(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: ListDuplicate
/**
* This class represents a list, where the function duplicate() can be used to
* duplicate all elements in the list.
* @author cotto
*/
public class ListDuplicate {
/**
* Walk through the list and, for each original element, copy it and append
* this copy after the original. This transforms abc to aabbcc.
*/
public static void duplicate(ObjectList list) {
ObjectList current = list;
boolean even = true;
while (current != null) {
// only copy the original elements!
if (even) {
final ObjectList copy =
new ObjectList(current.value, current.next);
current.next = copy;
}
current = current.next;
even = !even;
}
}

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


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();
}
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
ListDuplicate.main([Ljava/lang/String;)V: Graph of 30 nodes with 0 SCCs.

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

ListDuplicate.duplicate(LObjectList;)V: Graph of 83 nodes with 1 SCC.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:


Log for SCC 0:

Generated 77 rules for P and 2 rules for R.


Combined rules. Obtained 2 rules for P and 1 rules for R.


Filtered ground terms:


1144_0_duplicate_NULL(x1, x2, x3, x4) → 1144_0_duplicate_NULL(x2, x3, x4)
ObjectList(x1, x2, x3) → ObjectList(x2, x3)
1157_0_duplicate_Return(x1) → 1157_0_duplicate_Return

Filtered duplicate args:


1144_0_duplicate_NULL(x1, x2, x3) → 1144_0_duplicate_NULL(x2, x3)

Finished conversion. Obtained 2 rules for P and 1 rules for R. System has no predefined symbols.




Log for SCC 1:

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:


375_0_createList_LE(x1, x2, x3) → 375_0_createList_LE(x2, x3)
Cond_375_0_createList_LE(x1, x2, x3, x4) → Cond_375_0_createList_LE(x1, x3, x4)
385_0_createList_Return(x1) → 385_0_createList_Return

Filtered duplicate args:


375_0_createList_LE(x1, x2) → 375_0_createList_LE(x2)
Cond_375_0_createList_LE(x1, x2, x3) → Cond_375_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.


(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:
none


The ITRS R consists of the following rules:
1144_0_duplicate_NULL(x0, NULL) → 1157_0_duplicate_Return

The integer pair graph contains the following rules and edges:
(0): 1144_0_DUPLICATE_NULL(1, java.lang.Object(ObjectList(x0[0], x1[0]))) → 1144_0_DUPLICATE_NULL(0, java.lang.Object(ObjectList(x0[0], x1[0])))
(1): 1144_0_DUPLICATE_NULL(0, java.lang.Object(ObjectList(x0[1], x1[1]))) → 1144_0_DUPLICATE_NULL(1, x1[1])

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


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


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


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



The set Q consists of the following terms:
1144_0_duplicate_NULL(x0, NULL)

(6) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(7) Obligation:

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

1144_0_DUPLICATE_NULL(pos(s(01)), java.lang.Object(ObjectList(x0[0], x1[0]))) → 1144_0_DUPLICATE_NULL(pos(01), java.lang.Object(ObjectList(x0[0], x1[0])))
1144_0_DUPLICATE_NULL(pos(01), java.lang.Object(ObjectList(x0[1], x1[1]))) → 1144_0_DUPLICATE_NULL(pos(s(01)), x1[1])

The TRS R consists of the following rules:

1144_0_duplicate_NULL(x0, NULL) → 1157_0_duplicate_Return

The set Q consists of the following terms:

1144_0_duplicate_NULL(x0, NULL)

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

(8) 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.

(9) Obligation:

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

1144_0_DUPLICATE_NULL(pos(s(01)), java.lang.Object(ObjectList(x0[0], x1[0]))) → 1144_0_DUPLICATE_NULL(pos(01), java.lang.Object(ObjectList(x0[0], x1[0])))
1144_0_DUPLICATE_NULL(pos(01), java.lang.Object(ObjectList(x0[1], x1[1]))) → 1144_0_DUPLICATE_NULL(pos(s(01)), x1[1])

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

1144_0_duplicate_NULL(x0, NULL)

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

(10) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

1144_0_duplicate_NULL(x0, NULL)

(11) Obligation:

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

1144_0_DUPLICATE_NULL(pos(s(01)), java.lang.Object(ObjectList(x0[0], x1[0]))) → 1144_0_DUPLICATE_NULL(pos(01), java.lang.Object(ObjectList(x0[0], x1[0])))
1144_0_DUPLICATE_NULL(pos(01), java.lang.Object(ObjectList(x0[1], x1[1]))) → 1144_0_DUPLICATE_NULL(pos(s(01)), x1[1])

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

(12) 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:

  • 1144_0_DUPLICATE_NULL(pos(01), java.lang.Object(ObjectList(x0[1], x1[1]))) → 1144_0_DUPLICATE_NULL(pos(s(01)), x1[1])
    The graph contains the following edges 2 > 2

  • 1144_0_DUPLICATE_NULL(pos(s(01)), java.lang.Object(ObjectList(x0[0], x1[0]))) → 1144_0_DUPLICATE_NULL(pos(01), java.lang.Object(ObjectList(x0[0], x1[0])))
    The graph contains the following edges 2 >= 2

(13) YES

(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:
375_0_createList_LE(0) → 385_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 375_0_CREATELIST_LE(x0[0]) → COND_375_0_CREATELIST_LE(x0[0] > 0, x0[0])
(1): COND_375_0_CREATELIST_LE(TRUE, x0[1]) → 375_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:
375_0_createList_LE(0)

(15) 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 375_0_CREATELIST_LE(x0) → COND_375_0_CREATELIST_LE(>(x0, 0), x0) the following chains were created:
  • We consider the chain 375_0_CREATELIST_LE(x0[0]) → COND_375_0_CREATELIST_LE(>(x0[0], 0), x0[0]), COND_375_0_CREATELIST_LE(TRUE, x0[1]) → 375_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]375_0_CREATELIST_LE(x0[0])≥NonInfC∧375_0_CREATELIST_LE(x0[0])≥COND_375_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_375_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)=TRUE375_0_CREATELIST_LE(x0[0])≥NonInfC∧375_0_CREATELIST_LE(x0[0])≥COND_375_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_375_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 ⇒ (UIncreasing(COND_375_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 ⇒ (UIncreasing(COND_375_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 ⇒ (UIncreasing(COND_375_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 ⇒ (UIncreasing(COND_375_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_375_0_CREATELIST_LE(TRUE, x0) → 375_0_CREATELIST_LE(+(x0, -1)) the following chains were created:
  • We consider the chain COND_375_0_CREATELIST_LE(TRUE, x0[1]) → 375_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

    (7)    (COND_375_0_CREATELIST_LE(TRUE, x0[1])≥NonInfC∧COND_375_0_CREATELIST_LE(TRUE, x0[1])≥375_0_CREATELIST_LE(+(x0[1], -1))∧(UIncreasing(375_0_CREATELIST_LE(+(x0[1], -1))), ≥))



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

    (8)    ((UIncreasing(375_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)    ((UIncreasing(375_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)    ((UIncreasing(375_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)    ((UIncreasing(375_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.
  • 375_0_CREATELIST_LE(x0) → COND_375_0_CREATELIST_LE(>(x0, 0), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_375_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_375_0_CREATELIST_LE(TRUE, x0) → 375_0_CREATELIST_LE(+(x0, -1))
    • ((UIncreasing(375_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 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(375_0_createList_LE(x1)) = [-1]   
POL(0) = 0   
POL(385_0_createList_Return) = [-1]   
POL(375_0_CREATELIST_LE(x1)) = [2]x1   
POL(COND_375_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_375_0_CREATELIST_LE(TRUE, x0[1]) → 375_0_CREATELIST_LE(+(x0[1], -1))

The following pairs are in Pbound:

375_0_CREATELIST_LE(x0[0]) → COND_375_0_CREATELIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P:

375_0_CREATELIST_LE(x0[0]) → COND_375_0_CREATELIST_LE(>(x0[0], 0), 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


The ITRS R consists of the following rules:
375_0_createList_LE(0) → 385_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 375_0_CREATELIST_LE(x0[0]) → COND_375_0_CREATELIST_LE(x0[0] > 0, x0[0])


The set Q consists of the following terms:
375_0_createList_LE(0)

(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


The ITRS R consists of the following rules:
375_0_createList_LE(0) → 385_0_createList_Return

The integer pair graph contains the following rules and edges:
(1): COND_375_0_CREATELIST_LE(TRUE, x0[1]) → 375_0_CREATELIST_LE(x0[1] + -1)


The set Q consists of the following terms:
375_0_createList_LE(0)

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(22) TRUE