Termination of the given JBC could be proven:

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: RunningPointers

public class ObjectList {
  Object value;
  ObjectList next;

  public ObjectList(Object value, ObjectList next) {
    this.value = value;
    this.next = next;
  }

  public static ObjectList createList() {
    ObjectList result = null;
    int length = Random.random();
    while (length > 0) {
      result = new ObjectList(new Object(), result);
      length--;
    }
    return result;
  }
}


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

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


/**
 * Allegedly based on an interview question at Microsoft.
 */
public class RunningPointers {

  public static boolean isCyclic(ObjectList l) {
    if (l == null) {
      return false;
    }
    ObjectList l1, l2;
    l1 = l;
    l2 = l.next;
    while (l2 != null && l1 != l2) {
      l2 = l2.next;
      if (l2 == null) {
        return false;
      }
      else if (l2 == l1) {
        return true;
      }
      else {
        l2 = l2.next;
      }
      l1 = l1.next;
    }
    return l2 != null;
  }

  public static void main(String[] args) {
    Random.args = args;
    ObjectList list = ObjectList.createList();
    isCyclic(list);
  }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
RunningPointers.main([Ljava/lang/String;)V: Graph of 164 nodes with 1 SCC.

ObjectList.createList()LObjectList;: Graph of 97 nodes with 1 SCC.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 2 SCCss.

(4) Complex Obligation (AND)

(5) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: ObjectList.createList()LObjectList;
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(6) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 23 rules for P and 0 rules for R.

P rules:
354_0_createList_LE(EOS(STATIC_354), i37, i37) → 357_0_createList_LE(EOS(STATIC_357), i37, i37)
357_0_createList_LE(EOS(STATIC_357), i37, i37) → 362_0_createList_New(EOS(STATIC_362), i37) | >(i37, 0)
362_0_createList_New(EOS(STATIC_362), i37) → 366_0_createList_Duplicate(EOS(STATIC_366), i37)
366_0_createList_Duplicate(EOS(STATIC_366), i37) → 369_0_createList_New(EOS(STATIC_369), i37)
369_0_createList_New(EOS(STATIC_369), i37) → 373_0_createList_Duplicate(EOS(STATIC_373), i37)
373_0_createList_Duplicate(EOS(STATIC_373), i37) → 379_0_createList_InvokeMethod(EOS(STATIC_379), i37)
379_0_createList_InvokeMethod(EOS(STATIC_379), i37) → 383_0_createList_Load(EOS(STATIC_383), i37)
383_0_createList_Load(EOS(STATIC_383), i37) → 387_0_createList_InvokeMethod(EOS(STATIC_387), i37)
387_0_createList_InvokeMethod(EOS(STATIC_387), i37) → 391_0_<init>_Load(EOS(STATIC_391), i37)
391_0_<init>_Load(EOS(STATIC_391), i37) → 395_0_<init>_InvokeMethod(EOS(STATIC_395), i37)
395_0_<init>_InvokeMethod(EOS(STATIC_395), i37) → 399_0_<init>_Load(EOS(STATIC_399), i37)
399_0_<init>_Load(EOS(STATIC_399), i37) → 404_0_<init>_Load(EOS(STATIC_404), i37)
404_0_<init>_Load(EOS(STATIC_404), i37) → 411_0_<init>_FieldAccess(EOS(STATIC_411), i37)
411_0_<init>_FieldAccess(EOS(STATIC_411), i37) → 418_0_<init>_Load(EOS(STATIC_418), i37)
418_0_<init>_Load(EOS(STATIC_418), i37) → 424_0_<init>_Load(EOS(STATIC_424), i37)
424_0_<init>_Load(EOS(STATIC_424), i37) → 431_0_<init>_FieldAccess(EOS(STATIC_431), i37)
431_0_<init>_FieldAccess(EOS(STATIC_431), i37) → 438_0_<init>_Return(EOS(STATIC_438), i37)
438_0_<init>_Return(EOS(STATIC_438), i37) → 441_0_createList_Store(EOS(STATIC_441), i37)
441_0_createList_Store(EOS(STATIC_441), i37) → 445_0_createList_Inc(EOS(STATIC_445), i37)
445_0_createList_Inc(EOS(STATIC_445), i37) → 450_0_createList_JMP(EOS(STATIC_450), +(i37, -1)) | >(i37, 0)
450_0_createList_JMP(EOS(STATIC_450), i43) → 456_0_createList_Load(EOS(STATIC_456), i43)
456_0_createList_Load(EOS(STATIC_456), i43) → 351_0_createList_Load(EOS(STATIC_351), i43)
351_0_createList_Load(EOS(STATIC_351), i34) → 354_0_createList_LE(EOS(STATIC_354), i34, i34)
R rules:

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

P rules:
354_0_createList_LE(EOS(STATIC_354), x0, x0) → 354_0_createList_LE(EOS(STATIC_354), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

354_0_createList_LE(x1, x2, x3) → 354_0_createList_LE(x2, x3)
EOS(x1) → EOS
Cond_354_0_createList_LE(x1, x2, x3, x4) → Cond_354_0_createList_LE(x1, x3, x4)

Filtered duplicate args:

354_0_createList_LE(x1, x2) → 354_0_createList_LE(x2)
Cond_354_0_createList_LE(x1, x2, x3) → Cond_354_0_createList_LE(x1, x3)

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

P rules:
354_0_createList_LE(x0) → 354_0_createList_LE(+(x0, -1)) | >(x0, 0)
R rules:

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

P rules:
354_0_CREATELIST_LE(x0) → COND_354_0_CREATELIST_LE(>(x0, 0), x0)
COND_354_0_CREATELIST_LE(TRUE, x0) → 354_0_CREATELIST_LE(+(x0, -1))
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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 354_0_CREATELIST_LE(x0[0]) → COND_354_0_CREATELIST_LE(x0[0] > 0, x0[0])
(1): COND_354_0_CREATELIST_LE(TRUE, x0[1]) → 354_0_CREATELIST_LE(x0[1] + -1)

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

(1) -> (0), if (x0[1] + -1 →^* x0[0])

The set Q is empty.

(8) 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@532c734c 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 354_0_CREATELIST_LE(x0) → COND_354_0_CREATELIST_LE(>(x0, 0), x0) the following chains were created:

• We consider the chain 354_0_CREATELIST_LE(x0[0]) → COND_354_0_CREATELIST_LE(>(x0[0], 0), x0[0]), COND_354_0_CREATELIST_LE(TRUE, x0[1]) → 354_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:
  

  (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 354_0_CREATELIST_LE(x0[0])≥NonInfC∧354_0_CREATELIST_LE(x0[0])≥COND_354_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_354_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 ⇒ 354_0_CREATELIST_LE(x0[0])≥NonInfC∧354_0_CREATELIST_LE(x0[0])≥COND_354_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_354_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_354_0_CREATELIST_LE(>(x0[0], 0), 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] + [-1] ≥ 0 ⇒ (U^Increasing(COND_354_0_CREATELIST_LE(>(x0[0], 0), 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] + [-1] ≥ 0 ⇒ (U^Increasing(COND_354_0_CREATELIST_LE(>(x0[0], 0), 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_354_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

  
  
  

For Pair COND_354_0_CREATELIST_LE(TRUE, x0) → 354_0_CREATELIST_LE(+(x0, -1)) the following chains were created:

• We consider the chain COND_354_0_CREATELIST_LE(TRUE, x0[1]) → 354_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:
  

  (7)    (COND_354_0_CREATELIST_LE(TRUE, x0[1])≥NonInfC∧COND_354_0_CREATELIST_LE(TRUE, x0[1])≥354_0_CREATELIST_LE(+(x0[1], -1))∧(U^Increasing(354_0_CREATELIST_LE(+(x0[1], -1))), ≥))

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

  (8)    ((U^Increasing(354_0_CREATELIST_LE(+(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(354_0_CREATELIST_LE(+(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(354_0_CREATELIST_LE(+(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(354_0_CREATELIST_LE(+(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.

• 354_0_CREATELIST_LE(x0) → COND_354_0_CREATELIST_LE(>(x0, 0), x0)
  
  • (x0[0] ≥ 0 ⇒ (U^Increasing(COND_354_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)
    
  
  
• COND_354_0_CREATELIST_LE(TRUE, x0) → 354_0_CREATELIST_LE(+(x0, -1))
  
  • ((U^Increasing(354_0_CREATELIST_LE(+(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(354_0_CREATELIST_LE(x₁)) = [2]x₁   
POL(COND_354_0_CREATELIST_LE(x₁, x₂)) = [2]x₂   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   

The following pairs are in P_>:

COND_354_0_CREATELIST_LE(TRUE, x0[1]) → 354_0_CREATELIST_LE(+(x0[1], -1))

The following pairs are in P_bound:

354_0_CREATELIST_LE(x0[0]) → COND_354_0_CREATELIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

354_0_CREATELIST_LE(x0[0]) → COND_354_0_CREATELIST_LE(>(x0[0], 0), x0[0])

There are no usable rules.

(9) Complex Obligation (AND)

(10) 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): 354_0_CREATELIST_LE(x0[0]) → COND_354_0_CREATELIST_LE(x0[0] > 0, x0[0])


The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE

(13) 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_354_0_CREATELIST_LE(TRUE, x0[1]) → 354_0_CREATELIST_LE(x0[1] + -1)


The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

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

(17) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 35 rules for P and 0 rules for R.

P rules:
809_0_isCyclic_NULL(EOS(STATIC_809), java.lang.Object(o259sub), o257, java.lang.Object(o269sub), java.lang.Object(o269sub)) → 810_0_isCyclic_NULL(EOS(STATIC_810), java.lang.Object(o259sub), o257, java.lang.Object(o269sub), java.lang.Object(o269sub))
810_0_isCyclic_NULL(EOS(STATIC_810), java.lang.Object(o259sub), o257, java.lang.Object(o269sub), java.lang.Object(o269sub)) → 812_0_isCyclic_Load(EOS(STATIC_812), java.lang.Object(o259sub), o257, java.lang.Object(o269sub))
812_0_isCyclic_Load(EOS(STATIC_812), java.lang.Object(o259sub), o257, java.lang.Object(o269sub)) → 815_0_isCyclic_Load(EOS(STATIC_815), java.lang.Object(o259sub), o257, java.lang.Object(o269sub), o257)
815_0_isCyclic_Load(EOS(STATIC_815), java.lang.Object(o259sub), o257, java.lang.Object(o269sub), o257) → 819_0_isCyclic_EQ(EOS(STATIC_819), java.lang.Object(o259sub), o257, java.lang.Object(o269sub), o257, java.lang.Object(o269sub))
819_0_isCyclic_EQ(EOS(STATIC_819), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o269sub), java.lang.Object(o271sub), java.lang.Object(o269sub)) → 822_0_isCyclic_EQ(EOS(STATIC_822), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o269sub), java.lang.Object(o271sub), java.lang.Object(o269sub))
822_0_isCyclic_EQ(EOS(STATIC_822), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o269sub), java.lang.Object(o271sub), java.lang.Object(o269sub)) → 828_0_isCyclic_EQ(EOS(STATIC_828), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o269sub), java.lang.Object(o271sub), java.lang.Object(o269sub))
828_0_isCyclic_EQ(EOS(STATIC_828), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o269sub), java.lang.Object(o271sub), java.lang.Object(o269sub)) → 834_0_isCyclic_Load(EOS(STATIC_834), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o269sub))
834_0_isCyclic_Load(EOS(STATIC_834), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o269sub)) → 838_0_isCyclic_FieldAccess(EOS(STATIC_838), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o269sub))
838_0_isCyclic_FieldAccess(EOS(STATIC_838), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(ObjectList(EOC, o294))) → 844_0_isCyclic_FieldAccess(EOS(STATIC_844), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(ObjectList(EOC, o294)))
844_0_isCyclic_FieldAccess(EOS(STATIC_844), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(ObjectList(EOC, o294))) → 850_0_isCyclic_Store(EOS(STATIC_850), java.lang.Object(o259sub), java.lang.Object(o271sub), o294)
850_0_isCyclic_Store(EOS(STATIC_850), java.lang.Object(o259sub), java.lang.Object(o271sub), o294) → 855_0_isCyclic_Load(EOS(STATIC_855), java.lang.Object(o259sub), java.lang.Object(o271sub), o294)
855_0_isCyclic_Load(EOS(STATIC_855), java.lang.Object(o259sub), java.lang.Object(o271sub), o294) → 862_0_isCyclic_NONNULL(EOS(STATIC_862), java.lang.Object(o259sub), java.lang.Object(o271sub), o294, o294)
862_0_isCyclic_NONNULL(EOS(STATIC_862), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub), java.lang.Object(o308sub)) → 870_0_isCyclic_NONNULL(EOS(STATIC_870), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub), java.lang.Object(o308sub))
870_0_isCyclic_NONNULL(EOS(STATIC_870), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub), java.lang.Object(o308sub)) → 878_0_isCyclic_Load(EOS(STATIC_878), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub))
878_0_isCyclic_Load(EOS(STATIC_878), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub)) → 888_0_isCyclic_Load(EOS(STATIC_888), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub), java.lang.Object(o308sub))
888_0_isCyclic_Load(EOS(STATIC_888), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub), java.lang.Object(o308sub)) → 896_0_isCyclic_NE(EOS(STATIC_896), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub), java.lang.Object(o308sub), java.lang.Object(o271sub))
896_0_isCyclic_NE(EOS(STATIC_896), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub), java.lang.Object(o308sub), java.lang.Object(o271sub)) → 904_0_isCyclic_Load(EOS(STATIC_904), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub))
904_0_isCyclic_Load(EOS(STATIC_904), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub)) → 912_0_isCyclic_FieldAccess(EOS(STATIC_912), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(o308sub))
912_0_isCyclic_FieldAccess(EOS(STATIC_912), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(ObjectList(EOC, o345))) → 917_0_isCyclic_FieldAccess(EOS(STATIC_917), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(ObjectList(EOC, o345)))
917_0_isCyclic_FieldAccess(EOS(STATIC_917), java.lang.Object(o259sub), java.lang.Object(o271sub), java.lang.Object(ObjectList(EOC, o345))) → 920_0_isCyclic_Store(EOS(STATIC_920), java.lang.Object(o259sub), java.lang.Object(o271sub), o345)
920_0_isCyclic_Store(EOS(STATIC_920), java.lang.Object(o259sub), java.lang.Object(o271sub), o345) → 924_0_isCyclic_Load(EOS(STATIC_924), java.lang.Object(o259sub), java.lang.Object(o271sub), o345)
924_0_isCyclic_Load(EOS(STATIC_924), java.lang.Object(o259sub), java.lang.Object(o271sub), o345) → 929_0_isCyclic_FieldAccess(EOS(STATIC_929), java.lang.Object(o259sub), o345, java.lang.Object(o271sub))
929_0_isCyclic_FieldAccess(EOS(STATIC_929), java.lang.Object(o259sub), o345, java.lang.Object(o271sub)) → 932_0_isCyclic_FieldAccess(EOS(STATIC_932), java.lang.Object(o259sub), o345, java.lang.Object(o271sub))
929_0_isCyclic_FieldAccess(EOS(STATIC_929), java.lang.Object(o259sub), o345, java.lang.Object(o259sub)) → 933_0_isCyclic_FieldAccess(EOS(STATIC_933), java.lang.Object(o259sub), o345, java.lang.Object(o259sub))
932_0_isCyclic_FieldAccess(EOS(STATIC_932), java.lang.Object(o259sub), o345, java.lang.Object(ObjectList(EOC, o359))) → 936_0_isCyclic_FieldAccess(EOS(STATIC_936), java.lang.Object(o259sub), o345, java.lang.Object(ObjectList(EOC, o359)))
936_0_isCyclic_FieldAccess(EOS(STATIC_936), java.lang.Object(o259sub), o345, java.lang.Object(ObjectList(EOC, o359))) → 939_0_isCyclic_Store(EOS(STATIC_939), java.lang.Object(o259sub), o345, o359)
939_0_isCyclic_Store(EOS(STATIC_939), java.lang.Object(o259sub), o345, o359) → 944_0_isCyclic_JMP(EOS(STATIC_944), java.lang.Object(o259sub), o359, o345)
944_0_isCyclic_JMP(EOS(STATIC_944), java.lang.Object(o259sub), o359, o345) → 949_0_isCyclic_Load(EOS(STATIC_949), java.lang.Object(o259sub), o359, o345)
949_0_isCyclic_Load(EOS(STATIC_949), java.lang.Object(o259sub), o359, o345) → 806_0_isCyclic_Load(EOS(STATIC_806), java.lang.Object(o259sub), o359, o345)
806_0_isCyclic_Load(EOS(STATIC_806), java.lang.Object(o259sub), o257, o258) → 809_0_isCyclic_NULL(EOS(STATIC_809), java.lang.Object(o259sub), o257, o258, o258)
933_0_isCyclic_FieldAccess(EOS(STATIC_933), java.lang.Object(ObjectList(EOC, o362)), o345, java.lang.Object(ObjectList(EOC, o362))) → 937_0_isCyclic_FieldAccess(EOS(STATIC_937), java.lang.Object(ObjectList(EOC, o362)), o345, java.lang.Object(ObjectList(EOC, o362)))
937_0_isCyclic_FieldAccess(EOS(STATIC_937), java.lang.Object(ObjectList(EOC, o362)), o345, java.lang.Object(ObjectList(EOC, o362))) → 941_0_isCyclic_Store(EOS(STATIC_941), java.lang.Object(ObjectList(EOC, o362)), o345, o362)
941_0_isCyclic_Store(EOS(STATIC_941), java.lang.Object(ObjectList(EOC, o362)), o345, o362) → 945_0_isCyclic_JMP(EOS(STATIC_945), java.lang.Object(ObjectList(EOC, o362)), o362, o345)
945_0_isCyclic_JMP(EOS(STATIC_945), java.lang.Object(ObjectList(EOC, o362)), o362, o345) → 952_0_isCyclic_Load(EOS(STATIC_952), java.lang.Object(ObjectList(EOC, o362)), o362, o345)
952_0_isCyclic_Load(EOS(STATIC_952), java.lang.Object(ObjectList(EOC, o362)), o362, o345) → 806_0_isCyclic_Load(EOS(STATIC_806), java.lang.Object(ObjectList(EOC, o362)), o362, o345)
R rules:

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

P rules:
809_0_isCyclic_NULL(EOS(STATIC_809), java.lang.Object(x0), java.lang.Object(ObjectList(EOC, x1)), java.lang.Object(ObjectList(EOC, java.lang.Object(ObjectList(EOC, x2)))), java.lang.Object(ObjectList(EOC, java.lang.Object(ObjectList(EOC, x2))))) → 809_0_isCyclic_NULL(EOS(STATIC_809), java.lang.Object(x0), x1, x2, x2)
809_0_isCyclic_NULL(EOS(STATIC_809), java.lang.Object(ObjectList(EOC, x0)), java.lang.Object(ObjectList(EOC, x0)), java.lang.Object(ObjectList(EOC, java.lang.Object(ObjectList(EOC, x1)))), java.lang.Object(ObjectList(EOC, java.lang.Object(ObjectList(EOC, x1))))) → 809_0_isCyclic_NULL(EOS(STATIC_809), java.lang.Object(ObjectList(EOC, x0)), x0, x1, x1)
R rules:

Filtered ground terms:

809_0_isCyclic_NULL(x1, x2, x3, x4, x5) → 809_0_isCyclic_NULL(x2, x3, x4, x5)
ObjectList(x1, x2) → ObjectList(x2)
EOS(x1) → EOS

Filtered duplicate args:

809_0_isCyclic_NULL(x1, x2, x3, x4) → 809_0_isCyclic_NULL(x1, x2, x4)

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

P rules:
809_0_isCyclic_NULL(java.lang.Object(x0), java.lang.Object(ObjectList(x1)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x2))))) → 809_0_isCyclic_NULL(java.lang.Object(x0), x1, x2)
809_0_isCyclic_NULL(java.lang.Object(ObjectList(x0)), java.lang.Object(ObjectList(x0)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x1))))) → 809_0_isCyclic_NULL(java.lang.Object(ObjectList(x0)), x0, x1)
R rules:

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

P rules:
809_0_ISCYCLIC_NULL(java.lang.Object(x0), java.lang.Object(ObjectList(x1)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x2))))) → 809_0_ISCYCLIC_NULL(java.lang.Object(x0), x1, x2)
809_0_ISCYCLIC_NULL(java.lang.Object(ObjectList(x0)), java.lang.Object(ObjectList(x0)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x1))))) → 809_0_ISCYCLIC_NULL(java.lang.Object(ObjectList(x0)), x0, x1)
R rules:

(18) 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): 809_0_ISCYCLIC_NULL(java.lang.Object(x0[0]), java.lang.Object(ObjectList(x1[0])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x2[0]))))) → 809_0_ISCYCLIC_NULL(java.lang.Object(x0[0]), x1[0], x2[0])
(1): 809_0_ISCYCLIC_NULL(java.lang.Object(ObjectList(x0[1])), java.lang.Object(ObjectList(x0[1])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x1[1]))))) → 809_0_ISCYCLIC_NULL(java.lang.Object(ObjectList(x0[1])), x0[1], x1[1])

(0) -> (0), if (java.lang.Object(x0[0]) →^* java.lang.Object(x0[0]')∧x1[0] →^* java.lang.Object(ObjectList(x1[0]'))∧x2[0] →^* java.lang.Object(ObjectList(java.lang.Object(ObjectList(x2[0]')))))

(0) -> (1), if (java.lang.Object(x0[0]) →^* java.lang.Object(ObjectList(x0[1]))∧x1[0] →^* java.lang.Object(ObjectList(x0[1]))∧x2[0] →^* java.lang.Object(ObjectList(java.lang.Object(ObjectList(x1[1])))))

(1) -> (0), if (java.lang.Object(ObjectList(x0[1])) →^* java.lang.Object(x0[0])∧x0[1] →^* java.lang.Object(ObjectList(x1[0]))∧x1[1] →^* java.lang.Object(ObjectList(java.lang.Object(ObjectList(x2[0])))))

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

The set Q is empty.

(19) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(20) Obligation:

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

809_0_ISCYCLIC_NULL(java.lang.Object(x0[0]), java.lang.Object(ObjectList(x1[0])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x2[0]))))) → 809_0_ISCYCLIC_NULL(java.lang.Object(x0[0]), x1[0], x2[0])
809_0_ISCYCLIC_NULL(java.lang.Object(ObjectList(x0[1])), java.lang.Object(ObjectList(x0[1])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x1[1]))))) → 809_0_ISCYCLIC_NULL(java.lang.Object(ObjectList(x0[1])), x0[1], x1[1])

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

(21) 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:

• 809_0_ISCYCLIC_NULL(java.lang.Object(x0[0]), java.lang.Object(ObjectList(x1[0])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x2[0]))))) → 809_0_ISCYCLIC_NULL(java.lang.Object(x0[0]), x1[0], x2[0])
  The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3

• 809_0_ISCYCLIC_NULL(java.lang.Object(ObjectList(x0[1])), java.lang.Object(ObjectList(x0[1])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x1[1]))))) → 809_0_ISCYCLIC_NULL(java.lang.Object(ObjectList(x0[1])), x0[1], x1[1])
  The graph contains the following edges 1 >= 1, 2 >= 1, 1 > 2, 2 > 2, 3 > 3

(22) YES