### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: SearchTreeR/SearchTreeR
`package SearchTreeR;public class Random {  static String[] args;  static int index = 0;  public static int random() {    final String string = args[index];    index++;    return string.length();  }}package SearchTreeR;public class SearchTreeR {  public static void main(final String[] args) {    Random.args = args;    final Tree t = new Tree(Random.random());    final int numberOfVals = Random.random();    for (int i = 0; i < numberOfVals; i++) {        t.insert(Random.random());    }  }}package SearchTreeR;public class Tree {    Tree left;    Tree right;    int value;    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:
SearchTreeR.SearchTreeR.main([Ljava/lang/String;)V: Graph of 269 nodes with 1 SCC.

SearchTreeR.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:

SearchTreeR.Tree(x1, x2, x3, x4) → SearchTreeR.Tree(x2, x3, x4)
3577_0_insert_Return(x1) → 3577_0_insert_Return
3540_0_insert_Return(x1) → 3540_0_insert_Return
3470_0_insert_Return(x1) → 3470_0_insert_Return
3466_0_insert_Return(x1) → 3466_0_insert_Return

Filtered duplicate args:

Filtered unneeded arguments:

3393_1_insert_InvokeMethod(x1, x2, x3) → 3393_1_insert_InvokeMethod(x1, x2)
3401_1_insert_InvokeMethod(x1, x2, x3) → 3401_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 37 rules for P and 156 rules for R.

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

Filtered ground terms:

3655_1_main_InvokeMethod(x1, x2, x3, x4, x5) → 3655_1_main_InvokeMethod(x1, x3, x4)
SearchTreeR.Tree(x1) → SearchTreeR.Tree
3655_0_random_ArrayAccess(x1, x2, x3) → 3655_0_random_ArrayAccess(x2, x3)
Cond_3985_1_main_InvokeMethod3(x1, x2, x3, x4, x5, x6, x7) → Cond_3985_1_main_InvokeMethod3(x1, x4, x5, x7)
3577_0_insert_Return(x1) → 3577_0_insert_Return
3985_1_main_InvokeMethod(x1, x2, x3, x4, x5, x6) → 3985_1_main_InvokeMethod(x1, x3, x4, x6)
Cond_3985_1_main_InvokeMethod2(x1, x2, x3, x4, x5, x6, x7) → Cond_3985_1_main_InvokeMethod2(x1, x4, x5, x7)
3540_0_insert_Return(x1) → 3540_0_insert_Return
Cond_3985_1_main_InvokeMethod1(x1, x2, x3, x4, x5, x6, x7) → Cond_3985_1_main_InvokeMethod1(x1, x4, x5, x7)
3470_0_insert_Return(x1) → 3470_0_insert_Return
Cond_3985_1_main_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_3985_1_main_InvokeMethod(x1, x4, x5, x7)
3466_0_insert_Return(x1) → 3466_0_insert_Return
Cond_3710_1_main_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_3710_1_main_InvokeMethod(x1, x2, x4, x5)
3710_0_random_IntArithmetic(x1, x2, x3, x4) → 3710_0_random_IntArithmetic(x2, x3)
3710_1_main_InvokeMethod(x1, x2, x3, x4, x5) → 3710_1_main_InvokeMethod(x1, x3, x4)
Cond_3655_1_main_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_3655_1_main_InvokeMethod(x1, x2, x4, x5)
3215_0_insert_GT(x1, x2, x3, x4, x5) → 3215_0_insert_GT(x3, x4, x5)
Cond_3215_0_insert_GT3(x1, x2, x3, x4, x5, x6) → Cond_3215_0_insert_GT3(x1, x4, x5, x6)
Cond_3215_0_insert_GT2(x1, x2, x3, x4, x5, x6) → Cond_3215_0_insert_GT2(x1, x4, x5, x6)
Cond_3215_0_insert_GT1(x1, x2, x3, x4, x5, x6) → Cond_3215_0_insert_GT1(x1, x4, x5, x6)
Cond_3215_0_insert_GT(x1, x2, x3, x4, x5, x6) → Cond_3215_0_insert_GT(x1, x4, x5, x6)

Filtered duplicate args:

3215_0_insert_GT(x1, x2, x3) → 3215_0_insert_GT(x2, x3)
Cond_3215_0_insert_GT3(x1, x2, x3, x4) → Cond_3215_0_insert_GT3(x1, x3, x4)
Cond_3215_0_insert_GT2(x1, x2, x3, x4) → Cond_3215_0_insert_GT2(x1, x3, x4)
Cond_3215_0_insert_GT1(x1, x2, x3, x4) → Cond_3215_0_insert_GT1(x1, x3, x4)
Cond_3215_0_insert_GT(x1, x2, x3, x4) → Cond_3215_0_insert_GT(x1, x3, x4)

Filtered unneeded arguments:

3985_1_main_InvokeMethod(x1, x2, x3, x4) → 3985_1_main_InvokeMethod(x1, x2, x3)
Cond_3985_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_3985_1_main_InvokeMethod(x1, x2, x3)
Cond_3985_1_main_InvokeMethod1(x1, x2, x3, x4) → Cond_3985_1_main_InvokeMethod1(x1, x2, x3)
Cond_3985_1_main_InvokeMethod2(x1, x2, x3, x4) → Cond_3985_1_main_InvokeMethod2(x1, x2, x3)
Cond_3985_1_main_InvokeMethod3(x1, x2, x3, x4) → Cond_3985_1_main_InvokeMethod3(x1, x2, x3)
Cond_3215_0_insert_GT(x1, x2, x3) → Cond_3215_0_insert_GT(x1)
Cond_3215_0_insert_GT1(x1, x2, x3) → Cond_3215_0_insert_GT1(x1, x2)
3401_1_insert_InvokeMethod(x1, x2, x3) → 3401_1_insert_InvokeMethod(x1, x2)
Cond_3215_0_insert_GT2(x1, x2, x3) → Cond_3215_0_insert_GT2(x1)
Cond_3215_0_insert_GT3(x1, x2, x3) → Cond_3215_0_insert_GT3(x1, x2)
3393_1_insert_InvokeMethod(x1, x2, x3) → 3393_1_insert_InvokeMethod(x1, x2)

Filtered all free variables:

3710_1_main_InvokeMethod(x1, x2, x3) → 3710_1_main_InvokeMethod(x2, x3)
Cond_3710_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_3710_1_main_InvokeMethod(x1, x3, x4)
3985_1_main_InvokeMethod(x1, x2, x3) → 3985_1_main_InvokeMethod(x2, x3)
3655_1_main_InvokeMethod(x1, x2, x3) → 3655_1_main_InvokeMethod(x2, x3)
Cond_3655_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_3655_1_main_InvokeMethod(x1, x3, x4)
3215_0_insert_GT(x1, x2) → 3215_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:
3393_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3401_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3159_0_INSERT_LOAD(x3[0] > x0[0], java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
(2): 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3159_0_INSERT_LOAD1(x3[2] <= x0[2], java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])

(0) -> (1), if ((x3[0] > x0[0]* TRUE)∧(java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])) →* java.lang.Object(SearchTreeR.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(SearchTreeR.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(SearchTreeR.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(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))) →* java.lang.Object(SearchTreeR.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(SearchTreeR.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(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))))∧(x3[3]* x3[2]))

The set Q consists of the following terms:
3393_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3577_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 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0, java.lang.Object(x1), x2)), x3) → COND_3159_0_INSERT_LOAD(>(x3, x0), java.lang.Object(SearchTreeR.Tree(x0, java.lang.Object(x1), x2)), x3) the following chains were created:
• We consider the chain 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_3159_0_INSERT_LOAD(TRUE, java.lang.Object(SearchTreeR.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3159_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]) which results in the following constraint:

(1)    (>(x3[0], x0[0])=TRUEjava.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0]))=java.lang.Object(SearchTreeR.Tree(x0[1], java.lang.Object(x1[1]), x2[1]))∧x3[0]=x3[1]3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥NonInfC∧3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥COND_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])∧(UIncreasing(COND_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.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_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.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_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.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_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.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_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.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_3159_0_INSERT_LOAD(TRUE, java.lang.Object(SearchTreeR.Tree(x0, java.lang.Object(x1), x2)), x3) → 3159_0_INSERT_LOAD(java.lang.Object(x1), x3) the following chains were created:
• We consider the chain 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_3159_0_INSERT_LOAD(TRUE, java.lang.Object(SearchTreeR.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3159_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]), 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.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(3159_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(3159_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(3159_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(3159_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 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_3159_0_INSERT_LOAD(TRUE, java.lang.Object(SearchTreeR.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3159_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]), 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.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(3159_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(3159_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(3159_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(3159_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 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0, x1, java.lang.Object(x2))), x3) → COND_3159_0_INSERT_LOAD1(<=(x3, x0), java.lang.Object(SearchTreeR.Tree(x0, x1, java.lang.Object(x2))), x3) the following chains were created:
• We consider the chain 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_3159_0_INSERT_LOAD1(TRUE, java.lang.Object(SearchTreeR.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3159_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]) which results in the following constraint:

(19)    (<=(x3[2], x0[2])=TRUEjava.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2])))=java.lang.Object(SearchTreeR.Tree(x0[3], x1[3], java.lang.Object(x2[3])))∧x3[2]=x3[3]3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥NonInfC∧3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥COND_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])∧(UIncreasing(COND_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.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_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.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_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.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_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.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_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.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_3159_0_INSERT_LOAD1(TRUE, java.lang.Object(SearchTreeR.Tree(x0, x1, java.lang.Object(x2))), x3) → 3159_0_INSERT_LOAD(java.lang.Object(x2), x3) the following chains were created:
• We consider the chain 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_3159_0_INSERT_LOAD1(TRUE, java.lang.Object(SearchTreeR.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3159_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]), 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.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(3159_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(3159_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(3159_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(3159_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 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_3159_0_INSERT_LOAD1(TRUE, java.lang.Object(SearchTreeR.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3159_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]), 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.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(3159_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(3159_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(3159_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(3159_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.
• 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0, java.lang.Object(x1), x2)), x3) → COND_3159_0_INSERT_LOAD(>(x3, x0), java.lang.Object(SearchTreeR.Tree(x0, java.lang.Object(x1), x2)), x3)
• (0 ≥ 0 ⇒ (UIncreasing(COND_3159_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(SearchTreeR.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(3159_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(3159_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)

• 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0, x1, java.lang.Object(x2))), x3) → COND_3159_0_INSERT_LOAD1(<=(x3, x0), java.lang.Object(SearchTreeR.Tree(x0, x1, java.lang.Object(x2))), x3)
• (0 ≥ 0 ⇒ (UIncreasing(COND_3159_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(SearchTreeR.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(3159_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(3159_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(3393_1_insert_InvokeMethod(x1, x2)) = 0
POL(3466_0_insert_Return) = 0
POL(java.lang.Object(x1)) = [1] + x1
POL(3540_0_insert_Return) = 0
POL(3470_0_insert_Return) = 0
POL(3577_0_insert_Return) = 0
POL(3401_1_insert_InvokeMethod(x1, x2)) = 0
POL(3159_0_INSERT_LOAD(x1, x2)) = [-1] + x1
POL(SearchTreeR.Tree(x1, x2, x3)) = [2]x3 + x2 + x1
POL(COND_3159_0_INSERT_LOAD(x1, x2, x3)) = [-1] + x2
POL(>(x1, x2)) = 0
POL(COND_3159_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:
3393_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3401_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3159_0_INSERT_LOAD(x3[0] > x0[0], java.lang.Object(SearchTreeR.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
(2): 3159_0_INSERT_LOAD(java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3159_0_INSERT_LOAD1(x3[2] <= x0[2], java.lang.Object(SearchTreeR.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])

The set Q consists of the following terms:
3393_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3577_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:

Boolean, Integer

The ITRS R consists of the following rules:
3215_0_insert_GT(x0) → 3577_0_insert_Return
3215_0_insert_GT(x0) → 3401_1_insert_InvokeMethod(3215_0_insert_GT(x0), java.lang.Object(SearchTreeR.Tree))
3401_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3215_0_insert_GT(x0) → 3540_0_insert_Return
3215_0_insert_GT(x0) → 3393_1_insert_InvokeMethod(3215_0_insert_GT(x0), java.lang.Object(SearchTreeR.Tree))
3393_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 3655_1_MAIN_INVOKEMETHOD(x0[0], x1[0]) → COND_3655_1_MAIN_INVOKEMETHOD(x1[0] >= 0 && x0[0] > x1[0] + 1, x0[0], x1[0])
(1): COND_3655_1_MAIN_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 3655_1_MAIN_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:
3215_0_insert_GT(x0)
3401_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0))

### (11) 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 3655_1_MAIN_INVOKEMETHOD(x0, x1) → COND_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), x0, x1) the following chains were created:
• We consider the chain 3655_1_MAIN_INVOKEMETHOD(x0[0], x1[0]) → COND_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0]), COND_3655_1_MAIN_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 3655_1_MAIN_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]3655_1_MAIN_INVOKEMETHOD(x0[0], x1[0])≥NonInfC∧3655_1_MAIN_INVOKEMETHOD(x0[0], x1[0])≥COND_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])∧(UIncreasing(COND_3655_1_MAIN_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))=TRUE3655_1_MAIN_INVOKEMETHOD(x0[0], x1[0])≥NonInfC∧3655_1_MAIN_INVOKEMETHOD(x0[0], x1[0])≥COND_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])∧(UIncreasing(COND_3655_1_MAIN_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_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x1[0] + [(2)bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 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_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x1[0] + [(2)bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 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_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x1[0] + [(2)bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 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_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] + [(2)bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair COND_3655_1_MAIN_INVOKEMETHOD(TRUE, x0, x1) → 3655_1_MAIN_INVOKEMETHOD(x0, +(x1, 1)) the following chains were created:
• We consider the chain COND_3655_1_MAIN_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 3655_1_MAIN_INVOKEMETHOD(x0[1], +(x1[1], 1)) which results in the following constraint:

(7)    (COND_3655_1_MAIN_INVOKEMETHOD(TRUE, x0[1], x1[1])≥NonInfC∧COND_3655_1_MAIN_INVOKEMETHOD(TRUE, x0[1], x1[1])≥3655_1_MAIN_INVOKEMETHOD(x0[1], +(x1[1], 1))∧(UIncreasing(3655_1_MAIN_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥))

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

(8)    ((UIncreasing(3655_1_MAIN_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧[1 + (-1)bso_23] ≥ 0)

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

(9)    ((UIncreasing(3655_1_MAIN_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧[1 + (-1)bso_23] ≥ 0)

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

(10)    ((UIncreasing(3655_1_MAIN_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧[1 + (-1)bso_23] ≥ 0)

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

(11)    ((UIncreasing(3655_1_MAIN_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 3655_1_MAIN_INVOKEMETHOD(x0, x1) → COND_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), x0, x1)
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] + [(2)bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

• COND_3655_1_MAIN_INVOKEMETHOD(TRUE, x0, x1) → 3655_1_MAIN_INVOKEMETHOD(x0, +(x1, 1))
• ((UIncreasing(3655_1_MAIN_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 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(3215_0_insert_GT(x1)) = [-1]
POL(3577_0_insert_Return) = [-1]
POL(3401_1_insert_InvokeMethod(x1, x2)) = [-1]
POL(java.lang.Object(x1)) = [-1]
POL(SearchTreeR.Tree) = [-1]
POL(3466_0_insert_Return) = [-1]
POL(3470_0_insert_Return) = [-1]
POL(3540_0_insert_Return) = [-1]
POL(3393_1_insert_InvokeMethod(x1, x2)) = [-1]
POL(3655_1_MAIN_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2 + [2]x1
POL(COND_3655_1_MAIN_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_3655_1_MAIN_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 3655_1_MAIN_INVOKEMETHOD(x0[1], +(x1[1], 1))

The following pairs are in Pbound:

3655_1_MAIN_INVOKEMETHOD(x0[0], x1[0]) → COND_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])

The following pairs are in P:

3655_1_MAIN_INVOKEMETHOD(x0[0], x1[0]) → COND_3655_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])

There are no usable rules.

### (13) 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:
3215_0_insert_GT(x0) → 3577_0_insert_Return
3215_0_insert_GT(x0) → 3401_1_insert_InvokeMethod(3215_0_insert_GT(x0), java.lang.Object(SearchTreeR.Tree))
3401_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3215_0_insert_GT(x0) → 3540_0_insert_Return
3215_0_insert_GT(x0) → 3393_1_insert_InvokeMethod(3215_0_insert_GT(x0), java.lang.Object(SearchTreeR.Tree))
3393_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 3655_1_MAIN_INVOKEMETHOD(x0[0], x1[0]) → COND_3655_1_MAIN_INVOKEMETHOD(x1[0] >= 0 && x0[0] > x1[0] + 1, x0[0], x1[0])

The set Q consists of the following terms:
3215_0_insert_GT(x0)
3401_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0))

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (16) 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:
3215_0_insert_GT(x0) → 3577_0_insert_Return
3215_0_insert_GT(x0) → 3401_1_insert_InvokeMethod(3215_0_insert_GT(x0), java.lang.Object(SearchTreeR.Tree))
3401_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3401_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0)) → 3577_0_insert_Return
3215_0_insert_GT(x0) → 3540_0_insert_Return
3215_0_insert_GT(x0) → 3393_1_insert_InvokeMethod(3215_0_insert_GT(x0), java.lang.Object(SearchTreeR.Tree))
3393_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return
3393_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0)) → 3540_0_insert_Return

The integer pair graph contains the following rules and edges:
(1): COND_3655_1_MAIN_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 3655_1_MAIN_INVOKEMETHOD(x0[1], x1[1] + 1)

The set Q consists of the following terms:
3215_0_insert_GT(x0)
3401_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0))
3401_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3466_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3470_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3540_0_insert_Return, java.lang.Object(x0))
3393_1_insert_InvokeMethod(3577_0_insert_Return, java.lang.Object(x0))

### (17) IDependencyGraphProof (EQUIVALENT transformation)

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