(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: CountMetaList
`public class CountMetaList {	public static void main(String[] args) {		Random.args = args;		List l = createMetaList();		int count = countMetaList(l);	}	public static int countMetaList(List cur) {		int res = 0;		while (cur != null) {			if (cur.value instanceof List) {				List inner = (List) cur.value;				cur.value = inner.next;				cur = new List(inner.value, cur);			}			cur = cur.next;			res++;		}		return res;	}	public static List createMetaList() {		int count = Random.random();		List cur = null;		for (int i = 0; i < count; i++) {			int innerCount = Random.random();			List innerList = null;			for (int j = innerCount; j > 0; j--) {				innerList = new List(null, innerList);			}			cur = new List(innerList, cur);		}		return cur;	}}class List {	Object value;	List next;	public List(Object v, List n) {		this.value = v;		this.next = n;	}}public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}`

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
CountMetaList.main([Ljava/lang/String;)V: Graph of 39 nodes with 0 SCCs.

CountMetaList.createMetaList()LList;: Graph of 194 nodes with 1 SCC.

CountMetaList.countMetaList(LList;)I: Graph of 116 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 111 rules for P and 3 rules for R.

Combined rules. Obtained 11 rules for P and 1 rules for R.

Filtered ground terms:

6229_0_countMetaList_Store(x1, x2) → 6229_0_countMetaList_Store(x2)
5704_0_countMetaList_NULL(x1, x2, x3) → 5704_0_countMetaList_NULL(x2, x3)
List(x1, x2, x3) → List(x2, x3)
6266_0_countMetaList_Store(x1, x2) → 6266_0_countMetaList_Store(x2)
7041_0_countMetaList_Store(x1, x2) → 7041_0_countMetaList_Store(x2)
5733_0_countMetaList_Return(x1) → 5733_0_countMetaList_Return

Filtered duplicate args:

5704_0_countMetaList_NULL(x1, x2) → 5704_0_countMetaList_NULL(x2)

Filtered unneeded arguments:

Finished conversion. Obtained 11 rules for P and 1 rules for R. System has no predefined symbols.

Log for SCC 1:

Generated 69 rules for P and 49 rules for R.

Combined rules. Obtained 4 rules for P and 0 rules for R.

Filtered ground terms:

5728_0_createMetaList_LE(x1, x2, x3, x4, x5, x6, x7) → 5728_0_createMetaList_LE(x2, x3, x4, x5, x6, x7)
List(x1) → List
Cond_5728_0_createMetaList_LE1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_5728_0_createMetaList_LE1(x1, x3, x4, x5, x6, x7, x8)
4016_0_random_ArrayAccess(x1, x2, x3) → 4016_0_random_ArrayAccess(x2, x3)
Cond_5728_0_createMetaList_LE(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_5728_0_createMetaList_LE(x1, x3, x4, x5, x6)
4199_0_random_IntArithmetic(x1, x2, x3, x4) → 4199_0_random_IntArithmetic(x2, x3)

Filtered duplicate args:

5728_0_createMetaList_LE(x1, x2, x3, x4, x5, x6) → 5728_0_createMetaList_LE(x1, x2, x3, x4, x6)
Cond_5728_0_createMetaList_LE1(x1, x2, x3, x4, x5, x6, x7) → Cond_5728_0_createMetaList_LE1(x1, x2, x3, x4, x5, x7)

Filtered unneeded arguments:

4016_1_createMetaList_InvokeMethod(x1, x2, x3, x4) → 4016_1_createMetaList_InvokeMethod(x1, x2, x4)
Cond_4016_1_createMetaList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_4016_1_createMetaList_InvokeMethod(x1, x2, x3, x5)
4199_1_createMetaList_InvokeMethod(x1, x2, x3, x4) → 4199_1_createMetaList_InvokeMethod(x1, x2, x4)
Cond_4199_1_createMetaList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_4199_1_createMetaList_InvokeMethod(x1, x2, x3, x5)
5728_0_createMetaList_LE(x1, x2, x3, x4, x5) → 5728_0_createMetaList_LE(x1, x3, x5)
Cond_5728_0_createMetaList_LE(x1, x2, x3, x4, x5) → Cond_5728_0_createMetaList_LE(x1, x2, x4)
Cond_5728_0_createMetaList_LE1(x1, x2, x3, x4, x5, x6) → Cond_5728_0_createMetaList_LE1(x1, x2, x4, x6)

Combined rules. Obtained 4 rules for P and 0 rules for R.

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.

(5) Obligation:

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

The following domains are used:
none

The ITRS R consists of the following rules:
5704_0_countMetaList_NULL(NULL) → 5733_0_countMetaList_Return

The integer pair graph contains the following rules and edges:
(0): 7041_0_COUNTMETALIST_STORE(java.lang.Object(List(x0[0], x1[0]))) → 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(x0[0], x1[0])))
(1): 6266_0_COUNTMETALIST_STORE(x0[1]) → 5704_0_COUNTMETALIST_NULL(x0[1])
(2): 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(List(x0[2], x1[2])), x2[2]))) → 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(x1[2], x2[2])))
(3): 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(List(x0[3], x1[3])), x2[3]))) → 7041_0_COUNTMETALIST_STORE(java.lang.Object(List(x1[3], x2[3])))
(4): 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(x0[4]), x1[4]))) → 5704_0_COUNTMETALIST_NULL(x1[4])
(5): 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(NULL, x0[5]))) → 5704_0_COUNTMETALIST_NULL(x0[5])
(6): 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(x0[6]), x1[6]))) → 6266_0_COUNTMETALIST_STORE(x1[6])
(7): 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(NULL, x0[7]))) → 6266_0_COUNTMETALIST_STORE(x0[7])

(0) -> (2), if ((java.lang.Object(List(x0[0], x1[0])) →* java.lang.Object(List(java.lang.Object(List(x0[2], x1[2])), x2[2]))))

(0) -> (3), if ((java.lang.Object(List(x0[0], x1[0])) →* java.lang.Object(List(java.lang.Object(List(x0[3], x1[3])), x2[3]))))

(0) -> (4), if ((java.lang.Object(List(x0[0], x1[0])) →* java.lang.Object(List(java.lang.Object(x0[4]), x1[4]))))

(0) -> (5), if ((java.lang.Object(List(x0[0], x1[0])) →* java.lang.Object(List(NULL, x0[5]))))

(0) -> (6), if ((java.lang.Object(List(x0[0], x1[0])) →* java.lang.Object(List(java.lang.Object(x0[6]), x1[6]))))

(0) -> (7), if ((java.lang.Object(List(x0[0], x1[0])) →* java.lang.Object(List(NULL, x0[7]))))

(1) -> (2), if ((x0[1]* java.lang.Object(List(java.lang.Object(List(x0[2], x1[2])), x2[2]))))

(1) -> (3), if ((x0[1]* java.lang.Object(List(java.lang.Object(List(x0[3], x1[3])), x2[3]))))

(1) -> (4), if ((x0[1]* java.lang.Object(List(java.lang.Object(x0[4]), x1[4]))))

(1) -> (5), if ((x0[1]* java.lang.Object(List(NULL, x0[5]))))

(1) -> (6), if ((x0[1]* java.lang.Object(List(java.lang.Object(x0[6]), x1[6]))))

(1) -> (7), if ((x0[1]* java.lang.Object(List(NULL, x0[7]))))

(2) -> (2), if ((java.lang.Object(List(x1[2], x2[2])) →* java.lang.Object(List(java.lang.Object(List(x0[2]', x1[2]')), x2[2]'))))

(2) -> (3), if ((java.lang.Object(List(x1[2], x2[2])) →* java.lang.Object(List(java.lang.Object(List(x0[3], x1[3])), x2[3]))))

(2) -> (4), if ((java.lang.Object(List(x1[2], x2[2])) →* java.lang.Object(List(java.lang.Object(x0[4]), x1[4]))))

(2) -> (5), if ((java.lang.Object(List(x1[2], x2[2])) →* java.lang.Object(List(NULL, x0[5]))))

(2) -> (6), if ((java.lang.Object(List(x1[2], x2[2])) →* java.lang.Object(List(java.lang.Object(x0[6]), x1[6]))))

(2) -> (7), if ((java.lang.Object(List(x1[2], x2[2])) →* java.lang.Object(List(NULL, x0[7]))))

(3) -> (0), if ((java.lang.Object(List(x1[3], x2[3])) →* java.lang.Object(List(x0[0], x1[0]))))

(4) -> (2), if ((x1[4]* java.lang.Object(List(java.lang.Object(List(x0[2], x1[2])), x2[2]))))

(4) -> (3), if ((x1[4]* java.lang.Object(List(java.lang.Object(List(x0[3], x1[3])), x2[3]))))

(4) -> (4), if ((x1[4]* java.lang.Object(List(java.lang.Object(x0[4]'), x1[4]'))))

(4) -> (5), if ((x1[4]* java.lang.Object(List(NULL, x0[5]))))

(4) -> (6), if ((x1[4]* java.lang.Object(List(java.lang.Object(x0[6]), x1[6]))))

(4) -> (7), if ((x1[4]* java.lang.Object(List(NULL, x0[7]))))

(5) -> (2), if ((x0[5]* java.lang.Object(List(java.lang.Object(List(x0[2], x1[2])), x2[2]))))

(5) -> (3), if ((x0[5]* java.lang.Object(List(java.lang.Object(List(x0[3], x1[3])), x2[3]))))

(5) -> (4), if ((x0[5]* java.lang.Object(List(java.lang.Object(x0[4]), x1[4]))))

(5) -> (5), if ((x0[5]* java.lang.Object(List(NULL, x0[5]'))))

(5) -> (6), if ((x0[5]* java.lang.Object(List(java.lang.Object(x0[6]), x1[6]))))

(5) -> (7), if ((x0[5]* java.lang.Object(List(NULL, x0[7]))))

(6) -> (1), if ((x1[6]* x0[1]))

(7) -> (1), if ((x0[7]* x0[1]))

The set Q consists of the following terms:
5704_0_countMetaList_NULL(NULL)

(6) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(7) Obligation:

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

7041_0_COUNTMETALIST_STORE(java.lang.Object(List(x0[0], x1[0]))) → 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(x0[0], x1[0])))
6266_0_COUNTMETALIST_STORE(x0[1]) → 5704_0_COUNTMETALIST_NULL(x0[1])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(List(x0[2], x1[2])), x2[2]))) → 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(x1[2], x2[2])))
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(List(x0[3], x1[3])), x2[3]))) → 7041_0_COUNTMETALIST_STORE(java.lang.Object(List(x1[3], x2[3])))
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(x0[4]), x1[4]))) → 5704_0_COUNTMETALIST_NULL(x1[4])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(NULL, x0[5]))) → 5704_0_COUNTMETALIST_NULL(x0[5])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(x0[6]), x1[6]))) → 6266_0_COUNTMETALIST_STORE(x1[6])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(NULL, x0[7]))) → 6266_0_COUNTMETALIST_STORE(x0[7])

The TRS R consists of the following rules:

5704_0_countMetaList_NULL(NULL) → 5733_0_countMetaList_Return

The set Q consists of the following terms:

5704_0_countMetaList_NULL(NULL)

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

(8) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(9) Obligation:

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

7041_0_COUNTMETALIST_STORE(java.lang.Object(List(x0[0], x1[0]))) → 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(x0[0], x1[0])))
6266_0_COUNTMETALIST_STORE(x0[1]) → 5704_0_COUNTMETALIST_NULL(x0[1])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(List(x0[2], x1[2])), x2[2]))) → 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(x1[2], x2[2])))
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(List(x0[3], x1[3])), x2[3]))) → 7041_0_COUNTMETALIST_STORE(java.lang.Object(List(x1[3], x2[3])))
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(x0[4]), x1[4]))) → 5704_0_COUNTMETALIST_NULL(x1[4])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(NULL, x0[5]))) → 5704_0_COUNTMETALIST_NULL(x0[5])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(x0[6]), x1[6]))) → 6266_0_COUNTMETALIST_STORE(x1[6])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(NULL, x0[7]))) → 6266_0_COUNTMETALIST_STORE(x0[7])

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

5704_0_countMetaList_NULL(NULL)

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

(10) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

5704_0_countMetaList_NULL(NULL)

(11) Obligation:

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

7041_0_COUNTMETALIST_STORE(java.lang.Object(List(x0[0], x1[0]))) → 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(x0[0], x1[0])))
6266_0_COUNTMETALIST_STORE(x0[1]) → 5704_0_COUNTMETALIST_NULL(x0[1])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(List(x0[2], x1[2])), x2[2]))) → 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(x1[2], x2[2])))
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(List(x0[3], x1[3])), x2[3]))) → 7041_0_COUNTMETALIST_STORE(java.lang.Object(List(x1[3], x2[3])))
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(x0[4]), x1[4]))) → 5704_0_COUNTMETALIST_NULL(x1[4])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(NULL, x0[5]))) → 5704_0_COUNTMETALIST_NULL(x0[5])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(x0[6]), x1[6]))) → 6266_0_COUNTMETALIST_STORE(x1[6])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(NULL, x0[7]))) → 6266_0_COUNTMETALIST_STORE(x0[7])

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

(12) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

6266_0_COUNTMETALIST_STORE(x0[1]) → 5704_0_COUNTMETALIST_NULL(x0[1])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(List(x0[2], x1[2])), x2[2]))) → 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(x1[2], x2[2])))
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(List(x0[3], x1[3])), x2[3]))) → 7041_0_COUNTMETALIST_STORE(java.lang.Object(List(x1[3], x2[3])))
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(x0[4]), x1[4]))) → 5704_0_COUNTMETALIST_NULL(x1[4])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(NULL, x0[5]))) → 5704_0_COUNTMETALIST_NULL(x0[5])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(java.lang.Object(x0[6]), x1[6]))) → 6266_0_COUNTMETALIST_STORE(x1[6])
5704_0_COUNTMETALIST_NULL(java.lang.Object(List(NULL, x0[7]))) → 6266_0_COUNTMETALIST_STORE(x0[7])
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(5704_0_COUNTMETALIST_NULL(x1)) = 2·x1
POL(6266_0_COUNTMETALIST_STORE(x1)) = 1 + 2·x1
POL(7041_0_COUNTMETALIST_STORE(x1)) = 2·x1
POL(List(x1, x2)) = 2 + 2·x1 + x2
POL(NULL) = 2
POL(java.lang.Object(x1)) = 2 + 2·x1

(13) Obligation:

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

7041_0_COUNTMETALIST_STORE(java.lang.Object(List(x0[0], x1[0]))) → 5704_0_COUNTMETALIST_NULL(java.lang.Object(List(x0[0], x1[0])))

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

(14) DependencyGraphProof (EQUIVALENT transformation)

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

(16) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]) → COND_4016_1_CREATEMETALIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])
(1): COND_4016_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])
(2): 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]) → COND_4199_1_CREATEMETALIST_INVOKEMETHOD(x2[2] > 0, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])
(3): COND_4199_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])
(4): 5728_0_CREATEMETALIST_LE(x0[4], x2[4], 0) → COND_5728_0_CREATEMETALIST_LE(x2[4] >= 0 && x0[4] > x2[4] + 1, x0[4], x2[4], 0)
(5): COND_5728_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0) → 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], x2[5] + 1)
(6): 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])
(7): COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)

(0) -> (1), if ((x2[0] >= 1 && x2[0] < x0[0]* TRUE)∧(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →* 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0]* x3[1])∧(x5[0]* x5[1]))

(1) -> (2), if ((4199_0_random_IntArithmetic(x6[1], x7[1]) →* 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1]* x4[2])∧(x5[1]* x6[2]))

(2) -> (3), if ((x2[2] > 0* TRUE)∧(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]) →* 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]))∧(x4[2]* x4[3])∧(x6[2]* x6[3]))

(3) -> (4), if ((x4[3]* x0[4])∧(x6[3]* x2[4])∧(x1[3]* 0))

(3) -> (6), if ((x4[3]* x0[6])∧(x6[3]* x2[6])∧(x1[3]* x4[6]))

(4) -> (5), if ((x2[4] >= 0 && x0[4] > x2[4] + 1* TRUE)∧(x0[4]* x0[5])∧(x2[4]* x2[5]))

(5) -> (0), if ((4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]) →* 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x0[5]* x3[0])∧(x2[5] + 1* x5[0]))

(6) -> (7), if ((x4[6] > 0* TRUE)∧(x0[6]* x0[7])∧(x2[6]* x2[7])∧(x4[6]* x4[7]))

(7) -> (4), if ((x0[7]* x0[4])∧(x2[7]* x2[4])∧(x4[7] + -1* 0))

(7) -> (6), if ((x0[7]* x0[6])∧(x2[7]* x2[6])∧(x4[7] + -1* x4[6]))

The set Q is empty.

(17) 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 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5) → COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5) the following chains were created:
• We consider the chain 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]) → COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]), COND_4016_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1]) which results in the following constraint:

(1)    (&&(>=(x2[0], 1), <(x2[0], x0[0]))=TRUE4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])=4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1])∧x3[0]=x3[1]x5[0]=x5[1]4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])≥NonInfC∧4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])≥COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])∧(UIncreasing(COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥))

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

(2)    (>=(x2[0], 1)=TRUE<(x2[0], x0[0])=TRUE4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])≥NonInfC∧4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])≥COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])∧(UIncreasing(COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥))

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

(3)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]x5[0] + [bni_31]x3[0] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(4)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]x5[0] + [bni_31]x3[0] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(5)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]x5[0] + [bni_31]x3[0] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(6)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31] = 0∧[bni_31] = 0∧0 = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 0)

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

(7)    (x2[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31] = 0∧[bni_31] = 0∧0 = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 0)

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

(8)    (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31] = 0∧[bni_31] = 0∧0 = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 0)

For Pair COND_4016_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5) → 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6, x7), x3, x5) the following chains were created:
• We consider the chain COND_4016_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1]) which results in the following constraint:

(9)    (COND_4016_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1])≥NonInfC∧COND_4016_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1])≥4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])∧(UIncreasing(4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥))

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

(10)    ((UIncreasing(4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥)∧[(-1)bso_34] ≥ 0)

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

(11)    ((UIncreasing(4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥)∧[(-1)bso_34] ≥ 0)

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

(12)    ((UIncreasing(4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥)∧[(-1)bso_34] ≥ 0)

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

(13)    ((UIncreasing(4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_34] ≥ 0)

For Pair 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6) → COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2, 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6) the following chains were created:
• We consider the chain 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]) → COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]), COND_4199_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3]) which results in the following constraint:

(14)    (>(x2[2], 0)=TRUE4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2])=4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3])∧x4[2]=x4[3]x6[2]=x6[3]4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])≥NonInfC∧4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])≥COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])∧(UIncreasing(COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥))

We simplified constraint (14) using rules (I), (II), (IV) which results in the following new constraint:

(15)    (>(x2[2], 0)=TRUE4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])≥NonInfC∧4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])≥COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])∧(UIncreasing(COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥))

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

(16)    (x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]x6[2] + [bni_35]x4[2] ≥ 0∧[(-1)bso_36] ≥ 0)

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

(17)    (x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]x6[2] + [bni_35]x4[2] ≥ 0∧[(-1)bso_36] ≥ 0)

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

(18)    (x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]x6[2] + [bni_35]x4[2] ≥ 0∧[(-1)bso_36] ≥ 0)

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

(19)    (x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_35] = 0∧[bni_35] = 0∧[(-1)bni_35 + (-1)Bound*bni_35] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)

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

(20)    (x2[2] ≥ 0 ⇒ (UIncreasing(COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_35] = 0∧[bni_35] = 0∧[(-1)bni_35 + (-1)Bound*bni_35] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)

For Pair COND_4199_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6) → 5728_0_CREATEMETALIST_LE(x4, x6, x1) the following chains were created:
• We consider the chain COND_4199_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3]) which results in the following constraint:

(21)    (COND_4199_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3])≥NonInfC∧COND_4199_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3])≥5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])∧(UIncreasing(5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥))

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

(22)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥)∧[(-1)bso_38] ≥ 0)

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

(23)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥)∧[(-1)bso_38] ≥ 0)

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

(24)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥)∧[(-1)bso_38] ≥ 0)

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

(25)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

For Pair 5728_0_CREATEMETALIST_LE(x0, x2, 0) → COND_5728_0_CREATEMETALIST_LE(&&(>=(x2, 0), >(x0, +(x2, 1))), x0, x2, 0) the following chains were created:
• We consider the chain 5728_0_CREATEMETALIST_LE(x0[4], x2[4], 0) → COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0), COND_5728_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0) → 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1)) which results in the following constraint:

(26)    (&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1)))=TRUEx0[4]=x0[5]x2[4]=x2[5]5728_0_CREATEMETALIST_LE(x0[4], x2[4], 0)≥NonInfC∧5728_0_CREATEMETALIST_LE(x0[4], x2[4], 0)≥COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)∧(UIncreasing(COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥))

We simplified constraint (26) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(27)    (>=(x2[4], 0)=TRUE>(x0[4], +(x2[4], 1))=TRUE5728_0_CREATEMETALIST_LE(x0[4], x2[4], 0)≥NonInfC∧5728_0_CREATEMETALIST_LE(x0[4], x2[4], 0)≥COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)∧(UIncreasing(COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥))

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

(28)    (x2[4] ≥ 0∧x0[4] + [-2] + [-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]x2[4] + [bni_39]x0[4] ≥ 0∧[(-1)bso_40] ≥ 0)

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

(29)    (x2[4] ≥ 0∧x0[4] + [-2] + [-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]x2[4] + [bni_39]x0[4] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(30)    (x2[4] ≥ 0∧x0[4] + [-2] + [-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]x2[4] + [bni_39]x0[4] ≥ 0∧[(-1)bso_40] ≥ 0)

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

(31)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥)∧[bni_39 + (-1)Bound*bni_39] + [bni_39]x0[4] ≥ 0∧[(-1)bso_40] ≥ 0)

For Pair COND_5728_0_CREATEMETALIST_LE(TRUE, x0, x2, 0) → 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), x0, +(x2, 1)) the following chains were created:
• We consider the chain COND_5728_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0) → 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1)) which results in the following constraint:

(32)    (COND_5728_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0)≥NonInfC∧COND_5728_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0)≥4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))∧(UIncreasing(4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥))

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

(33)    ((UIncreasing(4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥)∧[1 + (-1)bso_42] ≥ 0)

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

(34)    ((UIncreasing(4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥)∧[1 + (-1)bso_42] ≥ 0)

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

(35)    ((UIncreasing(4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥)∧[1 + (-1)bso_42] ≥ 0)

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

(36)    ((UIncreasing(4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_42] ≥ 0)

For Pair 5728_0_CREATEMETALIST_LE(x0, x2, x4) → COND_5728_0_CREATEMETALIST_LE1(>(x4, 0), x0, x2, x4) the following chains were created:
• We consider the chain 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6]), COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:

(37)    (>(x4[6], 0)=TRUEx0[6]=x0[7]x2[6]=x2[7]x4[6]=x4[7]5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

We simplified constraint (37) using rule (IV) which results in the following new constraint:

(38)    (>(x4[6], 0)=TRUE5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

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

(39)    (x4[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]x2[6] + [bni_43]x0[6] ≥ 0∧[(-1)bso_44] ≥ 0)

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

(40)    (x4[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]x2[6] + [bni_43]x0[6] ≥ 0∧[(-1)bso_44] ≥ 0)

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

(41)    (x4[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]x2[6] + [bni_43]x0[6] ≥ 0∧[(-1)bso_44] ≥ 0)

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

(42)    (x4[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43] = 0∧[bni_43] = 0∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(43)    (x4[6] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43] = 0∧[bni_43] = 0∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_44] ≥ 0)

For Pair COND_5728_0_CREATEMETALIST_LE1(TRUE, x0, x2, x4) → 5728_0_CREATEMETALIST_LE(x0, x2, +(x4, -1)) the following chains were created:
• We consider the chain COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:

(44)    (COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥NonInfC∧COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))∧(UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥))

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

(45)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[(-1)bso_46] ≥ 0)

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

(46)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[(-1)bso_46] ≥ 0)

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

(47)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[(-1)bso_46] ≥ 0)

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

(48)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_46] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5) → COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5)
• (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])), ≥)∧[(-1)bni_31] = 0∧[bni_31] = 0∧0 = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 0)

• COND_4016_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3, x5) → 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6, x7), x3, x5)
• ((UIncreasing(4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_34] ≥ 0)

• 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6) → COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2, 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6)
• (x2[2] ≥ 0 ⇒ (UIncreasing(COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_35] = 0∧[bni_35] = 0∧[(-1)bni_35 + (-1)Bound*bni_35] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)

• COND_4199_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4, x6) → 5728_0_CREATEMETALIST_LE(x4, x6, x1)
• ((UIncreasing(5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

• 5728_0_CREATEMETALIST_LE(x0, x2, 0) → COND_5728_0_CREATEMETALIST_LE(&&(>=(x2, 0), >(x0, +(x2, 1))), x0, x2, 0)
• (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)), ≥)∧[bni_39 + (-1)Bound*bni_39] + [bni_39]x0[4] ≥ 0∧[(-1)bso_40] ≥ 0)

• COND_5728_0_CREATEMETALIST_LE(TRUE, x0, x2, 0) → 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), x0, +(x2, 1))
• ((UIncreasing(4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_42] ≥ 0)

• 5728_0_CREATEMETALIST_LE(x0, x2, x4) → COND_5728_0_CREATEMETALIST_LE1(>(x4, 0), x0, x2, x4)
• (x4[6] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_43] = 0∧[bni_43] = 0∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_44] ≥ 0)

• COND_5728_0_CREATEMETALIST_LE1(TRUE, x0, x2, x4) → 5728_0_CREATEMETALIST_LE(x0, x2, +(x4, -1))
• ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_46] ≥ 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(4016_1_CREATEMETALIST_INVOKEMETHOD(x1, x2, x3)) = [-1] + [-1]x3 + x2 + [-1]x1
POL(4016_0_random_ArrayAccess(x1, x2)) = [-1] + [-1]x1
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1, x2)) = [-1]
POL(COND_4016_1_CREATEMETALIST_INVOKEMETHOD(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3 + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(1) = [1]
POL(<(x1, x2)) = [-1]
POL(4199_1_CREATEMETALIST_INVOKEMETHOD(x1, x2, x3)) = [-1] + [-1]x1 + [-1]x3 + x2
POL(4199_0_random_IntArithmetic(x1, x2)) = 0
POL(java.lang.String(x1, x2)) = [-1] + [-1]x2 + x1
POL(COND_4199_1_CREATEMETALIST_INVOKEMETHOD(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3 + [-1]x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(5728_0_CREATEMETALIST_LE(x1, x2, x3)) = [-1] + [-1]x2 + x1
POL(COND_5728_0_CREATEMETALIST_LE(x1, x2, x3, x4)) = [-1] + [-1]x3 + x2
POL(+(x1, x2)) = x1 + x2
POL(COND_5728_0_CREATEMETALIST_LE1(x1, x2, x3, x4)) = [-1] + [-1]x3 + x2
POL(-1) = [-1]

The following pairs are in P>:

COND_5728_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0) → 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], +(x2[5], 1))

The following pairs are in Pbound:

5728_0_CREATEMETALIST_LE(x0[4], x2[4], 0) → COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)

The following pairs are in P:

4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]) → COND_4016_1_CREATEMETALIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])
COND_4016_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])
4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]) → COND_4199_1_CREATEMETALIST_INVOKEMETHOD(>(x2[2], 0), 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])
COND_4199_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])
5728_0_CREATEMETALIST_LE(x0[4], x2[4], 0) → COND_5728_0_CREATEMETALIST_LE(&&(>=(x2[4], 0), >(x0[4], +(x2[4], 1))), x0[4], x2[4], 0)
5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])
COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))

There are no usable rules.

(19) 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): 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]) → COND_4016_1_CREATEMETALIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])
(1): COND_4016_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])
(2): 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]) → COND_4199_1_CREATEMETALIST_INVOKEMETHOD(x2[2] > 0, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])
(3): COND_4199_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])
(4): 5728_0_CREATEMETALIST_LE(x0[4], x2[4], 0) → COND_5728_0_CREATEMETALIST_LE(x2[4] >= 0 && x0[4] > x2[4] + 1, x0[4], x2[4], 0)
(6): 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])
(7): COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)

(0) -> (1), if ((x2[0] >= 1 && x2[0] < x0[0]* TRUE)∧(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →* 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0]* x3[1])∧(x5[0]* x5[1]))

(1) -> (2), if ((4199_0_random_IntArithmetic(x6[1], x7[1]) →* 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1]* x4[2])∧(x5[1]* x6[2]))

(2) -> (3), if ((x2[2] > 0* TRUE)∧(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]) →* 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]))∧(x4[2]* x4[3])∧(x6[2]* x6[3]))

(3) -> (4), if ((x4[3]* x0[4])∧(x6[3]* x2[4])∧(x1[3]* 0))

(7) -> (4), if ((x0[7]* x0[4])∧(x2[7]* x2[4])∧(x4[7] + -1* 0))

(3) -> (6), if ((x4[3]* x0[6])∧(x6[3]* x2[6])∧(x1[3]* x4[6]))

(7) -> (6), if ((x0[7]* x0[6])∧(x2[7]* x2[6])∧(x4[7] + -1* x4[6]))

(6) -> (7), if ((x4[6] > 0* TRUE)∧(x0[6]* x0[7])∧(x2[6]* x2[7])∧(x4[6]* x4[7]))

The set Q is empty.

(20) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(21) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(7): COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)
(6): 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])

(7) -> (6), if ((x0[7]* x0[6])∧(x2[7]* x2[6])∧(x4[7] + -1* x4[6]))

(6) -> (7), if ((x4[6] > 0* TRUE)∧(x0[6]* x0[7])∧(x2[6]* x2[7])∧(x4[6]* x4[7]))

The set Q is empty.

(22) IDPNonInfProof (SOUND transformation)

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

For Pair COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) the following chains were created:
• We consider the chain COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:

(1)    (COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥NonInfC∧COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))∧(UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(2)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[2 + (-1)bso_8] ≥ 0)

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

(3)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[2 + (-1)bso_8] ≥ 0)

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

(4)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[2 + (-1)bso_8] ≥ 0)

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

(5)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧[2 + (-1)bso_8] ≥ 0)

For Pair 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6]) the following chains were created:
• We consider the chain 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6]), COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:

(6)    (>(x4[6], 0)=TRUEx0[6]=x0[7]x2[6]=x2[7]x4[6]=x4[7]5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

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

(7)    (>(x4[6], 0)=TRUE5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

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

(8)    (x4[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x4[6] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(9)    (x4[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x4[6] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(10)    (x4[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x4[6] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(11)    (x4[6] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x4[6] ≥ 0∧[(-1)bso_10] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))
• ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧[2 + (-1)bso_8] ≥ 0)

• 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])
• (x4[6] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x4[6] ≥ 0∧[(-1)bso_10] ≥ 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(COND_5728_0_CREATEMETALIST_LE1(x1, x2, x3, x4)) = [-1] + [2]x4
POL(5728_0_CREATEMETALIST_LE(x1, x2, x3)) = [-1] + [2]x3
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))

The following pairs are in Pbound:

5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])

The following pairs are in P:

5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])

There are no usable rules.

(24) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(6): 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])

The set Q is empty.

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(27) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(7): COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)

The set Q is empty.

(28) IDependencyGraphProof (EQUIVALENT transformation)

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

(30) 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): 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0]) → COND_4016_1_CREATEMETALIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0], x5[0])
(1): COND_4016_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1], x5[1]) → 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(x6[1], x7[1]), x3[1], x5[1])
(2): 4199_1_CREATEMETALIST_INVOKEMETHOD(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2]) → COND_4199_1_CREATEMETALIST_INVOKEMETHOD(x2[2] > 0, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2], x6[2])
(3): COND_4199_1_CREATEMETALIST_INVOKEMETHOD(TRUE, 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3], x6[3]) → 5728_0_CREATEMETALIST_LE(x4[3], x6[3], x1[3])
(5): COND_5728_0_CREATEMETALIST_LE(TRUE, x0[5], x2[5], 0) → 4016_1_CREATEMETALIST_INVOKEMETHOD(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]), x0[5], x2[5] + 1)
(6): 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])
(7): COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)

(5) -> (0), if ((4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[5], x7[5])), x8[5]) →* 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x0[5]* x3[0])∧(x2[5] + 1* x5[0]))

(0) -> (1), if ((x2[0] >= 1 && x2[0] < x0[0]* TRUE)∧(4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →* 4016_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0]* x3[1])∧(x5[0]* x5[1]))

(1) -> (2), if ((4199_0_random_IntArithmetic(x6[1], x7[1]) →* 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1]* x4[2])∧(x5[1]* x6[2]))

(2) -> (3), if ((x2[2] > 0* TRUE)∧(4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]) →* 4199_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]))∧(x4[2]* x4[3])∧(x6[2]* x6[3]))

(3) -> (6), if ((x4[3]* x0[6])∧(x6[3]* x2[6])∧(x1[3]* x4[6]))

(7) -> (6), if ((x0[7]* x0[6])∧(x2[7]* x2[6])∧(x4[7] + -1* x4[6]))

(6) -> (7), if ((x4[6] > 0* TRUE)∧(x0[6]* x0[7])∧(x2[6]* x2[7])∧(x4[6]* x4[7]))

The set Q is empty.

(31) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(32) 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:
(7): COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)
(6): 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])

(7) -> (6), if ((x0[7]* x0[6])∧(x2[7]* x2[6])∧(x4[7] + -1* x4[6]))

(6) -> (7), if ((x4[6] > 0* TRUE)∧(x0[6]* x0[7])∧(x2[6]* x2[7])∧(x4[6]* x4[7]))

The set Q is empty.

(33) 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 COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) the following chains were created:
• We consider the chain COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:

(1)    (COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥NonInfC∧COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7])≥5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))∧(UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(2)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(3)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(4)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(5)    ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

For Pair 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6]) the following chains were created:
• We consider the chain 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6]), COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1)) which results in the following constraint:

(6)    (>(x4[6], 0)=TRUEx0[6]=x0[7]x2[6]=x2[7]x4[6]=x4[7]5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

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

(7)    (>(x4[6], 0)=TRUE5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥NonInfC∧5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6])≥COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])∧(UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥))

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

(8)    (x4[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x4[6] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(9)    (x4[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x4[6] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(10)    (x4[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x4[6] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(11)    (x4[6] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x4[6] ≥ 0∧[(-1)bso_9] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))
• ((UIncreasing(5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

• 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])
• (x4[6] ≥ 0 ⇒ (UIncreasing(COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x4[6] ≥ 0∧[(-1)bso_9] ≥ 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(COND_5728_0_CREATEMETALIST_LE1(x1, x2, x3, x4)) = [1] + x4
POL(5728_0_CREATEMETALIST_LE(x1, x2, x3)) = [1] + x3
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], +(x4[7], -1))

The following pairs are in Pbound:

5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])

The following pairs are in P:

5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(>(x4[6], 0), x0[6], x2[6], x4[6])

There are no usable rules.

(35) 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:
(6): 5728_0_CREATEMETALIST_LE(x0[6], x2[6], x4[6]) → COND_5728_0_CREATEMETALIST_LE1(x4[6] > 0, x0[6], x2[6], x4[6])

The set Q is empty.

(36) IDependencyGraphProof (EQUIVALENT transformation)

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

(38) 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:
(7): COND_5728_0_CREATEMETALIST_LE1(TRUE, x0[7], x2[7], x4[7]) → 5728_0_CREATEMETALIST_LE(x0[7], x2[7], x4[7] + -1)

The set Q is empty.

(39) IDependencyGraphProof (EQUIVALENT transformation)

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