### (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)
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)
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.

### (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

### (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)
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)
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)
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)
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:

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)
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:

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:

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:

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:

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:

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.

### (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)))=TRUEx0[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])∧(UIncreasing(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))=TRUE16702_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])∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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))∧(UIncreasing(16702_1_CREATETREE_INVOKEMETHOD(+(x0[1], -1))), ≥))

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

(9)    ((UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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 ⇒ (UIncreasing(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))
• ((UIncreasing(16702_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(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 Pbound:

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.