Termination proof of /tmp/tmpgyKL6q/GCD3.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇐)

↳2 FIGraph

↳3 FIGtoITRSProof (⇐)

↳4 ITRS

↳5 DuplicateArgsRemoverProof (⇔)

↳6 ITRS

↳7 ITRStoIDPProof (⇔)

↳8 IDP

↳9 UsableRulesProof (⇔)

↳10 IDP

↳11 IDPNonInfProof (⇐)

↳12 AND

    ↳13 IDP

        ↳14 IDependencyGraphProof (⇔)

        ↳15 IDP

        ↳16 IDPNonInfProof (⇐)

        ↳17 AND

            ↳18 IDP

                ↳19 IDependencyGraphProof (⇔)

                ↳20 TRUE

            ↳21 IDP

                ↳22 IDependencyGraphProof (⇔)

                ↳23 TRUE

    ↳24 IDP

        ↳25 IDependencyGraphProof (⇔)

        ↳26 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: GCD3

public class GCD3 {
  public static int mod(int a, int b) {
    if(b == 0) {
      return b;
    }
    if(b < 0) {
      a = -a;
    }
    if(a > 0) {
      while(a>=b) {
        a -= b;
      }
      return a;
    } else {
      while(a < 0) {
        a -= b;
      }
      return a;
    }
  }

  public static int gcd(int a, int b) {
    int tmp;
    while(b > 0 && a > 0) {
      tmp = b;
      b = mod(a, b);
      a = tmp;
    }
    return a;
  }

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    gcd(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 222 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:
Load1366(i270, i268) → Cond_Load1366(i270 > 0 && i268 > 0, i270, i268)
Cond_Load1366(TRUE, i270, i268) → Load1489(i268, i268, i268, i270, i268)
Load1489(i268, i268, i268, i313, i268) → Cond_Load1489(i268 > 0 && i313 >= i268, i268, i268, i268, i313, i268)
Cond_Load1489(TRUE, i268, i268, i268, i313, i268) → Load1489(i268, i268, i268, i313 - i268, i268)
Load1489(i268, i268, i268, i313, i268) → Cond_Load14891(i313 < i268, i268, i268, i268, i313, i268)
Cond_Load14891(TRUE, i268, i268, i268, i313, i268) → Load1366(i268, i313)
The set Q consists of the following terms:
Load1366(x0, x1)
Cond_Load1366(TRUE, x0, x1)
Load1489(x0, x0, x0, x1, x0)
Cond_Load1489(TRUE, x0, x0, x0, x1, x0)
Cond_Load14891(TRUE, x0, x0, x0, x1, x0)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
We removed arguments according to the following replacements:

Cond_Load14891(x1, x2, x3, x4, x5, x6) → Cond_Load14891(x1, x5, x6)
Load1489(x1, x2, x3, x4, x5) → Load1489(x4, x5)
Cond_Load1489(x1, x2, x3, x4, x5, x6) → Cond_Load1489(x1, x5, x6)

(6) 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:
Load1366(i270, i268) → Cond_Load1366(i270 > 0 && i268 > 0, i270, i268)
Cond_Load1366(TRUE, i270, i268) → Load1489(i270, i268)
Load1489(i313, i268) → Cond_Load1489(i268 > 0 && i313 >= i268, i313, i268)
Cond_Load1489(TRUE, i313, i268) → Load1489(i313 - i268, i268)
Load1489(i313, i268) → Cond_Load14891(i313 < i268, i313, i268)
Cond_Load14891(TRUE, i313, i268) → Load1366(i268, i313)
The set Q consists of the following terms:
Load1366(x0, x1)
Cond_Load1366(TRUE, x0, x1)
Load1489(x0, x1)
Cond_Load1489(TRUE, x0, x1)
Cond_Load14891(TRUE, x0, x1)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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

The ITRS R consists of the following rules:
Load1366(i270, i268) → Cond_Load1366(i270 > 0 && i268 > 0, i270, i268)
Cond_Load1366(TRUE, i270, i268) → Load1489(i270, i268)
Load1489(i313, i268) → Cond_Load1489(i268 > 0 && i313 >= i268, i313, i268)
Cond_Load1489(TRUE, i313, i268) → Load1489(i313 - i268, i268)
Load1489(i313, i268) → Cond_Load14891(i313 < i268, i313, i268)
Cond_Load14891(TRUE, i313, i268) → Load1366(i268, i313)

The integer pair graph contains the following rules and edges:
(0): LOAD1366(i270[0], i268[0]) → COND_LOAD1366(i270[0] > 0 && i268[0] > 0, i270[0], i268[0])
(1): COND_LOAD1366(TRUE, i270[1], i268[1]) → LOAD1489(i270[1], i268[1])
(2): LOAD1489(i313[2], i268[2]) → COND_LOAD1489(i268[2] > 0 && i313[2] >= i268[2], i313[2], i268[2])
(3): COND_LOAD1489(TRUE, i313[3], i268[3]) → LOAD1489(i313[3] - i268[3], i268[3])
(4): LOAD1489(i313[4], i268[4]) → COND_LOAD14891(i313[4] < i268[4], i313[4], i268[4])
(5): COND_LOAD14891(TRUE, i313[5], i268[5]) → LOAD1366(i268[5], i313[5])

(0) -> (1), if ((i268[0] →^* i268[1])∧(i270[0] →^* i270[1])∧(i270[0] > 0 && i268[0] > 0 →^* TRUE))

(1) -> (2), if ((i270[1] →^* i313[2])∧(i268[1] →^* i268[2]))

(1) -> (4), if ((i270[1] →^* i313[4])∧(i268[1] →^* i268[4]))

(2) -> (3), if ((i313[2] →^* i313[3])∧(i268[2] > 0 && i313[2] >= i268[2] →^* TRUE)∧(i268[2] →^* i268[3]))

(3) -> (2), if ((i268[3] →^* i268[2])∧(i313[3] - i268[3] →^* i313[2]))

(3) -> (4), if ((i313[3] - i268[3] →^* i313[4])∧(i268[3] →^* i268[4]))

(4) -> (5), if ((i313[4] < i268[4] →^* TRUE)∧(i268[4] →^* i268[5])∧(i313[4] →^* i313[5]))

(5) -> (0), if ((i268[5] →^* i270[0])∧(i313[5] →^* i268[0]))

The set Q consists of the following terms:
Load1366(x0, x1)
Cond_Load1366(TRUE, x0, x1)
Load1489(x0, x1)
Cond_Load1489(TRUE, x0, x1)
Cond_Load14891(TRUE, x0, x1)

(9) 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.

(10) 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): LOAD1366(i270[0], i268[0]) → COND_LOAD1366(i270[0] > 0 && i268[0] > 0, i270[0], i268[0])
(1): COND_LOAD1366(TRUE, i270[1], i268[1]) → LOAD1489(i270[1], i268[1])
(2): LOAD1489(i313[2], i268[2]) → COND_LOAD1489(i268[2] > 0 && i313[2] >= i268[2], i313[2], i268[2])
(3): COND_LOAD1489(TRUE, i313[3], i268[3]) → LOAD1489(i313[3] - i268[3], i268[3])
(4): LOAD1489(i313[4], i268[4]) → COND_LOAD14891(i313[4] < i268[4], i313[4], i268[4])
(5): COND_LOAD14891(TRUE, i313[5], i268[5]) → LOAD1366(i268[5], i313[5])