL(9546_0_findR_Return(x1)) = 0   
POL(9551_0_findR_Return) = 0   
POL(7392_0_findR_Return(x1, x2)) = 0   
POL(7470_0_findR_Return(x1, x2)) = 0   
POL(8298_0_findR_Return) = 0   
POL(8362_0_findR_Return) = 0   
POL(11387_0_findR_Return) = 0   
POL(11292_0_findR_Return) = 0   
POL(11507_0_findR_Return) = 0   
POL(6491_0_FINDR_FIELDACCESS(x1, x2)) = [1] + x2 + [2]x1   
POL(COND_6491_0_FINDR_FIELDACCESS(x1, x2, x3)) = [-1] + x3 + [2]x2 + [-1]x1   
POL(!(x1)) = 0   
POL(=(x1, x2)) = 0   

The following pairs are in P>:

6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))) → COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))
COND_6491_0_FINDR_FIELDACCESS(TRUE, x2[1], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], java.lang.Object(x1[1])))) → 6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))

The following pairs are in Pbound:

6491_0_FINDR_FIELDACCESS(x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0])))) → COND_6491_0_FINDR_FIELDACCESS(!(=(x0[0], x2[0])), x2[0], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0], java.lang.Object(x1[0]))))
COND_6491_0_FINDR_FIELDACCESS(TRUE, x2[1], java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[1], java.lang.Object(x1[1])))) → 6491_0_FINDR_FIELDACCESS(x2[1], java.lang.Object(x1[1]))

The following pairs are in P≥:
none

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

FALSE1 → !(TRUE)1
TRUE1 → !(FALSE)1

(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:
none


The ITRS R consists of the following rules:
9154_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9154_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9583_0_findR_Return
9353_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9353_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9657_0_findR_Return
8848_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
8848_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9551_0_findR_Return
9154_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9154_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9154_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9583_0_findR_Return
9154_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9583_0_findR_Return
9154_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11387_0_findR_Return
9154_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11387_0_findR_Return
9353_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9353_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
9353_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9657_0_findR_Return
9353_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9657_0_findR_Return
9353_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11507_0_findR_Return
9353_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11507_0_findR_Return
8848_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
8848_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0) → 9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))
8848_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9551_0_findR_Return
8848_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1) → 9551_0_findR_Return
8848_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11292_0_findR_Return
8848_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4) → 11292_0_findR_Return

The integer pair graph is empty.

The set Q consists of the following terms:
9154_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9154_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9353_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9353_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
8848_1_findR_InvokeMethod(7332_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
8848_1_findR_InvokeMethod(8250_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9154_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9154_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9154_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9154_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9154_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9154_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9154_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9154_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9154_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9154_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9154_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
9154_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
9154_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
9353_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9353_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
9353_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9353_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
9353_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9353_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9353_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9353_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9353_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
9353_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
9353_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
9353_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
9353_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
8848_1_findR_InvokeMethod(7392_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
8848_1_findR_InvokeMethod(7470_0_findR_Return(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)), x0)
8848_1_findR_InvokeMethod(8298_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
8848_1_findR_InvokeMethod(8362_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, NULL)), x1)
8848_1_findR_InvokeMethod(9546_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
8848_1_findR_InvokeMethod(9551_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
8848_1_findR_InvokeMethod(9569_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
8848_1_findR_InvokeMethod(9583_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
8848_1_findR_InvokeMethod(9630_0_findR_Return(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1))), java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, x1)))), x0)
8848_1_findR_InvokeMethod(9657_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, NULL)))), x2)
8848_1_findR_InvokeMethod(11292_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
8848_1_findR_InvokeMethod(11387_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)
8848_1_findR_InvokeMethod(11507_0_findR_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x1, java.lang.Object(DoublyLinkedList.DoublyLinkedList(x2, x3)))))), x4)

(8) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
5079_1_getFirst_InvokeMethod(4999_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))) → 5132_0_getFirst_Return
5079_1_getFirst_InvokeMethod(5132_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))))) → 5429_0_getFirst_Return
5079_1_getFirst_InvokeMethod(5429_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0))))))) → 5429_0_getFirst_Return

The integer pair graph contains the following rules and edges:
(0): 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 4942_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
(1): 4942_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
(2): 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 4641_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:
5079_1_getFirst_InvokeMethod(4999_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
5079_1_getFirst_InvokeMethod(5132_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
5079_1_getFirst_InvokeMethod(5429_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:

4641_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 4942_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
4942_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
4641_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))

The TRS R consists of the following rules:

5079_1_getFirst_InvokeMethod(4999_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))) → 5132_0_getFirst_Return
5079_1_getFirst_InvokeMethod(5132_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL))))) → 5429_0_getFirst_Return
5079_1_getFirst_InvokeMethod(5429_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0))))))) → 5429_0_getFirst_Return

The set Q consists of the following terms:

5079_1_getFirst_InvokeMethod(4999_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
5079_1_getFirst_InvokeMethod(5132_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
5079_1_getFirst_InvokeMethod(5429_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:

4641_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 4942_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
4942_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
4641_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))

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

5079_1_getFirst_InvokeMethod(4999_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
5079_1_getFirst_InvokeMethod(5132_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
5079_1_getFirst_InvokeMethod(5429_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].

5079_1_getFirst_InvokeMethod(4999_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))
5079_1_getFirst_InvokeMethod(5132_0_getFirst_Return, java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(DoublyLinkedList.DoublyLinkedList(NULL)))))
5079_1_getFirst_InvokeMethod(5429_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:

4641_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 4942_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0])), x0[0])
4942_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
4641_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 4641_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:

  • 4942_0_GETFIRST_NONNULL(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[1]))), java.lang.Object(x0[1])) → 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[1]))
    The graph contains the following edges 1 > 1, 2 >= 1

  • 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(java.lang.Object(x0[2])))) → 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(x0[2]))
    The graph contains the following edges 1 > 1

  • 4641_0_GETFIRST_FIELDACCESS(java.lang.Object(DoublyLinkedList.DoublyLinkedList(x0[0]))) → 4942_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): 12751_0_CREATELIST_INC(x0[0], x1[0]) → 7090_0_CREATELIST_LOAD(x0[0], x1[0] + 1)
(1): 7090_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_7090_0_CREATELIST_LOAD(x1[1] < x0[1], x0[1], x1[1])
(2): COND_7090_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 12751_0_CREATELIST_INC(x0[2], x1[2])
(3): 7090_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_7090_0_CREATELIST_LOAD1(x1[3] < x0[3], x0[3], x1[3])
(4): COND_7090_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 7090_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 12751_0_CREATELIST_INC(x0, x1) → 7090_0_CREATELIST_LOAD(x0, +(x1, 1)) the following chains were created:
  • We consider the chain 12751_0_CREATELIST_INC(x0[0], x1[0]) → 7090_0_CREATELIST_LOAD(x0[0], +(x1[0], 1)) which results in the following constraint:

    (1)    (12751_0_CREATELIST_INC(x0[0], x1[0])≥NonInfC∧12751_0_CREATELIST_INC(x0[0], x1[0])≥7090_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))∧(UIncreasing(7090_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(7090_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(7090_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(7090_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(7090_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)







For Pair 7090_0_CREATELIST_LOAD(x0, x1) → COND_7090_0_CREATELIST_LOAD(<(x1, x0), x0, x1) the following chains were created:
  • We consider the chain 7090_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_7090_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1]), COND_7090_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 12751_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] ⇒ 7090_0_CREATELIST_LOAD(x0[1], x1[1])≥NonInfC∧7090_0_CREATELIST_LOAD(x0[1], x1[1])≥COND_7090_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])∧(UIncreasing(COND_7090_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 ⇒ 7090_0_CREATELIST_LOAD(x0[1], x1[1])≥NonInfC∧7090_0_CREATELIST_LOAD(x0[1], x1[1])≥COND_7090_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])∧(UIncreasing(COND_7090_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_7090_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_7090_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_7090_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_7090_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_7090_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_7090_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_7090_0_CREATELIST_LOAD(TRUE, x0, x1) → 12751_0_CREATELIST_INC(x0, x1) the following chains were created:
  • We consider the chain COND_7090_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 12751_0_CREATELIST_INC(x0[2], x1[2]), 12751_0_CREATELIST_INC(x0[0], x1[0]) → 7090_0_CREATELIST_LOAD(x0[0], +(x1[0], 1)) which results in the following constraint:

    (14)    (x0[2]=x0[0]∧x1[2]=x1[0] ⇒ COND_7090_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥NonInfC∧COND_7090_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥12751_0_CREATELIST_INC(x0[2], x1[2])∧(UIncreasing(12751_0_CREATELIST_INC(x0[2], x1[2])), ≥))



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

    (15)    (COND_7090_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥NonInfC∧COND_7090_0_CREATELIST_LOAD(TRUE, x0[2], x1[2])≥12751_0_CREATELIST_INC(x0[2], x1[2])∧(UIncreasing(12751_0_CREATELIST_INC(x0[2], x1[2])), ≥))



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

    (16)    ((UIncreasing(12751_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(12751_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(12751_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(12751_0_CREATELIST_INC(x0[2], x1[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)







For Pair 7090_0_CREATELIST_LOAD(x0, x1) → COND_7090_0_CREATELIST_LOAD1(<(x1, x0), x0, x1) the following chains were created:
  • We consider the chain 7090_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_7090_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3]), COND_7090_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 7090_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] ⇒ 7090_0_CREATELIST_LOAD(x0[3], x1[3])≥NonInfC∧7090_0_CREATELIST_LOAD(x0[3], x1[3])≥COND_7090_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])∧(UIncreasing(COND_7090_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 ⇒ 7090_0_CREATELIST_LOAD(x0[3], x1[3])≥NonInfC∧7090_0_CREATELIST_LOAD(x0[3], x1[3])≥COND_7090_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])∧(UIncreasing(COND_7090_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_7090_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_7090_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_7090_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_7090_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_7090_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_7090_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_7090_0_CREATELIST_LOAD1(TRUE, x0, x1) → 7090_0_CREATELIST_LOAD(x0, +(x1, 1)) the following chains were created:
  • We consider the chain COND_7090_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 7090_0_CREATELIST_LOAD(x0[4], +(x1[4], 1)) which results in the following constraint:

    (28)    (COND_7090_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4])≥NonInfC∧COND_7090_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4])≥7090_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))∧(UIncreasing(7090_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(7090_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(7090_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(7090_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(7090_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.
  • 12751_0_CREATELIST_INC(x0, x1) → 7090_0_CREATELIST_LOAD(x0, +(x1, 1))
    • ((UIncreasing(7090_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  • 7090_0_CREATELIST_LOAD(x0, x1) → COND_7090_0_CREATELIST_LOAD(<(x1, x0), x0, x1)
    • (x0[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_7090_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_7090_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_7090_0_CREATELIST_LOAD(TRUE, x0, x1) → 12751_0_CREATELIST_INC(x0, x1)
    • ((UIncreasing(12751_0_CREATELIST_INC(x0[2], x1[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

  • 7090_0_CREATELIST_LOAD(x0, x1) → COND_7090_0_CREATELIST_LOAD1(<(x1, x0), x0, x1)
    • (x0[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(COND_7090_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_7090_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_7090_0_CREATELIST_LOAD1(TRUE, x0, x1) → 7090_0_CREATELIST_LOAD(x0, +(x1, 1))
    • ((UIncreasing(7090_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(12751_0_CREATELIST_INC(x1, x2)) = [-1] + x1 + [-1]x2   
POL(7090_0_CREATELIST_LOAD(x1, x2)) = [-1] + [-1]x2 + x1   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(COND_7090_0_CREATELIST_LOAD(x1, x2, x3)) = [-1] + [-1]x3 + x2   
POL(<(x1, x2)) = [-1]   
POL(COND_7090_0_CREATELIST_LOAD1(x1, x2, x3)) = [-1] + [-1]x3 + x2   

The following pairs are in P>:

12751_0_CREATELIST_INC(x0[0], x1[0]) → 7090_0_CREATELIST_LOAD(x0[0], +(x1[0], 1))
COND_7090_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 7090_0_CREATELIST_LOAD(x0[4], +(x1[4], 1))

The following pairs are in Pbound:

7090_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_7090_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])
7090_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_7090_0_CREATELIST_LOAD1(<(x1[3], x0[3]), x0[3], x1[3])

The following pairs are in P≥:

7090_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_7090_0_CREATELIST_LOAD(<(x1[1], x0[1]), x0[1], x1[1])
COND_7090_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 12751_0_CREATELIST_INC(x0[2], x1[2])
7090_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_7090_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): 7090_0_CREATELIST_LOAD(x0[1], x1[1]) → COND_7090_0_CREATELIST_LOAD(x1[1] < x0[1], x0[1], x1[1])
(2): COND_7090_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 12751_0_CREATELIST_INC(x0[2], x1[2])
(3): 7090_0_CREATELIST_LOAD(x0[3], x1[3]) → COND_7090_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): 12751_0_CREATELIST_INC(x0[0], x1[0]) → 7090_0_CREATELIST_LOAD(x0[0], x1[0] + 1)
(2): COND_7090_0_CREATELIST_LOAD(TRUE, x0[2], x1[2]) → 12751_0_CREATELIST_INC(x0[2], x1[2])
(4): COND_7090_0_CREATELIST_LOAD1(TRUE, x0[4], x1[4]) → 7090_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.

(27) TRUE