Termination proof of /tmp/tmpNvfv7c/ListDuplicate.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPtoQDPProof (⇒)

        ↳7 QDP

        ↳8 UsableRulesProof (⇔)

        ↳9 QDP

        ↳10 QReductionProof (⇔)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 IDP

        ↳15 IDPNonInfProof (⇒)

        ↳16 AND

            ↳17 IDP

                ↳18 IDependencyGraphProof (⇔)

                ↳19 TRUE

            ↳20 IDP

                ↳21 IDependencyGraphProof (⇔)

                ↳22 TRUE

(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)=TRUE∧x0[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])∧(U^Increasing(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)=TRUE ⇒ 375_0_CREATELIST_LE(x0[0])≥NonInfC∧375_0_CREATELIST_LE(x0[0])≥COND_375_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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))∧(U^Increasing(375_0_CREATELIST_LE(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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 ⇒ (U^Increasing(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))
- ((U^Increasing(375_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(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 P_bound:

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.

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) IDPtoQDPProof (SOUND transformation)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) QReductionProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(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