### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: Gopher/Gopher
`package Gopher;public class Gopher {    // by Boyer & Moore (1979)  public static Tree gopher(Tree x) {    if (x == null) return null;    if (x.left == null) return x;    else return gopher(new Tree(x.left.left, new Tree(x.left.right, x.right)));  }  public static void main(String[] args) {    Random.args = args;    Tree x = Tree.createTree(Random.random());    Tree y = gopher(x);  }}package Gopher;public class Random {  static String[] args;  static int index = 0;  public static int random() {    final String string = args[index];    index++;    return string.length();  }}package Gopher;public class Tree {    Tree left;    Tree right;    int value;    public static Tree createTree(final int numElements) {        final Tree t = new Tree(Random.random());        for (int i = 0; i < numElements; i++) {            t.insert(Random.random());        }        return t;    }    public Tree(final Tree l, final Tree r) {        this.left = l;        this.right = r;    }    public Tree(final int val) {        this.value = val;    }    public void insert(final int v) {        if (v <= this.value) {            if (this.left == null) {                this.left = new Tree(v);            } else {                this.left.insert(v);            }        } else {            if (this.right == null) {                this.right = new Tree(v);            } else {                this.right.insert(v);            }        }    }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Gopher.Gopher.main([Ljava/lang/String;)V: Graph of 110 nodes with 0 SCCs.

Gopher.Tree.createTree(I)LGopher/Tree;: Graph of 193 nodes with 1 SCC.

Gopher.Gopher.gopher(LGopher/Tree;)LGopher/Tree;: Graph of 56 nodes with 0 SCCs.

Gopher.Tree.insert(I)V: Graph of 72 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 30 rules for P and 51 rules for R.

Combined rules. Obtained 2 rules for P and 8 rules for R.

Filtered ground terms:

Gopher.Tree(x1, x2, x3, x4) → Gopher.Tree(x2, x3, x4)
4020_0_insert_Return(x1) → 4020_0_insert_Return
3964_0_insert_Return(x1) → 3964_0_insert_Return
3880_0_insert_Return(x1) → 3880_0_insert_Return
3877_0_insert_Return(x1) → 3877_0_insert_Return

Filtered duplicate args:

Filtered unneeded arguments:

3730_1_insert_InvokeMethod(x1, x2, x3) → 3730_1_insert_InvokeMethod(x1, x2)
3737_1_insert_InvokeMethod(x1, x2, x3) → 3737_1_insert_InvokeMethod(x1, x2)

Combined rules. Obtained 2 rules for P and 8 rules for R.

Finished conversion. Obtained 2 rules for P and 8 rules for R. System has predefined symbols.

Log for SCC 1:

Generated 43 rules for P and 12 rules for R.

Combined rules. Obtained 1 rules for P and 3 rules for R.

Filtered ground terms:

Gopher.Tree(x1, x2, x3) → Gopher.Tree(x2, x3)
3105_0_gopher_NONNULL(x1, x2, x3) → 3105_0_gopher_NONNULL(x2, x3)
4668_0_gopher_Return(x1, x2) → 4668_0_gopher_Return(x2)
4022_0_gopher_Return(x1, x2) → 4022_0_gopher_Return(x2)
3173_0_gopher_Return(x1, x2, x3) → 3173_0_gopher_Return(x2, x3)

Filtered duplicate args:

3105_0_gopher_NONNULL(x1, x2) → 3105_0_gopher_NONNULL(x2)
3173_0_gopher_Return(x1, x2) → 3173_0_gopher_Return(x2)

Finished conversion. Obtained 1 rules for P and 3 rules for R. System has no predefined symbols.

Log for SCC 2:

Generated 37 rules for P and 157 rules for R.

Combined rules. Obtained 6 rules for P and 13 rules for R.

Filtered ground terms:

4773_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → 4773_1_createTree_InvokeMethod(x1, x2, x4)
Gopher.Tree(x1) → Gopher.Tree
4773_0_random_ArrayAccess(x1, x2, x3) → 4773_0_random_ArrayAccess(x2, x3)
Cond_4980_1_createTree_InvokeMethod3(x1, x2, x3, x4, x5, x6, x7) → Cond_4980_1_createTree_InvokeMethod3(x1, x3, x5, x7)
4020_0_insert_Return(x1) → 4020_0_insert_Return
4980_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6) → 4980_1_createTree_InvokeMethod(x1, x2, x4, x6)
Cond_4980_1_createTree_InvokeMethod2(x1, x2, x3, x4, x5, x6, x7) → Cond_4980_1_createTree_InvokeMethod2(x1, x3, x5, x7)
3964_0_insert_Return(x1) → 3964_0_insert_Return
Cond_4980_1_createTree_InvokeMethod1(x1, x2, x3, x4, x5, x6, x7) → Cond_4980_1_createTree_InvokeMethod1(x1, x3, x5, x7)
3880_0_insert_Return(x1) → 3880_0_insert_Return
Cond_4980_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_4980_1_createTree_InvokeMethod(x1, x3, x5, x7)
3877_0_insert_Return(x1) → 3877_0_insert_Return
Cond_4808_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_4808_1_createTree_InvokeMethod(x1, x2, x3, x5)
4808_0_random_IntArithmetic(x1, x2, x3, x4) → 4808_0_random_IntArithmetic(x2, x3)
4808_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → 4808_1_createTree_InvokeMethod(x1, x2, x4)
Cond_4773_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_4773_1_createTree_InvokeMethod(x1, x2, x3, x5)
3517_0_insert_GT(x1, x2, x3, x4, x5) → 3517_0_insert_GT(x3, x4, x5)
Cond_3517_0_insert_GT3(x1, x2, x3, x4, x5, x6) → Cond_3517_0_insert_GT3(x1, x4, x5, x6)
Cond_3517_0_insert_GT2(x1, x2, x3, x4, x5, x6) → Cond_3517_0_insert_GT2(x1, x4, x5, x6)
Cond_3517_0_insert_GT1(x1, x2, x3, x4, x5, x6) → Cond_3517_0_insert_GT1(x1, x4, x5, x6)
Cond_3517_0_insert_GT(x1, x2, x3, x4, x5, x6) → Cond_3517_0_insert_GT(x1, x4, x5, x6)

Filtered duplicate args:

3517_0_insert_GT(x1, x2, x3) → 3517_0_insert_GT(x2, x3)
Cond_3517_0_insert_GT3(x1, x2, x3, x4) → Cond_3517_0_insert_GT3(x1, x3, x4)
Cond_3517_0_insert_GT2(x1, x2, x3, x4) → Cond_3517_0_insert_GT2(x1, x3, x4)
Cond_3517_0_insert_GT1(x1, x2, x3, x4) → Cond_3517_0_insert_GT1(x1, x3, x4)
Cond_3517_0_insert_GT(x1, x2, x3, x4) → Cond_3517_0_insert_GT(x1, x3, x4)

Filtered unneeded arguments:

4980_1_createTree_InvokeMethod(x1, x2, x3, x4) → 4980_1_createTree_InvokeMethod(x1, x2, x3)
Cond_4980_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_4980_1_createTree_InvokeMethod(x1, x2, x3)
Cond_4980_1_createTree_InvokeMethod1(x1, x2, x3, x4) → Cond_4980_1_createTree_InvokeMethod1(x1, x2, x3)
Cond_4980_1_createTree_InvokeMethod2(x1, x2, x3, x4) → Cond_4980_1_createTree_InvokeMethod2(x1, x2, x3)
Cond_4980_1_createTree_InvokeMethod3(x1, x2, x3, x4) → Cond_4980_1_createTree_InvokeMethod3(x1, x2, x3)
Cond_3517_0_insert_GT(x1, x2, x3) → Cond_3517_0_insert_GT(x1)
Cond_3517_0_insert_GT1(x1, x2, x3) → Cond_3517_0_insert_GT1(x1, x2)
3737_1_insert_InvokeMethod(x1, x2, x3) → 3737_1_insert_InvokeMethod(x1, x2)
Cond_3517_0_insert_GT2(x1, x2, x3) → Cond_3517_0_insert_GT2(x1)
Cond_3517_0_insert_GT3(x1, x2, x3) → Cond_3517_0_insert_GT3(x1, x2)
3730_1_insert_InvokeMethod(x1, x2, x3) → 3730_1_insert_InvokeMethod(x1, x2)

Filtered all non-integer terms:

4808_0_random_IntArithmetic(x1, x2) → 4808_0_random_IntArithmetic(x2)
3737_1_insert_InvokeMethod(x1, x2) → 3737_1_insert_InvokeMethod(x1)
3730_1_insert_InvokeMethod(x1, x2) → 3730_1_insert_InvokeMethod(x1)

Filtered all free variables:

4808_1_createTree_InvokeMethod(x1, x2, x3) → 4808_1_createTree_InvokeMethod(x2, x3)
Cond_4808_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_4808_1_createTree_InvokeMethod(x1, x3, x4)
4980_1_createTree_InvokeMethod(x1, x2, x3) → 4980_1_createTree_InvokeMethod(x2, x3)
4773_1_createTree_InvokeMethod(x1, x2, x3) → 4773_1_createTree_InvokeMethod(x2, x3)
Cond_4773_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_4773_1_createTree_InvokeMethod(x1, x3, x4)
3517_0_insert_GT(x1, x2) → 3517_0_insert_GT(x1)

Combined rules. Obtained 1 rules for P and 13 rules for R.

Finished conversion. Obtained 1 rules for P and 13 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:

Integer

The ITRS R consists of the following rules:
3730_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3737_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(x3[0] > x0[0], java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
(2): 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(x3[2] <= x0[2], java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])

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

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

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

(2) -> (3), if ((x3[2] <= x0[2]* TRUE)∧(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))) →* java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))))∧(x3[2]* x3[3]))

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

(3) -> (2), if ((java.lang.Object(x2[3]) →* java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))))∧(x3[3]* x3[2]))

The set Q consists of the following terms:
3730_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0))

### (6) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3) → COND_3430_0_INSERT_LOAD(>(x3, x0), java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3) the following chains were created:
• We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]) which results in the following constraint:

(1)    (>(x3[0], x0[0])=TRUEjava.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0]))=java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1]))∧x3[0]=x3[1]3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥NonInfC∧3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])∧(UIncreasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥))

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

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

(3)    (0 ≥ 0 ⇒ (UIncreasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(2)bni_28]x2[0] + [bni_28]x1[0] + [bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(4)    (0 ≥ 0 ⇒ (UIncreasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(2)bni_28]x2[0] + [bni_28]x1[0] + [bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(5)    (0 ≥ 0 ⇒ (UIncreasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(2)bni_28]x2[0] + [bni_28]x1[0] + [bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(6)    (0 ≥ 0 ⇒ (UIncreasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥)∧0 ≥ 0∧[(2)bni_28] ≥ 0∧[bni_28] ≥ 0∧[bni_28] ≥ 0∧[bni_28 + (-1)Bound*bni_28] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_29] ≥ 0)

For Pair COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3) → 3430_0_INSERT_LOAD(java.lang.Object(x1), x3) the following chains were created:
• We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]), 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) which results in the following constraint:

We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:

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

(9)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[0]1 + [bni_30]x1[0]1 + [bni_30]x0[0]1 + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

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

(10)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[0]1 + [bni_30]x1[0]1 + [bni_30]x0[0]1 + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

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

(11)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[0]1 + [bni_30]x1[0]1 + [bni_30]x0[0]1 + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

We simplified constraint (11) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(12)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧0 ≥ 0∧[(2)bni_30] ≥ 0∧[(2)bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[(2)bni_30 + (-1)Bound*bni_30] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_31] ≥ 0∧[1] ≥ 0)

• We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]), 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) which results in the following constraint:

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

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

(15)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[2] + [bni_30]x1[2] + [bni_30]x0[2] + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

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

(16)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[2] + [bni_30]x1[2] + [bni_30]x0[2] + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

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

(17)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[2] + [bni_30]x1[2] + [bni_30]x0[2] + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

We simplified constraint (17) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(18)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧0 ≥ 0∧[(2)bni_30] ≥ 0∧[(2)bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[(3)bni_30 + (-1)Bound*bni_30] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_31] ≥ 0∧[1] ≥ 0)

For Pair 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3) → COND_3430_0_INSERT_LOAD1(<=(x3, x0), java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3) the following chains were created:
• We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]) which results in the following constraint:

(19)    (<=(x3[2], x0[2])=TRUEjava.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2])))=java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3])))∧x3[2]=x3[3]3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥NonInfC∧3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])∧(UIncreasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥))

We simplified constraint (19) using rules (I), (II), (IV) which results in the following new constraint:

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

(21)    (0 ≥ 0 ⇒ (UIncreasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥)∧[(2)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]x2[2] + [bni_32]x1[2] + [bni_32]x0[2] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(22)    (0 ≥ 0 ⇒ (UIncreasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥)∧[(2)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]x2[2] + [bni_32]x1[2] + [bni_32]x0[2] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(23)    (0 ≥ 0 ⇒ (UIncreasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥)∧[(2)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]x2[2] + [bni_32]x1[2] + [bni_32]x0[2] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(24)    (0 ≥ 0 ⇒ (UIncreasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥)∧0 ≥ 0∧[(2)bni_32] ≥ 0∧[bni_32] ≥ 0∧[bni_32] ≥ 0∧[(2)bni_32 + (-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)

For Pair COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3) → 3430_0_INSERT_LOAD(java.lang.Object(x2), x3) the following chains were created:
• We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]), 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) which results in the following constraint:

We simplified constraint (25) using rules (I), (II), (III), (IV) which results in the following new constraint:

We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(27)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(4)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[0] + [(2)bni_34]x1[0] + [(2)bni_34]x0[0] + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[3 + (-1)bso_35] + [2]x2[0] + x1[0] + x0[0] + x1[2] + x0[2] ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(28)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(4)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[0] + [(2)bni_34]x1[0] + [(2)bni_34]x0[0] + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[3 + (-1)bso_35] + [2]x2[0] + x1[0] + x0[0] + x1[2] + x0[2] ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(29)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(4)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[0] + [(2)bni_34]x1[0] + [(2)bni_34]x0[0] + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[3 + (-1)bso_35] + [2]x2[0] + x1[0] + x0[0] + x1[2] + x0[2] ≥ 0)

We simplified constraint (29) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(30)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧0 ≥ 0∧[(4)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[bni_34] ≥ 0∧[bni_34] ≥ 0∧[(4)bni_34 + (-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[3 + (-1)bso_35] ≥ 0∧[1] ≥ 0)

• We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]), 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) which results in the following constraint:

We simplified constraint (31) using rules (I), (II), (III), (IV) which results in the following new constraint:

We simplified constraint (32) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(33)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(6)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[2]1 + [(2)bni_34]x1[2]1 + [(2)bni_34]x0[2]1 + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[4 + (-1)bso_35] + [2]x2[2]1 + x1[2]1 + x0[2]1 + x1[2] + x0[2] ≥ 0)

We simplified constraint (33) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(34)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(6)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[2]1 + [(2)bni_34]x1[2]1 + [(2)bni_34]x0[2]1 + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[4 + (-1)bso_35] + [2]x2[2]1 + x1[2]1 + x0[2]1 + x1[2] + x0[2] ≥ 0)

We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(35)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(6)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[2]1 + [(2)bni_34]x1[2]1 + [(2)bni_34]x0[2]1 + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[4 + (-1)bso_35] + [2]x2[2]1 + x1[2]1 + x0[2]1 + x1[2] + x0[2] ≥ 0)

We simplified constraint (35) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(36)    (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧0 ≥ 0∧[(4)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[bni_34] ≥ 0∧[bni_34] ≥ 0∧[(6)bni_34 + (-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[4 + (-1)bso_35] ≥ 0∧[1] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3) → COND_3430_0_INSERT_LOAD(>(x3, x0), java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3)
• (0 ≥ 0 ⇒ (UIncreasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥)∧0 ≥ 0∧[(2)bni_28] ≥ 0∧[bni_28] ≥ 0∧[bni_28] ≥ 0∧[bni_28 + (-1)Bound*bni_28] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_29] ≥ 0)

• (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧0 ≥ 0∧[(2)bni_30] ≥ 0∧[(2)bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[(2)bni_30 + (-1)Bound*bni_30] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_31] ≥ 0∧[1] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧0 ≥ 0∧[(2)bni_30] ≥ 0∧[(2)bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[(3)bni_30 + (-1)Bound*bni_30] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_31] ≥ 0∧[1] ≥ 0)

• 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3) → COND_3430_0_INSERT_LOAD1(<=(x3, x0), java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3)
• (0 ≥ 0 ⇒ (UIncreasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥)∧0 ≥ 0∧[(2)bni_32] ≥ 0∧[bni_32] ≥ 0∧[bni_32] ≥ 0∧[(2)bni_32 + (-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)

• (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧0 ≥ 0∧[(4)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[bni_34] ≥ 0∧[bni_34] ≥ 0∧[(4)bni_34 + (-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[3 + (-1)bso_35] ≥ 0∧[1] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧0 ≥ 0∧[(4)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[bni_34] ≥ 0∧[bni_34] ≥ 0∧[(6)bni_34 + (-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[4 + (-1)bso_35] ≥ 0∧[1] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(3730_1_insert_InvokeMethod(x1, x2)) = 0
POL(3877_0_insert_Return) = 0
POL(java.lang.Object(x1)) = [1] + x1
POL(3964_0_insert_Return) = 0
POL(3880_0_insert_Return) = 0
POL(4020_0_insert_Return) = 0
POL(3737_1_insert_InvokeMethod(x1, x2)) = 0
POL(3430_0_INSERT_LOAD(x1, x2)) = [-1] + x1
POL(Gopher.Tree(x1, x2, x3)) = [2]x3 + x2 + x1
POL(COND_3430_0_INSERT_LOAD(x1, x2, x3)) = [-1] + x2
POL(>(x1, x2)) = 0
POL(COND_3430_0_INSERT_LOAD1(x1, x2, x3)) = [-1] + x2
POL(<=(x1, x2)) = 0

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (7) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

The ITRS R consists of the following rules:
3730_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3737_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(x3[0] > x0[0], java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
(2): 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(x3[2] <= x0[2], java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])

The set Q consists of the following terms:
3730_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0))

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))) → 4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2))))) → 4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2)))))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7))))) → 4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2)))))))

The integer pair graph contains the following rules and edges:
(0): 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0], x1[0])), x2[0]))) → 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))))

(0) -> (0), if ((java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))) →* java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0]', x1[0]')), x2[0]'))))

The set Q consists of the following terms:
3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2)))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7)))))

### (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:

3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0], x1[0])), x2[0]))) → 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))))

The TRS R consists of the following rules:

3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))) → 4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2))))) → 4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2)))))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7))))) → 4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2)))))))

The set Q consists of the following terms:

3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2)))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7)))))

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:

3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0], x1[0])), x2[0]))) → 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))))

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

3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2)))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7)))))

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].

3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2)))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7)))))

### (16) Obligation:

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

3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0], x1[0])), x2[0]))) → 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))))

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

### (17) 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:

3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0], x1[0])), x2[0]))) → 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))))

Used ordering: Polynomial interpretation [POLO]:

POL(3105_0_GOPHER_NONNULL(x1)) = 2·x1
POL(Gopher.Tree(x1, x2)) = 2 + 2·x1 + x2
POL(java.lang.Object(x1)) = 2 + x1

### (18) Obligation:

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

### (19) PisEmptyProof (EQUIVALENT transformation)

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

### (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

The ITRS R consists of the following rules:
3517_0_insert_GT(x0) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3737_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3737_1_insert_InvokeMethod(3877_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3880_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3964_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(4020_0_insert_Return) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3964_0_insert_Return
3517_0_insert_GT(x0) → 3730_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3730_1_insert_InvokeMethod(3877_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3880_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3964_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(4020_0_insert_Return) → 3964_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_4773_1_CREATETREE_INVOKEMETHOD(x1[0] >= 0 && x0[0] > x1[0] + 1, x0[0], x1[0])
(1): COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 4773_1_CREATETREE_INVOKEMETHOD(x0[1], x1[1] + 1)

(0) -> (1), if ((x1[0] >= 0 && x0[0] > x1[0] + 1* TRUE)∧(x0[0]* x0[1])∧(x1[0]* x1[1]))

(1) -> (0), if ((x0[1]* x0[0])∧(x1[1] + 1* x1[0]))

The set Q consists of the following terms:
3517_0_insert_GT(x0)
3737_1_insert_InvokeMethod(3877_0_insert_Return)
3737_1_insert_InvokeMethod(3880_0_insert_Return)
3737_1_insert_InvokeMethod(3964_0_insert_Return)
3737_1_insert_InvokeMethod(4020_0_insert_Return)
3730_1_insert_InvokeMethod(3877_0_insert_Return)
3730_1_insert_InvokeMethod(3880_0_insert_Return)
3730_1_insert_InvokeMethod(3964_0_insert_Return)
3730_1_insert_InvokeMethod(4020_0_insert_Return)

### (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 4773_1_CREATETREE_INVOKEMETHOD(x0, x1) → COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), x0, x1) the following chains were created:
• We consider the chain 4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0]), COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1)) which results in the following constraint:

(1)    (&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1)))=TRUEx0[0]=x0[1]x1[0]=x1[1]4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥NonInfC∧4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])∧(UIncreasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥))

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

(2)    (>=(x1[0], 0)=TRUE>(x0[0], +(x1[0], 1))=TRUE4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥NonInfC∧4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])∧(UIncreasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥))

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

(3)    (x1[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(4)    (x1[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(5)    (x1[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(6)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0, x1) → 4773_1_CREATETREE_INVOKEMETHOD(x0, +(x1, 1)) the following chains were created:
• We consider the chain COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1)) which results in the following constraint:

(7)    (COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1])≥NonInfC∧COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1])≥4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))∧(UIncreasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(8)    ((UIncreasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧[1 + (-1)bso_21] ≥ 0)

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

(9)    ((UIncreasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧[1 + (-1)bso_21] ≥ 0)

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

(10)    ((UIncreasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧[1 + (-1)bso_21] ≥ 0)

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

(11)    ((UIncreasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 4773_1_CREATETREE_INVOKEMETHOD(x0, x1) → COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), x0, x1)
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

• COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0, x1) → 4773_1_CREATETREE_INVOKEMETHOD(x0, +(x1, 1))
• ((UIncreasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 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(3517_0_insert_GT(x1)) = [-1]
POL(4020_0_insert_Return) = [-1]
POL(3737_1_insert_InvokeMethod(x1)) = [-1]
POL(3877_0_insert_Return) = [-1]
POL(3880_0_insert_Return) = [-1]
POL(3964_0_insert_Return) = [-1]
POL(3730_1_insert_InvokeMethod(x1)) = [-1]
POL(4773_1_CREATETREE_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2 + [2]x1
POL(COND_4773_1_CREATETREE_INVOKEMETHOD(x1, x2, x3)) = [-1] + [-1]x3 + [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>:

COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))

The following pairs are in Pbound:

4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])

The following pairs are in P:

4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])

There are no usable rules.

### (24) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
3517_0_insert_GT(x0) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3737_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3737_1_insert_InvokeMethod(3877_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3880_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3964_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(4020_0_insert_Return) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3964_0_insert_Return
3517_0_insert_GT(x0) → 3730_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3730_1_insert_InvokeMethod(3877_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3880_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3964_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(4020_0_insert_Return) → 3964_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_4773_1_CREATETREE_INVOKEMETHOD(x1[0] >= 0 && x0[0] > x1[0] + 1, x0[0], x1[0])

The set Q consists of the following terms:
3517_0_insert_GT(x0)
3737_1_insert_InvokeMethod(3877_0_insert_Return)
3737_1_insert_InvokeMethod(3880_0_insert_Return)
3737_1_insert_InvokeMethod(3964_0_insert_Return)
3737_1_insert_InvokeMethod(4020_0_insert_Return)
3730_1_insert_InvokeMethod(3877_0_insert_Return)
3730_1_insert_InvokeMethod(3880_0_insert_Return)
3730_1_insert_InvokeMethod(3964_0_insert_Return)
3730_1_insert_InvokeMethod(4020_0_insert_Return)

### (25) IDependencyGraphProof (EQUIVALENT transformation)

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

### (27) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

The ITRS R consists of the following rules:
3517_0_insert_GT(x0) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3737_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3737_1_insert_InvokeMethod(3877_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3880_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3964_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(4020_0_insert_Return) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3964_0_insert_Return
3517_0_insert_GT(x0) → 3730_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3730_1_insert_InvokeMethod(3877_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3880_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3964_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(4020_0_insert_Return) → 3964_0_insert_Return

The integer pair graph contains the following rules and edges:
(1): COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 4773_1_CREATETREE_INVOKEMETHOD(x0[1], x1[1] + 1)

The set Q consists of the following terms:
3517_0_insert_GT(x0)
3737_1_insert_InvokeMethod(3877_0_insert_Return)
3737_1_insert_InvokeMethod(3880_0_insert_Return)
3737_1_insert_InvokeMethod(3964_0_insert_Return)
3737_1_insert_InvokeMethod(4020_0_insert_Return)
3730_1_insert_InvokeMethod(3877_0_insert_Return)
3730_1_insert_InvokeMethod(3880_0_insert_Return)
3730_1_insert_InvokeMethod(3964_0_insert_Return)
3730_1_insert_InvokeMethod(4020_0_insert_Return)

### (28) IDependencyGraphProof (EQUIVALENT transformation)

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