Termination proof of /tmp/tmpLYLcky/RunningPointers.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPNonInfProof (⇒)

        ↳7 AND

            ↳8 IDP

                ↳9 IDependencyGraphProof (⇔)

                ↳10 TRUE

            ↳11 IDP

                ↳12 IDependencyGraphProof (⇔)

                ↳13 TRUE

    ↳14 IDP

        ↳15 IDPtoQDPProof (⇒)

        ↳16 QDP

        ↳17 MRRProof (⇔)

        ↳18 QDP

        ↳19 PisEmptyProof (⇔)

        ↳20 YES

(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) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
RunningPointers.main([Ljava/lang/String;)V: Graph of 134 nodes with 1 SCC.

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

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 23 rules for P and 3 rules for R.

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

Filtered ground terms:

325_0_createList_LE(x1, x2, x3) → 325_0_createList_LE(x2, x3)
Cond_325_0_createList_LE(x1, x2, x3, x4) → Cond_325_0_createList_LE(x1, x3, x4)
333_0_createList_Return(x1) → 333_0_createList_Return

Filtered duplicate args:

325_0_createList_LE(x1, x2) → 325_0_createList_LE(x2)
Cond_325_0_createList_LE(x1, x2, x3) → Cond_325_0_createList_LE(x1, x3)

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

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

Log for SCC 1:

Generated 33 rules for P and 66 rules for R.

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

Filtered ground terms:

ObjectList(x1, x2) → ObjectList(x2)
1055_0_isCyclic_NULL(x1, x2, x3, x4) → 1055_0_isCyclic_NULL(x2, x3, x4)

Filtered duplicate args:

1055_0_isCyclic_NULL(x1, x2, x3) → 1055_0_isCyclic_NULL(x1, x3)

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

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
325_0_createList_LE(0) → 333_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 325_0_CREATELIST_LE(x0[0]) → COND_325_0_CREATELIST_LE(x0[0] > 0, x0[0])
(1): COND_325_0_CREATELIST_LE(TRUE, x0[1]) → 325_0_CREATELIST_LE(x0[1] + -1)

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

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

The set Q consists of the following terms:
325_0_createList_LE(0)

(6) IDPNonInfProof (SOUND transformation)

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

For Pair 325_0_CREATELIST_LE(x0) → COND_325_0_CREATELIST_LE(>(x0, 0), x0) the following chains were created:

We consider the chain 325_0_CREATELIST_LE(x0[0]) → COND_325_0_CREATELIST_LE(>(x0[0], 0), x0[0]), COND_325_0_CREATELIST_LE(TRUE, x0[1]) → 325_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:
(1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 325_0_CREATELIST_LE(x0[0])≥NonInfC∧325_0_CREATELIST_LE(x0[0])≥COND_325_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_325_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 ⇒ 325_0_CREATELIST_LE(x0[0])≥NonInfC∧325_0_CREATELIST_LE(x0[0])≥COND_325_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_325_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_325_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_325_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_325_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_325_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_325_0_CREATELIST_LE(TRUE, x0) → 325_0_CREATELIST_LE(+(x0, -1)) the following chains were created:

We consider the chain COND_325_0_CREATELIST_LE(TRUE, x0[1]) → 325_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:
(7)    (COND_325_0_CREATELIST_LE(TRUE, x0[1])≥NonInfC∧COND_325_0_CREATELIST_LE(TRUE, x0[1])≥325_0_CREATELIST_LE(+(x0[1], -1))∧(U^Increasing(325_0_CREATELIST_LE(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(325_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(325_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(325_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(325_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 0)

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

325_0_CREATELIST_LE(x0) → COND_325_0_CREATELIST_LE(>(x0, 0), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_325_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)
COND_325_0_CREATELIST_LE(TRUE, x0) → 325_0_CREATELIST_LE(+(x0, -1))
- ((U^Increasing(325_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 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(325_0_createList_LE(x₁)) = [-1]
POL(0) = 0
POL(333_0_createList_Return) = [-1]
POL(325_0_CREATELIST_LE(x₁)) = [2]x₁
POL(COND_325_0_CREATELIST_LE(x₁, x₂)) = [2]x₂
POL(>(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_325_0_CREATELIST_LE(TRUE, x0[1]) → 325_0_CREATELIST_LE(+(x0[1], -1))

The following pairs are in P_bound:

325_0_CREATELIST_LE(x0[0]) → COND_325_0_CREATELIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

325_0_CREATELIST_LE(x0[0]) → COND_325_0_CREATELIST_LE(>(x0[0], 0), x0[0])

There are no usable rules.

(7) Complex Obligation (AND)

(8) Obligation:

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

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
325_0_createList_LE(0) → 333_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 325_0_CREATELIST_LE(x0[0]) → COND_325_0_CREATELIST_LE(x0[0] > 0, x0[0])

The set Q consists of the following terms:
325_0_createList_LE(0)

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) TRUE

(11) Obligation:

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

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
325_0_createList_LE(0) → 333_0_createList_Return

The integer pair graph contains the following rules and edges:
(1): COND_325_0_CREATELIST_LE(TRUE, x0[1]) → 325_0_CREATELIST_LE(x0[1] + -1)

The set Q consists of the following terms:
325_0_createList_LE(0)

(12) IDependencyGraphProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

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

The following domains are used:
none

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(java.lang.Object(ObjectList(x0[0])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(java.lang.Object(x1[0])))))), java.lang.Object(x2[0])) → 1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(x0[0], java.lang.Object(x1[0])), java.lang.Object(x2[0]))
(1): 1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(java.lang.Object(ObjectList(x0[1])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(java.lang.Object(x1[1])))))), java.lang.Object(ObjectList(x0[1]))) → 1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(x0[1], java.lang.Object(x1[1])), java.lang.Object(ObjectList(x0[1])))

(0) -> (0), if ((1055_0_isCyclic_NULL(x0[0], java.lang.Object(x1[0])) →^* 1055_0_isCyclic_NULL(java.lang.Object(ObjectList(x0[0]')), java.lang.Object(ObjectList(java.lang.Object(ObjectList(java.lang.Object(x1[0]')))))))∧(java.lang.Object(x2[0]) →^* java.lang.Object(x2[0]')))

(0) -> (1), if ((1055_0_isCyclic_NULL(x0[0], java.lang.Object(x1[0])) →^* 1055_0_isCyclic_NULL(java.lang.Object(ObjectList(x0[1])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(java.lang.Object(x1[1])))))))∧(java.lang.Object(x2[0]) →^* java.lang.Object(ObjectList(x0[1]))))

(1) -> (0), if ((1055_0_isCyclic_NULL(x0[1], java.lang.Object(x1[1])) →^* 1055_0_isCyclic_NULL(java.lang.Object(ObjectList(x0[0])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(java.lang.Object(x1[0])))))))∧(java.lang.Object(ObjectList(x0[1])) →^* java.lang.Object(x2[0])))

(1) -> (1), if ((1055_0_isCyclic_NULL(x0[1], java.lang.Object(x1[1])) →^* 1055_0_isCyclic_NULL(java.lang.Object(ObjectList(x0[1]')), java.lang.Object(ObjectList(java.lang.Object(ObjectList(java.lang.Object(x1[1]')))))))∧(java.lang.Object(ObjectList(x0[1])) →^* java.lang.Object(ObjectList(x0[1]'))))

The set Q is empty.

(15) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(16) Obligation:

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

1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(java.lang.Object(ObjectList(x0[0])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(java.lang.Object(x1[0])))))), java.lang.Object(x2[0])) → 1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(x0[0], java.lang.Object(x1[0])), java.lang.Object(x2[0]))
1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(java.lang.Object(ObjectList(x0[1])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(java.lang.Object(x1[1])))))), java.lang.Object(ObjectList(x0[1]))) → 1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(x0[1], java.lang.Object(x1[1])), java.lang.Object(ObjectList(x0[1])))

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

(17) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(java.lang.Object(ObjectList(x0[0])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(java.lang.Object(x1[0])))))), java.lang.Object(x2[0])) → 1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(x0[0], java.lang.Object(x1[0])), java.lang.Object(x2[0]))
1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(java.lang.Object(ObjectList(x0[1])), java.lang.Object(ObjectList(java.lang.Object(ObjectList(java.lang.Object(x1[1])))))), java.lang.Object(ObjectList(x0[1]))) → 1055_1_MAIN_INVOKEMETHOD(1055_0_isCyclic_NULL(x0[1], java.lang.Object(x1[1])), java.lang.Object(ObjectList(x0[1])))

Used ordering: Polynomial interpretation [POLO]:

POL(1055_0_isCyclic_NULL(x₁, x₂)) = x₁ + x₂
POL(1055_1_MAIN_INVOKEMETHOD(x₁, x₂)) = x₁ + x₂
POL(ObjectList(x₁)) = 1 + 2·x₁
POL(java.lang.Object(x₁)) = x₁

(18) Obligation:

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

(19) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) IDPNonInfProof (SOUND transformation)

(7) Complex Obligation (AND)

(8) Obligation:

(9) IDependencyGraphProof (EQUIVALENT transformation)

(10) TRUE

(11) Obligation:

(12) IDependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) IDPtoQDPProof (SOUND transformation)

(16) Obligation:

(17) MRRProof (EQUIVALENT transformation)

(18) Obligation:

(19) PisEmptyProof (EQUIVALENT transformation)

(20) YES