Termination proof of /tmp/tmplUPOl9/CAppE.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPNonInfProof (⇒)

        ↳7 AND

            ↳8 IDP

                ↳9 IDependencyGraphProof (⇔)

                ↳10 TRUE

            ↳11 IDP

                ↳12 IDependencyGraphProof (⇔)

                ↳13 TRUE

    ↳14 IDP

        ↳15 IDPNonInfProof (⇒)

        ↳16 AND

            ↳17 IDP

                ↳18 IDependencyGraphProof (⇔)

                ↳19 TRUE

            ↳20 IDP

                ↳21 IDependencyGraphProof (⇔)

                ↳22 TRUE

(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.

(4) Complex Obligation (AND)

(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)=TRUE∧x0[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])∧(U^Increasing(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)=TRUE ⇒ 757_0_APPE_FIELDACCESS(x0[0])≥NonInfC∧757_0_APPE_FIELDACCESS(x0[0])≥COND_757_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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))∧(U^Increasing(757_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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 ⇒ (U^Increasing(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))
- ((U^Increasing(757_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_15] ≥ 0)

The constraints for P_> respective P_bound 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 P_bound.
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 P_bound:

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.

(7) Complex Obligation (AND)

(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.

(10) TRUE

(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.

(13) TRUE

(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)=TRUE∧x0[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])∧(U^Increasing(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)=TRUE ⇒ 366_0_CREATELIST_LE(x0[0])≥NonInfC∧366_0_CREATELIST_LE(x0[0])≥COND_366_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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))∧(U^Increasing(366_0_CREATELIST_LE(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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 ⇒ (U^Increasing(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))
- ((U^Increasing(366_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 0)

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 P_bound:

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.

(16) Complex Obligation (AND)

(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.

(19) TRUE

(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.

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) IDPNonInfProof (SOUND transformation)

(7) Complex Obligation (AND)

(8) Obligation:

(9) IDependencyGraphProof (EQUIVALENT transformation)

(10) TRUE

(11) Obligation:

(12) IDependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) IDPNonInfProof (SOUND transformation)

(16) Complex Obligation (AND)

(17) Obligation:

(18) IDependencyGraphProof (EQUIVALENT transformation)

(19) TRUE

(20) Obligation:

(21) IDependencyGraphProof (EQUIVALENT transformation)

(22) TRUE