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 UsableRulesReductionPairsProof (⇔)
        ↳18 QDP
        ↳19 PisEmptyProof (⇔)
        ↳20 YES
    ↳21 IDP
        ↳22 IDPNonInfProof (⇒)
        ↳23 AND
            ↳24 IDP
                ↳25 IDependencyGraphProof (⇔)
                ↳26 TRUE
            ↳27 IDP
                ↳28 IDependencyGraphProof (⇔)
                ↳29 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Flatten 
public class Flatten {
  public static void main(String[] args) {
    Random.args = args;
    int listLength = Random.random();
    TreeList list = null;
    for (int i = listLength; i > 0; i--) {
      Tree tree = Tree.createTree();
      list = new TreeList(tree, list);
    }

    flatten(list);
  }

  public static ObjectList flatten(TreeList start) {
    ObjectList result = null;
    while (start != null) {
      Tree tree = start.value;
      if (tree != null) {
        result = new ObjectList(tree.value, result);
        start = start.next;
        start = new TreeList(tree.left, start);
        start = new TreeList(tree.right, start);
      } else {
        start = start.next;
      }
    }
    return result;
  }
}


public class ObjectList {
  Object value;
  ObjectList next;

  public ObjectList(Object value, ObjectList next) {
    this.value = value;
    this.next = next;
  }

  public static ObjectList createList() {
    ObjectList result = null;
    int length = Random.random();
    while (length > 0) {
      result = new ObjectList(new Object(), result);
      length--;
    }
    return result;
  }
}


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

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


public class Tree {
  Tree left;
  Tree right;
  Object value;

  public Tree(Tree l, Tree r) {
    this.left = l;
    this.right = r;
  }

  public Tree() {
  }

  public static Tree createNode() {
    Tree result = new Tree();
    result.value = new Object();
    return result;
  }

  public static Tree createTree() {
    int counter = Random.random();
    if (counter == 0) {
      return null;
    }
    Tree result = createNode();
    Tree t = result;

    while (counter > 0) {
      int branch = Random.random();
      if (branch > 0) {
        if (t.left == null) {
          t.left = createNode();
          t = result;
        } else {
          t = t.left;
        }
      } else {
        if (t.right == null) {
          t.right = createNode();
          t = result;
        } else {
          t = t.right;
        }
      }
      counter--;
    }

    return result;
  }
  public static void main(String[] args) {
    Random.args = args;
    createTree();
  }
}


public class TreeList {
  Tree value;
  TreeList next;

  public TreeList(Tree value, TreeList next) {
    this.value = value;
    this.next = next;
  }
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Flatten.main([Ljava/lang/String;)V: Graph of 153 nodes with 1 SCC.

Flatten.flatten(LTreeList;)LObjectList;: Graph of 110 nodes with 1 SCC.

Tree.createTree()LTree;: Graph of 444 nodes with 1 SCC.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 257 rules for P and 74 rules for R.


Combined rules. Obtained 15 rules for P and 0 rules for R.


Filtered ground terms:


Tree(x1, x2, x3) → Tree(x2, x3)
18401_0_random_ArrayAccess(x1, x2, x3) → 18401_0_random_ArrayAccess(x2, x3)
18597_0_random_IntArithmetic(x1, x2, x3, x4) → 18597_0_random_IntArithmetic(x2, x3)
Cond_18597_1_createTree_InvokeMethod9(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod9(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod2(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod2(x1, x2, x3)

Filtered all non-integer terms:


18401_1_createTree_InvokeMethod(x1, x2, x3, x4) → 18401_1_createTree_InvokeMethod(x1, x2)
Cond_18401_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → Cond_18401_1_createTree_InvokeMethod(x1, x2, x3)
18597_1_createTree_InvokeMethod(x1, x2, x3, x4) → 18597_1_createTree_InvokeMethod(x1, x2)
18597_0_random_IntArithmetic(x1, x2) → 18597_0_random_IntArithmetic(x2)
Tree(x1, x2) → Tree
Cond_18597_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod1(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod3(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod3(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod4(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod4(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod5(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod5(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod6(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod6(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod7(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod7(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod8(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod8(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod10(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod10(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod11(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod11(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod12(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod12(x1, x2, x3)
Cond_18597_1_createTree_InvokeMethod13(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod13(x1, x2, x3)

Filtered all free variables:


18597_1_createTree_InvokeMethod(x1, x2) → 18597_1_createTree_InvokeMethod(x2)
Cond_18597_1_createTree_InvokeMethod(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod(x1, x3)
18401_1_createTree_InvokeMethod(x1, x2) → 18401_1_createTree_InvokeMethod(x2)
Cond_18597_1_createTree_InvokeMethod1(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod1(x1, x3)
Cond_18597_1_createTree_InvokeMethod2(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod2(x1, x3)
Cond_18597_1_createTree_InvokeMethod3(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod3(x1, x3)
Cond_18597_1_createTree_InvokeMethod4(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod4(x1, x3)
Cond_18597_1_createTree_InvokeMethod5(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod5(x1, x3)
Cond_18597_1_createTree_InvokeMethod6(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod6(x1, x3)
Cond_18597_1_createTree_InvokeMethod7(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod7(x1, x3)
Cond_18597_1_createTree_InvokeMethod8(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod8(x1, x3)
Cond_18597_1_createTree_InvokeMethod9(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod9(x1, x3)
Cond_18597_1_createTree_InvokeMethod10(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod10(x1, x3)
Cond_18597_1_createTree_InvokeMethod11(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod11(x1, x3)
Cond_18597_1_createTree_InvokeMethod12(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod12(x1, x3)
Cond_18597_1_createTree_InvokeMethod13(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod13(x1, x3)
Cond_18401_1_createTree_InvokeMethod(x1, x2, x3) → Cond_18401_1_createTree_InvokeMethod(x1, x3)

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


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



Log for SCC 1:

Generated 104 rules for P and 3 rules for R.


Combined rules. Obtained 6 rules for P and 1 rules for R.


Filtered ground terms:


6786_0_flatten_Store(x1, x2, x3) → 6786_0_flatten_Store(x2, x3)
4752_0_flatten_NULL(x1, x2, x3, x4) → 4752_0_flatten_NULL(x2, x3, x4)
TreeList(x1, x2, x3) → TreeList(x2, x3)
7828_0_flatten_Store(x1, x2, x3, x4) → 7828_0_flatten_Store(x3, x4)
Tree(x1, x2, x3, x4) → Tree(x2, x3, x4)
ObjectList(x1) → ObjectList
5027_0_flatten_Return(x1, x2) → 5027_0_flatten_Return(x2)

Filtered duplicate args:


4752_0_flatten_NULL(x1, x2, x3) → 4752_0_flatten_NULL(x2, x3)

Filtered unneeded arguments:


4752_0_flatten_NULL(x1, x2) → 4752_0_flatten_NULL(x2)
6786_0_flatten_Store(x1, x2) → 6786_0_flatten_Store(x2)

Filtered all free variables:


5027_0_flatten_Return(x1) → 5027_0_flatten_Return

Finished conversion. Obtained 6 rules for P and 1 rules for R. System has no predefined symbols.



Log for SCC 2:

Generated 47 rules for P and 588 rules for R.


Combined rules. Obtained 2 rules for P and 49 rules for R.


Filtered ground terms:


TreeList(x1) → TreeList
9403_0_createTree_InvokeMethod(x1) → 9403_0_createTree_InvokeMethod
Cond_9403_1_main_InvokeMethod1(x1, x2, x3, x4) → Cond_9403_1_main_InvokeMethod1(x1, x3, x4)
Tree(x1) → Tree
18266_0_createTree_Return(x1, x2) → 18266_0_createTree_Return
Cond_9403_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_9403_1_main_InvokeMethod(x1, x3, x4)
1677_0_createTree_Return(x1, x2, x3) → 1677_0_createTree_Return
18239_0_createTree_LE(x1, x2, x3, x4, x5) → 18239_0_createTree_LE(x2, x4, x5)
Cond_21978_0_createTree_FieldAccess(x1, x2, x3, x4, x5) → Cond_21978_0_createTree_FieldAccess(x1, x3)
21978_0_createTree_FieldAccess(x1, x2, x3, x4) → 21978_0_createTree_FieldAccess(x2)
Cond_22035_0_createTree_Store(x1, x2, x3, x4, x5) → Cond_22035_0_createTree_Store(x1, x3, x5)
22035_0_createTree_Store(x1, x2, x3, x4) → 22035_0_createTree_Store(x2, x4)
Cond_26103_0_createTree_FieldAccess(x1, x2, x3, x4, x5, x6) → Cond_26103_0_createTree_FieldAccess(x1, x3)
26103_0_createTree_FieldAccess(x1, x2, x3, x4, x5) → 26103_0_createTree_FieldAccess(x2)
Cond_25902_0_createTree_Load(x1, x2, x3, x4) → Cond_25902_0_createTree_Load(x1, x3)
25902_0_createTree_Load(x1, x2, x3) → 25902_0_createTree_Load(x2)
Cond_26110_0_createTree_Load(x1, x2, x3, x4) → Cond_26110_0_createTree_Load(x1, x3)
26110_0_createTree_Load(x1, x2, x3) → 26110_0_createTree_Load(x2)
Cond_22085_0_createTree_FieldAccess(x1, x2, x3, x4, x5) → Cond_22085_0_createTree_FieldAccess(x1, x3)
22085_0_createTree_FieldAccess(x1, x2, x3, x4) → 22085_0_createTree_FieldAccess(x2)
Cond_22144_0_createTree_Store(x1, x2, x3, x4, x5) → Cond_22144_0_createTree_Store(x1, x3, x5)
22144_0_createTree_Store(x1, x2, x3, x4) → 22144_0_createTree_Store(x2, x4)
Cond_26165_0_createTree_FieldAccess(x1, x2, x3, x4, x5, x6) → Cond_26165_0_createTree_FieldAccess(x1, x3)
26165_0_createTree_FieldAccess(x1, x2, x3, x4, x5) → 26165_0_createTree_FieldAccess(x2)
Cond_26042_0_createTree_Load(x1, x2, x3, x4) → Cond_26042_0_createTree_Load(x1, x3)
26042_0_createTree_Load(x1, x2, x3) → 26042_0_createTree_Load(x2)
Cond_26173_0_createTree_Load(x1, x2, x3, x4) → Cond_26173_0_createTree_Load(x1, x3)
26173_0_createTree_Load(x1, x2, x3) → 26173_0_createTree_Load(x2)
1751_0_createTree_InvokeMethod(x1, x2) → 1751_0_createTree_InvokeMethod
java.lang.ArrayIndexOutOfBoundsException(x1) → java.lang.ArrayIndexOutOfBoundsException
java.lang.IndexOutOfBoundsException(x1) → java.lang.IndexOutOfBoundsException
1371_0_random_ArrayAccess(x1, x2, x3) → 1371_0_random_ArrayAccess(x2, x3)
20795_0_createTree_InvokeMethod(x1, x2, x3, x4, x5) → 20795_0_createTree_InvokeMethod(x3, x5)
Cond_18356_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → Cond_18356_1_createTree_InvokeMethod(x1, x2, x3, x5)
18356_0_random_ArrayAccess(x1, x2, x3) → 18356_0_random_ArrayAccess(x2, x3)
18356_1_createTree_InvokeMethod(x1, x2, x3, x4) → 18356_1_createTree_InvokeMethod(x1, x2, x4)
19835_0_createTree_LE(x1, x2, x3, x4, x5) → 19835_0_createTree_LE(x2, x4, x5)
Cond_19835_0_createTree_LE8(x1, x2, x3, x4, x5, x6) → Cond_19835_0_createTree_LE8(x1, x3)
Cond_19835_0_createTree_LE7(x1, x2, x3, x4, x5, x6) → Cond_19835_0_createTree_LE7(x1, x3)
Cond_19835_0_createTree_LE6(x1, x2, x3, x4, x5, x6) → Cond_19835_0_createTree_LE6(x1, x3, x6)
Cond_19835_0_createTree_LE5(x1, x2, x3, x4, x5, x6) → Cond_19835_0_createTree_LE5(x1, x3, x6)
Cond_19835_0_createTree_LE4(x1, x2, x3, x4, x5, x6) → Cond_19835_0_createTree_LE4(x1, x3, x6)
Cond_19835_0_createTree_LE3(x1, x2, x3, x4, x5, x6) → Cond_19835_0_createTree_LE3(x1, x3, x6)
Cond_19835_0_createTree_LE2(x1, x2, x3, x4, x5, x6) → Cond_19835_0_createTree_LE2(x1, x3, x6)
Cond_19835_0_createTree_LE1(x1, x2, x3, x4, x5, x6) → Cond_19835_0_createTree_LE1(x1, x3, x6)
Cond_19835_0_createTree_LE(x1, x2, x3, x4, x5, x6) → Cond_19835_0_createTree_LE(x1, x3, x6)
Cond_18597_1_createTree_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod1(x1, x2, x3, x5)
18597_0_random_IntArithmetic(x1, x2, x3, x4) → 18597_0_random_IntArithmetic(x2, x3)
18597_1_createTree_InvokeMethod(x1, x2, x3, x4) → 18597_1_createTree_InvokeMethod(x1, x2, x4)
22671_0_createTree_InvokeMethod(x1, x2, x3, x4, x5) → 22671_0_createTree_InvokeMethod(x3, x5)
java.lang.NullPointerException(x1) → java.lang.NullPointerException
Cond_18597_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → Cond_18597_1_createTree_InvokeMethod(x1, x2, x3, x5)
Cond_18401_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → Cond_18401_1_createTree_InvokeMethod(x1, x2, x3, x5)
18401_0_random_ArrayAccess(x1, x2, x3) → 18401_0_random_ArrayAccess(x2, x3)
18401_1_createTree_InvokeMethod(x1, x2, x3, x4) → 18401_1_createTree_InvokeMethod(x1, x2, x4)
21157_0_createTree_InvokeMethod(x1, x2, x3, x4, x5) → 21157_0_createTree_InvokeMethod(x3, x5)
Cond_18417_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → Cond_18417_1_createTree_InvokeMethod(x1, x2, x3, x5)
18417_0_random_ArrayAccess(x1, x2, x3) → 18417_0_random_ArrayAccess(x2, x3)
18417_1_createTree_InvokeMethod(x1, x2, x3, x4) → 18417_1_createTree_InvokeMethod(x1, x2, x4)
Cond_18239_0_createTree_LE2(x1, x2, x3, x4, x5, x6) → Cond_18239_0_createTree_LE2(x1, x3, x5, x6)
Cond_18239_0_createTree_LE1(x1, x2, x3, x4, x5, x6) → Cond_18239_0_createTree_LE1(x1, x3, x5, x6)
Cond_18239_0_createTree_LE(x1, x2, x3, x4, x5, x6) → Cond_18239_0_createTree_LE(x1, x3, x5, x6)
Cond_1604_0_createTree_NE(x1, x2, x3, x4) → Cond_1604_0_createTree_NE(x1, x3, x4)
1604_0_createTree_NE(x1, x2, x3) → 1604_0_createTree_NE(x2, x3)
1425_0_random_IntArithmetic(x1, x2, x3, x4) → 1425_0_random_IntArithmetic(x2, x3)
1894_0_createTree_InvokeMethod(x1, x2) → 1894_0_createTree_InvokeMethod
1381_0_random_ArrayAccess(x1, x2, x3) → 1381_0_random_ArrayAccess(x2, x3)
1798_0_createTree_InvokeMethod(x1, x2) → 1798_0_createTree_InvokeMethod
1383_0_random_ArrayAccess(x1, x2, x3) → 1383_0_random_ArrayAccess(x2, x3)
9627_0_main_InvokeMethod(x1, x2, x3, x4) → 9627_0_main_InvokeMethod(x2, x3, x4)

Filtered duplicate args:


18239_0_createTree_LE(x1, x2, x3) → 18239_0_createTree_LE(x2, x3)
Cond_18239_0_createTree_LE2(x1, x2, x3, x4) → Cond_18239_0_createTree_LE2(x1, x3, x4)
Cond_18239_0_createTree_LE1(x1, x2, x3, x4) → Cond_18239_0_createTree_LE1(x1, x3, x4)
Cond_18239_0_createTree_LE(x1, x2, x3, x4) → Cond_18239_0_createTree_LE(x1, x3, x4)
Cond_1604_0_createTree_NE(x1, x2, x3) → Cond_1604_0_createTree_NE(x1, x3)
1604_0_createTree_NE(x1, x2) → 1604_0_createTree_NE(x2)

Filtered unneeded arguments:


9403_1_main_InvokeMethod(x1, x2, x3) → 9403_1_main_InvokeMethod(x1, x3)
Cond_9403_1_main_InvokeMethod(x1, x2, x3) → Cond_9403_1_main_InvokeMethod(x1, x3)
Cond_9403_1_main_InvokeMethod1(x1, x2, x3) → Cond_9403_1_main_InvokeMethod1(x1, x3)
Cond_18239_0_createTree_LE(x1, x2, x3) → Cond_18239_0_createTree_LE(x1, x2)
Cond_18239_0_createTree_LE2(x1, x2, x3) → Cond_18239_0_createTree_LE2(x1, x2)
18417_1_createTree_InvokeMethod(x1, x2, x3) → 18417_1_createTree_InvokeMethod(x1, x3)
Cond_18417_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_18417_1_createTree_InvokeMethod(x1, x2, x4)
Cond_18597_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_18597_1_createTree_InvokeMethod(x1, x2, x4)
Cond_19835_0_createTree_LE(x1, x2, x3) → Cond_19835_0_createTree_LE(x1, x2)
Cond_19835_0_createTree_LE1(x1, x2, x3) → Cond_19835_0_createTree_LE1(x1, x2)
Cond_19835_0_createTree_LE2(x1, x2, x3) → Cond_19835_0_createTree_LE2(x1, x2)
Cond_19835_0_createTree_LE3(x1, x2, x3) → Cond_19835_0_createTree_LE3(x1, x2)
Cond_19835_0_createTree_LE4(x1, x2, x3) → Cond_19835_0_createTree_LE4(x1, x2)
Cond_19835_0_createTree_LE5(x1, x2, x3) → Cond_19835_0_createTree_LE5(x1, x2)
Cond_19835_0_createTree_LE6(x1, x2, x3) → Cond_19835_0_createTree_LE6(x1, x2)
18356_1_createTree_InvokeMethod(x1, x2, x3) → 18356_1_createTree_InvokeMethod(x1, x3)
Cond_18356_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_18356_1_createTree_InvokeMethod(x1, x2, x4)

Filtered all non-integer terms:


9627_0_main_InvokeMethod(x1, x2, x3) → 9627_0_main_InvokeMethod(x2, x3)
20795_0_createTree_InvokeMethod(x1, x2) → 20795_0_createTree_InvokeMethod(x1)
21157_0_createTree_InvokeMethod(x1, x2) → 21157_0_createTree_InvokeMethod(x1)
22671_0_createTree_InvokeMethod(x1, x2) → 22671_0_createTree_InvokeMethod(x1)
1425_0_random_IntArithmetic(x1, x2) → 1425_0_random_IntArithmetic(x2)
18239_0_createTree_LE(x1, x2) → 18239_0_createTree_LE(x2)
Cond_18239_0_createTree_LE(x1, x2) → Cond_18239_0_createTree_LE(x1)
18356_1_createTree_InvokeMethod(x1, x2) → 18356_1_createTree_InvokeMethod(x1)
Cond_18239_0_createTree_LE1(x1, x2, x3) → Cond_18239_0_createTree_LE1(x1, x3)
18401_1_createTree_InvokeMethod(x1, x2, x3) → 18401_1_createTree_InvokeMethod(x1, x2)
Cond_18239_0_createTree_LE2(x1, x2) → Cond_18239_0_createTree_LE2(x1)
18417_1_createTree_InvokeMethod(x1, x2) → 18417_1_createTree_InvokeMethod(x1)
Cond_18417_1_createTree_InvokeMethod(x1, x2, x3) → Cond_18417_1_createTree_InvokeMethod(x1, x2)
Cond_18401_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_18401_1_createTree_InvokeMethod(x1, x2, x3)
18597_1_createTree_InvokeMethod(x1, x2, x3) → 18597_1_createTree_InvokeMethod(x1, x2)
Cond_18597_1_createTree_InvokeMethod(x1, x2, x3) → Cond_18597_1_createTree_InvokeMethod(x1, x2)
18597_0_random_IntArithmetic(x1, x2) → 18597_0_random_IntArithmetic(x2)
Cond_18597_1_createTree_InvokeMethod1(x1, x2, x3, x4) → Cond_18597_1_createTree_InvokeMethod1(x1, x2, x3)
19835_0_createTree_LE(x1, x2, x3) → 19835_0_createTree_LE(x1, x3)
22144_0_createTree_Store(x1, x2) → 22144_0_createTree_Store(x1)
22035_0_createTree_Store(x1, x2) → 22035_0_createTree_Store(x1)
Cond_18356_1_createTree_InvokeMethod(x1, x2, x3) → Cond_18356_1_createTree_InvokeMethod(x1, x2)
Cond_22144_0_createTree_Store(x1, x2, x3) → Cond_22144_0_createTree_Store(x1, x2)
Cond_22035_0_createTree_Store(x1, x2, x3) → Cond_22035_0_createTree_Store(x1, x2)

Combined rules. Obtained 2 rules for P and 47 rules for R.


Finished conversion. Obtained 2 rules for P and 47 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


R is empty.

The integer pair graph contains the following rules and edges:
(0): 18401_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_18401_1_CREATETREE_INVOKEMETHOD(x0[0] > 0 && 0 < x0[0] + -1, x0[0])
(1): COND_18401_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 18401_1_CREATETREE_INVOKEMETHOD(x0[1] + -1)

(0) -> (1), if ((x0[0] > 0 && 0 < x0[0] + -1 →* TRUE)∧(x0[0] →* x0[1]))


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



The set Q is empty.

(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 18401_1_CREATETREE_INVOKEMETHOD(x0) → COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0, 0), <(0, +(x0, -1))), x0) the following chains were created:
  • We consider the chain 18401_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0]), COND_18401_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 18401_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1)) which results in the following constraint:

    (1)    (&&(>(x0[0], 0), <(0, +(x0[0], -1)))=TRUE∧x0[0]=x0[1] ⇒ 18401_1_CREATETREE_INVOKEMETHOD(x0[0])≥NonInfC∧18401_1_CREATETREE_INVOKEMETHOD(x0[0])≥COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])∧(UIncreasing(COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])), ≥))



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

    (2)    (>(x0[0], 0)=TRUE∧<(0, +(x0[0], -1))=TRUE ⇒ 18401_1_CREATETREE_INVOKEMETHOD(x0[0])≥NonInfC∧18401_1_CREATETREE_INVOKEMETHOD(x0[0])≥COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])∧(UIncreasing(COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])), ≥))



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

    (3)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[1 + (-1)bso_11] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[1 + (-1)bso_11] ≥ 0)



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

    (5)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[1 + (-1)bso_11] ≥ 0)



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

    (6)    (x0[0] ≥ 0∧[-1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[1 + (-1)bso_11] ≥ 0)



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

    (7)    ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])), ≥)∧[(-1)Bound*bni_10 + (4)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[1 + (-1)bso_11] ≥ 0)







For Pair COND_18401_1_CREATETREE_INVOKEMETHOD(TRUE, x0) → 18401_1_CREATETREE_INVOKEMETHOD(+(x0, -1)) the following chains were created:
  • We consider the chain COND_18401_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 18401_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1)) which results in the following constraint:

    (8)    (COND_18401_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1])≥NonInfC∧COND_18401_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1])≥18401_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))∧(UIncreasing(18401_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥))



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

    (9)    ((UIncreasing(18401_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥)∧[1 + (-1)bso_13] ≥ 0)



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

    (10)    ((UIncreasing(18401_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥)∧[1 + (-1)bso_13] ≥ 0)



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

    (11)    ((UIncreasing(18401_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥)∧[1 + (-1)bso_13] ≥ 0)



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

    (12)    ((UIncreasing(18401_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥)∧0 = 0∧[1 + (-1)bso_13] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 18401_1_CREATETREE_INVOKEMETHOD(x0) → COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0, 0), <(0, +(x0, -1))), x0)
    • ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])), ≥)∧[(-1)Bound*bni_10 + (4)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[1 + (-1)bso_11] ≥ 0)

  • COND_18401_1_CREATETREE_INVOKEMETHOD(TRUE, x0) → 18401_1_CREATETREE_INVOKEMETHOD(+(x0, -1))
    • ((UIncreasing(18401_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥)∧0 = 0∧[1 + (-1)bso_13] ≥ 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(18401_1_CREATETREE_INVOKEMETHOD(x1)) = [2]x1   
POL(COND_18401_1_CREATETREE_INVOKEMETHOD(x1, x2)) = [-1] + [2]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

18401_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])
COND_18401_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 18401_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))

The following pairs are in Pbound:

18401_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_18401_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])

The following pairs are in P≥:
none

There are no usable rules.

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_18401_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 18401_1_CREATETREE_INVOKEMETHOD(x0[1] + -1)


The set Q is empty.

(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:
4752_0_flatten_NULL(NULL) → 5027_0_flatten_Return

The integer pair graph contains the following rules and edges:
(0): 7828_0_FLATTEN_STORE(java.lang.Object(Tree(x0[0], x1[0], x2[0])), x3[0]) → 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(x2[0], java.lang.Object(TreeList(x1[0], x3[0])))))
(1): 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(java.lang.Object(Tree(x0[1], x1[1], x2[1])), x3[1]))) → 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(x2[1], java.lang.Object(TreeList(x1[1], x3[1])))))
(2): 6786_0_FLATTEN_STORE(x1[2]) → 4752_0_FLATTEN_NULL(x1[2])
(3): 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(NULL, x0[3]))) → 4752_0_FLATTEN_NULL(x0[3])
(4): 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4]))) → 7828_0_FLATTEN_STORE(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4])
(5): 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(NULL, x0[5]))) → 6786_0_FLATTEN_STORE(x0[5])

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


(0) -> (3), if ((java.lang.Object(TreeList(x2[0], java.lang.Object(TreeList(x1[0], x3[0])))) →* java.lang.Object(TreeList(NULL, x0[3]))))


(0) -> (4), if ((java.lang.Object(TreeList(x2[0], java.lang.Object(TreeList(x1[0], x3[0])))) →* java.lang.Object(TreeList(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4]))))


(0) -> (5), if ((java.lang.Object(TreeList(x2[0], java.lang.Object(TreeList(x1[0], x3[0])))) →* java.lang.Object(TreeList(NULL, x0[5]))))


(1) -> (1), if ((java.lang.Object(TreeList(x2[1], java.lang.Object(TreeList(x1[1], x3[1])))) →* java.lang.Object(TreeList(java.lang.Object(Tree(x0[1]', x1[1]', x2[1]')), x3[1]'))))


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


(1) -> (4), if ((java.lang.Object(TreeList(x2[1], java.lang.Object(TreeList(x1[1], x3[1])))) →* java.lang.Object(TreeList(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4]))))


(1) -> (5), if ((java.lang.Object(TreeList(x2[1], java.lang.Object(TreeList(x1[1], x3[1])))) →* java.lang.Object(TreeList(NULL, x0[5]))))


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


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


(2) -> (4), if ((x1[2] →* java.lang.Object(TreeList(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4]))))


(2) -> (5), if ((x1[2] →* java.lang.Object(TreeList(NULL, x0[5]))))


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


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


(3) -> (4), if ((x0[3] →* java.lang.Object(TreeList(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4]))))


(3) -> (5), if ((x0[3] →* java.lang.Object(TreeList(NULL, x0[5]))))


(4) -> (0), if ((java.lang.Object(Tree(x0[4], x1[4], x2[4])) →* java.lang.Object(Tree(x0[0], x1[0], x2[0])))∧(x3[4] →* x3[0]))


(5) -> (2), if ((x0[5] →* x1[2]))



The set Q consists of the following terms:
4752_0_flatten_NULL(NULL)

(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:

7828_0_FLATTEN_STORE(java.lang.Object(Tree(x0[0], x1[0], x2[0])), x3[0]) → 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(x2[0], java.lang.Object(TreeList(x1[0], x3[0])))))
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(java.lang.Object(Tree(x0[1], x1[1], x2[1])), x3[1]))) → 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(x2[1], java.lang.Object(TreeList(x1[1], x3[1])))))
6786_0_FLATTEN_STORE(x1[2]) → 4752_0_FLATTEN_NULL(x1[2])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(NULL, x0[3]))) → 4752_0_FLATTEN_NULL(x0[3])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4]))) → 7828_0_FLATTEN_STORE(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(NULL, x0[5]))) → 6786_0_FLATTEN_STORE(x0[5])

The TRS R consists of the following rules:

4752_0_flatten_NULL(NULL) → 5027_0_flatten_Return

The set Q consists of the following terms:

4752_0_flatten_NULL(NULL)

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:

7828_0_FLATTEN_STORE(java.lang.Object(Tree(x0[0], x1[0], x2[0])), x3[0]) → 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(x2[0], java.lang.Object(TreeList(x1[0], x3[0])))))
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(java.lang.Object(Tree(x0[1], x1[1], x2[1])), x3[1]))) → 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(x2[1], java.lang.Object(TreeList(x1[1], x3[1])))))
6786_0_FLATTEN_STORE(x1[2]) → 4752_0_FLATTEN_NULL(x1[2])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(NULL, x0[3]))) → 4752_0_FLATTEN_NULL(x0[3])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4]))) → 7828_0_FLATTEN_STORE(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(NULL, x0[5]))) → 6786_0_FLATTEN_STORE(x0[5])

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

4752_0_flatten_NULL(NULL)

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].

4752_0_flatten_NULL(NULL)

(16) Obligation:

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

7828_0_FLATTEN_STORE(java.lang.Object(Tree(x0[0], x1[0], x2[0])), x3[0]) → 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(x2[0], java.lang.Object(TreeList(x1[0], x3[0])))))
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(java.lang.Object(Tree(x0[1], x1[1], x2[1])), x3[1]))) → 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(x2[1], java.lang.Object(TreeList(x1[1], x3[1])))))
6786_0_FLATTEN_STORE(x1[2]) → 4752_0_FLATTEN_NULL(x1[2])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(NULL, x0[3]))) → 4752_0_FLATTEN_NULL(x0[3])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4]))) → 7828_0_FLATTEN_STORE(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(NULL, x0[5]))) → 6786_0_FLATTEN_STORE(x0[5])

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

(17) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

7828_0_FLATTEN_STORE(java.lang.Object(Tree(x0[0], x1[0], x2[0])), x3[0]) → 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(x2[0], java.lang.Object(TreeList(x1[0], x3[0])))))
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(java.lang.Object(Tree(x0[1], x1[1], x2[1])), x3[1]))) → 4752_0_FLATTEN_NULL(java.lang.Object(TreeList(x2[1], java.lang.Object(TreeList(x1[1], x3[1])))))
6786_0_FLATTEN_STORE(x1[2]) → 4752_0_FLATTEN_NULL(x1[2])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(NULL, x0[3]))) → 4752_0_FLATTEN_NULL(x0[3])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4]))) → 7828_0_FLATTEN_STORE(java.lang.Object(Tree(x0[4], x1[4], x2[4])), x3[4])
4752_0_FLATTEN_NULL(java.lang.Object(TreeList(NULL, x0[5]))) → 6786_0_FLATTEN_STORE(x0[5])
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(4752_0_FLATTEN_NULL(x1)) = 1 + x1   
POL(6786_0_FLATTEN_STORE(x1)) = 2 + x1   
POL(7828_0_FLATTEN_STORE(x1, x2)) = 1 + 2·x1 + x2   
POL(NULL) = 0   
POL(Tree(x1, x2, x3)) = 2 + x1 + 2·x2 + x3   
POL(TreeList(x1, x2)) = 2·x1 + x2   
POL(java.lang.Object(x1)) = 1 + x1   

(18) Obligation:

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

(19) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(20) YES

(21) 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, Boolean


The ITRS R consists of the following rules:
9403_0_createTree_InvokeMethod → 1371_1_createTree_InvokeMethod(1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
9403_0_createTree_InvokeMethod → 1381_1_createTree_InvokeMethod(1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
9403_0_createTree_InvokeMethod → 1383_1_createTree_InvokeMethod(1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
9403_1_main_InvokeMethod(1751_0_createTree_InvokeMethod, x1) → 9627_0_main_InvokeMethod(x0, x1)
9403_1_main_InvokeMethod(1798_0_createTree_InvokeMethod, x1) → 9627_0_main_InvokeMethod(x0, x1)
9403_1_main_InvokeMethod(1894_0_createTree_InvokeMethod, x1) → 9627_0_main_InvokeMethod(x0, x1)
9403_1_main_InvokeMethod(20795_0_createTree_InvokeMethod(x0), x3) → 9627_0_main_InvokeMethod(x2, x3)
9403_1_main_InvokeMethod(21157_0_createTree_InvokeMethod(x0), x3) → 9627_0_main_InvokeMethod(x2, x3)
9403_1_main_InvokeMethod(22671_0_createTree_InvokeMethod(x0), x3) → 9627_0_main_InvokeMethod(x2, x3)
1383_1_createTree_InvokeMethod(1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_1383_1_createTree_InvokeMethod(x2 >= x0, 1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1383_1_createTree_InvokeMethod(TRUE, 1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 1798_0_createTree_InvokeMethod
1381_1_createTree_InvokeMethod(1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_1381_1_createTree_InvokeMethod(x2 >= 0 && x2 < x0, 1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1381_1_createTree_InvokeMethod(TRUE, 1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x4))
1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x0)) → Cond_1425_1_createTree_InvokeMethod(x0 >= 0, 1425_0_random_IntArithmetic(x0))
Cond_1425_1_createTree_InvokeMethod(TRUE, 1425_0_random_IntArithmetic(x0)) → 1894_0_createTree_InvokeMethod
1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x1)) → Cond_1425_1_createTree_InvokeMethod1(x1 >= 0, 1425_0_random_IntArithmetic(x1))
Cond_1425_1_createTree_InvokeMethod1(TRUE, 1425_0_random_IntArithmetic(x1)) → 1604_0_createTree_NE(x3)
1604_0_createTree_NE(0) → 1677_0_createTree_Return
1604_0_createTree_NE(x0) → Cond_1604_0_createTree_NE(!(x0 = 0), x0)
Cond_1604_0_createTree_NE(TRUE, x0) → 18239_0_createTree_LE(x0)
18239_0_createTree_LE(0) → 18266_0_createTree_Return
18239_0_createTree_LE(x0) → Cond_18239_0_createTree_LE(x0 > 0, x0)
Cond_18239_0_createTree_LE(TRUE, x0) → 18356_1_createTree_InvokeMethod(18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4))
18239_0_createTree_LE(x0) → Cond_18239_0_createTree_LE1(x0 > 0, x0)
Cond_18239_0_createTree_LE1(TRUE, x0) → 18401_1_createTree_InvokeMethod(18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4), x0)
18239_0_createTree_LE(x0) → Cond_18239_0_createTree_LE2(x0 > 0, x0)
Cond_18239_0_createTree_LE2(TRUE, x0) → 18417_1_createTree_InvokeMethod(18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4))
18417_1_createTree_InvokeMethod(18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_18417_1_createTree_InvokeMethod(x2 >= x0, 18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18417_1_createTree_InvokeMethod(TRUE, 18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 21157_0_createTree_InvokeMethod(x3)
18401_1_createTree_InvokeMethod(18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → Cond_18401_1_createTree_InvokeMethod(x2 >= 0 && x2 < x0, 18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
Cond_18401_1_createTree_InvokeMethod(TRUE, 18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x6), x3)
18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x0), x2) → Cond_18597_1_createTree_InvokeMethod(x0 >= 0, 18597_0_random_IntArithmetic(x0), x2)
Cond_18597_1_createTree_InvokeMethod(TRUE, 18597_0_random_IntArithmetic(x0), x2) → 22671_0_createTree_InvokeMethod(x2)
18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x1), x3) → Cond_18597_1_createTree_InvokeMethod1(x1 >= 0, 18597_0_random_IntArithmetic(x1), x3)
Cond_18597_1_createTree_InvokeMethod1(TRUE, 18597_0_random_IntArithmetic(x1), x3) → 19835_0_createTree_LE(x3, x5)
19835_0_createTree_LE(x0, x1) → Cond_19835_0_createTree_LE(x1 > 0 && x0 > 0, x0, x1)
Cond_19835_0_createTree_LE(TRUE, x0, x1) → 18239_0_createTree_LE(x0 + -1)
19835_0_createTree_LE(x0, x1) → Cond_19835_0_createTree_LE1(x1 > 0, x0, x1)
Cond_19835_0_createTree_LE1(TRUE, x0, x1) → 26173_0_createTree_Load(x0)
19835_0_createTree_LE(x0, 0) → Cond_19835_0_createTree_LE6(x0 > 0, x0, 0)
Cond_19835_0_createTree_LE6(TRUE, x0, 0) → 18239_0_createTree_LE(x0 + -1)
19835_0_createTree_LE(x0, 0) → 26173_0_createTree_Load(x0)
18356_1_createTree_InvokeMethod(18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_18356_1_createTree_InvokeMethod(x2 <= -1, 18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18356_1_createTree_InvokeMethod(TRUE, 18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 20795_0_createTree_InvokeMethod(x3)
1371_1_createTree_InvokeMethod(1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_1371_1_createTree_InvokeMethod(x2 <= -1, 1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1371_1_createTree_InvokeMethod(TRUE, 1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 1751_0_createTree_InvokeMethod
26173_0_createTree_Load(x0) → Cond_26173_0_createTree_Load(x0 > 0, x0)
Cond_26173_0_createTree_Load(TRUE, x0) → 18239_0_createTree_LE(x0 + -1)

The integer pair graph contains the following rules and edges:
(0): 9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2[0]) → COND_9403_1_MAIN_INVOKEMETHOD(x2[0] > 0 && 0 < x2[0] + -1, 1677_0_createTree_Return, x2[0])
(1): COND_9403_1_MAIN_INVOKEMETHOD(TRUE, 1677_0_createTree_Return, x2[1]) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, x2[1] + -1)
(2): 9403_1_MAIN_INVOKEMETHOD(18266_0_createTree_Return, x1[2]) → COND_9403_1_MAIN_INVOKEMETHOD1(x1[2] > 0 && 0 < x1[2] + -1, 18266_0_createTree_Return, x1[2])
(3): COND_9403_1_MAIN_INVOKEMETHOD1(TRUE, 18266_0_createTree_Return, x1[3]) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, x1[3] + -1)

(0) -> (1), if ((x2[0] > 0 && 0 < x2[0] + -1 →* TRUE)∧(x2[0] →* x2[1]))


(1) -> (0), if ((9403_0_createTree_InvokeMethod →* 1677_0_createTree_Return)∧(x2[1] + -1 →* x2[0]))


(1) -> (2), if ((9403_0_createTree_InvokeMethod →* 18266_0_createTree_Return)∧(x2[1] + -1 →* x1[2]))


(2) -> (3), if ((x1[2] > 0 && 0 < x1[2] + -1 →* TRUE)∧(x1[2] →* x1[3]))


(3) -> (0), if ((9403_0_createTree_InvokeMethod →* 1677_0_createTree_Return)∧(x1[3] + -1 →* x2[0]))


(3) -> (2), if ((9403_0_createTree_InvokeMethod →* 18266_0_createTree_Return)∧(x1[3] + -1 →* x1[2]))



The set Q consists of the following terms:
9403_0_createTree_InvokeMethod
9403_1_main_InvokeMethod(1751_0_createTree_InvokeMethod, x0)
9403_1_main_InvokeMethod(1798_0_createTree_InvokeMethod, x0)
9403_1_main_InvokeMethod(1894_0_createTree_InvokeMethod, x0)
9403_1_main_InvokeMethod(20795_0_createTree_InvokeMethod(x0), x1)
9403_1_main_InvokeMethod(21157_0_createTree_InvokeMethod(x0), x1)
9403_1_main_InvokeMethod(22671_0_createTree_InvokeMethod(x0), x1)
1383_1_createTree_InvokeMethod(1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1383_1_createTree_InvokeMethod(TRUE, 1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
1381_1_createTree_InvokeMethod(1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1381_1_createTree_InvokeMethod(TRUE, 1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x0))
Cond_1425_1_createTree_InvokeMethod(TRUE, 1425_0_random_IntArithmetic(x0))
Cond_1425_1_createTree_InvokeMethod1(TRUE, 1425_0_random_IntArithmetic(x0))
1604_0_createTree_NE(x0)
Cond_1604_0_createTree_NE(TRUE, x0)
18239_0_createTree_LE(x0)
Cond_18239_0_createTree_LE(TRUE, x0)
Cond_18239_0_createTree_LE1(TRUE, x0)
Cond_18239_0_createTree_LE2(TRUE, x0)
18417_1_createTree_InvokeMethod(18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18417_1_createTree_InvokeMethod(TRUE, 18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
18401_1_createTree_InvokeMethod(18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
Cond_18401_1_createTree_InvokeMethod(TRUE, 18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x0), x1)
Cond_18597_1_createTree_InvokeMethod(TRUE, 18597_0_random_IntArithmetic(x0), x1)
Cond_18597_1_createTree_InvokeMethod1(TRUE, 18597_0_random_IntArithmetic(x0), x1)
19835_0_createTree_LE(x0, x1)
Cond_19835_0_createTree_LE(TRUE, x0, x1)
Cond_19835_0_createTree_LE1(TRUE, x0, x1)
Cond_19835_0_createTree_LE6(TRUE, x0, 0)
18356_1_createTree_InvokeMethod(18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18356_1_createTree_InvokeMethod(TRUE, 18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
1371_1_createTree_InvokeMethod(1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1371_1_createTree_InvokeMethod(TRUE, 1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
26173_0_createTree_Load(x0)
Cond_26173_0_createTree_Load(TRUE, x0)

(22) 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 9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2) → COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2, 0), <(0, +(x2, -1))), 1677_0_createTree_Return, x2) the following chains were created:
  • We consider the chain 9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2[0]) → COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0]), COND_9403_1_MAIN_INVOKEMETHOD(TRUE, 1677_0_createTree_Return, x2[1]) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2[1], -1)) which results in the following constraint:

    (1)    (&&(>(x2[0], 0), <(0, +(x2[0], -1)))=TRUE∧x2[0]=x2[1] ⇒ 9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2[0])≥NonInfC∧9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2[0])≥COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])∧(UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])), ≥))



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

    (2)    (>(x2[0], 0)=TRUE∧<(0, +(x2[0], -1))=TRUE ⇒ 9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2[0])≥NonInfC∧9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2[0])≥COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])∧(UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])), ≥))



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

    (3)    (x2[0] + [-1] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])), ≥)∧[(-1)Bound*bni_103] + [(2)bni_103]x2[0] ≥ 0∧[(-1)bso_104] ≥ 0)



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

    (4)    (x2[0] + [-1] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])), ≥)∧[(-1)Bound*bni_103] + [(2)bni_103]x2[0] ≥ 0∧[(-1)bso_104] ≥ 0)



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

    (5)    (x2[0] + [-1] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])), ≥)∧[(-1)Bound*bni_103] + [(2)bni_103]x2[0] ≥ 0∧[(-1)bso_104] ≥ 0)



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

    (6)    (x2[0] ≥ 0∧[-1] + x2[0] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])), ≥)∧[(-1)Bound*bni_103 + (2)bni_103] + [(2)bni_103]x2[0] ≥ 0∧[(-1)bso_104] ≥ 0)



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

    (7)    ([1] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])), ≥)∧[(-1)Bound*bni_103 + (4)bni_103] + [(2)bni_103]x2[0] ≥ 0∧[(-1)bso_104] ≥ 0)







For Pair COND_9403_1_MAIN_INVOKEMETHOD(TRUE, 1677_0_createTree_Return, x2) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2, -1)) the following chains were created:
  • We consider the chain COND_9403_1_MAIN_INVOKEMETHOD(TRUE, 1677_0_createTree_Return, x2[1]) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2[1], -1)) which results in the following constraint:

    (8)    (COND_9403_1_MAIN_INVOKEMETHOD(TRUE, 1677_0_createTree_Return, x2[1])≥NonInfC∧COND_9403_1_MAIN_INVOKEMETHOD(TRUE, 1677_0_createTree_Return, x2[1])≥9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2[1], -1))∧(UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2[1], -1))), ≥))



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

    (9)    ((UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2[1], -1))), ≥)∧[2 + (-1)bso_106] ≥ 0)



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

    (10)    ((UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2[1], -1))), ≥)∧[2 + (-1)bso_106] ≥ 0)



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

    (11)    ((UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2[1], -1))), ≥)∧[2 + (-1)bso_106] ≥ 0)



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

    (12)    ((UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_106] ≥ 0)







For Pair 9403_1_MAIN_INVOKEMETHOD(18266_0_createTree_Return, x1) → COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1, 0), <(0, +(x1, -1))), 18266_0_createTree_Return, x1) the following chains were created:
  • We consider the chain 9403_1_MAIN_INVOKEMETHOD(18266_0_createTree_Return, x1[2]) → COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2]), COND_9403_1_MAIN_INVOKEMETHOD1(TRUE, 18266_0_createTree_Return, x1[3]) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1[3], -1)) which results in the following constraint:

    (13)    (&&(>(x1[2], 0), <(0, +(x1[2], -1)))=TRUE∧x1[2]=x1[3] ⇒ 9403_1_MAIN_INVOKEMETHOD(18266_0_createTree_Return, x1[2])≥NonInfC∧9403_1_MAIN_INVOKEMETHOD(18266_0_createTree_Return, x1[2])≥COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])∧(UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])), ≥))



    We simplified constraint (13) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (14)    (>(x1[2], 0)=TRUE∧<(0, +(x1[2], -1))=TRUE ⇒ 9403_1_MAIN_INVOKEMETHOD(18266_0_createTree_Return, x1[2])≥NonInfC∧9403_1_MAIN_INVOKEMETHOD(18266_0_createTree_Return, x1[2])≥COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])∧(UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])), ≥))



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

    (15)    (x1[2] + [-1] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])), ≥)∧[(-1)Bound*bni_107] + [(2)bni_107]x1[2] ≥ 0∧[1 + (-1)bso_108] ≥ 0)



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

    (16)    (x1[2] + [-1] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])), ≥)∧[(-1)Bound*bni_107] + [(2)bni_107]x1[2] ≥ 0∧[1 + (-1)bso_108] ≥ 0)



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

    (17)    (x1[2] + [-1] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])), ≥)∧[(-1)Bound*bni_107] + [(2)bni_107]x1[2] ≥ 0∧[1 + (-1)bso_108] ≥ 0)



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

    (18)    (x1[2] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])), ≥)∧[(-1)Bound*bni_107 + (2)bni_107] + [(2)bni_107]x1[2] ≥ 0∧[1 + (-1)bso_108] ≥ 0)



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

    (19)    ([1] + x1[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])), ≥)∧[(-1)Bound*bni_107 + (4)bni_107] + [(2)bni_107]x1[2] ≥ 0∧[1 + (-1)bso_108] ≥ 0)







For Pair COND_9403_1_MAIN_INVOKEMETHOD1(TRUE, 18266_0_createTree_Return, x1) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1, -1)) the following chains were created:
  • We consider the chain COND_9403_1_MAIN_INVOKEMETHOD1(TRUE, 18266_0_createTree_Return, x1[3]) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1[3], -1)) which results in the following constraint:

    (20)    (COND_9403_1_MAIN_INVOKEMETHOD1(TRUE, 18266_0_createTree_Return, x1[3])≥NonInfC∧COND_9403_1_MAIN_INVOKEMETHOD1(TRUE, 18266_0_createTree_Return, x1[3])≥9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1[3], -1))∧(UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1[3], -1))), ≥))



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

    (21)    ((UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1[3], -1))), ≥)∧[1 + (-1)bso_110] ≥ 0)



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

    (22)    ((UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1[3], -1))), ≥)∧[1 + (-1)bso_110] ≥ 0)



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

    (23)    ((UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1[3], -1))), ≥)∧[1 + (-1)bso_110] ≥ 0)



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

    (24)    ((UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1[3], -1))), ≥)∧0 = 0∧[1 + (-1)bso_110] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2) → COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2, 0), <(0, +(x2, -1))), 1677_0_createTree_Return, x2)
    • ([1] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])), ≥)∧[(-1)Bound*bni_103 + (4)bni_103] + [(2)bni_103]x2[0] ≥ 0∧[(-1)bso_104] ≥ 0)

  • COND_9403_1_MAIN_INVOKEMETHOD(TRUE, 1677_0_createTree_Return, x2) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2, -1))
    • ((UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_106] ≥ 0)

  • 9403_1_MAIN_INVOKEMETHOD(18266_0_createTree_Return, x1) → COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1, 0), <(0, +(x1, -1))), 18266_0_createTree_Return, x1)
    • ([1] + x1[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])), ≥)∧[(-1)Bound*bni_107 + (4)bni_107] + [(2)bni_107]x1[2] ≥ 0∧[1 + (-1)bso_108] ≥ 0)

  • COND_9403_1_MAIN_INVOKEMETHOD1(TRUE, 18266_0_createTree_Return, x1) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1, -1))
    • ((UIncreasing(9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1[3], -1))), ≥)∧0 = 0∧[1 + (-1)bso_110] ≥ 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(9403_0_createTree_InvokeMethod) = [-1]   
POL(1371_1_createTree_InvokeMethod(x1)) = [-1] + [-1]x1   
POL(1371_0_random_ArrayAccess(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(java.lang.Object(x1)) = x1   
POL(ARRAY(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(1381_1_createTree_InvokeMethod(x1)) = [-1] + [-1]x1   
POL(1381_0_random_ArrayAccess(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(1383_1_createTree_InvokeMethod(x1)) = [-1] + [-1]x1   
POL(1383_0_random_ArrayAccess(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(9403_1_main_InvokeMethod(x1, x2)) = [-1]   
POL(1751_0_createTree_InvokeMethod) = [-1]   
POL(9627_0_main_InvokeMethod(x1, x2)) = [-1]   
POL(1798_0_createTree_InvokeMethod) = [-1]   
POL(1894_0_createTree_InvokeMethod) = [-1]   
POL(20795_0_createTree_InvokeMethod(x1)) = [-1]   
POL(21157_0_createTree_InvokeMethod(x1)) = [-1]   
POL(22671_0_createTree_InvokeMethod(x1)) = x1   
POL(Cond_1383_1_createTree_InvokeMethod(x1, x2)) = [-1] + [-1]x2   
POL(>=(x1, x2)) = [-1]   
POL(Cond_1381_1_createTree_InvokeMethod(x1, x2)) = [-1] + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   
POL(1425_1_createTree_InvokeMethod(x1)) = [-1] + [-1]x1   
POL(1425_0_random_IntArithmetic(x1)) = x1   
POL(Cond_1425_1_createTree_InvokeMethod(x1, x2)) = [-1] + [-1]x2   
POL(Cond_1425_1_createTree_InvokeMethod1(x1, x2)) = [-1] + [-1]x2   
POL(1604_0_createTree_NE(x1)) = [-1] + [-1]x1   
POL(1677_0_createTree_Return) = [-1]   
POL(Cond_1604_0_createTree_NE(x1, x2)) = [-1] + [-1]x2   
POL(!(x1)) = [-1]   
POL(=(x1, x2)) = [-1]   
POL(18239_0_createTree_LE(x1)) = [-1] + [-1]x1   
POL(18266_0_createTree_Return) = [-1]   
POL(Cond_18239_0_createTree_LE(x1, x2)) = [-1] + [-1]x2   
POL(>(x1, x2)) = [-1]   
POL(18356_1_createTree_InvokeMethod(x1)) = [-1] + [-1]x1   
POL(18356_0_random_ArrayAccess(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(Cond_18239_0_createTree_LE1(x1, x2)) = [-1] + [-1]x2   
POL(18401_1_createTree_InvokeMethod(x1, x2)) = [-1] + [-1]x1 + [-1]x2   
POL(18401_0_random_ArrayAccess(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(Cond_18239_0_createTree_LE2(x1, x2)) = [-1] + [-1]x2   
POL(18417_1_createTree_InvokeMethod(x1)) = [-1] + [-1]x1   
POL(18417_0_random_ArrayAccess(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(Cond_18417_1_createTree_InvokeMethod(x1, x2)) = [-1] + [-1]x2   
POL(Cond_18401_1_createTree_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2   
POL(18597_1_createTree_InvokeMethod(x1, x2)) = [-1] + [-1]x1 + [-1]x2   
POL(18597_0_random_IntArithmetic(x1)) = x1   
POL(Cond_18597_1_createTree_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2   
POL(Cond_18597_1_createTree_InvokeMethod1(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2   
POL(19835_0_createTree_LE(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(Cond_19835_0_createTree_LE(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(Cond_19835_0_createTree_LE1(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2   
POL(26173_0_createTree_Load(x1)) = [-1] + [-1]x1   
POL(Cond_19835_0_createTree_LE6(x1, x2, x3)) = [-1] + [-1]x2   
POL(Cond_18356_1_createTree_InvokeMethod(x1, x2)) = [-1] + [-1]x2   
POL(<=(x1, x2)) = [-1]   
POL(Cond_1371_1_createTree_InvokeMethod(x1, x2)) = [-1] + [-1]x2   
POL(Cond_26173_0_createTree_Load(x1, x2)) = [-1] + [-1]x2   
POL(9403_1_MAIN_INVOKEMETHOD(x1, x2)) = [2]x2   
POL(COND_9403_1_MAIN_INVOKEMETHOD(x1, x2, x3)) = [2]x3   
POL(COND_9403_1_MAIN_INVOKEMETHOD1(x1, x2, x3)) = [-1] + [2]x3   

The following pairs are in P>:

COND_9403_1_MAIN_INVOKEMETHOD(TRUE, 1677_0_createTree_Return, x2[1]) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x2[1], -1))
9403_1_MAIN_INVOKEMETHOD(18266_0_createTree_Return, x1[2]) → COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])
COND_9403_1_MAIN_INVOKEMETHOD1(TRUE, 18266_0_createTree_Return, x1[3]) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, +(x1[3], -1))

The following pairs are in Pbound:

9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2[0]) → COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])
9403_1_MAIN_INVOKEMETHOD(18266_0_createTree_Return, x1[2]) → COND_9403_1_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <(0, +(x1[2], -1))), 18266_0_createTree_Return, x1[2])

The following pairs are in P≥:

9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2[0]) → COND_9403_1_MAIN_INVOKEMETHOD(&&(>(x2[0], 0), <(0, +(x2[0], -1))), 1677_0_createTree_Return, x2[0])

There are no usable rules.

(23) Complex Obligation (AND)

(24) 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, Boolean


The ITRS R consists of the following rules:
9403_0_createTree_InvokeMethod → 1371_1_createTree_InvokeMethod(1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
9403_0_createTree_InvokeMethod → 1381_1_createTree_InvokeMethod(1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
9403_0_createTree_InvokeMethod → 1383_1_createTree_InvokeMethod(1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
9403_1_main_InvokeMethod(1751_0_createTree_InvokeMethod, x1) → 9627_0_main_InvokeMethod(x0, x1)
9403_1_main_InvokeMethod(1798_0_createTree_InvokeMethod, x1) → 9627_0_main_InvokeMethod(x0, x1)
9403_1_main_InvokeMethod(1894_0_createTree_InvokeMethod, x1) → 9627_0_main_InvokeMethod(x0, x1)
9403_1_main_InvokeMethod(20795_0_createTree_InvokeMethod(x0), x3) → 9627_0_main_InvokeMethod(x2, x3)
9403_1_main_InvokeMethod(21157_0_createTree_InvokeMethod(x0), x3) → 9627_0_main_InvokeMethod(x2, x3)
9403_1_main_InvokeMethod(22671_0_createTree_InvokeMethod(x0), x3) → 9627_0_main_InvokeMethod(x2, x3)
1383_1_createTree_InvokeMethod(1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_1383_1_createTree_InvokeMethod(x2 >= x0, 1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1383_1_createTree_InvokeMethod(TRUE, 1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 1798_0_createTree_InvokeMethod
1381_1_createTree_InvokeMethod(1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_1381_1_createTree_InvokeMethod(x2 >= 0 && x2 < x0, 1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1381_1_createTree_InvokeMethod(TRUE, 1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x4))
1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x0)) → Cond_1425_1_createTree_InvokeMethod(x0 >= 0, 1425_0_random_IntArithmetic(x0))
Cond_1425_1_createTree_InvokeMethod(TRUE, 1425_0_random_IntArithmetic(x0)) → 1894_0_createTree_InvokeMethod
1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x1)) → Cond_1425_1_createTree_InvokeMethod1(x1 >= 0, 1425_0_random_IntArithmetic(x1))
Cond_1425_1_createTree_InvokeMethod1(TRUE, 1425_0_random_IntArithmetic(x1)) → 1604_0_createTree_NE(x3)
1604_0_createTree_NE(0) → 1677_0_createTree_Return
1604_0_createTree_NE(x0) → Cond_1604_0_createTree_NE(!(x0 = 0), x0)
Cond_1604_0_createTree_NE(TRUE, x0) → 18239_0_createTree_LE(x0)
18239_0_createTree_LE(0) → 18266_0_createTree_Return
18239_0_createTree_LE(x0) → Cond_18239_0_createTree_LE(x0 > 0, x0)
Cond_18239_0_createTree_LE(TRUE, x0) → 18356_1_createTree_InvokeMethod(18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4))
18239_0_createTree_LE(x0) → Cond_18239_0_createTree_LE1(x0 > 0, x0)
Cond_18239_0_createTree_LE1(TRUE, x0) → 18401_1_createTree_InvokeMethod(18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4), x0)
18239_0_createTree_LE(x0) → Cond_18239_0_createTree_LE2(x0 > 0, x0)
Cond_18239_0_createTree_LE2(TRUE, x0) → 18417_1_createTree_InvokeMethod(18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4))
18417_1_createTree_InvokeMethod(18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_18417_1_createTree_InvokeMethod(x2 >= x0, 18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18417_1_createTree_InvokeMethod(TRUE, 18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 21157_0_createTree_InvokeMethod(x3)
18401_1_createTree_InvokeMethod(18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → Cond_18401_1_createTree_InvokeMethod(x2 >= 0 && x2 < x0, 18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
Cond_18401_1_createTree_InvokeMethod(TRUE, 18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x6), x3)
18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x0), x2) → Cond_18597_1_createTree_InvokeMethod(x0 >= 0, 18597_0_random_IntArithmetic(x0), x2)
Cond_18597_1_createTree_InvokeMethod(TRUE, 18597_0_random_IntArithmetic(x0), x2) → 22671_0_createTree_InvokeMethod(x2)
18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x1), x3) → Cond_18597_1_createTree_InvokeMethod1(x1 >= 0, 18597_0_random_IntArithmetic(x1), x3)
Cond_18597_1_createTree_InvokeMethod1(TRUE, 18597_0_random_IntArithmetic(x1), x3) → 19835_0_createTree_LE(x3, x5)
19835_0_createTree_LE(x0, x1) → Cond_19835_0_createTree_LE(x1 > 0 && x0 > 0, x0, x1)
Cond_19835_0_createTree_LE(TRUE, x0, x1) → 18239_0_createTree_LE(x0 + -1)
19835_0_createTree_LE(x0, x1) → Cond_19835_0_createTree_LE1(x1 > 0, x0, x1)
Cond_19835_0_createTree_LE1(TRUE, x0, x1) → 26173_0_createTree_Load(x0)
19835_0_createTree_LE(x0, 0) → Cond_19835_0_createTree_LE6(x0 > 0, x0, 0)
Cond_19835_0_createTree_LE6(TRUE, x0, 0) → 18239_0_createTree_LE(x0 + -1)
19835_0_createTree_LE(x0, 0) → 26173_0_createTree_Load(x0)
18356_1_createTree_InvokeMethod(18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_18356_1_createTree_InvokeMethod(x2 <= -1, 18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18356_1_createTree_InvokeMethod(TRUE, 18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 20795_0_createTree_InvokeMethod(x3)
1371_1_createTree_InvokeMethod(1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_1371_1_createTree_InvokeMethod(x2 <= -1, 1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1371_1_createTree_InvokeMethod(TRUE, 1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 1751_0_createTree_InvokeMethod
26173_0_createTree_Load(x0) → Cond_26173_0_createTree_Load(x0 > 0, x0)
Cond_26173_0_createTree_Load(TRUE, x0) → 18239_0_createTree_LE(x0 + -1)

The integer pair graph contains the following rules and edges:
(0): 9403_1_MAIN_INVOKEMETHOD(1677_0_createTree_Return, x2[0]) → COND_9403_1_MAIN_INVOKEMETHOD(x2[0] > 0 && 0 < x2[0] + -1, 1677_0_createTree_Return, x2[0])


The set Q consists of the following terms:
9403_0_createTree_InvokeMethod
9403_1_main_InvokeMethod(1751_0_createTree_InvokeMethod, x0)
9403_1_main_InvokeMethod(1798_0_createTree_InvokeMethod, x0)
9403_1_main_InvokeMethod(1894_0_createTree_InvokeMethod, x0)
9403_1_main_InvokeMethod(20795_0_createTree_InvokeMethod(x0), x1)
9403_1_main_InvokeMethod(21157_0_createTree_InvokeMethod(x0), x1)
9403_1_main_InvokeMethod(22671_0_createTree_InvokeMethod(x0), x1)
1383_1_createTree_InvokeMethod(1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1383_1_createTree_InvokeMethod(TRUE, 1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
1381_1_createTree_InvokeMethod(1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1381_1_createTree_InvokeMethod(TRUE, 1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x0))
Cond_1425_1_createTree_InvokeMethod(TRUE, 1425_0_random_IntArithmetic(x0))
Cond_1425_1_createTree_InvokeMethod1(TRUE, 1425_0_random_IntArithmetic(x0))
1604_0_createTree_NE(x0)
Cond_1604_0_createTree_NE(TRUE, x0)
18239_0_createTree_LE(x0)
Cond_18239_0_createTree_LE(TRUE, x0)
Cond_18239_0_createTree_LE1(TRUE, x0)
Cond_18239_0_createTree_LE2(TRUE, x0)
18417_1_createTree_InvokeMethod(18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18417_1_createTree_InvokeMethod(TRUE, 18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
18401_1_createTree_InvokeMethod(18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
Cond_18401_1_createTree_InvokeMethod(TRUE, 18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x0), x1)
Cond_18597_1_createTree_InvokeMethod(TRUE, 18597_0_random_IntArithmetic(x0), x1)
Cond_18597_1_createTree_InvokeMethod1(TRUE, 18597_0_random_IntArithmetic(x0), x1)
19835_0_createTree_LE(x0, x1)
Cond_19835_0_createTree_LE(TRUE, x0, x1)
Cond_19835_0_createTree_LE1(TRUE, x0, x1)
Cond_19835_0_createTree_LE6(TRUE, x0, 0)
18356_1_createTree_InvokeMethod(18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18356_1_createTree_InvokeMethod(TRUE, 18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
1371_1_createTree_InvokeMethod(1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1371_1_createTree_InvokeMethod(TRUE, 1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
26173_0_createTree_Load(x0)
Cond_26173_0_createTree_Load(TRUE, x0)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE

(27) 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, Boolean


The ITRS R consists of the following rules:
9403_0_createTree_InvokeMethod → 1371_1_createTree_InvokeMethod(1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
9403_0_createTree_InvokeMethod → 1381_1_createTree_InvokeMethod(1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
9403_0_createTree_InvokeMethod → 1383_1_createTree_InvokeMethod(1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
9403_1_main_InvokeMethod(1751_0_createTree_InvokeMethod, x1) → 9627_0_main_InvokeMethod(x0, x1)
9403_1_main_InvokeMethod(1798_0_createTree_InvokeMethod, x1) → 9627_0_main_InvokeMethod(x0, x1)
9403_1_main_InvokeMethod(1894_0_createTree_InvokeMethod, x1) → 9627_0_main_InvokeMethod(x0, x1)
9403_1_main_InvokeMethod(20795_0_createTree_InvokeMethod(x0), x3) → 9627_0_main_InvokeMethod(x2, x3)
9403_1_main_InvokeMethod(21157_0_createTree_InvokeMethod(x0), x3) → 9627_0_main_InvokeMethod(x2, x3)
9403_1_main_InvokeMethod(22671_0_createTree_InvokeMethod(x0), x3) → 9627_0_main_InvokeMethod(x2, x3)
1383_1_createTree_InvokeMethod(1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_1383_1_createTree_InvokeMethod(x2 >= x0, 1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1383_1_createTree_InvokeMethod(TRUE, 1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 1798_0_createTree_InvokeMethod
1381_1_createTree_InvokeMethod(1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_1381_1_createTree_InvokeMethod(x2 >= 0 && x2 < x0, 1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1381_1_createTree_InvokeMethod(TRUE, 1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x4))
1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x0)) → Cond_1425_1_createTree_InvokeMethod(x0 >= 0, 1425_0_random_IntArithmetic(x0))
Cond_1425_1_createTree_InvokeMethod(TRUE, 1425_0_random_IntArithmetic(x0)) → 1894_0_createTree_InvokeMethod
1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x1)) → Cond_1425_1_createTree_InvokeMethod1(x1 >= 0, 1425_0_random_IntArithmetic(x1))
Cond_1425_1_createTree_InvokeMethod1(TRUE, 1425_0_random_IntArithmetic(x1)) → 1604_0_createTree_NE(x3)
1604_0_createTree_NE(0) → 1677_0_createTree_Return
1604_0_createTree_NE(x0) → Cond_1604_0_createTree_NE(!(x0 = 0), x0)
Cond_1604_0_createTree_NE(TRUE, x0) → 18239_0_createTree_LE(x0)
18239_0_createTree_LE(0) → 18266_0_createTree_Return
18239_0_createTree_LE(x0) → Cond_18239_0_createTree_LE(x0 > 0, x0)
Cond_18239_0_createTree_LE(TRUE, x0) → 18356_1_createTree_InvokeMethod(18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4))
18239_0_createTree_LE(x0) → Cond_18239_0_createTree_LE1(x0 > 0, x0)
Cond_18239_0_createTree_LE1(TRUE, x0) → 18401_1_createTree_InvokeMethod(18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4), x0)
18239_0_createTree_LE(x0) → Cond_18239_0_createTree_LE2(x0 > 0, x0)
Cond_18239_0_createTree_LE2(TRUE, x0) → 18417_1_createTree_InvokeMethod(18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4))
18417_1_createTree_InvokeMethod(18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_18417_1_createTree_InvokeMethod(x2 >= x0, 18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18417_1_createTree_InvokeMethod(TRUE, 18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 21157_0_createTree_InvokeMethod(x3)
18401_1_createTree_InvokeMethod(18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → Cond_18401_1_createTree_InvokeMethod(x2 >= 0 && x2 < x0, 18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
Cond_18401_1_createTree_InvokeMethod(TRUE, 18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x6), x3)
18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x0), x2) → Cond_18597_1_createTree_InvokeMethod(x0 >= 0, 18597_0_random_IntArithmetic(x0), x2)
Cond_18597_1_createTree_InvokeMethod(TRUE, 18597_0_random_IntArithmetic(x0), x2) → 22671_0_createTree_InvokeMethod(x2)
18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x1), x3) → Cond_18597_1_createTree_InvokeMethod1(x1 >= 0, 18597_0_random_IntArithmetic(x1), x3)
Cond_18597_1_createTree_InvokeMethod1(TRUE, 18597_0_random_IntArithmetic(x1), x3) → 19835_0_createTree_LE(x3, x5)
19835_0_createTree_LE(x0, x1) → Cond_19835_0_createTree_LE(x1 > 0 && x0 > 0, x0, x1)
Cond_19835_0_createTree_LE(TRUE, x0, x1) → 18239_0_createTree_LE(x0 + -1)
19835_0_createTree_LE(x0, x1) → Cond_19835_0_createTree_LE1(x1 > 0, x0, x1)
Cond_19835_0_createTree_LE1(TRUE, x0, x1) → 26173_0_createTree_Load(x0)
19835_0_createTree_LE(x0, 0) → Cond_19835_0_createTree_LE6(x0 > 0, x0, 0)
Cond_19835_0_createTree_LE6(TRUE, x0, 0) → 18239_0_createTree_LE(x0 + -1)
19835_0_createTree_LE(x0, 0) → 26173_0_createTree_Load(x0)
18356_1_createTree_InvokeMethod(18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_18356_1_createTree_InvokeMethod(x2 <= -1, 18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18356_1_createTree_InvokeMethod(TRUE, 18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 20795_0_createTree_InvokeMethod(x3)
1371_1_createTree_InvokeMethod(1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → Cond_1371_1_createTree_InvokeMethod(x2 <= -1, 1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1371_1_createTree_InvokeMethod(TRUE, 1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2)) → 1751_0_createTree_InvokeMethod
26173_0_createTree_Load(x0) → Cond_26173_0_createTree_Load(x0 > 0, x0)
Cond_26173_0_createTree_Load(TRUE, x0) → 18239_0_createTree_LE(x0 + -1)

The integer pair graph contains the following rules and edges:
(1): COND_9403_1_MAIN_INVOKEMETHOD(TRUE, 1677_0_createTree_Return, x2[1]) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, x2[1] + -1)
(3): COND_9403_1_MAIN_INVOKEMETHOD1(TRUE, 18266_0_createTree_Return, x1[3]) → 9403_1_MAIN_INVOKEMETHOD(9403_0_createTree_InvokeMethod, x1[3] + -1)


The set Q consists of the following terms:
9403_0_createTree_InvokeMethod
9403_1_main_InvokeMethod(1751_0_createTree_InvokeMethod, x0)
9403_1_main_InvokeMethod(1798_0_createTree_InvokeMethod, x0)
9403_1_main_InvokeMethod(1894_0_createTree_InvokeMethod, x0)
9403_1_main_InvokeMethod(20795_0_createTree_InvokeMethod(x0), x1)
9403_1_main_InvokeMethod(21157_0_createTree_InvokeMethod(x0), x1)
9403_1_main_InvokeMethod(22671_0_createTree_InvokeMethod(x0), x1)
1383_1_createTree_InvokeMethod(1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1383_1_createTree_InvokeMethod(TRUE, 1383_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
1381_1_createTree_InvokeMethod(1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1381_1_createTree_InvokeMethod(TRUE, 1381_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
1425_1_createTree_InvokeMethod(1425_0_random_IntArithmetic(x0))
Cond_1425_1_createTree_InvokeMethod(TRUE, 1425_0_random_IntArithmetic(x0))
Cond_1425_1_createTree_InvokeMethod1(TRUE, 1425_0_random_IntArithmetic(x0))
1604_0_createTree_NE(x0)
Cond_1604_0_createTree_NE(TRUE, x0)
18239_0_createTree_LE(x0)
Cond_18239_0_createTree_LE(TRUE, x0)
Cond_18239_0_createTree_LE1(TRUE, x0)
Cond_18239_0_createTree_LE2(TRUE, x0)
18417_1_createTree_InvokeMethod(18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18417_1_createTree_InvokeMethod(TRUE, 18417_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
18401_1_createTree_InvokeMethod(18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
Cond_18401_1_createTree_InvokeMethod(TRUE, 18401_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
18597_1_createTree_InvokeMethod(18597_0_random_IntArithmetic(x0), x1)
Cond_18597_1_createTree_InvokeMethod(TRUE, 18597_0_random_IntArithmetic(x0), x1)
Cond_18597_1_createTree_InvokeMethod1(TRUE, 18597_0_random_IntArithmetic(x0), x1)
19835_0_createTree_LE(x0, x1)
Cond_19835_0_createTree_LE(TRUE, x0, x1)
Cond_19835_0_createTree_LE1(TRUE, x0, x1)
Cond_19835_0_createTree_LE6(TRUE, x0, 0)
18356_1_createTree_InvokeMethod(18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_18356_1_createTree_InvokeMethod(TRUE, 18356_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
1371_1_createTree_InvokeMethod(1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
Cond_1371_1_createTree_InvokeMethod(TRUE, 1371_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2))
26173_0_createTree_Load(x0)
Cond_26173_0_createTree_Load(TRUE, x0)

(28) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(29) TRUE