Termination proof of /tmp/tmpx7n_hU/CyclicAnalysis.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 630 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 AND

    ↳5 JBCTerminationSCC

        ↳6 SCCToIDPv1Proof (⇒, 80 ms)

        ↳7 IDP

        ↳8 IDPtoQDPProof (⇒, 10 ms)

        ↳9 QDP

        ↳10 QDPSizeChangeProof (⇔, 0 ms)

        ↳11 YES

    ↳12 JBCTerminationSCC

        ↳13 SCCToIDPv1Proof (⇒, 110 ms)

        ↳14 IDP

        ↳15 IDPNonInfProof (⇒, 140 ms)

        ↳16 AND

            ↳17 IDP

                ↳18 IDependencyGraphProof (⇔, 0 ms)

                ↳19 TRUE

            ↳20 IDP

                ↳21 IDependencyGraphProof (⇔, 0 ms)

                ↳22 TRUE

    ↳23 JBCTerminationSCC

        ↳24 SCCToIDPv1Proof (⇒, 90 ms)

        ↳25 IDP

        ↳26 IDPNonInfProof (⇒, 100 ms)

        ↳27 AND

            ↳28 IDP

                ↳29 IDependencyGraphProof (⇔, 0 ms)

                ↳30 TRUE

            ↳31 IDP

                ↳32 IDependencyGraphProof (⇔, 0 ms)

                ↳33 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_25 (Sun Microsystems Inc.) Main-Class: CyclicAnalysis/CyclicAnalysis

package CyclicAnalysis;

public class CyclicAnalysis {
	CyclicAnalysis field;

	public static void main(String[] args) {
		Random.args = args;
		CyclicAnalysis t = new CyclicAnalysis();
		t.field = new CyclicAnalysis();
		t.field.appendNewCyclicList(Random.random());
		t.appendNewList(Random.random());
		t.length();
	}

	public int length() {
		int length = 1;
		CyclicAnalysis cur = this.field;
		while (cur != null) {
			cur = cur.field;
			length++;
		}
		return length;
	}

	public void appendNewCyclicList(int i) {
		CyclicAnalysis last = this.appendNewList(i);
		last.field = this;
	}

	/**
 	 * @param i number of elements to append
 	 * @return the last list element appended
 	 */
	private CyclicAnalysis appendNewList(int i) {
		this.field = new CyclicAnalysis();
		CyclicAnalysis cur = this.field;
		while (i > 1) {
			i--;
			cur = cur.field = new CyclicAnalysis();
		}
		return cur;
	}
}


package CyclicAnalysis;

public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    final String string = args[index];
    index++;
    return string.length();
  }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
CyclicAnalysis.CyclicAnalysis.main([Ljava/lang/String;)V: Graph of 307 nodes with 3 SCCs.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 3 SCCss.

(4) Complex Obligation (AND)

(5) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: CyclicAnalysis.CyclicAnalysis.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(6) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 18 rules for P and 0 rules for R.

P rules:
1487_0_length_NULL(EOS(STATIC_1487), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(o365sub), java.lang.Object(o365sub)) → 1488_0_length_NULL(EOS(STATIC_1488), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(o365sub), java.lang.Object(o365sub))
1488_0_length_NULL(EOS(STATIC_1488), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(o365sub), java.lang.Object(o365sub)) → 1490_0_length_Load(EOS(STATIC_1490), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(o365sub))
1490_0_length_Load(EOS(STATIC_1490), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(o365sub)) → 1492_0_length_FieldAccess(EOS(STATIC_1492), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(o365sub))
1492_0_length_FieldAccess(EOS(STATIC_1492), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(o365sub)) → 1495_0_length_FieldAccess(EOS(STATIC_1495), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(o365sub))
1492_0_length_FieldAccess(EOS(STATIC_1492), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(o357sub)) → 1496_0_length_FieldAccess(EOS(STATIC_1496), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(o357sub))
1495_0_length_FieldAccess(EOS(STATIC_1495), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o367))) → 1499_0_length_FieldAccess(EOS(STATIC_1499), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o367)))
1499_0_length_FieldAccess(EOS(STATIC_1499), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o367))) → 1504_0_length_Store(EOS(STATIC_1504), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o367)
1504_0_length_Store(EOS(STATIC_1504), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o367) → 1509_0_length_Inc(EOS(STATIC_1509), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o367)
1509_0_length_Inc(EOS(STATIC_1509), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o367) → 1513_0_length_JMP(EOS(STATIC_1513), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o367)
1513_0_length_JMP(EOS(STATIC_1513), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o367) → 1519_0_length_Load(EOS(STATIC_1519), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o367)
1519_0_length_Load(EOS(STATIC_1519), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o367) → 1486_0_length_Load(EOS(STATIC_1486), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o367)
1486_0_length_Load(EOS(STATIC_1486), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o355) → 1487_0_length_NULL(EOS(STATIC_1487), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o357sub))), o355, o355)
1496_0_length_FieldAccess(EOS(STATIC_1496), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369))) → 1501_0_length_FieldAccess(EOS(STATIC_1501), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))
1501_0_length_FieldAccess(EOS(STATIC_1501), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369))) → 1506_0_length_Store(EOS(STATIC_1506), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), o369)
1506_0_length_Store(EOS(STATIC_1506), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), o369) → 1511_0_length_Inc(EOS(STATIC_1511), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), o369)
1511_0_length_Inc(EOS(STATIC_1511), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), o369) → 1515_0_length_JMP(EOS(STATIC_1515), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), o369)
1515_0_length_JMP(EOS(STATIC_1515), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), o369) → 1523_0_length_Load(EOS(STATIC_1523), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), o369)
1523_0_length_Load(EOS(STATIC_1523), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), o369) → 1486_0_length_Load(EOS(STATIC_1486), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o369)))), o369)
R rules:

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
1487_0_length_NULL(EOS(STATIC_1487), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(x0))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, x1)), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, x1))) → 1487_0_length_NULL(EOS(STATIC_1487), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(x0))), x1, x1)
1487_0_length_NULL(EOS(STATIC_1487), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, x0)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, x0)), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, x0))) → 1487_0_length_NULL(EOS(STATIC_1487), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, x0)))), x0, x0)
R rules:

Filtered ground terms:

1487_0_length_NULL(x1, x2, x3, x4) → 1487_0_length_NULL(x2, x3, x4)
CyclicAnalysis.CyclicAnalysis(x1, x2) → CyclicAnalysis.CyclicAnalysis(x2)
EOS(x1) → EOS

Filtered duplicate args:

1487_0_length_NULL(x1, x2, x3) → 1487_0_length_NULL(x1, x3)

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
1487_0_length_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1))) → 1487_0_length_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0))), x1)
1487_0_length_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0))) → 1487_0_length_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0)))), x0)
R rules:

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has no predefined symbols.

P rules:
1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1))) → 1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0))), x1)
1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0))) → 1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0)))), x0)
R rules:

(7) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:
none

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1[0]))) → 1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), x1[0])
(1): 1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1]))) → 1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))), x0[1])

(0) -> (0), if (java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))) →^* java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]')))∧x1[0] →^* java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1[0]')))

(0) -> (1), if (java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))) →^* java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1]))))∧x1[0] →^* java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))

(1) -> (0), if (java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))) →^* java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0])))∧x0[1] →^* java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1[0])))

(1) -> (1), if (java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))) →^* java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1]'))))∧x0[1] →^* java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1]')))

The set Q is empty.

(8) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(9) Obligation:

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

1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1[0]))) → 1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), x1[0])
1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1]))) → 1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))), x0[1])

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

(10) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1[0]))) → 1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), x1[0])
The graph contains the following edges 1 >= 1, 2 > 2
1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1]))) → 1487_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))), x0[1])
The graph contains the following edges 1 >= 1, 1 > 2, 2 > 2

(11) YES

(12) Obligation:

(13) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 23 rules for P and 0 rules for R.

P rules:
1084_0_appendNewList_ConstantStackPush(EOS(STATIC_1084), i128, i128) → 1088_0_appendNewList_LE(EOS(STATIC_1088), i128, i128, 1)
1088_0_appendNewList_LE(EOS(STATIC_1088), i138, i138, matching1) → 1091_0_appendNewList_LE(EOS(STATIC_1091), i138, i138, 1) | =(matching1, 1)
1091_0_appendNewList_LE(EOS(STATIC_1091), i138, i138, matching1) → 1095_0_appendNewList_Inc(EOS(STATIC_1095), i138) | &&(>(i138, 1), =(matching1, 1))
1095_0_appendNewList_Inc(EOS(STATIC_1095), i138) → 1100_0_appendNewList_Load(EOS(STATIC_1100), +(i138, -1)) | >(i138, 0)
1100_0_appendNewList_Load(EOS(STATIC_1100), i139) → 1104_0_appendNewList_New(EOS(STATIC_1104), i139)
1104_0_appendNewList_New(EOS(STATIC_1104), i139) → 1107_0_appendNewList_Duplicate(EOS(STATIC_1107), i139)
1107_0_appendNewList_Duplicate(EOS(STATIC_1107), i139) → 1109_0_appendNewList_InvokeMethod(EOS(STATIC_1109), i139)
1109_0_appendNewList_InvokeMethod(EOS(STATIC_1109), i139) → 1111_0_<init>_Load(EOS(STATIC_1111), i139)
1111_0_<init>_Load(EOS(STATIC_1111), i139) → 1116_0_<init>_InvokeMethod(EOS(STATIC_1116), i139)
1116_0_<init>_InvokeMethod(EOS(STATIC_1116), i139) → 1120_0_<init>_Return(EOS(STATIC_1120), i139)
1120_0_<init>_Return(EOS(STATIC_1120), i139) → 1124_0_appendNewList_Duplicate(EOS(STATIC_1124), i139)
1124_0_appendNewList_Duplicate(EOS(STATIC_1124), i139) → 1129_0_appendNewList_FieldAccess(EOS(STATIC_1129), i139)
1129_0_appendNewList_FieldAccess(EOS(STATIC_1129), i139) → 1133_0_appendNewList_FieldAccess(EOS(STATIC_1133), i139)
1129_0_appendNewList_FieldAccess(EOS(STATIC_1129), i139) → 1134_0_appendNewList_FieldAccess(EOS(STATIC_1134), i139)
1133_0_appendNewList_FieldAccess(EOS(STATIC_1133), i139) → 1138_0_appendNewList_Store(EOS(STATIC_1138), i139)
1138_0_appendNewList_Store(EOS(STATIC_1138), i139) → 1146_0_appendNewList_JMP(EOS(STATIC_1146), i139)
1146_0_appendNewList_JMP(EOS(STATIC_1146), i139) → 1150_0_appendNewList_Load(EOS(STATIC_1150), i139)
1150_0_appendNewList_Load(EOS(STATIC_1150), i139) → 1082_0_appendNewList_Load(EOS(STATIC_1082), i139)
1082_0_appendNewList_Load(EOS(STATIC_1082), i128) → 1084_0_appendNewList_ConstantStackPush(EOS(STATIC_1084), i128, i128)
1134_0_appendNewList_FieldAccess(EOS(STATIC_1134), i139) → 1142_0_appendNewList_Store(EOS(STATIC_1142), i139)
1142_0_appendNewList_Store(EOS(STATIC_1142), i139) → 1147_0_appendNewList_JMP(EOS(STATIC_1147), i139)
1147_0_appendNewList_JMP(EOS(STATIC_1147), i139) → 1156_0_appendNewList_Load(EOS(STATIC_1156), i139)
1156_0_appendNewList_Load(EOS(STATIC_1156), i139) → 1082_0_appendNewList_Load(EOS(STATIC_1082), i139)
R rules:

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

P rules:
1084_0_appendNewList_ConstantStackPush(EOS(STATIC_1084), x0, x0) → 1084_0_appendNewList_ConstantStackPush(EOS(STATIC_1084), +(x0, -1), +(x0, -1)) | >(x0, 1)
R rules:

Filtered ground terms:

1084_0_appendNewList_ConstantStackPush(x1, x2, x3) → 1084_0_appendNewList_ConstantStackPush(x2, x3)
EOS(x1) → EOS
Cond_1084_0_appendNewList_ConstantStackPush(x1, x2, x3, x4) → Cond_1084_0_appendNewList_ConstantStackPush(x1, x3, x4)

Filtered duplicate args:

1084_0_appendNewList_ConstantStackPush(x1, x2) → 1084_0_appendNewList_ConstantStackPush(x2)
Cond_1084_0_appendNewList_ConstantStackPush(x1, x2, x3) → Cond_1084_0_appendNewList_ConstantStackPush(x1, x3)

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

P rules:
1084_0_appendNewList_ConstantStackPush(x0) → 1084_0_appendNewList_ConstantStackPush(+(x0, -1)) | >(x0, 1)
R rules:

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

P rules:
1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0)
COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0, -1))
R rules:

(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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[1] + -1)

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

(1) -> (0), if (x0[1] + -1 →^* x0[0])

The set Q is empty.

(15) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@781d57c7 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:

We consider the chain 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1)) which results in the following constraint:
(1)    (>(x0[0], 1)=TRUE∧x0[0]=x0[1] ⇒ 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x0[0], 1)=TRUE ⇒ 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_8 + (4)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0, -1)) the following chains were created:

We consider the chain COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1)) which results in the following constraint:
(7)    (COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1])≥1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))∧(U^Increasing(1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_8 + (4)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)
COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0, -1))
- ((U^Increasing(1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 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(1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x₁)) = [2]x₁
POL(COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x₁, x₂)) = [2]x₂
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))

The following pairs are in P_bound:

1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

The following pairs are in P_≥:

1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), 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

(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

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 1084_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[1] + -1)

The set Q is empty.

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(22) TRUE

(23) Obligation:

(24) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 23 rules for P and 0 rules for R.

P rules:
385_0_appendNewList_ConstantStackPush(EOS(STATIC_385), i36, i36) → 388_0_appendNewList_LE(EOS(STATIC_388), i36, i36, 1)
388_0_appendNewList_LE(EOS(STATIC_388), i50, i50, matching1) → 392_0_appendNewList_LE(EOS(STATIC_392), i50, i50, 1) | =(matching1, 1)
392_0_appendNewList_LE(EOS(STATIC_392), i50, i50, matching1) → 396_0_appendNewList_Inc(EOS(STATIC_396), i50) | &&(>(i50, 1), =(matching1, 1))
396_0_appendNewList_Inc(EOS(STATIC_396), i50) → 400_0_appendNewList_Load(EOS(STATIC_400), +(i50, -1)) | >(i50, 0)
400_0_appendNewList_Load(EOS(STATIC_400), i51) → 402_0_appendNewList_New(EOS(STATIC_402), i51)
402_0_appendNewList_New(EOS(STATIC_402), i51) → 406_0_appendNewList_Duplicate(EOS(STATIC_406), i51)
406_0_appendNewList_Duplicate(EOS(STATIC_406), i51) → 410_0_appendNewList_InvokeMethod(EOS(STATIC_410), i51)
410_0_appendNewList_InvokeMethod(EOS(STATIC_410), i51) → 417_0_<init>_Load(EOS(STATIC_417), i51)
417_0_<init>_Load(EOS(STATIC_417), i51) → 440_0_<init>_InvokeMethod(EOS(STATIC_440), i51)
440_0_<init>_InvokeMethod(EOS(STATIC_440), i51) → 449_0_<init>_Return(EOS(STATIC_449), i51)
449_0_<init>_Return(EOS(STATIC_449), i51) → 456_0_appendNewList_Duplicate(EOS(STATIC_456), i51)
456_0_appendNewList_Duplicate(EOS(STATIC_456), i51) → 462_0_appendNewList_FieldAccess(EOS(STATIC_462), i51)
462_0_appendNewList_FieldAccess(EOS(STATIC_462), i51) → 467_0_appendNewList_FieldAccess(EOS(STATIC_467), i51)
462_0_appendNewList_FieldAccess(EOS(STATIC_462), i51) → 468_0_appendNewList_FieldAccess(EOS(STATIC_468), i51)
467_0_appendNewList_FieldAccess(EOS(STATIC_467), i51) → 476_0_appendNewList_Store(EOS(STATIC_476), i51)
476_0_appendNewList_Store(EOS(STATIC_476), i51) → 516_0_appendNewList_JMP(EOS(STATIC_516), i51)
516_0_appendNewList_JMP(EOS(STATIC_516), i51) → 538_0_appendNewList_Load(EOS(STATIC_538), i51)
538_0_appendNewList_Load(EOS(STATIC_538), i51) → 381_0_appendNewList_Load(EOS(STATIC_381), i51)
381_0_appendNewList_Load(EOS(STATIC_381), i36) → 385_0_appendNewList_ConstantStackPush(EOS(STATIC_385), i36, i36)
468_0_appendNewList_FieldAccess(EOS(STATIC_468), i51) → 482_0_appendNewList_Store(EOS(STATIC_482), i51)
482_0_appendNewList_Store(EOS(STATIC_482), i51) → 519_0_appendNewList_JMP(EOS(STATIC_519), i51)
519_0_appendNewList_JMP(EOS(STATIC_519), i51) → 551_0_appendNewList_Load(EOS(STATIC_551), i51)
551_0_appendNewList_Load(EOS(STATIC_551), i51) → 381_0_appendNewList_Load(EOS(STATIC_381), i51)
R rules:

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

P rules:
385_0_appendNewList_ConstantStackPush(EOS(STATIC_385), x0, x0) → 385_0_appendNewList_ConstantStackPush(EOS(STATIC_385), +(x0, -1), +(x0, -1)) | >(x0, 1)
R rules:

Filtered ground terms:

385_0_appendNewList_ConstantStackPush(x1, x2, x3) → 385_0_appendNewList_ConstantStackPush(x2, x3)
EOS(x1) → EOS
Cond_385_0_appendNewList_ConstantStackPush(x1, x2, x3, x4) → Cond_385_0_appendNewList_ConstantStackPush(x1, x3, x4)

Filtered duplicate args:

385_0_appendNewList_ConstantStackPush(x1, x2) → 385_0_appendNewList_ConstantStackPush(x2)
Cond_385_0_appendNewList_ConstantStackPush(x1, x2, x3) → Cond_385_0_appendNewList_ConstantStackPush(x1, x3)

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

P rules:
385_0_appendNewList_ConstantStackPush(x0) → 385_0_appendNewList_ConstantStackPush(+(x0, -1)) | >(x0, 1)
R rules:

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

P rules:
385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0)
COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0, -1))
R rules:

(25) 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:
(0): 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[1] + -1)

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

(1) -> (0), if (x0[1] + -1 →^* x0[0])

The set Q is empty.

(26) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@781d57c7 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:

We consider the chain 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1)) which results in the following constraint:
(1)    (>(x0[0], 1)=TRUE∧x0[0]=x0[1] ⇒ 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x0[0], 1)=TRUE ⇒ 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_8 + (4)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0, -1)) the following chains were created:

We consider the chain COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1)) which results in the following constraint:
(7)    (COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1])≥385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))∧(U^Increasing(385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_8 + (4)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)
COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0, -1))
- ((U^Increasing(385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x₁)) = [2]x₁
POL(COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x₁, x₂)) = [2]x₂
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))

The following pairs are in P_bound:

385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

The following pairs are in P_≥:

385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

There are no usable rules.

(27) Complex Obligation (AND)

(28) 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

(29) IDependencyGraphProof (EQUIVALENT transformation)

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

(30) TRUE

(31) 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:
(1): COND_385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 385_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[1] + -1)

The set Q is empty.

(32) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) SCCToIDPv1Proof (SOUND transformation)

(7) Obligation:

(8) IDPtoQDPProof (SOUND transformation)

(9) Obligation:

(10) QDPSizeChangeProof (EQUIVALENT transformation)

(11) YES

(12) Obligation:

(13) SCCToIDPv1Proof (SOUND transformation)

(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

(23) Obligation:

(24) SCCToIDPv1Proof (SOUND transformation)

(25) Obligation:

(26) IDPNonInfProof (SOUND transformation)

(27) Complex Obligation (AND)

(28) Obligation:

(29) IDependencyGraphProof (EQUIVALENT transformation)

(30) TRUE

(31) Obligation:

(32) IDependencyGraphProof (EQUIVALENT transformation)

(33) TRUE