(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.

(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)

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:

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.

(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)))=TRUEx0[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])∧(UIncreasing(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))=TRUE15741_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])∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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))∧(UIncreasing(15741_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥))

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

(9)    ((UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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 ⇒ (UIncreasing(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))
• ((UIncreasing(15741_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(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 Pbound:

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.