(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:
Graph of 269 nodes with 2 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) Complex Obligation (AND)

(5) Obligation:

ITRS 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 TRS R consists of the following rules:
Load2975(java.lang.Object(ObjectList(o2219)), java.lang.Object(ObjectList(o2219)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o2899))))) → JMP3744(java.lang.Object(ObjectList(o2219)), o2219, o2899)
JMP3744(java.lang.Object(ObjectList(o2219)), o3107, o2899) → Load2975(java.lang.Object(ObjectList(o2219)), o3107, o2899)
Load2975(java.lang.Object(ObjectList(o2219)), java.lang.Object(ObjectList(o3107)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o2899))))) → Load2975(java.lang.Object(ObjectList(o2219)), o3107, o2899)
The set Q consists of the following terms:
JMP3744(java.lang.Object(ObjectList(x0)), x1, x2)
Load2975(java.lang.Object(ObjectList(x0)), java.lang.Object(ObjectList(x1)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x2)))))

(6) ITRStoQTRSProof (EQUIVALENT transformation)

Represented integers and predefined function symbols by Terms

(7) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

Load2975(java.lang.Object(ObjectList(o2219)), java.lang.Object(ObjectList(o2219)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o2899))))) → JMP3744(java.lang.Object(ObjectList(o2219)), o2219, o2899)
JMP3744(java.lang.Object(ObjectList(o2219)), o3107, o2899) → Load2975(java.lang.Object(ObjectList(o2219)), o3107, o2899)
Load2975(java.lang.Object(ObjectList(o2219)), java.lang.Object(ObjectList(o3107)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o2899))))) → Load2975(java.lang.Object(ObjectList(o2219)), o3107, o2899)

The set Q consists of the following terms:

JMP3744(java.lang.Object(ObjectList(x0)), x1, x2)
Load2975(java.lang.Object(ObjectList(x0)), java.lang.Object(ObjectList(x1)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x2)))))

(8) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(JMP3744(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(Load2975(x1, x2, x3)) = x1 + x2 + x3   
POL(ObjectList(x1)) = 1 + x1   
POL(java.lang.Object(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

Load2975(java.lang.Object(ObjectList(o2219)), java.lang.Object(ObjectList(o2219)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o2899))))) → JMP3744(java.lang.Object(ObjectList(o2219)), o2219, o2899)
JMP3744(java.lang.Object(ObjectList(o2219)), o3107, o2899) → Load2975(java.lang.Object(ObjectList(o2219)), o3107, o2899)
Load2975(java.lang.Object(ObjectList(o2219)), java.lang.Object(ObjectList(o3107)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o2899))))) → Load2975(java.lang.Object(ObjectList(o2219)), o3107, o2899)


(9) Obligation:

Q restricted rewrite system:
R is empty.
The set Q consists of the following terms:

JMP3744(java.lang.Object(ObjectList(x0)), x1, x2)
Load2975(java.lang.Object(ObjectList(x0)), java.lang.Object(ObjectList(x1)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x2)))))

(10) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(11) TRUE

(12) Obligation:

ITRS 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 TRS R consists of the following rules:
Load876(i34) → Cond_Load876(i34 > 0, i34)
Cond_Load876(TRUE, i34) → Load876(i34 + -1)
The set Q consists of the following terms:
Load876(x0)
Cond_Load876(TRUE, x0)

(13) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(14) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Integer


The ITRS R consists of the following rules:
Load876(i34) → Cond_Load876(i34 > 0, i34)
Cond_Load876(TRUE, i34) → Load876(i34 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD876(i34[0]) → COND_LOAD876(i34[0] > 0, i34[0])
(1): COND_LOAD876(TRUE, i34[1]) → LOAD876(i34[1] + -1)

(0) -> (1), if ((i34[0]* i34[1])∧(i34[0] > 0* TRUE))


(1) -> (0), if ((i34[1] + -1* i34[0]))



The set Q consists of the following terms:
Load876(x0)
Cond_Load876(TRUE, x0)

(15) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(16) 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): LOAD876(i34[0]) → COND_LOAD876(i34[0] > 0, i34[0])
(1): COND_LOAD876(TRUE, i34[1]) → LOAD876(i34[1] + -1)

(0) -> (1), if ((i34[0]* i34[1])∧(i34[0] > 0* TRUE))


(1) -> (0), if ((i34[1] + -1* i34[0]))



The set Q consists of the following terms:
Load876(x0)
Cond_Load876(TRUE, x0)

(17) 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 LOAD876(i34) → COND_LOAD876(>(i34, 0), i34) the following chains were created:
  • We consider the chain LOAD876(i34[0]) → COND_LOAD876(>(i34[0], 0), i34[0]), COND_LOAD876(TRUE, i34[1]) → LOAD876(+(i34[1], -1)) which results in the following constraint:

    (1)    (i34[0]=i34[1]>(i34[0], 0)=TRUELOAD876(i34[0])≥NonInfC∧LOAD876(i34[0])≥COND_LOAD876(>(i34[0], 0), i34[0])∧(UIncreasing(COND_LOAD876(>(i34[0], 0), i34[0])), ≥))



    We simplified constraint (1) using rule (IV) which results in the following new constraint:

    (2)    (>(i34[0], 0)=TRUELOAD876(i34[0])≥NonInfC∧LOAD876(i34[0])≥COND_LOAD876(>(i34[0], 0), i34[0])∧(UIncreasing(COND_LOAD876(>(i34[0], 0), i34[0])), ≥))



    We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (3)    (i34[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD876(>(i34[0], 0), i34[0])), ≥)∧[(-1)Bound*bni_9] + [(2)bni_9]i34[0] ≥ 0∧[1 + (-1)bso_10] ≥ 0)



    We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (4)    (i34[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD876(>(i34[0], 0), i34[0])), ≥)∧[(-1)Bound*bni_9] + [(2)bni_9]i34[0] ≥ 0∧[1 + (-1)bso_10] ≥ 0)



    We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (5)    (i34[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD876(>(i34[0], 0), i34[0])), ≥)∧[(-1)Bound*bni_9] + [(2)bni_9]i34[0] ≥ 0∧[1 + (-1)bso_10] ≥ 0)



    We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (6)    (i34[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD876(>(i34[0], 0), i34[0])), ≥)∧[(-1)Bound*bni_9 + (2)bni_9] + [(2)bni_9]i34[0] ≥ 0∧[1 + (-1)bso_10] ≥ 0)







For Pair COND_LOAD876(TRUE, i34) → LOAD876(+(i34, -1)) the following chains were created:
  • We consider the chain LOAD876(i34[0]) → COND_LOAD876(>(i34[0], 0), i34[0]), COND_LOAD876(TRUE, i34[1]) → LOAD876(+(i34[1], -1)), LOAD876(i34[0]) → COND_LOAD876(>(i34[0], 0), i34[0]) which results in the following constraint:

    (7)    (i34[0]=i34[1]>(i34[0], 0)=TRUE+(i34[1], -1)=i34[0]1COND_LOAD876(TRUE, i34[1])≥NonInfC∧COND_LOAD876(TRUE, i34[1])≥LOAD876(+(i34[1], -1))∧(UIncreasing(LOAD876(+(i34[1], -1))), ≥))



    We simplified constraint (7) using rules (III), (IV) which results in the following new constraint:

    (8)    (>(i34[0], 0)=TRUECOND_LOAD876(TRUE, i34[0])≥NonInfC∧COND_LOAD876(TRUE, i34[0])≥LOAD876(+(i34[0], -1))∧(UIncreasing(LOAD876(+(i34[1], -1))), ≥))



    We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (9)    (i34[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD876(+(i34[1], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i34[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



    We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (10)    (i34[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD876(+(i34[1], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i34[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



    We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (11)    (i34[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD876(+(i34[1], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i34[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



    We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (12)    (i34[0] ≥ 0 ⇒ (UIncreasing(LOAD876(+(i34[1], -1))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i34[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • LOAD876(i34) → COND_LOAD876(>(i34, 0), i34)
    • (i34[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD876(>(i34[0], 0), i34[0])), ≥)∧[(-1)Bound*bni_9 + (2)bni_9] + [(2)bni_9]i34[0] ≥ 0∧[1 + (-1)bso_10] ≥ 0)

  • COND_LOAD876(TRUE, i34) → LOAD876(+(i34, -1))
    • (i34[0] ≥ 0 ⇒ (UIncreasing(LOAD876(+(i34[1], -1))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i34[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(LOAD876(x1)) = [2]x1   
POL(COND_LOAD876(x1, x2)) = [-1] + [2]x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

LOAD876(i34[0]) → COND_LOAD876(>(i34[0], 0), i34[0])
COND_LOAD876(TRUE, i34[1]) → LOAD876(+(i34[1], -1))

The following pairs are in Pbound:

LOAD876(i34[0]) → COND_LOAD876(>(i34[0], 0), i34[0])
COND_LOAD876(TRUE, i34[1]) → LOAD876(+(i34[1], -1))

The following pairs are in P:
none

There are no usable 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 is empty.

The set Q consists of the following terms:
Load876(x0)
Cond_Load876(TRUE, x0)

(19) IDependencyGraphProof (EQUIVALENT transformation)

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

(20) TRUE