Termination proof of /tmp/tmpJV9HzF/FlattenTree.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 UsableRulesProof (⇔)

        ↳9 QDP

        ↳10 QReductionProof (⇔)

        ↳11 QDP

        ↳12 MRRProof (⇔)

        ↳13 QDP

        ↳14 DependencyGraphProof (⇔)

        ↳15 QDP

        ↳16 MRRProof (⇔)

        ↳17 QDP

        ↳18 DependencyGraphProof (⇔)

        ↳19 TRUE

    ↳20 IDP

        ↳21 IDPNonInfProof (⇒)

        ↳22 IDP

        ↳23 IDependencyGraphProof (⇔)

        ↳24 TRUE

(0) Obligation:

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

public class FlattenTree {
  public static void main(String[] args) {
    Random.args = args;
    Tree tree = Tree.createTree();
    flatten(tree);
  }

  public static Tree flatten(Tree start) {
    Tree result = null;
    Tree s,t,u;

    while (start != null) {

      if (start.left == null) {

        result = new Tree(null,result);
        start = start.right;
      }
      else {
        s = start.left.left;
        t = start.left.right;
        u = start.right;
        start = new Tree(s, new Tree(t,u));
      }
    }

    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();
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
FlattenTree.main([Ljava/lang/String;)V: Graph of 54 nodes with 0 SCCs.

Tree.createTree()LTree;: Graph of 419 nodes with 1 SCC.

FlattenTree.flatten(LTree;)LTree;: Graph of 114 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 110 rules for P and 3 rules for R.

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

Filtered ground terms:

5117_0_flatten_Store(x1, x2, x3) → 5117_0_flatten_Store(x2, x3)
Tree(x1, x2, x3) → Tree(x2, x3)
4005_0_flatten_NULL(x1, x2, x3, x4) → 4005_0_flatten_NULL(x2, x3, x4)
4733_0_flatten_Store(x1, x2, x3, x4, x5) → 4733_0_flatten_Store(x2, x3, x4, x5)
4054_0_flatten_Return(x1, x2) → 4054_0_flatten_Return(x2)

Filtered duplicate args:

4005_0_flatten_NULL(x1, x2, x3) → 4005_0_flatten_NULL(x2, x3)

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

Log for SCC 1:

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)
15741_0_random_ArrayAccess(x1, x2, x3) → 15741_0_random_ArrayAccess(x2, x3)
15902_0_random_IntArithmetic(x1, x2, x3, x4) → 15902_0_random_IntArithmetic(x2, x3)
Cond_15902_1_createTree_InvokeMethod9(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod9(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod2(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod2(x1, x2, x3)

Filtered all non-integer terms:

15741_1_createTree_InvokeMethod(x1, x2, x3, x4) → 15741_1_createTree_InvokeMethod(x1, x2)
Cond_15741_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → Cond_15741_1_createTree_InvokeMethod(x1, x2, x3)
15902_1_createTree_InvokeMethod(x1, x2, x3, x4) → 15902_1_createTree_InvokeMethod(x1, x2)
15902_0_random_IntArithmetic(x1, x2) → 15902_0_random_IntArithmetic(x2)
Tree(x1, x2) → Tree
Cond_15902_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod1(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod3(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod3(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod4(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod4(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod5(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod5(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod6(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod6(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod7(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod7(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod8(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod8(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod10(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod10(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod11(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod11(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod12(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod12(x1, x2, x3)
Cond_15902_1_createTree_InvokeMethod13(x1, x2, x3, x4, x5) → Cond_15902_1_createTree_InvokeMethod13(x1, x2, x3)

Filtered all free variables:

15902_1_createTree_InvokeMethod(x1, x2) → 15902_1_createTree_InvokeMethod(x2)
Cond_15902_1_createTree_InvokeMethod(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod(x1, x3)
15741_1_createTree_InvokeMethod(x1, x2) → 15741_1_createTree_InvokeMethod(x2)
Cond_15902_1_createTree_InvokeMethod1(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod1(x1, x3)
Cond_15902_1_createTree_InvokeMethod2(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod2(x1, x3)
Cond_15902_1_createTree_InvokeMethod3(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod3(x1, x3)
Cond_15902_1_createTree_InvokeMethod4(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod4(x1, x3)
Cond_15902_1_createTree_InvokeMethod5(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod5(x1, x3)
Cond_15902_1_createTree_InvokeMethod6(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod6(x1, x3)
Cond_15902_1_createTree_InvokeMethod7(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod7(x1, x3)
Cond_15902_1_createTree_InvokeMethod8(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod8(x1, x3)
Cond_15902_1_createTree_InvokeMethod9(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod9(x1, x3)
Cond_15902_1_createTree_InvokeMethod10(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod10(x1, x3)
Cond_15902_1_createTree_InvokeMethod11(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod11(x1, x3)
Cond_15902_1_createTree_InvokeMethod12(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod12(x1, x3)
Cond_15902_1_createTree_InvokeMethod13(x1, x2, x3) → Cond_15902_1_createTree_InvokeMethod13(x1, x3)
Cond_15741_1_createTree_InvokeMethod(x1, x2, x3) → Cond_15741_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

The ITRS R consists of the following rules:
4005_0_flatten_NULL(x0, NULL) → 4054_0_flatten_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 4733_0_FLATTEN_STORE(x0[0], x1[0], x2[0], x3[0]) → 4005_0_FLATTEN_NULL(x0[0], java.lang.Object(Tree(x1[0], java.lang.Object(Tree(x2[0], x3[0])))))
(1): 4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(java.lang.Object(Tree(x0[1], x1[1])), x2[1]))) → 4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(x0[1], java.lang.Object(Tree(x1[1], x2[1])))))
(2): 5117_0_FLATTEN_STORE(java.lang.Object(Tree(NULL, x0[2])), x1[2]) → 4005_0_FLATTEN_NULL(java.lang.Object(Tree(NULL, x0[2])), x1[2])
(3): 4005_0_FLATTEN_NULL(x1[3], java.lang.Object(Tree(NULL, x0[3]))) → 4005_0_FLATTEN_NULL(java.lang.Object(Tree(NULL, x1[3])), x0[3])
(4): 4005_0_FLATTEN_NULL(x3[4], java.lang.Object(Tree(java.lang.Object(Tree(x0[4], x1[4])), x2[4]))) → 4733_0_FLATTEN_STORE(x3[4], x0[4], x1[4], x2[4])
(5): 4005_0_FLATTEN_NULL(x1[5], java.lang.Object(Tree(NULL, x0[5]))) → 5117_0_FLATTEN_STORE(java.lang.Object(Tree(NULL, x1[5])), x0[5])

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(3) -> (5), if ((java.lang.Object(Tree(NULL, x1[3])) →^* x1[5])∧(x0[3] →^* java.lang.Object(Tree(NULL, x0[5]))))

(4) -> (0), if ((x3[4] →^* x0[0])∧(x0[4] →^* x1[0])∧(x1[4] →^* x2[0])∧(x2[4] →^* x3[0]))

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

The set Q consists of the following terms:
4005_0_flatten_NULL(x0, NULL)

(6) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(7) Obligation:

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

4733_0_FLATTEN_STORE(x0[0], x1[0], x2[0], x3[0]) → 4005_0_FLATTEN_NULL(x0[0], java.lang.Object(Tree(x1[0], java.lang.Object(Tree(x2[0], x3[0])))))
4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(java.lang.Object(Tree(x0[1], x1[1])), x2[1]))) → 4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(x0[1], java.lang.Object(Tree(x1[1], x2[1])))))
5117_0_FLATTEN_STORE(java.lang.Object(Tree(NULL, x0[2])), x1[2]) → 4005_0_FLATTEN_NULL(java.lang.Object(Tree(NULL, x0[2])), x1[2])
4005_0_FLATTEN_NULL(x1[3], java.lang.Object(Tree(NULL, x0[3]))) → 4005_0_FLATTEN_NULL(java.lang.Object(Tree(NULL, x1[3])), x0[3])
4005_0_FLATTEN_NULL(x3[4], java.lang.Object(Tree(java.lang.Object(Tree(x0[4], x1[4])), x2[4]))) → 4733_0_FLATTEN_STORE(x3[4], x0[4], x1[4], x2[4])
4005_0_FLATTEN_NULL(x1[5], java.lang.Object(Tree(NULL, x0[5]))) → 5117_0_FLATTEN_STORE(java.lang.Object(Tree(NULL, x1[5])), x0[5])

The TRS R consists of the following rules:

4005_0_flatten_NULL(x0, NULL) → 4054_0_flatten_Return(x0)

The set Q consists of the following terms:

4005_0_flatten_NULL(x0, NULL)

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

(8) 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.

(9) Obligation:

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

4733_0_FLATTEN_STORE(x0[0], x1[0], x2[0], x3[0]) → 4005_0_FLATTEN_NULL(x0[0], java.lang.Object(Tree(x1[0], java.lang.Object(Tree(x2[0], x3[0])))))
4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(java.lang.Object(Tree(x0[1], x1[1])), x2[1]))) → 4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(x0[1], java.lang.Object(Tree(x1[1], x2[1])))))
5117_0_FLATTEN_STORE(java.lang.Object(Tree(NULL, x0[2])), x1[2]) → 4005_0_FLATTEN_NULL(java.lang.Object(Tree(NULL, x0[2])), x1[2])
4005_0_FLATTEN_NULL(x1[3], java.lang.Object(Tree(NULL, x0[3]))) → 4005_0_FLATTEN_NULL(java.lang.Object(Tree(NULL, x1[3])), x0[3])
4005_0_FLATTEN_NULL(x3[4], java.lang.Object(Tree(java.lang.Object(Tree(x0[4], x1[4])), x2[4]))) → 4733_0_FLATTEN_STORE(x3[4], x0[4], x1[4], x2[4])
4005_0_FLATTEN_NULL(x1[5], java.lang.Object(Tree(NULL, x0[5]))) → 5117_0_FLATTEN_STORE(java.lang.Object(Tree(NULL, x1[5])), x0[5])

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

4005_0_flatten_NULL(x0, NULL)

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

(10) 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].

4005_0_flatten_NULL(x0, NULL)

(11) Obligation:

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

4733_0_FLATTEN_STORE(x0[0], x1[0], x2[0], x3[0]) → 4005_0_FLATTEN_NULL(x0[0], java.lang.Object(Tree(x1[0], java.lang.Object(Tree(x2[0], x3[0])))))
4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(java.lang.Object(Tree(x0[1], x1[1])), x2[1]))) → 4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(x0[1], java.lang.Object(Tree(x1[1], x2[1])))))
5117_0_FLATTEN_STORE(java.lang.Object(Tree(NULL, x0[2])), x1[2]) → 4005_0_FLATTEN_NULL(java.lang.Object(Tree(NULL, x0[2])), x1[2])
4005_0_FLATTEN_NULL(x1[3], java.lang.Object(Tree(NULL, x0[3]))) → 4005_0_FLATTEN_NULL(java.lang.Object(Tree(NULL, x1[3])), x0[3])
4005_0_FLATTEN_NULL(x3[4], java.lang.Object(Tree(java.lang.Object(Tree(x0[4], x1[4])), x2[4]))) → 4733_0_FLATTEN_STORE(x3[4], x0[4], x1[4], x2[4])
4005_0_FLATTEN_NULL(x1[5], java.lang.Object(Tree(NULL, x0[5]))) → 5117_0_FLATTEN_STORE(java.lang.Object(Tree(NULL, x1[5])), x0[5])

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

(12) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

5117_0_FLATTEN_STORE(java.lang.Object(Tree(NULL, x0[2])), x1[2]) → 4005_0_FLATTEN_NULL(java.lang.Object(Tree(NULL, x0[2])), x1[2])
4005_0_FLATTEN_NULL(x1[3], java.lang.Object(Tree(NULL, x0[3]))) → 4005_0_FLATTEN_NULL(java.lang.Object(Tree(NULL, x1[3])), x0[3])

Used ordering: Polynomial interpretation [POLO]:

POL(4005_0_FLATTEN_NULL(x₁, x₂)) = x₁ + 2·x₂
POL(4733_0_FLATTEN_STORE(x₁, x₂, x₃, x₄)) = x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(5117_0_FLATTEN_STORE(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(NULL) = 2
POL(Tree(x₁, x₂)) = x₁ + x₂
POL(java.lang.Object(x₁)) = x₁

(13) Obligation:

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

4733_0_FLATTEN_STORE(x0[0], x1[0], x2[0], x3[0]) → 4005_0_FLATTEN_NULL(x0[0], java.lang.Object(Tree(x1[0], java.lang.Object(Tree(x2[0], x3[0])))))
4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(java.lang.Object(Tree(x0[1], x1[1])), x2[1]))) → 4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(x0[1], java.lang.Object(Tree(x1[1], x2[1])))))
4005_0_FLATTEN_NULL(x3[4], java.lang.Object(Tree(java.lang.Object(Tree(x0[4], x1[4])), x2[4]))) → 4733_0_FLATTEN_STORE(x3[4], x0[4], x1[4], x2[4])
4005_0_FLATTEN_NULL(x1[5], java.lang.Object(Tree(NULL, x0[5]))) → 5117_0_FLATTEN_STORE(java.lang.Object(Tree(NULL, x1[5])), x0[5])

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

(14) DependencyGraphProof (EQUIVALENT transformation)

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

(15) Obligation:

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

4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(java.lang.Object(Tree(x0[1], x1[1])), x2[1]))) → 4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(x0[1], java.lang.Object(Tree(x1[1], x2[1])))))
4005_0_FLATTEN_NULL(x3[4], java.lang.Object(Tree(java.lang.Object(Tree(x0[4], x1[4])), x2[4]))) → 4733_0_FLATTEN_STORE(x3[4], x0[4], x1[4], x2[4])
4733_0_FLATTEN_STORE(x0[0], x1[0], x2[0], x3[0]) → 4005_0_FLATTEN_NULL(x0[0], java.lang.Object(Tree(x1[0], java.lang.Object(Tree(x2[0], x3[0])))))

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

(16) MRRProof (EQUIVALENT transformation)

4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(java.lang.Object(Tree(x0[1], x1[1])), x2[1]))) → 4005_0_FLATTEN_NULL(x3[1], java.lang.Object(Tree(x0[1], java.lang.Object(Tree(x1[1], x2[1])))))
4005_0_FLATTEN_NULL(x3[4], java.lang.Object(Tree(java.lang.Object(Tree(x0[4], x1[4])), x2[4]))) → 4733_0_FLATTEN_STORE(x3[4], x0[4], x1[4], x2[4])

Used ordering: Polynomial interpretation [POLO]:

POL(4005_0_FLATTEN_NULL(x₁, x₂)) = x₁ + x₂
POL(4733_0_FLATTEN_STORE(x₁, x₂, x₃, x₄)) = 2 + x₁ + 2·x₂ + 2·x₃ + x₄
POL(Tree(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(java.lang.Object(x₁)) = x₁

(17) Obligation:

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

4733_0_FLATTEN_STORE(x0[0], x1[0], x2[0], x3[0]) → 4005_0_FLATTEN_NULL(x0[0], java.lang.Object(Tree(x1[0], java.lang.Object(Tree(x2[0], x3[0])))))

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

(18) DependencyGraphProof (EQUIVALENT transformation)

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

(19) TRUE

(20) 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): 15741_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_15741_1_CREATETREE_INVOKEMETHOD(x0[0] > 0 && 0 < x0[0] + -1, x0[0])
(1): COND_15741_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 15741_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.

(21) 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 15741_1_CREATETREE_INVOKEMETHOD(x0) → COND_15741_1_CREATETREE_INVOKEMETHOD(&&(>(x0, 0), <(0, +(x0, -1))), x0) the following chains were created:

We consider the chain 15741_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_15741_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0]), COND_15741_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 15741_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] ⇒ 15741_1_CREATETREE_INVOKEMETHOD(x0[0])≥NonInfC∧15741_1_CREATETREE_INVOKEMETHOD(x0[0])≥COND_15741_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])∧(U^Increasing(COND_15741_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 ⇒ 15741_1_CREATETREE_INVOKEMETHOD(x0[0])≥NonInfC∧15741_1_CREATETREE_INVOKEMETHOD(x0[0])≥COND_15741_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])∧(U^Increasing(COND_15741_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_15741_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_15741_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_15741_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_15741_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_15741_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_15741_1_CREATETREE_INVOKEMETHOD(TRUE, x0) → 15741_1_CREATETREE_INVOKEMETHOD(+(x0, -1)) the following chains were created:

We consider the chain COND_15741_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 15741_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1)) which results in the following constraint:
(8)    (COND_15741_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1])≥NonInfC∧COND_15741_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1])≥15741_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))∧(U^Increasing(15741_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(15741_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(15741_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(15741_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(15741_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.

15741_1_CREATETREE_INVOKEMETHOD(x0) → COND_15741_1_CREATETREE_INVOKEMETHOD(&&(>(x0, 0), <(0, +(x0, -1))), x0)
- ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_15741_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_15741_1_CREATETREE_INVOKEMETHOD(TRUE, x0) → 15741_1_CREATETREE_INVOKEMETHOD(+(x0, -1))
- ((U^Increasing(15741_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(15741_1_CREATETREE_INVOKEMETHOD(x₁)) = [2]x₁
POL(COND_15741_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_>:

15741_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_15741_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])
COND_15741_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 15741_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))

The following pairs are in P_bound:

15741_1_CREATETREE_INVOKEMETHOD(x0[0]) → COND_15741_1_CREATETREE_INVOKEMETHOD(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0])

The following pairs are in P_≥:
none

There are no usable rules.

(22) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_15741_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1]) → 15741_1_CREATETREE_INVOKEMETHOD(x0[1] + -1)

The set Q is empty.

(23) 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) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) QReductionProof (EQUIVALENT transformation)

(11) Obligation:

(12) MRRProof (EQUIVALENT transformation)

(13) Obligation:

(14) DependencyGraphProof (EQUIVALENT transformation)

(15) Obligation:

(16) MRRProof (EQUIVALENT transformation)

(17) Obligation:

(18) DependencyGraphProof (EQUIVALENT transformation)

(19) TRUE

(20) Obligation:

(21) IDPNonInfProof (SOUND transformation)

(22) Obligation:

(23) IDependencyGraphProof (EQUIVALENT transformation)

(24) TRUE