Termination proof of /tmp/tmpagGHBT/Convert.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇐)

↳2 FIGraph

↳3 FIGtoITRSProof (⇐)

↳4 AND

    ↳5 ITRS

        ↳6 ITRStoIDPProof (⇔)

        ↳7 IDP

        ↳8 UsableRulesProof (⇔)

        ↳9 IDP

        ↳10 ItpfGraphProof (⇔)

        ↳11 IDP

        ↳12 IDPNonInfProof (⇐)

        ↳13 AND

            ↳14 IDP

                ↳15 IDependencyGraphProof (⇔)

                ↳16 IDP

                ↳17 IDPNonInfProof (⇐)

                ↳18 AND

                    ↳19 IDP

                        ↳20 IDPNonInfProof (⇐)

                        ↳21 AND

                            ↳22 IDP

                                ↳23 IDependencyGraphProof (⇔)

                                ↳24 TRUE

                            ↳25 IDP

                                ↳26 IDependencyGraphProof (⇔)

                                ↳27 TRUE

                    ↳28 IDP

                        ↳29 IDependencyGraphProof (⇔)

                        ↳30 TRUE

            ↳31 IDP

                ↳32 IDependencyGraphProof (⇔)

                ↳33 TRUE

    ↳34 ITRS

        ↳35 ITRSFilterProcessorProof (⇐)

        ↳36 ITRS

        ↳37 ITRStoIDPProof (⇔)

        ↳38 IDP

        ↳39 UsableRulesProof (⇔)

        ↳40 IDP

        ↳41 ItpfGraphProof (⇔)

        ↳42 IDP

        ↳43 IDPNonInfProof (⇐)

        ↳44 AND

            ↳45 IDP

                ↳46 IDependencyGraphProof (⇔)

                ↳47 TRUE

            ↳48 IDP

                ↳49 IDependencyGraphProof (⇔)

                ↳50 TRUE

(0) Obligation:

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

public class Convert{

  // adapted from [Giesl, 95]
  // converts a number to decimal notation

  public static void main(String[] args) {
    Random.args = args;
    IntList l = IntList.createIntList();

    int b = Random.random();
    int res = 0;

    while (l != null) {

      if (l.value <= 0) {
        l = l.next;
        if (l != null) res = res * b;
      }
      else {
        l.value--;
        res++;
      }
    }
  }
}

class IntList {
  int value;
  IntList next;

  public IntList(int value, IntList next) {
    this.value = value;
    this.next = next;
  }

  public static IntList createIntList() {

    int i = Random.random();
    IntList l = null;

    while (i > 0) {
      l = new IntList(Random.random(), l);
      i--;
    }

    return l;
  }
}


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:
Graph of 307 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:
Load2958(java.lang.Object(IntList(i466, o4023)), i164, i463) → Cond_Load2958(i466 > 0, java.lang.Object(IntList(i466, o4023)), i164, i463)
Cond_Load2958(TRUE, java.lang.Object(IntList(i466, o4023)), i164, i463) → Load2958(java.lang.Object(IntList(i466 - 1, o4023)), i164, i463 + 1)
Load2958(java.lang.Object(IntList(i467, java.lang.Object(IntList(o4048Field0, o4048Field1)))), i164, i463) → Cond_Load29581(i467 <= 0, java.lang.Object(IntList(i467, java.lang.Object(IntList(o4048Field0, o4048Field1)))), i164, i463)
Cond_Load29581(TRUE, java.lang.Object(IntList(i467, java.lang.Object(IntList(o4048Field0, o4048Field1)))), i164, i463) → Load2958(java.lang.Object(IntList(o4048Field0, o4048Field1)), i164, i463 * i164)
Load2958(java.lang.Object(IntList(i467, NULL)), i164, i463) → Cond_Load29582(i467 <= 0, java.lang.Object(IntList(i467, NULL)), i164, i463)
Cond_Load29582(TRUE, java.lang.Object(IntList(i467, NULL)), i164, i463) → Load2958(NULL, i164, i463)
The set Q consists of the following terms:
Load2958(java.lang.Object(IntList(x0, x1)), x2, x3)
Cond_Load2958(TRUE, java.lang.Object(IntList(x0, x1)), x2, x3)
Cond_Load29581(TRUE, java.lang.Object(IntList(x0, java.lang.Object(IntList(x1, x2)))), x3, x4)
Cond_Load29582(TRUE, java.lang.Object(IntList(x0, NULL)), x1, x2)

(6) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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

The ITRS R consists of the following rules:
Load2958(java.lang.Object(IntList(i466, o4023)), i164, i463) → Cond_Load2958(i466 > 0, java.lang.Object(IntList(i466, o4023)), i164, i463)
Cond_Load2958(TRUE, java.lang.Object(IntList(i466, o4023)), i164, i463) → Load2958(java.lang.Object(IntList(i466 - 1, o4023)), i164, i463 + 1)
Load2958(java.lang.Object(IntList(i467, java.lang.Object(IntList(o4048Field0, o4048Field1)))), i164, i463) → Cond_Load29581(i467 <= 0, java.lang.Object(IntList(i467, java.lang.Object(IntList(o4048Field0, o4048Field1)))), i164, i463)
Cond_Load29581(TRUE, java.lang.Object(IntList(i467, java.lang.Object(IntList(o4048Field0, o4048Field1)))), i164, i463) → Load2958(java.lang.Object(IntList(o4048Field0, o4048Field1)), i164, i463 * i164)
Load2958(java.lang.Object(IntList(i467, NULL)), i164, i463) → Cond_Load29582(i467 <= 0, java.lang.Object(IntList(i467, NULL)), i164, i463)
Cond_Load29582(TRUE, java.lang.Object(IntList(i467, NULL)), i164, i463) → Load2958(NULL, i164, i463)

The integer pair graph contains the following rules and edges:
(0): LOAD2958(java.lang.Object(IntList(i466[0], o4023[0])), i164[0], i463[0]) → COND_LOAD2958(i466[0] > 0, java.lang.Object(IntList(i466[0], o4023[0])), i164[0], i463[0])
(1): COND_LOAD2958(TRUE, java.lang.Object(IntList(i466[1], o4023[1])), i164[1], i463[1]) → LOAD2958(java.lang.Object(IntList(i466[1] - 1, o4023[1])), i164[1], i463[1] + 1)
(2): LOAD2958(java.lang.Object(IntList(i467[2], java.lang.Object(IntList(o4048Field0[2], o4048Field1[2])))), i164[2], i463[2]) → COND_LOAD29581(i467[2] <= 0, java.lang.Object(IntList(i467[2], java.lang.Object(IntList(o4048Field0[2], o4048Field1[2])))), i164[2], i463[2])
(3): COND_LOAD29581(TRUE, java.lang.Object(IntList(i467[3], java.lang.Object(IntList(o4048Field0[3], o4048Field1[3])))), i164[3], i463[3]) → LOAD2958(java.lang.Object(IntList(o4048Field0[3], o4048Field1[3])), i164[3], i463[3] * i164[3])
(4): LOAD2958(java.lang.Object(IntList(i467[4], NULL)), i164[4], i463[4]) → COND_LOAD29582(i467[4] <= 0, java.lang.Object(IntList(i467[4], NULL)), i164[4], i463[4])
(5): COND_LOAD29582(TRUE, java.lang.Object(IntList(i467[5], NULL)), i164[5], i463[5]) → LOAD2958(NULL, i164[5], i463[5])

(0) -> (1), if ((i466[0] > 0 →^* TRUE)∧(i463[0] →^* i463[1])∧(i164[0] →^* i164[1])∧(java.lang.Object(IntList(i466[0], o4023[0])) →^* java.lang.Object(IntList(i466[1], o4023[1]))))

(1) -> (0), if ((i463[1] + 1 →^* i463[0])∧(java.lang.Object(IntList(i466[1] - 1, o4023[1])) →^* java.lang.Object(IntList(i466[0], o4023[0])))∧(i164[1] →^* i164[0]))

(1) -> (2), if ((i463[1] + 1 →^* i463[2])∧(i164[1] →^* i164[2])∧(java.lang.Object(IntList(i466[1] - 1, o4023[1])) →^* java.lang.Object(IntList(i467[2], java.lang.Object(IntList(o4048Field0[2], o4048Field1[2]))))))

(1) -> (4), if ((i463[1] + 1 →^* i463[4])∧(java.lang.Object(IntList(i466[1] - 1, o4023[1])) →^* java.lang.Object(IntList(i467[4], NULL)))∧(i164[1] →^* i164[4]))

(2) -> (3), if ((i164[2] →^* i164[3])∧(i467[2] <= 0 →^* TRUE)∧(java.lang.Object(IntList(i467[2], java.lang.Object(IntList(o4048Field0[2], o4048Field1[2])))) →^* java.lang.Object(IntList(i467[3], java.lang.Object(IntList(o4048Field0[3], o4048Field1[3])))))∧(i463[2] →^* i463[3]))

(3) -> (0), if ((i463[3] * i164[3] →^* i463[0])∧(i164[3] →^* i164[0])∧(java.lang.Object(IntList(o4048Field0[3], o4048Field1[3])) →^* java.lang.Object(IntList(i466[0], o4023[0]))))

(3) -> (2), if ((i463[3] * i164[3] →^* i463[2])∧(java.lang.Object(IntList(o4048Field0[3], o4048Field1[3])) →^* java.lang.Object(IntList(i467[2], java.lang.Object(IntList(o4048Field0[2], o4048Field1[2])))))∧(i164[3] →^* i164[2]))

(3) -> (4), if ((i463[3] * i164[3] →^* i463[4])∧(java.lang.Object(IntList(o4048Field0[3], o4048Field1[3])) →^* java.lang.Object(IntList(i467[4], NULL)))∧(i164[3] →^* i164[4]))

(4) -> (5), if ((java.lang.Object(IntList(i467[4], NULL)) →^* java.lang.Object(IntList(i467[5], NULL)))∧(i467[4] <= 0 →^* TRUE)∧(i164[4] →^* i164[5])∧(i463[4] →^* i463[5]))

(5) -> (0), if ((i164[5] →^* i164[0])∧(i463[5] →^* i463[0])∧(NULL →^* java.lang.Object(IntList(i466[0], o4023[0]))))

(5) -> (2), if ((NULL →^* java.lang.Object(IntList(i467[2], java.lang.Object(IntList(o4048Field0[2], o4048Field1[2])))))∧(i463[5] →^* i463[2])∧(i164[5] →^* i164[2]))

(5) -> (4), if ((i463[5] →^* i463[4])∧(NULL →^* java.lang.Object(IntList(i467[4], NULL)))∧(i164[5] →^* i164[4]))

The set Q consists of the following terms:
Load2958(java.lang.Object(IntList(x0, x1)), x2, x3)
Cond_Load2958(TRUE, java.lang.Object(IntList(x0, x1)), x2, x3)
Cond_Load29581(TRUE, java.lang.Object(IntList(x0, java.lang.Object(IntList(x1, x2)))), x3, x4)
Cond_Load29582(TRUE, java.lang.Object(IntList(x0, NULL)), x1, x2)

(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:

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): LOAD2958(java.lang.Object(IntList(i466[0], o4023[0])), i164[0], i463[0]) → COND_LOAD2958(i466[0] > 0, java.lang.Object(IntList(i466[0], o4023[0])), i164[0], i463[0])
(1): COND_LOAD2958(TRUE, java.lang.Object(IntList(i466[1], o4023[1])), i164[1], i463[1]) → LOAD2958(java.lang.Object(IntList(i466[1] - 1, o4023[1])), i164[1], i463[1] + 1)
(2): LOAD2958(java.lang.Object(IntList(i467[2], java.lang.Object(IntList(o4048Field0[2], o4048Field1[2])))), i164[2], i463[2]) → COND_LOAD29581(i467[2] <= 0, java.lang.Object(IntList(i467[2], java.lang.Object(IntList(o4048Field0[2], o4048Field1[2])))), i164[2], i463[2])
(3): COND_LOAD29581(TRUE, java.lang.Object(IntList(i467[3], java.lang.Object(IntList(o4048Field0[3], o4048Field1[3])))), i164[3], i463[3]) → LOAD2958(java.lang.Object(IntList(o4048Field0[3], o4048Field1[3])), i164[3], i463[3] * i164[3])
(4): LOAD2958(java.lang.Object(IntList(i467[4], NULL)), i164[4], i463[4]) → COND_LOAD29582(i467[4] <= 0, java.lang.Object(IntList(i467[4], NULL)), i164[4], i463[4])
(5): COND_LOAD29582(TRUE, java.lang.Object(IntList(i467[5], NULL)), i164[5], i463[5]) → LOAD2958(NULL, i164[5], i463[5])