### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: BubbleSortR
`class BubbleSortR {    public static void main(String[] args) {	Random.args = args;	int dim = Random.random()*Random.random();	int[] t = new int[dim] ;	for(int i = 0; i < t.length; i++)	    t[i] = Random.random();	sort(t);    }    public static void sort(int[] x) {	int n = x.length;	for (int pass=1; pass < n; pass++)  // count how many times	    // This next loop becomes shorter and shorter	    aux(0,n-pass,x);	//for (int i = 0; i < x.length; i++)	//    System.out.print(x[i]+ " ");		}    private static void aux(int min, int max, int[] x) {	if (min >= max) return;	int i = min;	if (x[i] > x[i+1]) {	    // exchange elements	    int temp = x[i]; x[i] = x[i+1];  x[i+1] = temp;	}	aux(min+1,max,x);    }}public class Random {  static String[] args;  static int index = 0;  public static int random() {      if (index >= args.length)	  return 0;      String string = args[index];      index++;      return string.length();  }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
BubbleSortR.main([Ljava/lang/String;)V: Graph of 414 nodes with 2 SCCs.

BubbleSortR.aux(II[I)V: Graph of 217 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 51 rules for P and 165 rules for R.

Combined rules. Obtained 4 rules for P and 9 rules for R.

Filtered ground terms:

Cond_4854_0_aux_LE(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_4854_0_aux_LE(x1, x3, x4, x5, x6, x7, x8)
4854_0_aux_LE(x1, x2, x3, x4, x5, x6, x7) → 4854_0_aux_LE(x2, x3, x4, x5, x6, x7)
Cond_4853_0_aux_LE(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_4853_0_aux_LE(x1, x3, x4, x5, x6, x7, x8)
4853_0_aux_LE(x1, x2, x3, x4, x5, x6, x7) → 4853_0_aux_LE(x2, x3, x4, x5, x6, x7)
Cond_4199_0_aux_Load1(x1, x2, x3, x4, x5, x6) → Cond_4199_0_aux_Load1(x1, x3, x4, x5, x6)
Cond_4199_0_aux_Load(x1, x2, x3, x4, x5, x6) → Cond_4199_0_aux_Load(x1, x3, x4, x5, x6)
5778_0_aux_InvokeMethod(x1, x2, x3, x4) → 5778_0_aux_InvokeMethod(x2, x3, x4)
6211_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7) → 6211_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7)
6115_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7, x8, x9) → 6115_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7, x8, x9)
6068_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) → 6068_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7, x8, x9, x10)
5775_0_aux_Return(x1) → 5775_0_aux_Return
5579_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7) → 5579_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7)
5338_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7, x8) → 5338_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7, x8)
5100_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7) → 5100_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7)
4295_0_aux_Return(x1, x2, x3, x4) → 4295_0_aux_Return(x2, x3, x4)

Filtered duplicate args:

Cond_4854_0_aux_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_4854_0_aux_LE(x1, x3, x4, x5, x6, x7)
4854_0_aux_LE(x1, x2, x3, x4, x5, x6) → 4854_0_aux_LE(x2, x3, x4, x5, x6)
Cond_4853_0_aux_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_4853_0_aux_LE(x1, x3, x4, x5, x6, x7)
4853_0_aux_LE(x1, x2, x3, x4, x5, x6) → 4853_0_aux_LE(x2, x3, x4, x5, x6)
6211_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6) → 6211_0_aux_ArrayAccess(x1, x2, x4, x5, x6)
6115_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7, x8) → 6115_0_aux_ArrayAccess(x2, x5, x6, x7, x8)
6068_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7, x8, x9) → 6068_0_aux_ArrayAccess(x2, x5, x7, x8, x9)
5579_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6) → 5579_0_aux_ArrayAccess(x2, x5, x6)
5338_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7) → 5338_0_aux_ArrayAccess(x2, x4, x5, x6, x7)
5100_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6) → 5100_0_aux_ArrayAccess(x2, x5, x6)

Filtered unneeded arguments:

Cond_4853_0_aux_LE(x1, x2, x3, x4, x5, x6) → Cond_4853_0_aux_LE(x1, x2, x3, x4)
Cond_4854_0_aux_LE(x1, x2, x3, x4, x5, x6) → Cond_4854_0_aux_LE(x1, x2, x3, x4)

Combined rules. Obtained 4 rules for P and 9 rules for R.

Finished conversion. Obtained 4 rules for P and 9 rules for R. System has predefined symbols.

Log for SCC 1:

Generated 18 rules for P and 242 rules for R.

Combined rules. Obtained 2 rules for P and 28 rules for R.

Filtered ground terms:

6187_1_sort_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → 6187_1_sort_InvokeMethod(x1, x2, x3, x4, x6, x7)
5775_0_aux_Return(x1) → 5775_0_aux_Return
4295_0_aux_Return(x1, x2, x3, x4) → 4295_0_aux_Return(x2, x3, x4)
5778_0_aux_InvokeMethod(x1, x2, x3, x4) → 5778_0_aux_InvokeMethod(x2, x3, x4)
6211_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7) → 6211_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7)
6115_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7, x8, x9) → 6115_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7, x8, x9)
6068_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) → 6068_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7, x8, x9, x10)
5579_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7) → 5579_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7)
5338_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7, x8) → 5338_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7, x8)
5100_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7) → 5100_0_aux_ArrayAccess(x2, x3, x4, x5, x6, x7)
4204_0_aux_LT(x1, x2, x3, x4, x5, x6) → 4204_0_aux_LT(x2, x3, x4, x5, x6)
Cond_4854_0_aux_LE2(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_4854_0_aux_LE2(x1, x3, x4, x5, x6, x7, x8)
4854_0_aux_LE(x1, x2, x3, x4, x5, x6, x7) → 4854_0_aux_LE(x2, x3, x4, x5, x6, x7)
Cond_4853_0_aux_LE(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_4853_0_aux_LE(x1, x3, x4, x5, x6, x7, x8)
4853_0_aux_LE(x1, x2, x3, x4, x5, x6, x7) → 4853_0_aux_LE(x2, x3, x4, x5, x6, x7)
Cond_4854_0_aux_LE1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_4854_0_aux_LE1(x1, x3, x4, x5, x6, x7, x8)
Cond_4854_0_aux_LE(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_4854_0_aux_LE(x1, x3, x4, x5, x6, x7, x8)
Cond_4204_0_aux_LT4(x1, x2, x3, x4, x5, x6, x7) → Cond_4204_0_aux_LT4(x1, x3, x4, x5, x6, x7)
Cond_4204_0_aux_LT3(x1, x2, x3, x4, x5, x6, x7) → Cond_4204_0_aux_LT3(x1, x3, x4, x5, x6, x7)
Cond_4204_0_aux_LT2(x1, x2, x3, x4, x5, x6, x7) → Cond_4204_0_aux_LT2(x1, x3, x4, x5, x6, x7)
Cond_4204_0_aux_LT1(x1, x2, x3, x4, x5, x6, x7) → Cond_4204_0_aux_LT1(x1, x3, x4, x5, x6, x7)
Cond_4204_0_aux_LT(x1, x2, x3, x4, x5, x6, x7) → Cond_4204_0_aux_LT(x1, x3, x4, x5, x6, x7)
6255_0_main_InvokeMethod(x1, x2) → 6255_0_main_InvokeMethod(x2)

Filtered duplicate args:

6187_1_sort_InvokeMethod(x1, x2, x3, x4, x5, x6) → 6187_1_sort_InvokeMethod(x1, x3, x4, x5, x6)
6211_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6) → 6211_0_aux_ArrayAccess(x1, x2, x4, x5, x6)
6115_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7, x8) → 6115_0_aux_ArrayAccess(x2, x5, x6, x7, x8)
6068_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7, x8, x9) → 6068_0_aux_ArrayAccess(x2, x5, x7, x8, x9)
5579_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6) → 5579_0_aux_ArrayAccess(x2, x5, x6)
5338_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6, x7) → 5338_0_aux_ArrayAccess(x2, x4, x5, x6, x7)
5100_0_aux_ArrayAccess(x1, x2, x3, x4, x5, x6) → 5100_0_aux_ArrayAccess(x2, x5, x6)
4204_0_aux_LT(x1, x2, x3, x4, x5) → 4204_0_aux_LT(x3, x4, x5)
Cond_4854_0_aux_LE2(x1, x2, x3, x4, x5, x6, x7) → Cond_4854_0_aux_LE2(x1, x3, x4, x5, x6, x7)
4854_0_aux_LE(x1, x2, x3, x4, x5, x6) → 4854_0_aux_LE(x2, x3, x4, x5, x6)
Cond_4853_0_aux_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_4853_0_aux_LE(x1, x3, x4, x5, x6, x7)
4853_0_aux_LE(x1, x2, x3, x4, x5, x6) → 4853_0_aux_LE(x2, x3, x4, x5, x6)
Cond_4854_0_aux_LE1(x1, x2, x3, x4, x5, x6, x7) → Cond_4854_0_aux_LE1(x1, x3, x4, x5, x6, x7)
Cond_4854_0_aux_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_4854_0_aux_LE(x1, x3, x4, x5, x6, x7)
Cond_4204_0_aux_LT4(x1, x2, x3, x4, x5, x6) → Cond_4204_0_aux_LT4(x1, x4, x5, x6)
Cond_4204_0_aux_LT3(x1, x2, x3, x4, x5, x6) → Cond_4204_0_aux_LT3(x1, x4, x5, x6)
Cond_4204_0_aux_LT2(x1, x2, x3, x4, x5, x6) → Cond_4204_0_aux_LT2(x1, x4, x5, x6)
Cond_4204_0_aux_LT1(x1, x2, x3, x4, x5, x6) → Cond_4204_0_aux_LT1(x1, x4, x5, x6)
Cond_4204_0_aux_LT(x1, x2, x3, x4, x5, x6) → Cond_4204_0_aux_LT(x1, x4, x5, x6)

Filtered unneeded arguments:

Cond_4204_0_aux_LT(x1, x2, x3, x4) → Cond_4204_0_aux_LT(x1, x2, x4)
Cond_4204_0_aux_LT1(x1, x2, x3, x4) → Cond_4204_0_aux_LT1(x1, x2)
Cond_4204_0_aux_LT4(x1, x2, x3, x4) → Cond_4204_0_aux_LT4(x1, x2)
Cond_4854_0_aux_LE(x1, x2, x3, x4, x5, x6) → Cond_4854_0_aux_LE(x1, x3)
Cond_4854_0_aux_LE1(x1, x2, x3, x4, x5, x6) → Cond_4854_0_aux_LE1(x1, x3)
Cond_4853_0_aux_LE(x1, x2, x3, x4, x5, x6) → Cond_4853_0_aux_LE(x1, x2, x3, x4)
Cond_4854_0_aux_LE2(x1, x2, x3, x4, x5, x6) → Cond_4854_0_aux_LE2(x1, x2, x3, x4)

Combined rules. Obtained 2 rules for P and 26 rules for R.

Finished conversion. Obtained 2 rules for P and 26 rules for R. System has predefined symbols.

Log for SCC 2:

Generated 45 rules for P and 438 rules for R.

Combined rules. Obtained 6 rules for P and 0 rules for R.

Filtered ground terms:

2680_0_random_LT(x1, x2, x3) → 2680_0_random_LT(x2, x3)
2777_0_random_IntArithmetic(x1, x2, x3, x4) → 2777_0_random_IntArithmetic(x2, x3)
2720_0_random_ArrayAccess(x1, x2, x3) → 2720_0_random_ArrayAccess(x2, x3)
2678_0_random_LT(x1, x2, x3) → 2678_0_random_LT(x2, x3)

Filtered duplicate args:

Cond_2680_1_main_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2680_1_main_InvokeMethod(x1, x2, x5, x6)
2680_1_main_InvokeMethod(x1, x2, x3, x4, x5) → 2680_1_main_InvokeMethod(x1, x4, x5)
Cond_2777_1_main_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2777_1_main_InvokeMethod(x1, x2, x5, x6)
2777_1_main_InvokeMethod(x1, x2, x3, x4, x5) → 2777_1_main_InvokeMethod(x1, x4, x5)
Cond_2720_1_main_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2720_1_main_InvokeMethod(x1, x2, x5, x6)
2720_1_main_InvokeMethod(x1, x2, x3, x4, x5) → 2720_1_main_InvokeMethod(x1, x4, x5)
Cond_2678_1_main_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2678_1_main_InvokeMethod(x1, x2, x5, x6)
2678_1_main_InvokeMethod(x1, x2, x3, x4, x5) → 2678_1_main_InvokeMethod(x1, x4, x5)

Filtered all free variables:

2678_1_main_InvokeMethod(x1, x2, x3) → 2678_1_main_InvokeMethod(x2, x3)
2680_1_main_InvokeMethod(x1, x2, x3) → 2680_1_main_InvokeMethod(x2, x3)
Cond_2678_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_2678_1_main_InvokeMethod(x1, x3, x4)
2720_1_main_InvokeMethod(x1, x2, x3) → 2720_1_main_InvokeMethod(x2, x3)
Cond_2720_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_2720_1_main_InvokeMethod(x1, x3, x4)
2777_1_main_InvokeMethod(x1, x2, x3) → 2777_1_main_InvokeMethod(x2, x3)
Cond_2777_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_2777_1_main_InvokeMethod(x1, x3, x4)
Cond_2680_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_2680_1_main_InvokeMethod(x1, x3, x4)

Combined rules. Obtained 1 rules for P and 0 rules for R.

Finished conversion. Obtained 1 rules for P and 0 rules for R. System has predefined symbols.

### (5) Obligation:

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

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x1, x0, x4, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6))) → 5778_0_aux_InvokeMethod(x4, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x1, x4, x0, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x1, x4, java.lang.Object(ARRAY(x2, x3)), x0, x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))

The integer pair graph contains the following rules and edges:
(0): 4199_0_AUX_LOAD(x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0]) → COND_4199_0_AUX_LOAD(x2[0] > x0[0] && x2[0] > x0[0] + 1 && x1[0] > x0[0] && x0[0] >= 0 && 1 <= x0[0] + 1, x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])
(1): COND_4199_0_AUX_LOAD(TRUE, x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1]) → 4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])
(2): 4199_0_AUX_LOAD(x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2]) → COND_4199_0_AUX_LOAD1(x2[2] > x0[2] && x2[2] > x0[2] + 1 && x1[2] > x0[2] && x0[2] >= 0 && 1 <= x0[2] + 1, x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])
(3): COND_4199_0_AUX_LOAD1(TRUE, x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3]) → 4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])
(4): 4853_0_AUX_LE(x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4]) → COND_4853_0_AUX_LE(x5[4] >= x4[4] && x0[4] >= 0, x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])
(5): COND_4853_0_AUX_LE(TRUE, x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), x0[5], x4[5], x5[5]) → 4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), x0[5] + 1)
(6): 4854_0_AUX_LE(x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6]) → COND_4853_0_AUX_LE(x5[6] < x4[6] && x2[6] > x0[6] && x2[6] > x0[6] + 1 && x0[6] >= 0 && 1 <= x0[6] + 1, x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])

(0) -> (1), if ((x2[0] > x0[0] && x2[0] > x0[0] + 1 && x1[0] > x0[0] && x0[0] >= 0 && 1 <= x0[0] + 1* TRUE)∧(x1[0]* x1[1])∧(java.lang.Object(ARRAY(x2[0], x3[0])) →* java.lang.Object(ARRAY(x2[1], x3[1])))∧(x0[0]* x0[1]))

(1) -> (4), if ((x1[1]* x1[4])∧(java.lang.Object(ARRAY(x2[1], x3[1])) →* java.lang.Object(ARRAY(x2[4], x3[4])))∧(x0[1]* x0[4])∧(x4[1]* x4[4])∧(x5[1]* x5[4]))

(2) -> (3), if ((x2[2] > x0[2] && x2[2] > x0[2] + 1 && x1[2] > x0[2] && x0[2] >= 0 && 1 <= x0[2] + 1* TRUE)∧(x1[2]* x1[3])∧(java.lang.Object(ARRAY(x2[2], x3[2])) →* java.lang.Object(ARRAY(x2[3], x3[3])))∧(x0[2]* x0[3]))

(3) -> (6), if ((x1[3]* x1[6])∧(java.lang.Object(ARRAY(x2[3], x3[3])) →* java.lang.Object(ARRAY(x2[6], x3[6])))∧(x0[3]* x0[6])∧(x4[3]* x4[6])∧(x5[3]* x5[6]))

(4) -> (5), if ((x5[4] >= x4[4] && x0[4] >= 0* TRUE)∧(x1[4]* x1[5])∧(java.lang.Object(ARRAY(x2[4], x3[4])) →* java.lang.Object(ARRAY(x2[5], x3[5])))∧(x0[4]* x0[5])∧(x4[4]* x4[5])∧(x5[4]* x5[5]))

(5) -> (0), if ((x1[5]* x1[0])∧(java.lang.Object(ARRAY(x2[5], x3[5])) →* java.lang.Object(ARRAY(x2[0], x3[0])))∧(x0[5] + 1* x0[0]))

(5) -> (2), if ((x1[5]* x1[2])∧(java.lang.Object(ARRAY(x2[5], x3[5])) →* java.lang.Object(ARRAY(x2[2], x3[2])))∧(x0[5] + 1* x0[2]))

(6) -> (5), if ((x5[6] < x4[6] && x2[6] > x0[6] && x2[6] > x0[6] + 1 && x0[6] >= 0 && 1 <= x0[6] + 1* TRUE)∧(x1[6]* x1[5])∧(java.lang.Object(ARRAY(x2[6], x3[6])) →* java.lang.Object(ARRAY(x2[5], x3[5])))∧(x0[6]* x0[5])∧(x4[6]* x4[5])∧(x5[6]* x5[5]))

The set Q consists of the following terms:
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x1, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x2, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x4, x0, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7)))

### (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 4199_0_AUX_LOAD(x1, java.lang.Object(ARRAY(x2, x3)), x0) → COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2, x0), >(x2, +(x0, 1))), >(x1, x0)), >=(x0, 0)), <=(1, +(x0, 1))), x1, java.lang.Object(ARRAY(x2, x3)), x0) the following chains were created:
• We consider the chain 4199_0_AUX_LOAD(x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0]) → COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0]), COND_4199_0_AUX_LOAD(TRUE, x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1]) → 4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1]) which results in the following constraint:

(1)    (&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1)))=TRUEx1[0]=x1[1]java.lang.Object(ARRAY(x2[0], x3[0]))=java.lang.Object(ARRAY(x2[1], x3[1]))∧x0[0]=x0[1]4199_0_AUX_LOAD(x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])≥NonInfC∧4199_0_AUX_LOAD(x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])≥COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])∧(UIncreasing(COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])), ≥))

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

(2)    (<=(1, +(x0[0], 1))=TRUE>=(x0[0], 0)=TRUE>(x1[0], x0[0])=TRUE>(x2[0], x0[0])=TRUE>(x2[0], +(x0[0], 1))=TRUE4199_0_AUX_LOAD(x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])≥NonInfC∧4199_0_AUX_LOAD(x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])≥COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])∧(UIncreasing(COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])), ≥))

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

(3)    (x0[0] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] + [-2] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])), ≥)∧[(-1)Bound*bni_32] + [(-1)bni_32]x0[0] + [bni_32]x2[0] + [(2)bni_32]x1[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(4)    (x0[0] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] + [-2] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])), ≥)∧[(-1)Bound*bni_32] + [(-1)bni_32]x0[0] + [bni_32]x2[0] + [(2)bni_32]x1[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(5)    (x0[0] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] + [-2] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])), ≥)∧[(-1)Bound*bni_32] + [(-1)bni_32]x0[0] + [bni_32]x2[0] + [(2)bni_32]x1[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(6)    (x0[0] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] + [-2] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])), ≥)∧0 = 0∧[(-1)Bound*bni_32] + [(-1)bni_32]x0[0] + [bni_32]x2[0] + [(2)bni_32]x1[0] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(7)    (x0[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0∧x2[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] + [-2] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])), ≥)∧0 = 0∧[(-1)Bound*bni_32 + (2)bni_32] + [bni_32]x0[0] + [bni_32]x2[0] + [(2)bni_32]x1[0] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

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

(8)    (x0[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0∧x2[0] ≥ 0∧[-1] + x2[0] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])), ≥)∧0 = 0∧[(-1)Bound*bni_32 + (3)bni_32] + [(2)bni_32]x0[0] + [bni_32]x2[0] + [(2)bni_32]x1[0] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

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

(9)    (x0[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])), ≥)∧0 = 0∧[(-1)Bound*bni_32 + (4)bni_32] + [(2)bni_32]x0[0] + [bni_32]x2[0] + [(2)bni_32]x1[0] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

For Pair COND_4199_0_AUX_LOAD(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0) → 4853_0_AUX_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) the following chains were created:
• We consider the chain COND_4199_0_AUX_LOAD(TRUE, x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1]) → 4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1]) which results in the following constraint:

(10)    (COND_4199_0_AUX_LOAD(TRUE, x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1])≥NonInfC∧COND_4199_0_AUX_LOAD(TRUE, x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1])≥4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])∧(UIncreasing(4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])), ≥))

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

(11)    ((UIncreasing(4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])), ≥)∧[(-1)bso_35] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(12)    ((UIncreasing(4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])), ≥)∧[(-1)bso_35] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(13)    ((UIncreasing(4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])), ≥)∧[(-1)bso_35] ≥ 0)

We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(14)    ((UIncreasing(4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)

For Pair 4199_0_AUX_LOAD(x1, java.lang.Object(ARRAY(x2, x3)), x0) → COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2, x0), >(x2, +(x0, 1))), >(x1, x0)), >=(x0, 0)), <=(1, +(x0, 1))), x1, java.lang.Object(ARRAY(x2, x3)), x0) the following chains were created:
• We consider the chain 4199_0_AUX_LOAD(x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2]) → COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2]), COND_4199_0_AUX_LOAD1(TRUE, x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3]) → 4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3]) which results in the following constraint:

(15)    (&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1)))=TRUEx1[2]=x1[3]java.lang.Object(ARRAY(x2[2], x3[2]))=java.lang.Object(ARRAY(x2[3], x3[3]))∧x0[2]=x0[3]4199_0_AUX_LOAD(x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])≥NonInfC∧4199_0_AUX_LOAD(x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])≥COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])∧(UIncreasing(COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])), ≥))

We simplified constraint (15) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(16)    (<=(1, +(x0[2], 1))=TRUE>=(x0[2], 0)=TRUE>(x1[2], x0[2])=TRUE>(x2[2], x0[2])=TRUE>(x2[2], +(x0[2], 1))=TRUE4199_0_AUX_LOAD(x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])≥NonInfC∧4199_0_AUX_LOAD(x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])≥COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])∧(UIncreasing(COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])), ≥))

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

(17)    (x0[2] ≥ 0∧x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x2[2] + [-2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])), ≥)∧[(-1)Bound*bni_36] + [(-1)bni_36]x0[2] + [bni_36]x2[2] + [(2)bni_36]x1[2] ≥ 0∧[(-1)bso_37] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(18)    (x0[2] ≥ 0∧x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x2[2] + [-2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])), ≥)∧[(-1)Bound*bni_36] + [(-1)bni_36]x0[2] + [bni_36]x2[2] + [(2)bni_36]x1[2] ≥ 0∧[(-1)bso_37] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(19)    (x0[2] ≥ 0∧x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x2[2] + [-2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])), ≥)∧[(-1)Bound*bni_36] + [(-1)bni_36]x0[2] + [bni_36]x2[2] + [(2)bni_36]x1[2] ≥ 0∧[(-1)bso_37] ≥ 0)

We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(20)    (x0[2] ≥ 0∧x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x2[2] + [-2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [(-1)bni_36]x0[2] + [bni_36]x2[2] + [(2)bni_36]x1[2] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

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

(21)    (x0[2] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x2[2] + [-2] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_36 + (2)bni_36] + [bni_36]x0[2] + [bni_36]x2[2] + [(2)bni_36]x1[2] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

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

(22)    (x0[2] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0∧x2[2] ≥ 0∧[-1] + x2[2] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_36 + (3)bni_36] + [(2)bni_36]x0[2] + [bni_36]x2[2] + [(2)bni_36]x1[2] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

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

(23)    (x0[2] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0∧[1] + x2[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_36 + (4)bni_36] + [(2)bni_36]x0[2] + [bni_36]x2[2] + [(2)bni_36]x1[2] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

For Pair COND_4199_0_AUX_LOAD1(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0) → 4854_0_AUX_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) the following chains were created:
• We consider the chain COND_4199_0_AUX_LOAD1(TRUE, x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3]) → 4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3]) which results in the following constraint:

(24)    (COND_4199_0_AUX_LOAD1(TRUE, x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3])≥NonInfC∧COND_4199_0_AUX_LOAD1(TRUE, x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3])≥4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])∧(UIncreasing(4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])), ≥))

We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(25)    ((UIncreasing(4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])), ≥)∧[(-1)bso_39] ≥ 0)

We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(26)    ((UIncreasing(4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])), ≥)∧[(-1)bso_39] ≥ 0)

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

(27)    ((UIncreasing(4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])), ≥)∧[(-1)bso_39] ≥ 0)

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

(28)    ((UIncreasing(4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_39] ≥ 0)

For Pair 4853_0_AUX_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → COND_4853_0_AUX_LE(&&(>=(x5, x4), >=(x0, 0)), x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) the following chains were created:
• We consider the chain 4853_0_AUX_LE(x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4]) → COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4]), COND_4853_0_AUX_LE(TRUE, x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), x0[5], x4[5], x5[5]) → 4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1)) which results in the following constraint:

(29)    (&&(>=(x5[4], x4[4]), >=(x0[4], 0))=TRUEx1[4]=x1[5]java.lang.Object(ARRAY(x2[4], x3[4]))=java.lang.Object(ARRAY(x2[5], x3[5]))∧x0[4]=x0[5]x4[4]=x4[5]x5[4]=x5[5]4853_0_AUX_LE(x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])≥NonInfC∧4853_0_AUX_LE(x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])≥COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])∧(UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥))

We simplified constraint (29) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(30)    (>=(x5[4], x4[4])=TRUE>=(x0[4], 0)=TRUE4853_0_AUX_LE(x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])≥NonInfC∧4853_0_AUX_LE(x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])≥COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])∧(UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(31)    (x5[4] + [-1]x4[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥)∧[(-1)Bound*bni_40] + [(-1)bni_40]x0[4] + [bni_40]x2[4] + [(2)bni_40]x1[4] ≥ 0∧[(-1)bso_41] ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(32)    (x5[4] + [-1]x4[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥)∧[(-1)Bound*bni_40] + [(-1)bni_40]x0[4] + [bni_40]x2[4] + [(2)bni_40]x1[4] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(33)    (x5[4] + [-1]x4[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥)∧[(-1)Bound*bni_40] + [(-1)bni_40]x0[4] + [bni_40]x2[4] + [(2)bni_40]x1[4] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(34)    (x5[4] + [-1]x4[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥)∧0 = 0∧[bni_40] = 0∧[(2)bni_40] = 0∧[(-1)Bound*bni_40] + [(-1)bni_40]x0[4] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_41] ≥ 0)

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

(35)    (x5[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥)∧0 = 0∧[bni_40] = 0∧[(2)bni_40] = 0∧[(-1)Bound*bni_40] + [(-1)bni_40]x0[4] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_41] ≥ 0)

We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(36)    (x5[4] ≥ 0∧x0[4] ≥ 0∧x4[4] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥)∧0 = 0∧[bni_40] = 0∧[(2)bni_40] = 0∧[(-1)Bound*bni_40] + [(-1)bni_40]x0[4] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_41] ≥ 0)

(37)    (x5[4] ≥ 0∧x0[4] ≥ 0∧x4[4] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥)∧0 = 0∧[bni_40] = 0∧[(2)bni_40] = 0∧[(-1)Bound*bni_40] + [(-1)bni_40]x0[4] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_41] ≥ 0)

For Pair COND_4853_0_AUX_LE(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 4199_0_AUX_LOAD(x1, java.lang.Object(ARRAY(x2, x3)), +(x0, 1)) the following chains were created:
• We consider the chain COND_4853_0_AUX_LE(TRUE, x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), x0[5], x4[5], x5[5]) → 4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1)) which results in the following constraint:

(38)    (COND_4853_0_AUX_LE(TRUE, x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), x0[5], x4[5], x5[5])≥NonInfC∧COND_4853_0_AUX_LE(TRUE, x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), x0[5], x4[5], x5[5])≥4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1))∧(UIncreasing(4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1))), ≥))

We simplified constraint (38) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(39)    ((UIncreasing(4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1))), ≥)∧[1 + (-1)bso_43] ≥ 0)

We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(40)    ((UIncreasing(4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1))), ≥)∧[1 + (-1)bso_43] ≥ 0)

We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(41)    ((UIncreasing(4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1))), ≥)∧[1 + (-1)bso_43] ≥ 0)

We simplified constraint (41) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(42)    ((UIncreasing(4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_43] ≥ 0)

For Pair 4854_0_AUX_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5, x4), >(x2, x0)), >(x2, +(x0, 1))), >=(x0, 0)), <=(1, +(x0, 1))), x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) the following chains were created:
• We consider the chain 4854_0_AUX_LE(x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6]) → COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6]), COND_4853_0_AUX_LE(TRUE, x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), x0[5], x4[5], x5[5]) → 4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1)) which results in the following constraint:

(43)    (&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1)))=TRUEx1[6]=x1[5]java.lang.Object(ARRAY(x2[6], x3[6]))=java.lang.Object(ARRAY(x2[5], x3[5]))∧x0[6]=x0[5]x4[6]=x4[5]x5[6]=x5[5]4854_0_AUX_LE(x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])≥NonInfC∧4854_0_AUX_LE(x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])≥COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])∧(UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥))

We simplified constraint (43) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(44)    (<=(1, +(x0[6], 1))=TRUE>=(x0[6], 0)=TRUE>(x2[6], +(x0[6], 1))=TRUE<(x5[6], x4[6])=TRUE>(x2[6], x0[6])=TRUE4854_0_AUX_LE(x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])≥NonInfC∧4854_0_AUX_LE(x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])≥COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])∧(UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥))

We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(45)    (x0[6] ≥ 0∧x0[6] ≥ 0∧x2[6] + [-2] + [-1]x0[6] ≥ 0∧x4[6] + [-1] + [-1]x5[6] ≥ 0∧x2[6] + [-1] + [-1]x0[6] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥)∧[(-1)Bound*bni_44] + [(-1)bni_44]x0[6] + [bni_44]x2[6] + [(2)bni_44]x1[6] ≥ 0∧[(-1)bso_45] ≥ 0)

We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(46)    (x0[6] ≥ 0∧x0[6] ≥ 0∧x2[6] + [-2] + [-1]x0[6] ≥ 0∧x4[6] + [-1] + [-1]x5[6] ≥ 0∧x2[6] + [-1] + [-1]x0[6] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥)∧[(-1)Bound*bni_44] + [(-1)bni_44]x0[6] + [bni_44]x2[6] + [(2)bni_44]x1[6] ≥ 0∧[(-1)bso_45] ≥ 0)

We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(47)    (x0[6] ≥ 0∧x0[6] ≥ 0∧x2[6] + [-2] + [-1]x0[6] ≥ 0∧x4[6] + [-1] + [-1]x5[6] ≥ 0∧x2[6] + [-1] + [-1]x0[6] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥)∧[(-1)Bound*bni_44] + [(-1)bni_44]x0[6] + [bni_44]x2[6] + [(2)bni_44]x1[6] ≥ 0∧[(-1)bso_45] ≥ 0)

We simplified constraint (47) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(48)    (x0[6] ≥ 0∧x0[6] ≥ 0∧x2[6] + [-2] + [-1]x0[6] ≥ 0∧x4[6] + [-1] + [-1]x5[6] ≥ 0∧x2[6] + [-1] + [-1]x0[6] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥)∧0 = 0∧[(2)bni_44] = 0∧[(-1)Bound*bni_44] + [(-1)bni_44]x0[6] + [bni_44]x2[6] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_45] ≥ 0)

We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(49)    (x0[6] ≥ 0∧x0[6] ≥ 0∧x2[6] ≥ 0∧x4[6] + [-1] + [-1]x5[6] ≥ 0∧[1] + x2[6] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥)∧0 = 0∧[(2)bni_44] = 0∧[(-1)Bound*bni_44 + (2)bni_44] + [bni_44]x2[6] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_45] ≥ 0)

We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(50)    (x0[6] ≥ 0∧x0[6] ≥ 0∧x2[6] ≥ 0∧x4[6] ≥ 0∧[1] + x2[6] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥)∧0 = 0∧[(2)bni_44] = 0∧[(-1)Bound*bni_44 + (2)bni_44] + [bni_44]x2[6] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_45] ≥ 0)

We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(51)    (x0[6] ≥ 0∧x0[6] ≥ 0∧x2[6] ≥ 0∧x4[6] ≥ 0∧[1] + x2[6] ≥ 0∧x5[6] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥)∧0 = 0∧[(2)bni_44] = 0∧[(-1)Bound*bni_44 + (2)bni_44] + [bni_44]x2[6] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_45] ≥ 0)

(52)    (x0[6] ≥ 0∧x0[6] ≥ 0∧x2[6] ≥ 0∧x4[6] ≥ 0∧[1] + x2[6] ≥ 0∧x5[6] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥)∧0 = 0∧[(2)bni_44] = 0∧[(-1)Bound*bni_44 + (2)bni_44] + [bni_44]x2[6] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_45] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 4199_0_AUX_LOAD(x1, java.lang.Object(ARRAY(x2, x3)), x0) → COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2, x0), >(x2, +(x0, 1))), >(x1, x0)), >=(x0, 0)), <=(1, +(x0, 1))), x1, java.lang.Object(ARRAY(x2, x3)), x0)
• (x0[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])), ≥)∧0 = 0∧[(-1)Bound*bni_32 + (4)bni_32] + [(2)bni_32]x0[0] + [bni_32]x2[0] + [(2)bni_32]x1[0] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

• COND_4199_0_AUX_LOAD(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0) → 4853_0_AUX_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
• ((UIncreasing(4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)

• 4199_0_AUX_LOAD(x1, java.lang.Object(ARRAY(x2, x3)), x0) → COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2, x0), >(x2, +(x0, 1))), >(x1, x0)), >=(x0, 0)), <=(1, +(x0, 1))), x1, java.lang.Object(ARRAY(x2, x3)), x0)
• (x0[2] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0∧[1] + x2[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_36 + (4)bni_36] + [(2)bni_36]x0[2] + [bni_36]x2[2] + [(2)bni_36]x1[2] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

• COND_4199_0_AUX_LOAD1(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0) → 4854_0_AUX_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
• ((UIncreasing(4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_39] ≥ 0)

• 4853_0_AUX_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → COND_4853_0_AUX_LE(&&(>=(x5, x4), >=(x0, 0)), x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
• (x5[4] ≥ 0∧x0[4] ≥ 0∧x4[4] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥)∧0 = 0∧[bni_40] = 0∧[(2)bni_40] = 0∧[(-1)Bound*bni_40] + [(-1)bni_40]x0[4] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_41] ≥ 0)
• (x5[4] ≥ 0∧x0[4] ≥ 0∧x4[4] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])), ≥)∧0 = 0∧[bni_40] = 0∧[(2)bni_40] = 0∧[(-1)Bound*bni_40] + [(-1)bni_40]x0[4] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_41] ≥ 0)

• COND_4853_0_AUX_LE(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 4199_0_AUX_LOAD(x1, java.lang.Object(ARRAY(x2, x3)), +(x0, 1))
• ((UIncreasing(4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_43] ≥ 0)

• 4854_0_AUX_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5, x4), >(x2, x0)), >(x2, +(x0, 1))), >=(x0, 0)), <=(1, +(x0, 1))), x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
• (x0[6] ≥ 0∧x0[6] ≥ 0∧x2[6] ≥ 0∧x4[6] ≥ 0∧[1] + x2[6] ≥ 0∧x5[6] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥)∧0 = 0∧[(2)bni_44] = 0∧[(-1)Bound*bni_44 + (2)bni_44] + [bni_44]x2[6] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_45] ≥ 0)
• (x0[6] ≥ 0∧x0[6] ≥ 0∧x2[6] ≥ 0∧x4[6] ≥ 0∧[1] + x2[6] ≥ 0∧x5[6] ≥ 0 ⇒ (UIncreasing(COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])), ≥)∧0 = 0∧[(2)bni_44] = 0∧[(-1)Bound*bni_44 + (2)bni_44] + [bni_44]x2[6] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_45] ≥ 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(5704_1_aux_InvokeMethod(x1, x2, x3, x4)) = [-1]
POL(4295_0_aux_Return(x1, x2, x3)) = [-1]
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1, x2)) = [-1] + [-1]x1
POL(5775_0_aux_Return) = [-1]
POL(5100_0_aux_ArrayAccess(x1, x2, x3)) = [-1]
POL(5778_0_aux_InvokeMethod(x1, x2, x3)) = [-1]
POL(5338_0_aux_ArrayAccess(x1, x2, x3, x4, x5)) = [-1]
POL(5579_0_aux_ArrayAccess(x1, x2, x3)) = [-1]
POL(6068_0_aux_ArrayAccess(x1, x2, x3, x4, x5)) = [-1]
POL(6115_0_aux_ArrayAccess(x1, x2, x3, x4, x5)) = [-1]
POL(6211_0_aux_ArrayAccess(x1, x2, x3, x4, x5)) = [-1]
POL(4199_0_AUX_LOAD(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [2]x1
POL(COND_4199_0_AUX_LOAD(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [2]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(<=(x1, x2)) = [-1]
POL(4853_0_AUX_LE(x1, x2, x3, x4, x5)) = [-1] + [-1]x3 + [-1]x2 + [2]x1
POL(COND_4199_0_AUX_LOAD1(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [2]x2
POL(4854_0_AUX_LE(x1, x2, x3, x4, x5)) = [-1] + [-1]x3 + [-1]x2 + [2]x1
POL(COND_4853_0_AUX_LE(x1, x2, x3, x4, x5, x6)) = [-1] + [-1]x4 + [-1]x3 + [2]x2
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

COND_4853_0_AUX_LE(TRUE, x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), x0[5], x4[5], x5[5]) → 4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), +(x0[5], 1))

The following pairs are in Pbound:

4199_0_AUX_LOAD(x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0]) → COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])
4199_0_AUX_LOAD(x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2]) → COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])

The following pairs are in P:

4199_0_AUX_LOAD(x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0]) → COND_4199_0_AUX_LOAD(&&(&&(&&(&&(>(x2[0], x0[0]), >(x2[0], +(x0[0], 1))), >(x1[0], x0[0])), >=(x0[0], 0)), <=(1, +(x0[0], 1))), x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])
COND_4199_0_AUX_LOAD(TRUE, x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1]) → 4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])
4199_0_AUX_LOAD(x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2]) → COND_4199_0_AUX_LOAD1(&&(&&(&&(&&(>(x2[2], x0[2]), >(x2[2], +(x0[2], 1))), >(x1[2], x0[2])), >=(x0[2], 0)), <=(1, +(x0[2], 1))), x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])
COND_4199_0_AUX_LOAD1(TRUE, x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3]) → 4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])
4853_0_AUX_LE(x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4]) → COND_4853_0_AUX_LE(&&(>=(x5[4], x4[4]), >=(x0[4], 0)), x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])
4854_0_AUX_LE(x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6]) → COND_4853_0_AUX_LE(&&(&&(&&(&&(<(x5[6], x4[6]), >(x2[6], x0[6])), >(x2[6], +(x0[6], 1))), >=(x0[6], 0)), <=(1, +(x0[6], 1))), x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])

There are no usable rules.

### (8) 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:
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x1, x0, x4, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6))) → 5778_0_aux_InvokeMethod(x4, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x1, x4, x0, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x1, x4, java.lang.Object(ARRAY(x2, x3)), x0, x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))

The integer pair graph contains the following rules and edges:
(0): 4199_0_AUX_LOAD(x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0]) → COND_4199_0_AUX_LOAD(x2[0] > x0[0] && x2[0] > x0[0] + 1 && x1[0] > x0[0] && x0[0] >= 0 && 1 <= x0[0] + 1, x1[0], java.lang.Object(ARRAY(x2[0], x3[0])), x0[0])
(1): COND_4199_0_AUX_LOAD(TRUE, x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1]) → 4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])
(2): 4199_0_AUX_LOAD(x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2]) → COND_4199_0_AUX_LOAD1(x2[2] > x0[2] && x2[2] > x0[2] + 1 && x1[2] > x0[2] && x0[2] >= 0 && 1 <= x0[2] + 1, x1[2], java.lang.Object(ARRAY(x2[2], x3[2])), x0[2])
(3): COND_4199_0_AUX_LOAD1(TRUE, x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3]) → 4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])
(4): 4853_0_AUX_LE(x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4]) → COND_4853_0_AUX_LE(x5[4] >= x4[4] && x0[4] >= 0, x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])
(6): 4854_0_AUX_LE(x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6]) → COND_4853_0_AUX_LE(x5[6] < x4[6] && x2[6] > x0[6] && x2[6] > x0[6] + 1 && x0[6] >= 0 && 1 <= x0[6] + 1, x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])

(0) -> (1), if ((x2[0] > x0[0] && x2[0] > x0[0] + 1 && x1[0] > x0[0] && x0[0] >= 0 && 1 <= x0[0] + 1* TRUE)∧(x1[0]* x1[1])∧(java.lang.Object(ARRAY(x2[0], x3[0])) →* java.lang.Object(ARRAY(x2[1], x3[1])))∧(x0[0]* x0[1]))

(2) -> (3), if ((x2[2] > x0[2] && x2[2] > x0[2] + 1 && x1[2] > x0[2] && x0[2] >= 0 && 1 <= x0[2] + 1* TRUE)∧(x1[2]* x1[3])∧(java.lang.Object(ARRAY(x2[2], x3[2])) →* java.lang.Object(ARRAY(x2[3], x3[3])))∧(x0[2]* x0[3]))

(1) -> (4), if ((x1[1]* x1[4])∧(java.lang.Object(ARRAY(x2[1], x3[1])) →* java.lang.Object(ARRAY(x2[4], x3[4])))∧(x0[1]* x0[4])∧(x4[1]* x4[4])∧(x5[1]* x5[4]))

(3) -> (6), if ((x1[3]* x1[6])∧(java.lang.Object(ARRAY(x2[3], x3[3])) →* java.lang.Object(ARRAY(x2[6], x3[6])))∧(x0[3]* x0[6])∧(x4[3]* x4[6])∧(x5[3]* x5[6]))

The set Q consists of the following terms:
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x1, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x2, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x4, x0, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7)))

### (9) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes.

### (11) 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:
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x1, x0, x4, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6))) → 5778_0_aux_InvokeMethod(x4, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x1, x4, x0, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x1, x4, java.lang.Object(ARRAY(x2, x3)), x0, x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))

The integer pair graph contains the following rules and edges:
(1): COND_4199_0_AUX_LOAD(TRUE, x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1]) → 4853_0_AUX_LE(x1[1], java.lang.Object(ARRAY(x2[1], x3[1])), x0[1], x4[1], x5[1])
(3): COND_4199_0_AUX_LOAD1(TRUE, x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3]) → 4854_0_AUX_LE(x1[3], java.lang.Object(ARRAY(x2[3], x3[3])), x0[3], x4[3], x5[3])
(4): 4853_0_AUX_LE(x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4]) → COND_4853_0_AUX_LE(x5[4] >= x4[4] && x0[4] >= 0, x1[4], java.lang.Object(ARRAY(x2[4], x3[4])), x0[4], x4[4], x5[4])
(5): COND_4853_0_AUX_LE(TRUE, x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), x0[5], x4[5], x5[5]) → 4199_0_AUX_LOAD(x1[5], java.lang.Object(ARRAY(x2[5], x3[5])), x0[5] + 1)
(6): 4854_0_AUX_LE(x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6]) → COND_4853_0_AUX_LE(x5[6] < x4[6] && x2[6] > x0[6] && x2[6] > x0[6] + 1 && x0[6] >= 0 && 1 <= x0[6] + 1, x1[6], java.lang.Object(ARRAY(x2[6], x3[6])), x0[6], x4[6], x5[6])

(1) -> (4), if ((x1[1]* x1[4])∧(java.lang.Object(ARRAY(x2[1], x3[1])) →* java.lang.Object(ARRAY(x2[4], x3[4])))∧(x0[1]* x0[4])∧(x4[1]* x4[4])∧(x5[1]* x5[4]))

(4) -> (5), if ((x5[4] >= x4[4] && x0[4] >= 0* TRUE)∧(x1[4]* x1[5])∧(java.lang.Object(ARRAY(x2[4], x3[4])) →* java.lang.Object(ARRAY(x2[5], x3[5])))∧(x0[4]* x0[5])∧(x4[4]* x4[5])∧(x5[4]* x5[5]))

(6) -> (5), if ((x5[6] < x4[6] && x2[6] > x0[6] && x2[6] > x0[6] + 1 && x0[6] >= 0 && 1 <= x0[6] + 1* TRUE)∧(x1[6]* x1[5])∧(java.lang.Object(ARRAY(x2[6], x3[6])) →* java.lang.Object(ARRAY(x2[5], x3[5])))∧(x0[6]* x0[5])∧(x4[6]* x4[5])∧(x5[6]* x5[5]))

(3) -> (6), if ((x1[3]* x1[6])∧(java.lang.Object(ARRAY(x2[3], x3[3])) →* java.lang.Object(ARRAY(x2[6], x3[6])))∧(x0[3]* x0[6])∧(x4[3]* x4[6])∧(x5[3]* x5[6]))

The set Q consists of the following terms:
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x1, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x2, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x4, x0, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7)))

### (12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.

### (14) 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, Boolean

The ITRS R consists of the following rules:
6187_0_aux_Load(x1, java.lang.Object(ARRAY(x2, x3))) → 4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), 0, x1)
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), 0), x6, x7, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5338_0_aux_ArrayAccess(x1, 0, x5, java.lang.Object(ARRAY(x2, x3)), x6), x7, x8, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), 0), x6, x7, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x6, x7, x1, java.lang.Object(ARRAY(x4, x5))), java.lang.Object(ARRAY(x4, x5))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6068_0_aux_ArrayAccess(x1, x5, 0, java.lang.Object(ARRAY(x2, x3)), x7), x8, x9, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6115_0_aux_ArrayAccess(x1, x5, java.lang.Object(ARRAY(x2, x3)), 0, x7), x8, x9, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6211_0_aux_ArrayAccess(0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x8, x9, x1, java.lang.Object(ARRAY(x6, x7))), java.lang.Object(ARRAY(x6, x7))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT(x1 <= x0, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3)))
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT1(x2 <= x0 && x1 > x0, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT1(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0)
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT2(x2 > x0 && x2 > x0 + 1 && x1 > x0 && x0 >= 0 && 1 <= x0 + 1, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT2(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 4853_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT3(x2 > x0 && x2 > x0 + 1 && x1 > x0 && x0 >= 0 && 1 <= x0 + 1, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT3(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT4(x2 > x0 && x2 <= x0 + 1 && x1 > x0 && x0 >= 0, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT4(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 5338_0_aux_ArrayAccess(x1, x0, x4, java.lang.Object(ARRAY(x2, x3)), x0 + 1)
4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4854_0_aux_LE(x5 < x4 && x2 <= x0, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
Cond_4854_0_aux_LE(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0)
4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4854_0_aux_LE1(x5 < x4 && x2 > x0 && x2 <= x0 + 1 && x0 >= 0, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
Cond_4854_0_aux_LE1(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 6068_0_aux_ArrayAccess(x1, x6, x0, java.lang.Object(ARRAY(x2, x3)), x0 + 1)
4853_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4853_0_aux_LE(x5 >= x4 && x0 >= 0, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
Cond_4853_0_aux_LE(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 5704_1_aux_InvokeMethod(4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0 + 1, x1), x0 + 1, x1, java.lang.Object(ARRAY(x2, x3)))
4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4853_0_aux_LE(x5 < x4 && x2 > x0 && x2 > x0 + 1 && x0 >= 0 && 1 <= x0 + 1, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x1, x0, x4, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6))) → 5778_0_aux_InvokeMethod(x4, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x1, x4, x0, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x1, x4, java.lang.Object(ARRAY(x2, x3)), x0, x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))

The integer pair graph contains the following rules and edges:
(0): 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0]))) → COND_6187_2_MAIN_INVOKEMETHOD(x4[0] > x5[0] + 1, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))
(1): COND_6187_2_MAIN_INVOKEMETHOD(TRUE, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1], x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1]))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(x4[1] - x5[1] + 1, java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1] + 1, x4[1] - x5[1] + 1, java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))
(2): 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2]))) → COND_6187_2_MAIN_INVOKEMETHOD1(x2[2] > x3[2] + 1, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))
(3): COND_6187_2_MAIN_INVOKEMETHOD1(TRUE, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[3], x3[3], x5[3], java.lang.Object(ARRAY(x0[3], x1[3]))), java.lang.Object(ARRAY(x0[3], x1[3]))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(x2[3] - x3[3] + 1, java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], x3[3] + 1, x2[3] - x3[3] + 1, java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3])))

(0) -> (1), if ((x4[0] > x5[0] + 1* TRUE)∧(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))) →* 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1], x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))))∧(java.lang.Object(ARRAY(x2[0], x3[0])) →* java.lang.Object(ARRAY(x2[1], x3[1]))))

(1) -> (0), if ((6187_1_sort_InvokeMethod(6187_0_aux_Load(x4[1] - x5[1] + 1, java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1] + 1, x4[1] - x5[1] + 1, java.lang.Object(ARRAY(x2[1], x3[1]))) →* 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))))∧(java.lang.Object(ARRAY(x2[1], x3[1])) →* java.lang.Object(ARRAY(x2[0], x3[0]))))

(1) -> (2), if ((6187_1_sort_InvokeMethod(6187_0_aux_Load(x4[1] - x5[1] + 1, java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1] + 1, x4[1] - x5[1] + 1, java.lang.Object(ARRAY(x2[1], x3[1]))) →* 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))))∧(java.lang.Object(ARRAY(x2[1], x3[1])) →* java.lang.Object(ARRAY(x0[2], x1[2]))))

(2) -> (3), if ((x2[2] > x3[2] + 1* TRUE)∧(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))) →* 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[3], x3[3], x5[3], java.lang.Object(ARRAY(x0[3], x1[3]))))∧(java.lang.Object(ARRAY(x0[2], x1[2])) →* java.lang.Object(ARRAY(x0[3], x1[3]))))

(3) -> (0), if ((6187_1_sort_InvokeMethod(6187_0_aux_Load(x2[3] - x3[3] + 1, java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], x3[3] + 1, x2[3] - x3[3] + 1, java.lang.Object(ARRAY(x6[3], x7[3]))) →* 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))))∧(java.lang.Object(ARRAY(x6[3], x7[3])) →* java.lang.Object(ARRAY(x2[0], x3[0]))))

(3) -> (2), if ((6187_1_sort_InvokeMethod(6187_0_aux_Load(x2[3] - x3[3] + 1, java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], x3[3] + 1, x2[3] - x3[3] + 1, java.lang.Object(ARRAY(x6[3], x7[3]))) →* 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))))∧(java.lang.Object(ARRAY(x6[3], x7[3])) →* java.lang.Object(ARRAY(x0[2], x1[2]))))

The set Q consists of the following terms:
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5100_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), 0), x3, x4, x0, java.lang.Object(ARRAY(x1, x2))), java.lang.Object(ARRAY(x1, x2)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5338_0_aux_ArrayAccess(x0, 0, x1, java.lang.Object(ARRAY(x2, x3)), x4), x5, x6, x0, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5579_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), 0), x3, x4, x0, java.lang.Object(ARRAY(x1, x2))), java.lang.Object(ARRAY(x1, x2)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x5, x1, java.lang.Object(ARRAY(x6, x7))), java.lang.Object(ARRAY(x6, x7)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6068_0_aux_ArrayAccess(x0, x1, 0, java.lang.Object(ARRAY(x2, x3)), x4), x5, x6, x0, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6115_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), 0, x4), x5, x6, x0, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6211_0_aux_ArrayAccess(0, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4), x5, x6, x0, java.lang.Object(ARRAY(x7, x8))), java.lang.Object(ARRAY(x7, x8)))
4204_0_aux_LT(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT2(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT3(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT4(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
4854_0_aux_LE(x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
Cond_4854_0_aux_LE(TRUE, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
Cond_4854_0_aux_LE1(TRUE, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
4853_0_aux_LE(x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
Cond_4853_0_aux_LE(TRUE, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x1, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x2, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x4, x0, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7)))

### (15) 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 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x5, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → COND_6187_2_MAIN_INVOKEMETHOD(>(x4, +(x5, 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x5, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) the following chains were created:
• We consider the chain 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0]))) → COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0]))), COND_6187_2_MAIN_INVOKEMETHOD(TRUE, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1], x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1]))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], +(x5[1], 1), -(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1]))) which results in the following constraint:

(1)    (>(x4[0], +(x5[0], 1))=TRUE6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0])))=6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1], x1[1], java.lang.Object(ARRAY(x2[1], x3[1])))∧java.lang.Object(ARRAY(x2[0], x3[0]))=java.lang.Object(ARRAY(x2[1], x3[1])) ⇒ 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))≥NonInfC∧6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))≥COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))∧(UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥))

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

(2)    (>(x4[0], +(x5[0], 1))=TRUE6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))≥NonInfC∧6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))≥COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))∧(UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥))

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

(3)    (x4[0] + [-2] + [-1]x5[0] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥)∧[(-1)bni_70 + (-1)Bound*bni_70] + [(-1)bni_70]x5[0] + [bni_70]x4[0] ≥ 0∧[(-1)bso_71] ≥ 0)

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

(4)    (x4[0] + [-2] + [-1]x5[0] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥)∧[(-1)bni_70 + (-1)Bound*bni_70] + [(-1)bni_70]x5[0] + [bni_70]x4[0] ≥ 0∧[(-1)bso_71] ≥ 0)

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

(5)    (x4[0] + [-2] + [-1]x5[0] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥)∧[(-1)bni_70 + (-1)Bound*bni_70] + [(-1)bni_70]x5[0] + [bni_70]x4[0] ≥ 0∧[(-1)bso_71] ≥ 0)

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

(6)    (x4[0] + [-2] + [-1]x5[0] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bni_70 + (-1)Bound*bni_70] + [(-1)bni_70]x5[0] + [bni_70]x4[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_71] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(7)    (x4[0] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_70 + (-1)Bound*bni_70] + [bni_70]x4[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_71] ≥ 0)

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

(8)    (x4[0] ≥ 0∧x5[0] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_70 + (-1)Bound*bni_70] + [bni_70]x4[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_71] ≥ 0)

(9)    (x4[0] ≥ 0∧x5[0] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_70 + (-1)Bound*bni_70] + [bni_70]x4[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_71] ≥ 0)

For Pair COND_6187_2_MAIN_INVOKEMETHOD(TRUE, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x5, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4, +(x5, 1)), java.lang.Object(ARRAY(x2, x3))), x4, +(x5, 1), -(x4, +(x5, 1)), java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) the following chains were created:
• We consider the chain COND_6187_2_MAIN_INVOKEMETHOD(TRUE, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1], x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1]))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], +(x5[1], 1), -(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1]))) which results in the following constraint:

(10)    (COND_6187_2_MAIN_INVOKEMETHOD(TRUE, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1], x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))≥NonInfC∧COND_6187_2_MAIN_INVOKEMETHOD(TRUE, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1], x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))≥6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], +(x5[1], 1), -(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))∧(UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], +(x5[1], 1), -(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))), ≥))

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

(11)    ((UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], +(x5[1], 1), -(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))), ≥)∧[1 + (-1)bso_73] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(12)    ((UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], +(x5[1], 1), -(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))), ≥)∧[1 + (-1)bso_73] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(13)    ((UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], +(x5[1], 1), -(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))), ≥)∧[1 + (-1)bso_73] ≥ 0)

We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(14)    ((UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], +(x5[1], 1), -(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_73] ≥ 0)

For Pair 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2, x3, x5, java.lang.Object(ARRAY(x0, x1))), java.lang.Object(ARRAY(x0, x1))) → COND_6187_2_MAIN_INVOKEMETHOD1(>(x2, +(x3, 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2, x3, x5, java.lang.Object(ARRAY(x0, x1))), java.lang.Object(ARRAY(x0, x1))) the following chains were created:
• We consider the chain 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2]))) → COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2]))), COND_6187_2_MAIN_INVOKEMETHOD1(TRUE, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[3], x3[3], x5[3], java.lang.Object(ARRAY(x0[3], x1[3]))), java.lang.Object(ARRAY(x0[3], x1[3]))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], +(x3[3], 1), -(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3]))) which results in the following constraint:

(15)    (>(x2[2], +(x3[2], 1))=TRUE6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2])))=6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[3], x3[3], x5[3], java.lang.Object(ARRAY(x0[3], x1[3])))∧java.lang.Object(ARRAY(x0[2], x1[2]))=java.lang.Object(ARRAY(x0[3], x1[3])) ⇒ 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))≥NonInfC∧6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))≥COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))∧(UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥))

We simplified constraint (15) using rules (I), (II), (IV), (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint:

(16)    (>(x2[2], +(x3[2], 1))=TRUE6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))≥NonInfC∧6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))≥COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))∧(UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥))

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

(17)    (x2[2] + [-2] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥)∧[(-1)bni_74 + (-1)Bound*bni_74] + [(-1)bni_74]x3[2] + [bni_74]x2[2] ≥ 0∧[(-1)bso_75] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(18)    (x2[2] + [-2] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥)∧[(-1)bni_74 + (-1)Bound*bni_74] + [(-1)bni_74]x3[2] + [bni_74]x2[2] ≥ 0∧[(-1)bso_75] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(19)    (x2[2] + [-2] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥)∧[(-1)bni_74 + (-1)Bound*bni_74] + [(-1)bni_74]x3[2] + [bni_74]x2[2] ≥ 0∧[(-1)bso_75] ≥ 0)

We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(20)    (x2[2] + [-2] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bni_74 + (-1)Bound*bni_74] + [(-1)bni_74]x3[2] + [bni_74]x2[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_75] ≥ 0)

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

(21)    (x2[2] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_74 + (-1)Bound*bni_74] + [bni_74]x2[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_75] ≥ 0)

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

(22)    (x2[2] ≥ 0∧x3[2] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_74 + (-1)Bound*bni_74] + [bni_74]x2[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_75] ≥ 0)

(23)    (x2[2] ≥ 0∧x3[2] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_74 + (-1)Bound*bni_74] + [bni_74]x2[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_75] ≥ 0)

For Pair COND_6187_2_MAIN_INVOKEMETHOD1(TRUE, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2, x3, x5, java.lang.Object(ARRAY(x0, x1))), java.lang.Object(ARRAY(x0, x1))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2, +(x3, 1)), java.lang.Object(ARRAY(x6, x7))), x2, +(x3, 1), -(x2, +(x3, 1)), java.lang.Object(ARRAY(x6, x7))), java.lang.Object(ARRAY(x6, x7))) the following chains were created:
• We consider the chain COND_6187_2_MAIN_INVOKEMETHOD1(TRUE, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[3], x3[3], x5[3], java.lang.Object(ARRAY(x0[3], x1[3]))), java.lang.Object(ARRAY(x0[3], x1[3]))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], +(x3[3], 1), -(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3]))) which results in the following constraint:

(24)    (COND_6187_2_MAIN_INVOKEMETHOD1(TRUE, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[3], x3[3], x5[3], java.lang.Object(ARRAY(x0[3], x1[3]))), java.lang.Object(ARRAY(x0[3], x1[3])))≥NonInfC∧COND_6187_2_MAIN_INVOKEMETHOD1(TRUE, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[3], x3[3], x5[3], java.lang.Object(ARRAY(x0[3], x1[3]))), java.lang.Object(ARRAY(x0[3], x1[3])))≥6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], +(x3[3], 1), -(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3])))∧(UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], +(x3[3], 1), -(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3])))), ≥))

We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(25)    ((UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], +(x3[3], 1), -(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3])))), ≥)∧[1 + (-1)bso_77] ≥ 0)

We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(26)    ((UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], +(x3[3], 1), -(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3])))), ≥)∧[1 + (-1)bso_77] ≥ 0)

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

(27)    ((UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], +(x3[3], 1), -(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3])))), ≥)∧[1 + (-1)bso_77] ≥ 0)

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

(28)    ((UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], +(x3[3], 1), -(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_77] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x5, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → COND_6187_2_MAIN_INVOKEMETHOD(>(x4, +(x5, 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x5, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
• (x4[0] ≥ 0∧x5[0] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_70 + (-1)Bound*bni_70] + [bni_70]x4[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_71] ≥ 0)
• (x4[0] ≥ 0∧x5[0] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_70 + (-1)Bound*bni_70] + [bni_70]x4[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_71] ≥ 0)

• COND_6187_2_MAIN_INVOKEMETHOD(TRUE, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x5, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4, +(x5, 1)), java.lang.Object(ARRAY(x2, x3))), x4, +(x5, 1), -(x4, +(x5, 1)), java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
• ((UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], +(x5[1], 1), -(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_73] ≥ 0)

• 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2, x3, x5, java.lang.Object(ARRAY(x0, x1))), java.lang.Object(ARRAY(x0, x1))) → COND_6187_2_MAIN_INVOKEMETHOD1(>(x2, +(x3, 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2, x3, x5, java.lang.Object(ARRAY(x0, x1))), java.lang.Object(ARRAY(x0, x1)))
• (x2[2] ≥ 0∧x3[2] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_74 + (-1)Bound*bni_74] + [bni_74]x2[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_75] ≥ 0)
• (x2[2] ≥ 0∧x3[2] ≥ 0 ⇒ (UIncreasing(COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_74 + (-1)Bound*bni_74] + [bni_74]x2[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_75] ≥ 0)

• COND_6187_2_MAIN_INVOKEMETHOD1(TRUE, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2, x3, x5, java.lang.Object(ARRAY(x0, x1))), java.lang.Object(ARRAY(x0, x1))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2, +(x3, 1)), java.lang.Object(ARRAY(x6, x7))), x2, +(x3, 1), -(x2, +(x3, 1)), java.lang.Object(ARRAY(x6, x7))), java.lang.Object(ARRAY(x6, x7)))
• ((UIncreasing(6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], +(x3[3], 1), -(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_77] ≥ 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(6187_0_aux_Load(x1, x2)) = [-1] + [-1]x2
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1, x2)) = [-1] + [-1]x2
POL(4204_0_aux_LT(x1, x2, x3)) = x2 + [-1]x1
POL(0) = 0
POL(6187_2_main_InvokeMethod(x1, x2)) = [-1]
POL(6187_1_sort_InvokeMethod(x1, x2, x3, x4, x5)) = [-1]x5 + x3 + [-1]x2
POL(5100_0_aux_ArrayAccess(x1, x2, x3)) = [-1] + [-1]x1 + x3 + [-1]x2
POL(6255_0_main_InvokeMethod(x1)) = [-1]
POL(5338_0_aux_ArrayAccess(x1, x2, x3, x4, x5)) = [-1] + [2]x5 + x1 + x2 + [-1]x4
POL(5579_0_aux_ArrayAccess(x1, x2, x3)) = [-1] + x3 + [-1]x2
POL(5778_0_aux_InvokeMethod(x1, x2, x3)) = [-1]x3 + x1
POL(6068_0_aux_ArrayAccess(x1, x2, x3, x4, x5)) = [-1] + [-1]x5 + [-1]x2 + x3 + [-1]x4
POL(6115_0_aux_ArrayAccess(x1, x2, x3, x4, x5)) = [-1] + x5 + [-1]x4 + [-1]x3 + x2 + [-1]x1
POL(6211_0_aux_ArrayAccess(x1, x2, x3, x4, x5)) = [-1] + x5 + x4 + [-1]x3 + x2 + [2]x1
POL(Cond_4204_0_aux_LT(x1, x2, x3, x4)) = [1] + x3 + [-1]x2
POL(<=(x1, x2)) = [-1]
POL(4295_0_aux_Return(x1, x2, x3)) = x1 + [-1]x3
POL(Cond_4204_0_aux_LT1(x1, x2, x3, x4)) = [-1] + x3 + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(Cond_4204_0_aux_LT2(x1, x2, x3, x4)) = [-1] + x3 + [-1]x2
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(>=(x1, x2)) = [-1]
POL(4853_0_aux_LE(x1, x2, x3, x4, x5)) = [2] + [2]x1 + x3 + [-1]x2 + [2]x5 + [2]x4
POL(Cond_4204_0_aux_LT3(x1, x2, x3, x4)) = [1] + x3 + [-1]x2
POL(4854_0_aux_LE(x1, x2, x3, x4, x5)) = [-1] + [2]x5 + [2]x4 + x3 + [-1]x2
POL(Cond_4204_0_aux_LT4(x1, x2, x3, x4)) = [-1] + x3 + [-1]x2
POL(Cond_4854_0_aux_LE(x1, x2, x3, x4, x5, x6)) = [-1] + [2]x6 + [2]x5 + x4 + [-1]x3
POL(<(x1, x2)) = [-1]
POL(Cond_4854_0_aux_LE1(x1, x2, x3, x4, x5, x6)) = [2] + [2]x6 + [2]x5 + x4 + [-1]x3
POL(Cond_4853_0_aux_LE(x1, x2, x3, x4, x5, x6)) = [1] + x4 + [-1]x3 + [2]x6 + [2]x5
POL(5704_1_aux_InvokeMethod(x1, x2, x3, x4)) = x2 + [-1]x1 + [-1]x4
POL(5775_0_aux_Return) = 0
POL(6187_2_MAIN_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(COND_6187_2_MAIN_INVOKEMETHOD(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(-(x1, x2)) = x1 + [-1]x2
POL(COND_6187_2_MAIN_INVOKEMETHOD1(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2

The following pairs are in P>:

COND_6187_2_MAIN_INVOKEMETHOD(TRUE, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1], x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1]))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], +(x5[1], 1), -(x4[1], +(x5[1], 1)), java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))
COND_6187_2_MAIN_INVOKEMETHOD1(TRUE, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[3], x3[3], x5[3], java.lang.Object(ARRAY(x0[3], x1[3]))), java.lang.Object(ARRAY(x0[3], x1[3]))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(-(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], +(x3[3], 1), -(x2[3], +(x3[3], 1)), java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3])))

The following pairs are in Pbound:

6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0]))) → COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))
6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2]))) → COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))

The following pairs are in P:

6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0]))) → COND_6187_2_MAIN_INVOKEMETHOD(>(x4[0], +(x5[0], 1)), 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))
6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2]))) → COND_6187_2_MAIN_INVOKEMETHOD1(>(x2[2], +(x3[2], 1)), 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))

At least the following rules have been oriented under context sensitive arithmetic replacement:

Cond_4204_0_aux_LT(<=(x1, x0), java.lang.Object(ARRAY(x2, x3)), x0, x1)14204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1)1
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1)1Cond_4204_0_aux_LT2(&&(&&(&&(&&(>(x2, x0), >(x2, +(x0, 1))), >(x1, x0)), >=(x0, 0)), <=(1, +(x0, 1))), java.lang.Object(ARRAY(x2, x3)), x0, x1)1
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1)1Cond_4204_0_aux_LT4(&&(&&(&&(>(x2, x0), <=(x2, +(x0, 1))), >(x1, x0)), >=(x0, 0)), java.lang.Object(ARRAY(x2, x3)), x0, x1)1
Cond_4853_0_aux_LE(&&(&&(&&(&&(<(x5, x4), >(x2, x0)), >(x2, +(x0, 1))), >=(x0, 0)), <=(1, +(x0, 1))), x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)14854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)1
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3)))15775_0_aux_Return1
4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)1Cond_4854_0_aux_LE(&&(<(x5, x4), <=(x2, x0)), x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)1
Cond_4854_0_aux_LE1(&&(&&(&&(<(x5, x4), >(x2, x0)), <=(x2, +(x0, 1))), >=(x0, 0)), x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)14854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)1

### (17) 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, Boolean

The ITRS R consists of the following rules:
6187_0_aux_Load(x1, java.lang.Object(ARRAY(x2, x3))) → 4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), 0, x1)
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), 0), x6, x7, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5338_0_aux_ArrayAccess(x1, 0, x5, java.lang.Object(ARRAY(x2, x3)), x6), x7, x8, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), 0), x6, x7, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x6, x7, x1, java.lang.Object(ARRAY(x4, x5))), java.lang.Object(ARRAY(x4, x5))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6068_0_aux_ArrayAccess(x1, x5, 0, java.lang.Object(ARRAY(x2, x3)), x7), x8, x9, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6115_0_aux_ArrayAccess(x1, x5, java.lang.Object(ARRAY(x2, x3)), 0, x7), x8, x9, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6211_0_aux_ArrayAccess(0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x8, x9, x1, java.lang.Object(ARRAY(x6, x7))), java.lang.Object(ARRAY(x6, x7))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT(x1 <= x0, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3)))
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT1(x2 <= x0 && x1 > x0, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT1(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0)
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT2(x2 > x0 && x2 > x0 + 1 && x1 > x0 && x0 >= 0 && 1 <= x0 + 1, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT2(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 4853_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT3(x2 > x0 && x2 > x0 + 1 && x1 > x0 && x0 >= 0 && 1 <= x0 + 1, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT3(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT4(x2 > x0 && x2 <= x0 + 1 && x1 > x0 && x0 >= 0, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT4(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 5338_0_aux_ArrayAccess(x1, x0, x4, java.lang.Object(ARRAY(x2, x3)), x0 + 1)
4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4854_0_aux_LE(x5 < x4 && x2 <= x0, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
Cond_4854_0_aux_LE(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0)
4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4854_0_aux_LE1(x5 < x4 && x2 > x0 && x2 <= x0 + 1 && x0 >= 0, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
Cond_4854_0_aux_LE1(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 6068_0_aux_ArrayAccess(x1, x6, x0, java.lang.Object(ARRAY(x2, x3)), x0 + 1)
4853_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4853_0_aux_LE(x5 >= x4 && x0 >= 0, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
Cond_4853_0_aux_LE(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 5704_1_aux_InvokeMethod(4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0 + 1, x1), x0 + 1, x1, java.lang.Object(ARRAY(x2, x3)))
4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4853_0_aux_LE(x5 < x4 && x2 > x0 && x2 > x0 + 1 && x0 >= 0 && 1 <= x0 + 1, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x1, x0, x4, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6))) → 5778_0_aux_InvokeMethod(x4, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x1, x4, x0, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x1, x4, java.lang.Object(ARRAY(x2, x3)), x0, x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))

The integer pair graph contains the following rules and edges:
(0): 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0]))) → COND_6187_2_MAIN_INVOKEMETHOD(x4[0] > x5[0] + 1, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), x4[0], x5[0], x1[0], java.lang.Object(ARRAY(x2[0], x3[0]))), java.lang.Object(ARRAY(x2[0], x3[0])))
(2): 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2]))) → COND_6187_2_MAIN_INVOKEMETHOD1(x2[2] > x3[2] + 1, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[2], x3[2], x5[2], java.lang.Object(ARRAY(x0[2], x1[2]))), java.lang.Object(ARRAY(x0[2], x1[2])))

The set Q consists of the following terms:
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5100_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), 0), x3, x4, x0, java.lang.Object(ARRAY(x1, x2))), java.lang.Object(ARRAY(x1, x2)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5338_0_aux_ArrayAccess(x0, 0, x1, java.lang.Object(ARRAY(x2, x3)), x4), x5, x6, x0, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5579_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), 0), x3, x4, x0, java.lang.Object(ARRAY(x1, x2))), java.lang.Object(ARRAY(x1, x2)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x5, x1, java.lang.Object(ARRAY(x6, x7))), java.lang.Object(ARRAY(x6, x7)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6068_0_aux_ArrayAccess(x0, x1, 0, java.lang.Object(ARRAY(x2, x3)), x4), x5, x6, x0, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6115_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), 0, x4), x5, x6, x0, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6211_0_aux_ArrayAccess(0, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4), x5, x6, x0, java.lang.Object(ARRAY(x7, x8))), java.lang.Object(ARRAY(x7, x8)))
4204_0_aux_LT(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT2(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT3(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT4(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
4854_0_aux_LE(x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
Cond_4854_0_aux_LE(TRUE, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
Cond_4854_0_aux_LE1(TRUE, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
4853_0_aux_LE(x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
Cond_4853_0_aux_LE(TRUE, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x1, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x2, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x4, x0, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7)))

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (20) Obligation:

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

The following domains are used:

Integer, Boolean

The ITRS R consists of the following rules:
6187_0_aux_Load(x1, java.lang.Object(ARRAY(x2, x3))) → 4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), 0, x1)
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), 0), x6, x7, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5338_0_aux_ArrayAccess(x1, 0, x5, java.lang.Object(ARRAY(x2, x3)), x6), x7, x8, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), 0), x6, x7, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x6, x7, x1, java.lang.Object(ARRAY(x4, x5))), java.lang.Object(ARRAY(x4, x5))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6068_0_aux_ArrayAccess(x1, x5, 0, java.lang.Object(ARRAY(x2, x3)), x7), x8, x9, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6115_0_aux_ArrayAccess(x1, x5, java.lang.Object(ARRAY(x2, x3)), 0, x7), x8, x9, x1, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6211_0_aux_ArrayAccess(0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x8, x9, x1, java.lang.Object(ARRAY(x6, x7))), java.lang.Object(ARRAY(x6, x7))) → 6255_0_main_InvokeMethod(java.lang.Object(ARRAY(x2, x3)))
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT(x1 <= x0, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3)))
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT1(x2 <= x0 && x1 > x0, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT1(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0)
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT2(x2 > x0 && x2 > x0 + 1 && x1 > x0 && x0 >= 0 && 1 <= x0 + 1, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT2(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 4853_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT3(x2 > x0 && x2 > x0 + 1 && x1 > x0 && x0 >= 0 && 1 <= x0 + 1, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT3(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0, x1) → Cond_4204_0_aux_LT4(x2 > x0 && x2 <= x0 + 1 && x1 > x0 && x0 >= 0, java.lang.Object(ARRAY(x2, x3)), x0, x1)
Cond_4204_0_aux_LT4(TRUE, java.lang.Object(ARRAY(x2, x3)), x0, x1) → 5338_0_aux_ArrayAccess(x1, x0, x4, java.lang.Object(ARRAY(x2, x3)), x0 + 1)
4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4854_0_aux_LE(x5 < x4 && x2 <= x0, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
Cond_4854_0_aux_LE(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0)
4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4854_0_aux_LE1(x5 < x4 && x2 > x0 && x2 <= x0 + 1 && x0 >= 0, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
Cond_4854_0_aux_LE1(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 6068_0_aux_ArrayAccess(x1, x6, x0, java.lang.Object(ARRAY(x2, x3)), x0 + 1)
4853_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4853_0_aux_LE(x5 >= x4 && x0 >= 0, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
Cond_4853_0_aux_LE(TRUE, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → 5704_1_aux_InvokeMethod(4204_0_aux_LT(java.lang.Object(ARRAY(x2, x3)), x0 + 1, x1), x0 + 1, x1, java.lang.Object(ARRAY(x2, x3)))
4854_0_aux_LE(x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5) → Cond_4853_0_aux_LE(x5 < x4 && x2 > x0 && x2 > x0 + 1 && x0 >= 0 && 1 <= x0 + 1, x1, java.lang.Object(ARRAY(x2, x3)), x0, x4, x5)
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x1, x0, x4, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x1, java.lang.Object(ARRAY(x2, x3)), x0), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5775_0_aux_Return
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6))) → 5778_0_aux_InvokeMethod(x4, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x1, x4, x0, java.lang.Object(ARRAY(x2, x3)), x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x1, x4, java.lang.Object(ARRAY(x2, x3)), x0, x5), x0, x1, java.lang.Object(ARRAY(x2, x3))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7))) → 5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3)))

The integer pair graph contains the following rules and edges:
(1): COND_6187_2_MAIN_INVOKEMETHOD(TRUE, 6187_1_sort_InvokeMethod(4295_0_aux_Return(0, x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1], x1[1], java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1]))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(x4[1] - x5[1] + 1, java.lang.Object(ARRAY(x2[1], x3[1]))), x4[1], x5[1] + 1, x4[1] - x5[1] + 1, java.lang.Object(ARRAY(x2[1], x3[1]))), java.lang.Object(ARRAY(x2[1], x3[1])))
(3): COND_6187_2_MAIN_INVOKEMETHOD1(TRUE, 6187_1_sort_InvokeMethod(5775_0_aux_Return, x2[3], x3[3], x5[3], java.lang.Object(ARRAY(x0[3], x1[3]))), java.lang.Object(ARRAY(x0[3], x1[3]))) → 6187_2_MAIN_INVOKEMETHOD(6187_1_sort_InvokeMethod(6187_0_aux_Load(x2[3] - x3[3] + 1, java.lang.Object(ARRAY(x6[3], x7[3]))), x2[3], x3[3] + 1, x2[3] - x3[3] + 1, java.lang.Object(ARRAY(x6[3], x7[3]))), java.lang.Object(ARRAY(x6[3], x7[3])))

The set Q consists of the following terms:
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5100_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), 0), x3, x4, x0, java.lang.Object(ARRAY(x1, x2))), java.lang.Object(ARRAY(x1, x2)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5338_0_aux_ArrayAccess(x0, 0, x1, java.lang.Object(ARRAY(x2, x3)), x4), x5, x6, x0, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5579_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), 0), x3, x4, x0, java.lang.Object(ARRAY(x1, x2))), java.lang.Object(ARRAY(x1, x2)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x5, x1, java.lang.Object(ARRAY(x6, x7))), java.lang.Object(ARRAY(x6, x7)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6068_0_aux_ArrayAccess(x0, x1, 0, java.lang.Object(ARRAY(x2, x3)), x4), x5, x6, x0, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6115_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), 0, x4), x5, x6, x0, java.lang.Object(ARRAY(x2, x3))), java.lang.Object(ARRAY(x2, x3)))
6187_2_main_InvokeMethod(6187_1_sort_InvokeMethod(6211_0_aux_ArrayAccess(0, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4), x5, x6, x0, java.lang.Object(ARRAY(x7, x8))), java.lang.Object(ARRAY(x7, x8)))
4204_0_aux_LT(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT2(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT3(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_4204_0_aux_LT4(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)
4854_0_aux_LE(x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
Cond_4854_0_aux_LE(TRUE, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
Cond_4854_0_aux_LE1(TRUE, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
4853_0_aux_LE(x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
Cond_4853_0_aux_LE(TRUE, x0, java.lang.Object(ARRAY(x1, x2)), x3, x4, x5)
5704_1_aux_InvokeMethod(4295_0_aux_Return(x0, x1, java.lang.Object(ARRAY(x2, x3))), x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5100_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5338_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x1, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(5579_0_aux_ArrayAccess(x0, java.lang.Object(ARRAY(x1, x2)), x3), x3, x0, java.lang.Object(ARRAY(x1, x2)))
5704_1_aux_InvokeMethod(5775_0_aux_Return, x0, x1, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(5778_0_aux_InvokeMethod(x0, x1, java.lang.Object(ARRAY(x2, x3))), x4, x1, java.lang.Object(ARRAY(x5, x6)))
5704_1_aux_InvokeMethod(6068_0_aux_ArrayAccess(x0, x1, x2, java.lang.Object(ARRAY(x3, x4)), x5), x2, x0, java.lang.Object(ARRAY(x3, x4)))
5704_1_aux_InvokeMethod(6115_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x4, x0, java.lang.Object(ARRAY(x2, x3)))
5704_1_aux_InvokeMethod(6211_0_aux_ArrayAccess(x0, x1, java.lang.Object(ARRAY(x2, x3)), x4, x5), x0, x1, java.lang.Object(ARRAY(x6, x7)))

### (21) IDependencyGraphProof (EQUIVALENT transformation)

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

### (23) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) → COND_2614_0_MAIN_LOAD(x2[0] >= 0 && x2[0] < x0[0], java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])

(0) -> (1), if ((x2[0] >= 0 && x2[0] < x0[0]* TRUE)∧(java.lang.Object(ARRAY(x0[0], x1[0])) →* java.lang.Object(ARRAY(x0[1], x1[1])))∧(x2[0]* x2[1]))

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

The set Q is empty.

### (24) 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 2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), x2) → COND_2614_0_MAIN_LOAD(&&(>=(x2, 0), <(x2, x0)), java.lang.Object(ARRAY(x0, x1)), x2) the following chains were created:
• We consider the chain 2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) → COND_2614_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), COND_2614_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]) → 2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1)) which results in the following constraint:

(1)    (&&(>=(x2[0], 0), <(x2[0], x0[0]))=TRUEjava.lang.Object(ARRAY(x0[0], x1[0]))=java.lang.Object(ARRAY(x0[1], x1[1]))∧x2[0]=x2[1]2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])≥NonInfC∧2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])≥COND_2614_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])∧(UIncreasing(COND_2614_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥))

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

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

(3)    (x2[0] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_2614_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]x2[0] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(4)    (x2[0] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_2614_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]x2[0] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(5)    (x2[0] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_2614_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]x2[0] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(6)    (x2[0] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_2614_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]x2[0] + [bni_11]x0[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(7)    (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_2614_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

For Pair COND_2614_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0, x1)), x2) → 2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), +(x2, 1)) the following chains were created:
• We consider the chain COND_2614_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]) → 2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1)) which results in the following constraint:

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

(9)    ((UIncreasing(2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥)∧[1 + (-1)bso_14] ≥ 0)

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

(10)    ((UIncreasing(2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥)∧[1 + (-1)bso_14] ≥ 0)

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

(11)    ((UIncreasing(2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥)∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(12)    ((UIncreasing(2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), x2) → COND_2614_0_MAIN_LOAD(&&(>=(x2, 0), <(x2, x0)), java.lang.Object(ARRAY(x0, x1)), x2)
• (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_2614_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

• ((UIncreasing(2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 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(2614_0_MAIN_LOAD(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1, x2)) = [-1]x1
POL(COND_2614_0_MAIN_LOAD(x1, x2, x3)) = [-1] + [-1]x3 + [-1]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>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (26) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 2614_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) → COND_2614_0_MAIN_LOAD(x2[0] >= 0 && x2[0] < x0[0], java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])

The set Q is empty.

### (27) IDependencyGraphProof (EQUIVALENT transformation)

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

### (29) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges: