Termination proof of /tmp/tmpJVbCi9/LessLeaves.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPtoQDPProof (⇒)

        ↳7 QDP

        ↳8 QDPSizeChangeProof (⇔)

        ↳9 YES

    ↳10 IDP

        ↳11 IDPtoQDPProof (⇒)

        ↳12 QDP

        ↳13 UsableRulesProof (⇔)

        ↳14 QDP

        ↳15 QReductionProof (⇔)

        ↳16 QDP

        ↳17 QDPOrderProof (⇔)

        ↳18 QDP

        ↳19 DependencyGraphProof (⇔)

        ↳20 TRUE

    ↳21 IDP

        ↳22 IDPNonInfProof (⇒)

        ↳23 IDP

        ↳24 IDependencyGraphProof (⇔)

        ↳25 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: LessLeaves

public class LessLeaves {
  // Checks if a binary tree
  // has less leaves than another one

  public static void main(String[] args) {
    Random.args = args;
    Tree tree1 = Tree.createTree(); 
    Tree tree2 = Tree.createTree();
    boolean b = less_leaves(tree1,tree2);
  }



  public static Tree append(Tree t1, Tree t2) {

    Tree t;

    if (t1 == null) return t2;
    else {
      t = t1;

      while (t.right != null) {
        t = t.right;
      }

      t.right = t2;
      return t1;
    }
  }

  public static boolean less_leaves(Tree t1, Tree t2) {


    while ((t1 != null) && (t2 != null)) {
      t1 = append(t1.left,t1.right);
      t2 = append(t2.left,t2.right);
    }

    if (t2 == null) return false;
    else            return true;

  }
}


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();
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
LessLeaves.main([Ljava/lang/String;)V: Graph of 134 nodes with 0 SCCs.

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

LessLeaves.less_leaves(LTree;LTree;)Z: Graph of 52 nodes with 1 SCC.

LessLeaves.append(LTree;LTree;)LTree;: Graph of 40 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 19 rules for P and 12 rules for R.

Combined rules. Obtained 2 rules for P and 0 rules for R.

Filtered ground terms:

9606_0_append_FieldAccess(x1, x2, x3, x4, x5) → 9606_0_append_FieldAccess(x2, x3, x4, x5)
Tree(x1, x2) → Tree(x2)

Filtered duplicate args:

9606_0_append_FieldAccess(x1, x2, x3, x4) → 9606_0_append_FieldAccess(x1, x2, x4)

Filtered unneeded arguments:

9606_0_append_FieldAccess(x1, x2, x3) → 9606_0_append_FieldAccess(x1, x3)

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has no predefined symbols.

Log for SCC 1:

Generated 39 rules for P and 56 rules for R.

Combined rules. Obtained 7 rules for P and 11 rules for R.

Filtered ground terms:

12323_0_less_leaves_InvokeMethod(x1, x2, x3, x4) → 12323_0_less_leaves_InvokeMethod(x2, x3, x4)
10023_0_less_leaves_NULL(x1, x2, x3, x4) → 10023_0_less_leaves_NULL(x2, x3, x4)
Tree(x1, x2, x3) → Tree(x2, x3)
12577_0_append_Load(x1, x2, x3) → 12577_0_append_Load(x2, x3)
11545_0_append_Return(x1, x2) → 11545_0_append_Return(x2)
7398_0_append_Return(x1, x2, x3, x4) → 7398_0_append_Return(x3, x4)
12372_0_append_Load(x1, x2, x3) → 12372_0_append_Load(x2, x3)
11349_0_less_leaves_Return(x1) → 11349_0_less_leaves_Return
10691_0_less_leaves_NONNULL(x1, x2) → 10691_0_less_leaves_NONNULL(x2)
11348_0_less_leaves_Return(x1) → 11348_0_less_leaves_Return
9606_0_append_FieldAccess(x1, x2, x3, x4, x5) → 9606_0_append_FieldAccess(x2, x3, x4, x5)
7010_0_append_NONNULL(x1, x2, x3, x4) → 7010_0_append_NONNULL(x2, x3, x4)

Filtered duplicate args:

10023_0_less_leaves_NULL(x1, x2, x3) → 10023_0_less_leaves_NULL(x2, x3)
7398_0_append_Return(x1, x2) → 7398_0_append_Return(x2)
9606_0_append_FieldAccess(x1, x2, x3, x4) → 9606_0_append_FieldAccess(x1, x2, x4)
7010_0_append_NONNULL(x1, x2, x3) → 7010_0_append_NONNULL(x2, x3)

Finished conversion. Obtained 7 rules for P and 11 rules for R. System has no predefined symbols.

Log for SCC 2:

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)
16702_0_random_ArrayAccess(x1, x2, x3) → 16702_0_random_ArrayAccess(x2, x3)
16890_0_random_IntArithmetic(x1, x2, x3, x4) → 16890_0_random_IntArithmetic(x2, x3)
Cond_16890_1_createTree_InvokeMethod9(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod9(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod2(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod2(x1, x2, x3)

Filtered all non-integer terms:

16702_1_createTree_InvokeMethod(x1, x2, x3, x4) → 16702_1_createTree_InvokeMethod(x1, x2)
Cond_16702_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → Cond_16702_1_createTree_InvokeMethod(x1, x2, x3)
16890_1_createTree_InvokeMethod(x1, x2, x3, x4) → 16890_1_createTree_InvokeMethod(x1, x2)
16890_0_random_IntArithmetic(x1, x2) → 16890_0_random_IntArithmetic(x2)
Tree(x1, x2) → Tree
Cond_16890_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod1(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod3(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod3(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod4(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod4(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod5(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod5(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod6(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod6(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod7(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod7(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod8(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod8(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod10(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod10(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod11(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod11(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod12(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod12(x1, x2, x3)
Cond_16890_1_createTree_InvokeMethod13(x1, x2, x3, x4, x5) → Cond_16890_1_createTree_InvokeMethod13(x1, x2, x3)

Filtered all free variables:

16890_1_createTree_InvokeMethod(x1, x2) → 16890_1_createTree_InvokeMethod(x2)
Cond_16890_1_createTree_InvokeMethod(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod(x1, x3)
16702_1_createTree_InvokeMethod(x1, x2) → 16702_1_createTree_InvokeMethod(x2)
Cond_16890_1_createTree_InvokeMethod1(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod1(x1, x3)
Cond_16890_1_createTree_InvokeMethod2(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod2(x1, x3)
Cond_16890_1_createTree_InvokeMethod3(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod3(x1, x3)
Cond_16890_1_createTree_InvokeMethod4(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod4(x1, x3)
Cond_16890_1_createTree_InvokeMethod5(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod5(x1, x3)
Cond_16890_1_createTree_InvokeMethod6(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod6(x1, x3)
Cond_16890_1_createTree_InvokeMethod7(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod7(x1, x3)
Cond_16890_1_createTree_InvokeMethod8(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod8(x1, x3)
Cond_16890_1_createTree_InvokeMethod9(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod9(x1, x3)
Cond_16890_1_createTree_InvokeMethod10(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod10(x1, x3)
Cond_16890_1_createTree_InvokeMethod11(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod11(x1, x3)
Cond_16890_1_createTree_InvokeMethod12(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod12(x1, x3)
Cond_16890_1_createTree_InvokeMethod13(x1, x2, x3) → Cond_16890_1_createTree_InvokeMethod13(x1, x3)
Cond_16702_1_createTree_InvokeMethod(x1, x2, x3) → Cond_16702_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.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The following domains are used:
none

R is empty.

The integer pair graph contains the following rules and edges:
(0): 9606_0_APPEND_FIELDACCESS(java.lang.Object(x0[0]), java.lang.Object(Tree(java.lang.Object(x2[0])))) → 9606_0_APPEND_FIELDACCESS(java.lang.Object(x0[0]), java.lang.Object(x2[0]))
(1): 9606_0_APPEND_FIELDACCESS(java.lang.Object(Tree(java.lang.Object(x0[1]))), java.lang.Object(Tree(java.lang.Object(x0[1])))) → 9606_0_APPEND_FIELDACCESS(java.lang.Object(Tree(java.lang.Object(x0[1]))), java.lang.Object(x0[1]))

(0) -> (0), if ((java.lang.Object(x0[0]) →^* java.lang.Object(x0[0]'))∧(java.lang.Object(x2[0]) →^* java.lang.Object(Tree(java.lang.Object(x2[0]')))))

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

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

(1) -> (1), if ((java.lang.Object(Tree(java.lang.Object(x0[1]))) →^* java.lang.Object(Tree(java.lang.Object(x0[1]'))))∧(java.lang.Object(x0[1]) →^* java.lang.Object(Tree(java.lang.Object(x0[1]')))))

The set Q is empty.

(6) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(7) Obligation:

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

9606_0_APPEND_FIELDACCESS(java.lang.Object(x0[0]), java.lang.Object(Tree(java.lang.Object(x2[0])))) → 9606_0_APPEND_FIELDACCESS(java.lang.Object(x0[0]), java.lang.Object(x2[0]))
9606_0_APPEND_FIELDACCESS(java.lang.Object(Tree(java.lang.Object(x0[1]))), java.lang.Object(Tree(java.lang.Object(x0[1])))) → 9606_0_APPEND_FIELDACCESS(java.lang.Object(Tree(java.lang.Object(x0[1]))), java.lang.Object(x0[1]))

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

(8) 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:

9606_0_APPEND_FIELDACCESS(java.lang.Object(x0[0]), java.lang.Object(Tree(java.lang.Object(x2[0])))) → 9606_0_APPEND_FIELDACCESS(java.lang.Object(x0[0]), java.lang.Object(x2[0]))
The graph contains the following edges 1 >= 1, 2 > 2
9606_0_APPEND_FIELDACCESS(java.lang.Object(Tree(java.lang.Object(x0[1]))), java.lang.Object(Tree(java.lang.Object(x0[1])))) → 9606_0_APPEND_FIELDACCESS(java.lang.Object(Tree(java.lang.Object(x0[1]))), java.lang.Object(x0[1]))
The graph contains the following edges 1 >= 1, 2 >= 1, 1 > 2, 2 > 2

(9) YES

(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:
10023_0_less_leaves_NULL(x0, NULL) → 10691_0_less_leaves_NONNULL(x0)
12372_0_append_Load(x0, x1) → 7010_0_append_NONNULL(x1, x0)
7010_0_append_NONNULL(x0, NULL) → 7398_0_append_Return(x0)
7010_0_append_NONNULL(x1, java.lang.Object(x0)) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x0))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, NULL)), x1, java.lang.Object(Tree(x0, NULL))) → 11545_0_append_Return(java.lang.Object(Tree(x0, x1)))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(Tree(x0, java.lang.Object(x1)))) → 9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL))) → 11545_0_append_Return(java.lang.Object(x0))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3)))) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x3))
10691_0_less_leaves_NONNULL(java.lang.Object(x0)) → 11348_0_less_leaves_Return
10691_0_less_leaves_NONNULL(NULL) → 11349_0_less_leaves_Return

The integer pair graph contains the following rules and edges:
(0): 12323_0_LESS_LEAVES_INVOKEMETHOD(java.lang.Object(x0[0]), x1[0], x2[0]) → 12372_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[0], x2[0]), java.lang.Object(x0[0]), x1[0], x2[0])
(1): 10023_0_LESS_LEAVES_NULL(java.lang.Object(x2[1]), java.lang.Object(Tree(x0[1], x1[1]))) → 12372_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x0[1], x1[1]), java.lang.Object(x2[1]), x0[1], x1[1])
(2): 12372_1_LESS_LEAVES_INVOKEMETHOD(7398_0_append_Return(x0[2]), java.lang.Object(Tree(x1[2], x2[2])), NULL, x0[2]) → 12577_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[2], x2[2]), x0[2], x1[2], x2[2])
(3): 12577_1_LESS_LEAVES_INVOKEMETHOD(7398_0_append_Return(x0[3]), x1[3], NULL, x0[3]) → 10023_0_LESS_LEAVES_NULL(x0[3], x1[3])
(4): 12577_1_LESS_LEAVES_INVOKEMETHOD(11545_0_append_Return(java.lang.Object(x0[4])), x1[4], x2[4], x3[4]) → 10023_0_LESS_LEAVES_NULL(java.lang.Object(x0[4]), x1[4])
(5): 12372_1_LESS_LEAVES_INVOKEMETHOD(11545_0_append_Return(java.lang.Object(x0[5])), java.lang.Object(Tree(x1[5], x2[5])), x3[5], x4[5]) → 12577_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[5], x2[5]), java.lang.Object(x0[5]), x1[5], x2[5])
(6): 10023_0_LESS_LEAVES_NULL(java.lang.Object(x2[6]), java.lang.Object(Tree(x0[6], x1[6]))) → 12323_0_LESS_LEAVES_INVOKEMETHOD(java.lang.Object(x2[6]), x0[6], x1[6])

(0) -> (2), if ((12372_0_append_Load(x1[0], x2[0]) →^* 7398_0_append_Return(x0[2]))∧(java.lang.Object(x0[0]) →^* java.lang.Object(Tree(x1[2], x2[2])))∧(x1[0] →^* NULL)∧(x2[0] →^* x0[2]))

(0) -> (5), if ((12372_0_append_Load(x1[0], x2[0]) →^* 11545_0_append_Return(java.lang.Object(x0[5])))∧(java.lang.Object(x0[0]) →^* java.lang.Object(Tree(x1[5], x2[5])))∧(x1[0] →^* x3[5])∧(x2[0] →^* x4[5]))

(1) -> (2), if ((12372_0_append_Load(x0[1], x1[1]) →^* 7398_0_append_Return(x0[2]))∧(java.lang.Object(x2[1]) →^* java.lang.Object(Tree(x1[2], x2[2])))∧(x0[1] →^* NULL)∧(x1[1] →^* x0[2]))

(1) -> (5), if ((12372_0_append_Load(x0[1], x1[1]) →^* 11545_0_append_Return(java.lang.Object(x0[5])))∧(java.lang.Object(x2[1]) →^* java.lang.Object(Tree(x1[5], x2[5])))∧(x0[1] →^* x3[5])∧(x1[1] →^* x4[5]))

(2) -> (3), if ((12372_0_append_Load(x1[2], x2[2]) →^* 7398_0_append_Return(x0[3]))∧(x0[2] →^* x1[3])∧(x1[2] →^* NULL)∧(x2[2] →^* x0[3]))

(2) -> (4), if ((12372_0_append_Load(x1[2], x2[2]) →^* 11545_0_append_Return(java.lang.Object(x0[4])))∧(x0[2] →^* x1[4])∧(x1[2] →^* x2[4])∧(x2[2] →^* x3[4]))

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

(3) -> (6), if ((x0[3] →^* java.lang.Object(x2[6]))∧(x1[3] →^* java.lang.Object(Tree(x0[6], x1[6]))))

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

(4) -> (6), if ((java.lang.Object(x0[4]) →^* java.lang.Object(x2[6]))∧(x1[4] →^* java.lang.Object(Tree(x0[6], x1[6]))))

(5) -> (3), if ((12372_0_append_Load(x1[5], x2[5]) →^* 7398_0_append_Return(x0[3]))∧(java.lang.Object(x0[5]) →^* x1[3])∧(x1[5] →^* NULL)∧(x2[5] →^* x0[3]))

(5) -> (4), if ((12372_0_append_Load(x1[5], x2[5]) →^* 11545_0_append_Return(java.lang.Object(x0[4])))∧(java.lang.Object(x0[5]) →^* x1[4])∧(x1[5] →^* x2[4])∧(x2[5] →^* x3[4]))

(6) -> (0), if ((java.lang.Object(x2[6]) →^* java.lang.Object(x0[0]))∧(x0[6] →^* x1[0])∧(x1[6] →^* x2[0]))

The set Q consists of the following terms:
10023_0_less_leaves_NULL(x0, NULL)
12372_0_append_Load(x0, x1)
7010_0_append_NONNULL(x0, NULL)
7010_0_append_NONNULL(x0, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL)))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3))))
10691_0_less_leaves_NONNULL(java.lang.Object(x0))
10691_0_less_leaves_NONNULL(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:

12323_0_LESS_LEAVES_INVOKEMETHOD(java.lang.Object(x0[0]), x1[0], x2[0]) → 12372_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[0], x2[0]), java.lang.Object(x0[0]), x1[0], x2[0])
10023_0_LESS_LEAVES_NULL(java.lang.Object(x2[1]), java.lang.Object(Tree(x0[1], x1[1]))) → 12372_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x0[1], x1[1]), java.lang.Object(x2[1]), x0[1], x1[1])
12372_1_LESS_LEAVES_INVOKEMETHOD(7398_0_append_Return(x0[2]), java.lang.Object(Tree(x1[2], x2[2])), NULL, x0[2]) → 12577_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[2], x2[2]), x0[2], x1[2], x2[2])
12577_1_LESS_LEAVES_INVOKEMETHOD(7398_0_append_Return(x0[3]), x1[3], NULL, x0[3]) → 10023_0_LESS_LEAVES_NULL(x0[3], x1[3])
12577_1_LESS_LEAVES_INVOKEMETHOD(11545_0_append_Return(java.lang.Object(x0[4])), x1[4], x2[4], x3[4]) → 10023_0_LESS_LEAVES_NULL(java.lang.Object(x0[4]), x1[4])
12372_1_LESS_LEAVES_INVOKEMETHOD(11545_0_append_Return(java.lang.Object(x0[5])), java.lang.Object(Tree(x1[5], x2[5])), x3[5], x4[5]) → 12577_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[5], x2[5]), java.lang.Object(x0[5]), x1[5], x2[5])
10023_0_LESS_LEAVES_NULL(java.lang.Object(x2[6]), java.lang.Object(Tree(x0[6], x1[6]))) → 12323_0_LESS_LEAVES_INVOKEMETHOD(java.lang.Object(x2[6]), x0[6], x1[6])

The TRS R consists of the following rules:

10023_0_less_leaves_NULL(x0, NULL) → 10691_0_less_leaves_NONNULL(x0)
12372_0_append_Load(x0, x1) → 7010_0_append_NONNULL(x1, x0)
7010_0_append_NONNULL(x0, NULL) → 7398_0_append_Return(x0)
7010_0_append_NONNULL(x1, java.lang.Object(x0)) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x0))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, NULL)), x1, java.lang.Object(Tree(x0, NULL))) → 11545_0_append_Return(java.lang.Object(Tree(x0, x1)))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(Tree(x0, java.lang.Object(x1)))) → 9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL))) → 11545_0_append_Return(java.lang.Object(x0))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3)))) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x3))
10691_0_less_leaves_NONNULL(java.lang.Object(x0)) → 11348_0_less_leaves_Return
10691_0_less_leaves_NONNULL(NULL) → 11349_0_less_leaves_Return

The set Q consists of the following terms:

10023_0_less_leaves_NULL(x0, NULL)
12372_0_append_Load(x0, x1)
7010_0_append_NONNULL(x0, NULL)
7010_0_append_NONNULL(x0, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL)))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3))))
10691_0_less_leaves_NONNULL(java.lang.Object(x0))
10691_0_less_leaves_NONNULL(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:

12323_0_LESS_LEAVES_INVOKEMETHOD(java.lang.Object(x0[0]), x1[0], x2[0]) → 12372_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[0], x2[0]), java.lang.Object(x0[0]), x1[0], x2[0])
10023_0_LESS_LEAVES_NULL(java.lang.Object(x2[1]), java.lang.Object(Tree(x0[1], x1[1]))) → 12372_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x0[1], x1[1]), java.lang.Object(x2[1]), x0[1], x1[1])
12372_1_LESS_LEAVES_INVOKEMETHOD(7398_0_append_Return(x0[2]), java.lang.Object(Tree(x1[2], x2[2])), NULL, x0[2]) → 12577_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[2], x2[2]), x0[2], x1[2], x2[2])
12577_1_LESS_LEAVES_INVOKEMETHOD(7398_0_append_Return(x0[3]), x1[3], NULL, x0[3]) → 10023_0_LESS_LEAVES_NULL(x0[3], x1[3])
12577_1_LESS_LEAVES_INVOKEMETHOD(11545_0_append_Return(java.lang.Object(x0[4])), x1[4], x2[4], x3[4]) → 10023_0_LESS_LEAVES_NULL(java.lang.Object(x0[4]), x1[4])
12372_1_LESS_LEAVES_INVOKEMETHOD(11545_0_append_Return(java.lang.Object(x0[5])), java.lang.Object(Tree(x1[5], x2[5])), x3[5], x4[5]) → 12577_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[5], x2[5]), java.lang.Object(x0[5]), x1[5], x2[5])
10023_0_LESS_LEAVES_NULL(java.lang.Object(x2[6]), java.lang.Object(Tree(x0[6], x1[6]))) → 12323_0_LESS_LEAVES_INVOKEMETHOD(java.lang.Object(x2[6]), x0[6], x1[6])

The TRS R consists of the following rules:

12372_0_append_Load(x0, x1) → 7010_0_append_NONNULL(x1, x0)
7010_0_append_NONNULL(x0, NULL) → 7398_0_append_Return(x0)
7010_0_append_NONNULL(x1, java.lang.Object(x0)) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x0))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, NULL)), x1, java.lang.Object(Tree(x0, NULL))) → 11545_0_append_Return(java.lang.Object(Tree(x0, x1)))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(Tree(x0, java.lang.Object(x1)))) → 9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3)))) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x3))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL))) → 11545_0_append_Return(java.lang.Object(x0))

The set Q consists of the following terms:

10023_0_less_leaves_NULL(x0, NULL)
12372_0_append_Load(x0, x1)
7010_0_append_NONNULL(x0, NULL)
7010_0_append_NONNULL(x0, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL)))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3))))
10691_0_less_leaves_NONNULL(java.lang.Object(x0))
10691_0_less_leaves_NONNULL(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].

10023_0_less_leaves_NULL(x0, NULL)
10691_0_less_leaves_NONNULL(java.lang.Object(x0))
10691_0_less_leaves_NONNULL(NULL)

(16) Obligation:

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

12323_0_LESS_LEAVES_INVOKEMETHOD(java.lang.Object(x0[0]), x1[0], x2[0]) → 12372_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[0], x2[0]), java.lang.Object(x0[0]), x1[0], x2[0])
10023_0_LESS_LEAVES_NULL(java.lang.Object(x2[1]), java.lang.Object(Tree(x0[1], x1[1]))) → 12372_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x0[1], x1[1]), java.lang.Object(x2[1]), x0[1], x1[1])
12372_1_LESS_LEAVES_INVOKEMETHOD(7398_0_append_Return(x0[2]), java.lang.Object(Tree(x1[2], x2[2])), NULL, x0[2]) → 12577_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[2], x2[2]), x0[2], x1[2], x2[2])
12577_1_LESS_LEAVES_INVOKEMETHOD(7398_0_append_Return(x0[3]), x1[3], NULL, x0[3]) → 10023_0_LESS_LEAVES_NULL(x0[3], x1[3])
12577_1_LESS_LEAVES_INVOKEMETHOD(11545_0_append_Return(java.lang.Object(x0[4])), x1[4], x2[4], x3[4]) → 10023_0_LESS_LEAVES_NULL(java.lang.Object(x0[4]), x1[4])
12372_1_LESS_LEAVES_INVOKEMETHOD(11545_0_append_Return(java.lang.Object(x0[5])), java.lang.Object(Tree(x1[5], x2[5])), x3[5], x4[5]) → 12577_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[5], x2[5]), java.lang.Object(x0[5]), x1[5], x2[5])
10023_0_LESS_LEAVES_NULL(java.lang.Object(x2[6]), java.lang.Object(Tree(x0[6], x1[6]))) → 12323_0_LESS_LEAVES_INVOKEMETHOD(java.lang.Object(x2[6]), x0[6], x1[6])

The TRS R consists of the following rules:

12372_0_append_Load(x0, x1) → 7010_0_append_NONNULL(x1, x0)
7010_0_append_NONNULL(x0, NULL) → 7398_0_append_Return(x0)
7010_0_append_NONNULL(x1, java.lang.Object(x0)) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x0))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, NULL)), x1, java.lang.Object(Tree(x0, NULL))) → 11545_0_append_Return(java.lang.Object(Tree(x0, x1)))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(Tree(x0, java.lang.Object(x1)))) → 9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3)))) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x3))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL))) → 11545_0_append_Return(java.lang.Object(x0))

The set Q consists of the following terms:

12372_0_append_Load(x0, x1)
7010_0_append_NONNULL(x0, NULL)
7010_0_append_NONNULL(x0, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL)))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3))))

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

12372_1_LESS_LEAVES_INVOKEMETHOD(7398_0_append_Return(x0[2]), java.lang.Object(Tree(x1[2], x2[2])), NULL, x0[2]) → 12577_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[2], x2[2]), x0[2], x1[2], x2[2])
12372_1_LESS_LEAVES_INVOKEMETHOD(11545_0_append_Return(java.lang.Object(x0[5])), java.lang.Object(Tree(x1[5], x2[5])), x3[5], x4[5]) → 12577_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[5], x2[5]), java.lang.Object(x0[5]), x1[5], x2[5])

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(10023_0_LESS_LEAVES_NULL(x₁, x₂)) = x₁
POL(11545_0_append_Return(x₁)) = x₁
POL(12323_0_LESS_LEAVES_INVOKEMETHOD(x₁, x₂, x₃)) = x₁
POL(12372_0_append_Load(x₁, x₂)) = x₁ + x₂
POL(12372_1_LESS_LEAVES_INVOKEMETHOD(x₁, x₂, x₃, x₄)) = x₂
POL(12577_1_LESS_LEAVES_INVOKEMETHOD(x₁, x₂, x₃, x₄)) = x₁
POL(7010_0_append_NONNULL(x₁, x₂)) = x₁ + x₂
POL(7398_0_append_Return(x₁)) = x₁
POL(9606_0_append_FieldAccess(x₁, x₂, x₃)) = x₁ + x₂
POL(NULL) = 0
POL(Tree(x₁, x₂)) = 1 + x₁ + x₂
POL(java.lang.Object(x₁)) = x₁

The following usable rules [FROCOS05] were oriented:

12372_0_append_Load(x0, x1) → 7010_0_append_NONNULL(x1, x0)
7010_0_append_NONNULL(x0, NULL) → 7398_0_append_Return(x0)
7010_0_append_NONNULL(x1, java.lang.Object(x0)) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x0))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, NULL)), x1, java.lang.Object(Tree(x0, NULL))) → 11545_0_append_Return(java.lang.Object(Tree(x0, x1)))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(Tree(x0, java.lang.Object(x1)))) → 9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3)))) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x3))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL))) → 11545_0_append_Return(java.lang.Object(x0))

(18) Obligation:

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

12323_0_LESS_LEAVES_INVOKEMETHOD(java.lang.Object(x0[0]), x1[0], x2[0]) → 12372_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x1[0], x2[0]), java.lang.Object(x0[0]), x1[0], x2[0])
10023_0_LESS_LEAVES_NULL(java.lang.Object(x2[1]), java.lang.Object(Tree(x0[1], x1[1]))) → 12372_1_LESS_LEAVES_INVOKEMETHOD(12372_0_append_Load(x0[1], x1[1]), java.lang.Object(x2[1]), x0[1], x1[1])
12577_1_LESS_LEAVES_INVOKEMETHOD(7398_0_append_Return(x0[3]), x1[3], NULL, x0[3]) → 10023_0_LESS_LEAVES_NULL(x0[3], x1[3])
12577_1_LESS_LEAVES_INVOKEMETHOD(11545_0_append_Return(java.lang.Object(x0[4])), x1[4], x2[4], x3[4]) → 10023_0_LESS_LEAVES_NULL(java.lang.Object(x0[4]), x1[4])
10023_0_LESS_LEAVES_NULL(java.lang.Object(x2[6]), java.lang.Object(Tree(x0[6], x1[6]))) → 12323_0_LESS_LEAVES_INVOKEMETHOD(java.lang.Object(x2[6]), x0[6], x1[6])

The TRS R consists of the following rules:

12372_0_append_Load(x0, x1) → 7010_0_append_NONNULL(x1, x0)
7010_0_append_NONNULL(x0, NULL) → 7398_0_append_Return(x0)
7010_0_append_NONNULL(x1, java.lang.Object(x0)) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x0))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, NULL)), x1, java.lang.Object(Tree(x0, NULL))) → 11545_0_append_Return(java.lang.Object(Tree(x0, x1)))
9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(Tree(x0, java.lang.Object(x1)))) → 9606_0_append_FieldAccess(java.lang.Object(Tree(x0, java.lang.Object(x1))), x2, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3)))) → 9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(x3))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL))) → 11545_0_append_Return(java.lang.Object(x0))

The set Q consists of the following terms:

12372_0_append_Load(x0, x1)
7010_0_append_NONNULL(x0, NULL)
7010_0_append_NONNULL(x0, java.lang.Object(x1))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, NULL)))
9606_0_append_FieldAccess(java.lang.Object(x0), x1, java.lang.Object(Tree(x2, java.lang.Object(x3))))

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.

(20) TRUE

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 16702_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_16702_1_CREATETREE_INVOKEMETHOD(x0[0] > 0 && 0 < x0[0] + -1, x0[0])
(1): COND_16702_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 16702_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.

(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 16702_1_CREATETREE_INVOKEMETHOD(x0) → COND_16702_1_CREATETREE_INVOKEMETHOD(&&(>(x0, 0), <(0, +(x0, -1))), x0) the following chains were created:

We consider the chain 16702_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_16702_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0]), COND_16702_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 16702_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] ⇒ 16702_1_CREATETREE_INVOKEMETHOD(x0[0])≥NonInfC∧16702_1_CREATETREE_INVOKEMETHOD(x0[0])≥COND_16702_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])∧(U^Increasing(COND_16702_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 ⇒ 16702_1_CREATETREE_INVOKEMETHOD(x0[0])≥NonInfC∧16702_1_CREATETREE_INVOKEMETHOD(x0[0])≥COND_16702_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])∧(U^Increasing(COND_16702_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 ⇒ (U^Increasing(COND_16702_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 ⇒ (U^Increasing(COND_16702_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 ⇒ (U^Increasing(COND_16702_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 ⇒ (U^Increasing(COND_16702_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 ⇒ (U^Increasing(COND_16702_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_16702_1_CREATETREE_INVOKEMETHOD(TRUE, x0) → 16702_1_CREATETREE_INVOKEMETHOD(+(x0, -1)) the following chains were created:

We consider the chain COND_16702_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 16702_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1)) which results in the following constraint:
(8)    (COND_16702_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1])≥NonInfC∧COND_16702_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1])≥16702_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))∧(U^Increasing(16702_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(16702_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)    ((U^Increasing(16702_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)    ((U^Increasing(16702_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)    ((U^Increasing(16702_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.

16702_1_CREATETREE_INVOKEMETHOD(x0) → COND_16702_1_CREATETREE_INVOKEMETHOD(&&(>(x0, 0), <(0, +(x0, -1))), x0)
- ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_16702_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_16702_1_CREATETREE_INVOKEMETHOD(TRUE, x0) → 16702_1_CREATETREE_INVOKEMETHOD(+(x0, -1))
- ((U^Increasing(16702_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥)∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(16702_1_CREATETREE_INVOKEMETHOD(x₁)) = [2]x₁
POL(COND_16702_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_>:

16702_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_16702_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])
COND_16702_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 16702_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))

The following pairs are in P_bound:

16702_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_16702_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])

The following pairs are in P_≥:
none

There are no usable rules.

(23) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_16702_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 16702_1_CREATETREE_INVOKEMETHOD(x0[1] + -1)

The set Q is empty.

(24) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) IDPtoQDPProof (SOUND transformation)

(7) Obligation:

(8) QDPSizeChangeProof (EQUIVALENT transformation)

(9) YES

(10) Obligation:

(11) IDPtoQDPProof (SOUND transformation)

(12) Obligation:

(13) UsableRulesProof (EQUIVALENT transformation)

(14) Obligation:

(15) QReductionProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) DependencyGraphProof (EQUIVALENT transformation)

(20) TRUE

(21) Obligation:

(22) IDPNonInfProof (SOUND transformation)

(23) Obligation:

(24) IDependencyGraphProof (EQUIVALENT transformation)

(25) TRUE