Termination proof of /tmp/tmpHU8Cpn/CyclicAnalysis.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 260 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 AND

    ↳5 JBCTerminationSCC

        ↳6 SCCToIDPv1Proof (⇒, 180 ms)

        ↳7 IDP

        ↳8 IDPtoQDPProof (⇒, 0 ms)

        ↳9 QDP

        ↳10 QDPSizeChangeProof (⇔, 0 ms)

        ↳11 YES

    ↳12 JBCTerminationSCC

        ↳13 SCCToIDPv1Proof (⇒, 100 ms)

        ↳14 IDP

        ↳15 IDPNonInfProof (⇒, 90 ms)

        ↳16 AND

            ↳17 IDP

                ↳18 IDependencyGraphProof (⇔, 0 ms)

                ↳19 TRUE

            ↳20 IDP

                ↳21 IDependencyGraphProof (⇔, 0 ms)

                ↳22 TRUE

    ↳23 JBCTerminationSCC

        ↳24 SCCToIDPv1Proof (⇒, 80 ms)

        ↳25 IDP

        ↳26 IDPNonInfProof (⇒, 80 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:
1035_0_length_NULL(EOS(STATIC_1035), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(o265sub), java.lang.Object(o265sub)) → 1037_0_length_NULL(EOS(STATIC_1037), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(o265sub), java.lang.Object(o265sub))
1037_0_length_NULL(EOS(STATIC_1037), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(o265sub), java.lang.Object(o265sub)) → 1040_0_length_Load(EOS(STATIC_1040), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(o265sub))
1040_0_length_Load(EOS(STATIC_1040), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(o265sub)) → 1046_0_length_FieldAccess(EOS(STATIC_1046), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(o265sub))
1046_0_length_FieldAccess(EOS(STATIC_1046), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(o265sub)) → 1050_0_length_FieldAccess(EOS(STATIC_1050), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(o265sub))
1046_0_length_FieldAccess(EOS(STATIC_1046), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(o258sub)) → 1052_0_length_FieldAccess(EOS(STATIC_1052), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(o258sub))
1050_0_length_FieldAccess(EOS(STATIC_1050), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o267))) → 1056_0_length_FieldAccess(EOS(STATIC_1056), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o267)))
1056_0_length_FieldAccess(EOS(STATIC_1056), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o267))) → 1061_0_length_Store(EOS(STATIC_1061), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o267)
1061_0_length_Store(EOS(STATIC_1061), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o267) → 1067_0_length_Inc(EOS(STATIC_1067), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o267)
1067_0_length_Inc(EOS(STATIC_1067), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o267) → 1072_0_length_JMP(EOS(STATIC_1072), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o267)
1072_0_length_JMP(EOS(STATIC_1072), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o267) → 1080_0_length_Load(EOS(STATIC_1080), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o267)
1080_0_length_Load(EOS(STATIC_1080), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o267) → 1031_0_length_Load(EOS(STATIC_1031), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o267)
1031_0_length_Load(EOS(STATIC_1031), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o256) → 1035_0_length_NULL(EOS(STATIC_1035), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(o258sub))), o256, o256)
1052_0_length_FieldAccess(EOS(STATIC_1052), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269))) → 1057_0_length_FieldAccess(EOS(STATIC_1057), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))
1057_0_length_FieldAccess(EOS(STATIC_1057), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269))) → 1064_0_length_Store(EOS(STATIC_1064), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), o269)
1064_0_length_Store(EOS(STATIC_1064), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), o269) → 1070_0_length_Inc(EOS(STATIC_1070), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), o269)
1070_0_length_Inc(EOS(STATIC_1070), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), o269) → 1075_0_length_JMP(EOS(STATIC_1075), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), o269)
1075_0_length_JMP(EOS(STATIC_1075), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), o269) → 1083_0_length_Load(EOS(STATIC_1083), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), o269)
1083_0_length_Load(EOS(STATIC_1083), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), o269) → 1031_0_length_Load(EOS(STATIC_1031), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, o269)))), o269)
R rules:

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

P rules:
1035_0_length_NULL(EOS(STATIC_1035), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(x0))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, x1)), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, x1))) → 1035_0_length_NULL(EOS(STATIC_1035), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(x0))), x1, x1)
1035_0_length_NULL(EOS(STATIC_1035), 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))) → 1035_0_length_NULL(EOS(STATIC_1035), java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, java.lang.Object(CyclicAnalysis.CyclicAnalysis(EOC, x0)))), x0, x0)
R rules:

Filtered ground terms:

1035_0_length_NULL(x1, x2, x3, x4) → 1035_0_length_NULL(x2, x3, x4)
CyclicAnalysis.CyclicAnalysis(x1, x2) → CyclicAnalysis.CyclicAnalysis(x2)
EOS(x1) → EOS

Filtered duplicate args:

1035_0_length_NULL(x1, x2, x3) → 1035_0_length_NULL(x1, x3)

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

P rules:
1035_0_length_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1))) → 1035_0_length_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0))), x1)
1035_0_length_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0))) → 1035_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:
1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1))) → 1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0))), x1)
1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0)))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0))) → 1035_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): 1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1[0]))) → 1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), x1[0])
(1): 1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1]))) → 1035_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:

1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1[0]))) → 1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), x1[0])
1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1]))) → 1035_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:

1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(x0[0]))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x1[0]))) → 1035_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
1035_0_LENGTH_NULL(java.lang.Object(CyclicAnalysis.CyclicAnalysis(java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1])))), java.lang.Object(CyclicAnalysis.CyclicAnalysis(x0[1]))) → 1035_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:
772_0_appendNewList_ConstantStackPush(EOS(STATIC_772), i106, i106) → 775_0_appendNewList_LE(EOS(STATIC_775), i106, i106, 1)
775_0_appendNewList_LE(EOS(STATIC_775), i121, i121, matching1) → 778_0_appendNewList_LE(EOS(STATIC_778), i121, i121, 1) | =(matching1, 1)
778_0_appendNewList_LE(EOS(STATIC_778), i121, i121, matching1) → 783_0_appendNewList_Inc(EOS(STATIC_783), i121) | &&(>(i121, 1), =(matching1, 1))
783_0_appendNewList_Inc(EOS(STATIC_783), i121) → 788_0_appendNewList_Load(EOS(STATIC_788), +(i121, -1)) | >(i121, 0)
788_0_appendNewList_Load(EOS(STATIC_788), i122) → 792_0_appendNewList_New(EOS(STATIC_792), i122)
792_0_appendNewList_New(EOS(STATIC_792), i122) → 797_0_appendNewList_Duplicate(EOS(STATIC_797), i122)
797_0_appendNewList_Duplicate(EOS(STATIC_797), i122) → 802_0_appendNewList_InvokeMethod(EOS(STATIC_802), i122)
802_0_appendNewList_InvokeMethod(EOS(STATIC_802), i122) → 806_0_<init>_Load(EOS(STATIC_806), i122)
806_0_<init>_Load(EOS(STATIC_806), i122) → 812_0_<init>_InvokeMethod(EOS(STATIC_812), i122)
812_0_<init>_InvokeMethod(EOS(STATIC_812), i122) → 819_0_<init>_Return(EOS(STATIC_819), i122)
819_0_<init>_Return(EOS(STATIC_819), i122) → 823_0_appendNewList_Duplicate(EOS(STATIC_823), i122)
823_0_appendNewList_Duplicate(EOS(STATIC_823), i122) → 831_0_appendNewList_FieldAccess(EOS(STATIC_831), i122)
831_0_appendNewList_FieldAccess(EOS(STATIC_831), i122) → 835_0_appendNewList_FieldAccess(EOS(STATIC_835), i122)
831_0_appendNewList_FieldAccess(EOS(STATIC_831), i122) → 836_0_appendNewList_FieldAccess(EOS(STATIC_836), i122)
835_0_appendNewList_FieldAccess(EOS(STATIC_835), i122) → 840_0_appendNewList_Store(EOS(STATIC_840), i122)
840_0_appendNewList_Store(EOS(STATIC_840), i122) → 849_0_appendNewList_JMP(EOS(STATIC_849), i122)
849_0_appendNewList_JMP(EOS(STATIC_849), i122) → 861_0_appendNewList_Load(EOS(STATIC_861), i122)
861_0_appendNewList_Load(EOS(STATIC_861), i122) → 770_0_appendNewList_Load(EOS(STATIC_770), i122)
770_0_appendNewList_Load(EOS(STATIC_770), i106) → 772_0_appendNewList_ConstantStackPush(EOS(STATIC_772), i106, i106)
836_0_appendNewList_FieldAccess(EOS(STATIC_836), i122) → 845_0_appendNewList_Store(EOS(STATIC_845), i122)
845_0_appendNewList_Store(EOS(STATIC_845), i122) → 853_0_appendNewList_JMP(EOS(STATIC_853), i122)
853_0_appendNewList_JMP(EOS(STATIC_853), i122) → 866_0_appendNewList_Load(EOS(STATIC_866), i122)
866_0_appendNewList_Load(EOS(STATIC_866), i122) → 770_0_appendNewList_Load(EOS(STATIC_770), i122)
R rules:

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

P rules:
772_0_appendNewList_ConstantStackPush(EOS(STATIC_772), x0, x0) → 772_0_appendNewList_ConstantStackPush(EOS(STATIC_772), +(x0, -1), +(x0, -1)) | >(x0, 1)
R rules:

Filtered ground terms:

772_0_appendNewList_ConstantStackPush(x1, x2, x3) → 772_0_appendNewList_ConstantStackPush(x2, x3)
EOS(x1) → EOS
Cond_772_0_appendNewList_ConstantStackPush(x1, x2, x3, x4) → Cond_772_0_appendNewList_ConstantStackPush(x1, x3, x4)

Filtered duplicate args:

772_0_appendNewList_ConstantStackPush(x1, x2) → 772_0_appendNewList_ConstantStackPush(x2)
Cond_772_0_appendNewList_ConstantStackPush(x1, x2, x3) → Cond_772_0_appendNewList_ConstantStackPush(x1, x3)

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

P rules:
772_0_appendNewList_ConstantStackPush(x0) → 772_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:
772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0)
COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 772_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): 772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 772_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@5c382de5 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 772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:

We consider the chain 772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1)) which results in the following constraint:
(1)    (>(x0[0], 1)=TRUE∧x0[0]=x0[1] ⇒ 772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_772_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 ⇒ 772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_772_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_772_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_772_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_772_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_772_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_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0, -1)) the following chains were created:

We consider the chain COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1)) which results in the following constraint:
(7)    (COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1])≥772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))∧(U^Increasing(772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(772_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(772_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(772_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(772_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.

772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_772_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_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0, -1))
- ((U^Increasing(772_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(772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x₁)) = [2]x₁
POL(COND_772_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_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))

The following pairs are in P_bound:

772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

The following pairs are in P_≥:

772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_772_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_772_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 772_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:
377_0_appendNewList_ConstantStackPush(EOS(STATIC_377), i36, i36) → 378_0_appendNewList_LE(EOS(STATIC_378), i36, i36, 1)
378_0_appendNewList_LE(EOS(STATIC_378), i49, i49, matching1) → 382_0_appendNewList_LE(EOS(STATIC_382), i49, i49, 1) | =(matching1, 1)
382_0_appendNewList_LE(EOS(STATIC_382), i49, i49, matching1) → 386_0_appendNewList_Inc(EOS(STATIC_386), i49) | &&(>(i49, 1), =(matching1, 1))
386_0_appendNewList_Inc(EOS(STATIC_386), i49) → 390_0_appendNewList_Load(EOS(STATIC_390), +(i49, -1)) | >(i49, 0)
390_0_appendNewList_Load(EOS(STATIC_390), i50) → 395_0_appendNewList_New(EOS(STATIC_395), i50)
395_0_appendNewList_New(EOS(STATIC_395), i50) → 399_0_appendNewList_Duplicate(EOS(STATIC_399), i50)
399_0_appendNewList_Duplicate(EOS(STATIC_399), i50) → 403_0_appendNewList_InvokeMethod(EOS(STATIC_403), i50)
403_0_appendNewList_InvokeMethod(EOS(STATIC_403), i50) → 408_0_<init>_Load(EOS(STATIC_408), i50)
408_0_<init>_Load(EOS(STATIC_408), i50) → 426_0_<init>_InvokeMethod(EOS(STATIC_426), i50)
426_0_<init>_InvokeMethod(EOS(STATIC_426), i50) → 435_0_<init>_Return(EOS(STATIC_435), i50)
435_0_<init>_Return(EOS(STATIC_435), i50) → 441_0_appendNewList_Duplicate(EOS(STATIC_441), i50)
441_0_appendNewList_Duplicate(EOS(STATIC_441), i50) → 449_0_appendNewList_FieldAccess(EOS(STATIC_449), i50)
449_0_appendNewList_FieldAccess(EOS(STATIC_449), i50) → 454_0_appendNewList_FieldAccess(EOS(STATIC_454), i50)
449_0_appendNewList_FieldAccess(EOS(STATIC_449), i50) → 455_0_appendNewList_FieldAccess(EOS(STATIC_455), i50)
454_0_appendNewList_FieldAccess(EOS(STATIC_454), i50) → 464_0_appendNewList_Store(EOS(STATIC_464), i50)
464_0_appendNewList_Store(EOS(STATIC_464), i50) → 491_0_appendNewList_JMP(EOS(STATIC_491), i50)
491_0_appendNewList_JMP(EOS(STATIC_491), i50) → 505_0_appendNewList_Load(EOS(STATIC_505), i50)
505_0_appendNewList_Load(EOS(STATIC_505), i50) → 372_0_appendNewList_Load(EOS(STATIC_372), i50)
372_0_appendNewList_Load(EOS(STATIC_372), i36) → 377_0_appendNewList_ConstantStackPush(EOS(STATIC_377), i36, i36)
455_0_appendNewList_FieldAccess(EOS(STATIC_455), i50) → 467_0_appendNewList_Store(EOS(STATIC_467), i50)
467_0_appendNewList_Store(EOS(STATIC_467), i50) → 494_0_appendNewList_JMP(EOS(STATIC_494), i50)
494_0_appendNewList_JMP(EOS(STATIC_494), i50) → 514_0_appendNewList_Load(EOS(STATIC_514), i50)
514_0_appendNewList_Load(EOS(STATIC_514), i50) → 372_0_appendNewList_Load(EOS(STATIC_372), i50)
R rules:

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

P rules:
377_0_appendNewList_ConstantStackPush(EOS(STATIC_377), x0, x0) → 377_0_appendNewList_ConstantStackPush(EOS(STATIC_377), +(x0, -1), +(x0, -1)) | >(x0, 1)
R rules:

Filtered ground terms:

377_0_appendNewList_ConstantStackPush(x1, x2, x3) → 377_0_appendNewList_ConstantStackPush(x2, x3)
EOS(x1) → EOS
Cond_377_0_appendNewList_ConstantStackPush(x1, x2, x3, x4) → Cond_377_0_appendNewList_ConstantStackPush(x1, x3, x4)

Filtered duplicate args:

377_0_appendNewList_ConstantStackPush(x1, x2) → 377_0_appendNewList_ConstantStackPush(x2)
Cond_377_0_appendNewList_ConstantStackPush(x1, x2, x3) → Cond_377_0_appendNewList_ConstantStackPush(x1, x3)

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

P rules:
377_0_appendNewList_ConstantStackPush(x0) → 377_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:
377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0)
COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 377_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): 377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 377_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@5c382de5 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 377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:

We consider the chain 377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1)) which results in the following constraint:
(1)    (>(x0[0], 1)=TRUE∧x0[0]=x0[1] ⇒ 377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_377_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 ⇒ 377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0])≥COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_377_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_377_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_377_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_377_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_377_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_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0, -1)) the following chains were created:

We consider the chain COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1)) which results in the following constraint:
(7)    (COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1])≥377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))∧(U^Increasing(377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(377_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(377_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(377_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(377_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.

377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0) → COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0, 1), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_377_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_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0) → 377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0, -1))
- ((U^Increasing(377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x₁)) = [2]x₁
POL(COND_377_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_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(+(x0[1], -1))

The following pairs are in P_bound:

377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

The following pairs are in P_≥:

377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(x0[0]) → COND_377_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_377_0_APPENDNEWLIST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 377_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