Termination proof of /tmp/tmphMZ

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇐)

↳2 FIGraph

↳3 FIGtoITRSProof (⇐)

↳4 ITRS

↳5 ITRStoIDPProof (⇔)

↳6 IDP

↳7 UsableRulesProof (⇔)

↳8 IDP

↳9 IDPNonInfProof (⇐)

↳10 AND

    ↳11 IDP

        ↳12 IDependencyGraphProof (⇔)

        ↳13 TRUE

    ↳14 IDP

        ↳15 IDependencyGraphProof (⇔)

        ↳16 TRUE

(0) Obligation:

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

/**
 * Abstract class to provide some additional mathematical functions
 * which are not provided by java.lang.Math.
 *
 * @author fuhs
 */
public abstract class AProVEMath {

  /**
   * Returns <code>base<sup>exponent</sup></code>.
   * Works considerably faster than java.lang.Math.pow(double, double).
   *
   * @param base base of the power
   * @param exponent non-negative exponent of the power
   * @return base<sup>exponent</sup>
   */
  public static int power (int base, int exponent) {
    if (exponent == 0) {
      return 1;
    }
    else if (exponent == 1) {
      return base;
    }
    else if (base == 2) {
      return base << (exponent-1);
    }
    else {
      int result = 1;
      while (exponent > 0) {
        if (exponent % 2 == 1) {
          result *= base;
        }
        base *= base;
        exponent /= 2;
      }
      return result;
    }
  }

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    power(x, y);
  }
}


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 238 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) 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:
Load1494(i358, i362, i360) → Cond_Load1494(i362 > 0 && 1 = i362 % 2, i358, i362, i360)
Cond_Load1494(TRUE, i358, i362, i360) → Load1533(i358, i362, i360 * i358)
Load1533(i358, i362, i360) → Load1494(i358 * i358, i362 / 2, i360)
Load1494(i358, i362, i360) → Cond_Load14941(i362 % 2 >= 0 && !(i362 % 2 = 1) && i362 > 0, i358, i362, i360)
Cond_Load14941(TRUE, i358, i362, i360) → Load1494(i358 * i358, i362 / 2, i360)
The set Q consists of the following terms:
Load1494(x0, x1, x2)
Cond_Load1494(TRUE, x0, x1, x2)
Load1533(x0, x1, x2)
Cond_Load14941(TRUE, x0, x1, x2)

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(6) 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:

Boolean, Integer

The ITRS R consists of the following rules:
Load1494(i358, i362, i360) → Cond_Load1494(i362 > 0 && 1 = i362 % 2, i358, i362, i360)
Cond_Load1494(TRUE, i358, i362, i360) → Load1533(i358, i362, i360 * i358)
Load1533(i358, i362, i360) → Load1494(i358 * i358, i362 / 2, i360)
Load1494(i358, i362, i360) → Cond_Load14941(i362 % 2 >= 0 && !(i362 % 2 = 1) && i362 > 0, i358, i362, i360)
Cond_Load14941(TRUE, i358, i362, i360) → Load1494(i358 * i358, i362 / 2, i360)

The integer pair graph contains the following rules and edges:
(0): LOAD1494(i358[0], i362[0], i360[0]) → COND_LOAD1494(i362[0] > 0 && 1 = i362[0] % 2, i358[0], i362[0], i360[0])
(1): COND_LOAD1494(TRUE, i358[1], i362[1], i360[1]) → LOAD1533(i358[1], i362[1], i360[1] * i358[1])
(2): LOAD1533(i358[2], i362[2], i360[2]) → LOAD1494(i358[2] * i358[2], i362[2] / 2, i360[2])
(3): LOAD1494(i358[3], i362[3], i360[3]) → COND_LOAD14941(i362[3] % 2 >= 0 && !(i362[3] % 2 = 1) && i362[3] > 0, i358[3], i362[3], i360[3])
(4): COND_LOAD14941(TRUE, i358[4], i362[4], i360[4]) → LOAD1494(i358[4] * i358[4], i362[4] / 2, i360[4])

(0) -> (1), if ((i362[0] →^* i362[1])∧(i362[0] > 0 && 1 = i362[0] % 2 →^* TRUE)∧(i358[0] →^* i358[1])∧(i360[0] →^* i360[1]))

(1) -> (2), if ((i360[1] * i358[1] →^* i360[2])∧(i358[1] →^* i358[2])∧(i362[1] →^* i362[2]))

(2) -> (0), if ((i358[2] * i358[2] →^* i358[0])∧(i362[2] / 2 →^* i362[0])∧(i360[2] →^* i360[0]))

(2) -> (3), if ((i360[2] →^* i360[3])∧(i358[2] * i358[2] →^* i358[3])∧(i362[2] / 2 →^* i362[3]))

(3) -> (4), if ((i358[3] →^* i358[4])∧(i360[3] →^* i360[4])∧(i362[3] % 2 >= 0 && !(i362[3] % 2 = 1) && i362[3] > 0 →^* TRUE)∧(i362[3] →^* i362[4]))

(4) -> (0), if ((i360[4] →^* i360[0])∧(i362[4] / 2 →^* i362[0])∧(i358[4] * i358[4] →^* i358[0]))

(4) -> (3), if ((i358[4] * i358[4] →^* i358[3])∧(i360[4] →^* i360[3])∧(i362[4] / 2 →^* i362[3]))

The set Q consists of the following terms:
Load1494(x0, x1, x2)
Cond_Load1494(TRUE, x0, x1, x2)
Load1533(x0, x1, x2)
Cond_Load14941(TRUE, x0, x1, x2)

(7) 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.

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1494(i358[0], i362[0], i360[0]) → COND_LOAD1494(i362[0] > 0 && 1 = i362[0] % 2, i358[0], i362[0], i360[0])
(1): COND_LOAD1494(TRUE, i358[1], i362[1], i360[1]) → LOAD1533(i358[1], i362[1], i360[1] * i358[1])
(2): LOAD1533(i358[2], i362[2], i360[2]) → LOAD1494(i358[2] * i358[2], i362[2] / 2, i360[2])
(3): LOAD1494(i358[3], i362[3], i360[3]) → COND_LOAD14941(i362[3] % 2 >= 0 && !(i362[3] % 2 = 1) && i362[3] > 0, i358[3], i362[3], i360[3])
(4): COND_LOAD14941(TRUE, i358[4], i362[4], i360[4]) → LOAD1494(i358[4] * i358[4], i362[4] / 2, i360[4])