### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: CAppE
`public class CAppE {	CAppE next;	public static void main(String[] args) {		Random.args = args;		CAppE list = createList();		cappE(Random.random());	}	public static void cappE(int j) {		CAppE a = new CAppE();		if (j > 0) {			a.appE(j);			while (a.next == null) {}		}	}	public void appE(int i) {		if (next == null) {			if (i <= 0) {				return;			} else {				next = new CAppE();			}			i--;	 	}		next.appE(i);	}	public static CAppE createList() {		CAppE result = null;		int length = Random.random();		while (length > 0) {			result = new CAppE(result);			length--;		}		return result;	}	public CAppE() {		this.next = null;	}	public CAppE(CAppE n) {		this.next = n;	}}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:
CAppE.main([Ljava/lang/String;)V: Graph of 118 nodes with 0 SCCs.

CAppE.createList()LCAppE;: Graph of 91 nodes with 1 SCC.

CAppE.appE(I)V: Graph of 35 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 25 rules for P and 9 rules for R.

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

Filtered ground terms:

757_0_appE_FieldAccess(x1, x2) → 757_0_appE_FieldAccess(x2)
Cond_757_0_appE_FieldAccess(x1, x2, x3) → Cond_757_0_appE_FieldAccess(x1, x3)
874_0_appE_Return(x1) → 874_0_appE_Return
821_0_appE_Return(x1) → 821_0_appE_Return
768_0_appE_Return(x1, x2) → 768_0_appE_Return

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

Finished conversion. Obtained 1 rules for P and 3 rules for R. System has predefined symbols.

Log for SCC 1:

Generated 17 rules for P and 3 rules for R.

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

Filtered ground terms:

366_0_createList_LE(x1, x2, x3) → 366_0_createList_LE(x2, x3)
Cond_366_0_createList_LE(x1, x2, x3, x4) → Cond_366_0_createList_LE(x1, x3, x4)
395_0_createList_Return(x1) → 395_0_createList_Return

Filtered duplicate args:

366_0_createList_LE(x1, x2) → 366_0_createList_LE(x2)
Cond_366_0_createList_LE(x1, x2, x3) → Cond_366_0_createList_LE(x1, x3)

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

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

### (5) Obligation:

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
807_1_appE_InvokeMethod(768_0_appE_Return, 0) → 821_0_appE_Return
807_1_appE_InvokeMethod(821_0_appE_Return, x0) → 874_0_appE_Return
807_1_appE_InvokeMethod(874_0_appE_Return, x0) → 874_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 757_0_APPE_FIELDACCESS(x0[0]) → COND_757_0_APPE_FIELDACCESS(x0[0] > 0, x0[0])
(1): COND_757_0_APPE_FIELDACCESS(TRUE, x0[1]) → 757_0_APPE_FIELDACCESS(x0[1] + -1)

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

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

The set Q consists of the following terms:
807_1_appE_InvokeMethod(768_0_appE_Return, 0)
807_1_appE_InvokeMethod(821_0_appE_Return, x0)
807_1_appE_InvokeMethod(874_0_appE_Return, x0)

### (6) IDPNonInfProof (SOUND transformation)

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

For Pair 757_0_APPE_FIELDACCESS(x0) → COND_757_0_APPE_FIELDACCESS(>(x0, 0), x0) the following chains were created:
• We consider the chain 757_0_APPE_FIELDACCESS(x0[0]) → COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0]), COND_757_0_APPE_FIELDACCESS(TRUE, x0[1]) → 757_0_APPE_FIELDACCESS(+(x0[1], -1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]757_0_APPE_FIELDACCESS(x0[0])≥NonInfC∧757_0_APPE_FIELDACCESS(x0[0])≥COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])∧(UIncreasing(COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE757_0_APPE_FIELDACCESS(x0[0])≥NonInfC∧757_0_APPE_FIELDACCESS(x0[0])≥COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])∧(UIncreasing(COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_12 + (2)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_757_0_APPE_FIELDACCESS(TRUE, x0) → 757_0_APPE_FIELDACCESS(+(x0, -1)) the following chains were created:
• We consider the chain COND_757_0_APPE_FIELDACCESS(TRUE, x0[1]) → 757_0_APPE_FIELDACCESS(+(x0[1], -1)) which results in the following constraint:

(7)    (COND_757_0_APPE_FIELDACCESS(TRUE, x0[1])≥NonInfC∧COND_757_0_APPE_FIELDACCESS(TRUE, x0[1])≥757_0_APPE_FIELDACCESS(+(x0[1], -1))∧(UIncreasing(757_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥))

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

(8)    ((UIncreasing(757_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[2 + (-1)bso_15] ≥ 0)

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

(9)    ((UIncreasing(757_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[2 + (-1)bso_15] ≥ 0)

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

(10)    ((UIncreasing(757_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[2 + (-1)bso_15] ≥ 0)

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

(11)    ((UIncreasing(757_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_15] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 757_0_APPE_FIELDACCESS(x0) → COND_757_0_APPE_FIELDACCESS(>(x0, 0), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_12 + (2)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

• COND_757_0_APPE_FIELDACCESS(TRUE, x0) → 757_0_APPE_FIELDACCESS(+(x0, -1))
• ((UIncreasing(757_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_15] ≥ 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(807_1_appE_InvokeMethod(x1, x2)) = [-1]
POL(768_0_appE_Return) = [-1]
POL(0) = 0
POL(821_0_appE_Return) = [-1]
POL(874_0_appE_Return) = [-1]
POL(757_0_APPE_FIELDACCESS(x1)) = [2]x1
POL(COND_757_0_APPE_FIELDACCESS(x1, x2)) = [2]x2
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

COND_757_0_APPE_FIELDACCESS(TRUE, x0[1]) → 757_0_APPE_FIELDACCESS(+(x0[1], -1))

The following pairs are in Pbound:

757_0_APPE_FIELDACCESS(x0[0]) → COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])

The following pairs are in P:

757_0_APPE_FIELDACCESS(x0[0]) → COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])

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:

Integer

The ITRS R consists of the following rules:
807_1_appE_InvokeMethod(768_0_appE_Return, 0) → 821_0_appE_Return
807_1_appE_InvokeMethod(821_0_appE_Return, x0) → 874_0_appE_Return
807_1_appE_InvokeMethod(874_0_appE_Return, x0) → 874_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 757_0_APPE_FIELDACCESS(x0[0]) → COND_757_0_APPE_FIELDACCESS(x0[0] > 0, x0[0])

The set Q consists of the following terms:
807_1_appE_InvokeMethod(768_0_appE_Return, 0)
807_1_appE_InvokeMethod(821_0_appE_Return, x0)
807_1_appE_InvokeMethod(874_0_appE_Return, x0)

### (9) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:

Integer

The ITRS R consists of the following rules:
807_1_appE_InvokeMethod(768_0_appE_Return, 0) → 821_0_appE_Return
807_1_appE_InvokeMethod(821_0_appE_Return, x0) → 874_0_appE_Return
807_1_appE_InvokeMethod(874_0_appE_Return, x0) → 874_0_appE_Return

The integer pair graph contains the following rules and edges:
(1): COND_757_0_APPE_FIELDACCESS(TRUE, x0[1]) → 757_0_APPE_FIELDACCESS(x0[1] + -1)

The set Q consists of the following terms:
807_1_appE_InvokeMethod(768_0_appE_Return, 0)
807_1_appE_InvokeMethod(821_0_appE_Return, x0)
807_1_appE_InvokeMethod(874_0_appE_Return, x0)

### (12) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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

The ITRS R consists of the following rules:
366_0_createList_LE(0) → 395_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 366_0_CREATELIST_LE(x0[0]) → COND_366_0_CREATELIST_LE(x0[0] > 0, x0[0])
(1): COND_366_0_CREATELIST_LE(TRUE, x0[1]) → 366_0_CREATELIST_LE(x0[1] + -1)

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

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

The set Q consists of the following terms:
366_0_createList_LE(0)

### (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 366_0_CREATELIST_LE(x0) → COND_366_0_CREATELIST_LE(>(x0, 0), x0) the following chains were created:
• We consider the chain 366_0_CREATELIST_LE(x0[0]) → COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0]), COND_366_0_CREATELIST_LE(TRUE, x0[1]) → 366_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]366_0_CREATELIST_LE(x0[0])≥NonInfC∧366_0_CREATELIST_LE(x0[0])≥COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE366_0_CREATELIST_LE(x0[0])≥NonInfC∧366_0_CREATELIST_LE(x0[0])≥COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_366_0_CREATELIST_LE(TRUE, x0) → 366_0_CREATELIST_LE(+(x0, -1)) the following chains were created:
• We consider the chain COND_366_0_CREATELIST_LE(TRUE, x0[1]) → 366_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

(7)    (COND_366_0_CREATELIST_LE(TRUE, x0[1])≥NonInfC∧COND_366_0_CREATELIST_LE(TRUE, x0[1])≥366_0_CREATELIST_LE(+(x0[1], -1))∧(UIncreasing(366_0_CREATELIST_LE(+(x0[1], -1))), ≥))

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

(8)    ((UIncreasing(366_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)

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

(9)    ((UIncreasing(366_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(366_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(366_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 366_0_CREATELIST_LE(x0) → COND_366_0_CREATELIST_LE(>(x0, 0), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

• COND_366_0_CREATELIST_LE(TRUE, x0) → 366_0_CREATELIST_LE(+(x0, -1))
• ((UIncreasing(366_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 0)

The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(366_0_createList_LE(x1)) = [-1]
POL(0) = 0
POL(395_0_createList_Return) = [-1]
POL(366_0_CREATELIST_LE(x1)) = [2]x1
POL(COND_366_0_CREATELIST_LE(x1, x2)) = [2]x2
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

COND_366_0_CREATELIST_LE(TRUE, x0[1]) → 366_0_CREATELIST_LE(+(x0[1], -1))

The following pairs are in Pbound:

366_0_CREATELIST_LE(x0[0]) → COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P:

366_0_CREATELIST_LE(x0[0]) → COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])

There are no usable rules.

### (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

The ITRS R consists of the following rules:
366_0_createList_LE(0) → 395_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 366_0_CREATELIST_LE(x0[0]) → COND_366_0_CREATELIST_LE(x0[0] > 0, x0[0])

The set Q consists of the following terms:
366_0_createList_LE(0)

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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

The ITRS R consists of the following rules:
366_0_createList_LE(0) → 395_0_createList_Return

The integer pair graph contains the following rules and edges:
(1): COND_366_0_CREATELIST_LE(TRUE, x0[1]) → 366_0_CREATELIST_LE(x0[1] + -1)

The set Q consists of the following terms:
366_0_createList_LE(0)

### (21) IDependencyGraphProof (EQUIVALENT transformation)

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