Termination proof of /tmp/tmpMFbx5S/GCD4.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: GCD4

public class GCD4 {
  public static int mod(int a, int b) {
    while(a>=b && b > 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 218 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:
Load1303(i427, i421) → Cond_Load1303(i427 > 0 && i421 > 0, i427, i421)
Cond_Load1303(TRUE, i427, i421) → Load1438(i421, i421, i421, i427, i421)
Load1438(i421, i421, i421, i507, i421) → Cond_Load1438(i421 > 0 && i507 >= i421, i421, i421, i421, i507, i421)
Cond_Load1438(TRUE, i421, i421, i421, i507, i421) → Load1438(i421, i421, i421, i507 - i421, i421)
Load1438(i421, i421, i421, i507, i421) → Cond_Load14381(i507 < i421, i421, i421, i421, i507, i421)
Cond_Load14381(TRUE, i421, i421, i421, i507, i421) → Load1303(i421, i507)
The set Q consists of the following terms:
Load1303(x0, x1)
Cond_Load1303(TRUE, x0, x1)
Load1438(x0, x0, x0, x1, x0)
Cond_Load1438(TRUE, x0, x0, x0, x1, x0)
Cond_Load14381(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_Load14381(x1, x2, x3, x4, x5, x6) → Cond_Load14381(x1, x5, x6)
Load1438(x1, x2, x3, x4, x5) → Load1438(x4, x5)
Cond_Load1438(x1, x2, x3, x4, x5, x6) → Cond_Load1438(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:
Load1303(i427, i421) → Cond_Load1303(i427 > 0 && i421 > 0, i427, i421)
Cond_Load1303(TRUE, i427, i421) → Load1438(i427, i421)
Load1438(i507, i421) → Cond_Load1438(i421 > 0 && i507 >= i421, i507, i421)
Cond_Load1438(TRUE, i507, i421) → Load1438(i507 - i421, i421)
Load1438(i507, i421) → Cond_Load14381(i507 < i421, i507, i421)
Cond_Load14381(TRUE, i507, i421) → Load1303(i421, i507)
The set Q consists of the following terms:
Load1303(x0, x1)
Cond_Load1303(TRUE, x0, x1)
Load1438(x0, x1)
Cond_Load1438(TRUE, x0, x1)
Cond_Load14381(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:
Load1303(i427, i421) → Cond_Load1303(i427 > 0 && i421 > 0, i427, i421)
Cond_Load1303(TRUE, i427, i421) → Load1438(i427, i421)
Load1438(i507, i421) → Cond_Load1438(i421 > 0 && i507 >= i421, i507, i421)
Cond_Load1438(TRUE, i507, i421) → Load1438(i507 - i421, i421)
Load1438(i507, i421) → Cond_Load14381(i507 < i421, i507, i421)
Cond_Load14381(TRUE, i507, i421) → Load1303(i421, i507)

The integer pair graph contains the following rules and edges:
(0): LOAD1303(i427[0], i421[0]) → COND_LOAD1303(i427[0] > 0 && i421[0] > 0, i427[0], i421[0])
(1): COND_LOAD1303(TRUE, i427[1], i421[1]) → LOAD1438(i427[1], i421[1])
(2): LOAD1438(i507[2], i421[2]) → COND_LOAD1438(i421[2] > 0 && i507[2] >= i421[2], i507[2], i421[2])
(3): COND_LOAD1438(TRUE, i507[3], i421[3]) → LOAD1438(i507[3] - i421[3], i421[3])
(4): LOAD1438(i507[4], i421[4]) → COND_LOAD14381(i507[4] < i421[4], i507[4], i421[4])
(5): COND_LOAD14381(TRUE, i507[5], i421[5]) → LOAD1303(i421[5], i507[5])

(0) -> (1), if ((i427[0] > 0 && i421[0] > 0 →^* TRUE)∧(i421[0] →^* i421[1])∧(i427[0] →^* i427[1]))

(1) -> (2), if ((i427[1] →^* i507[2])∧(i421[1] →^* i421[2]))

(1) -> (4), if ((i421[1] →^* i421[4])∧(i427[1] →^* i507[4]))

(2) -> (3), if ((i507[2] →^* i507[3])∧(i421[2] > 0 && i507[2] >= i421[2] →^* TRUE)∧(i421[2] →^* i421[3]))

(3) -> (2), if ((i421[3] →^* i421[2])∧(i507[3] - i421[3] →^* i507[2]))

(3) -> (4), if ((i421[3] →^* i421[4])∧(i507[3] - i421[3] →^* i507[4]))

(4) -> (5), if ((i421[4] →^* i421[5])∧(i507[4] →^* i507[5])∧(i507[4] < i421[4] →^* TRUE))

(5) -> (0), if ((i507[5] →^* i421[0])∧(i421[5] →^* i427[0]))

The set Q consists of the following terms:
Load1303(x0, x1)
Cond_Load1303(TRUE, x0, x1)
Load1438(x0, x1)
Cond_Load1438(TRUE, x0, x1)
Cond_Load14381(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): LOAD1303(i427[0], i421[0]) → COND_LOAD1303(i427[0] > 0 && i421[0] > 0, i427[0], i421[0])
(1): COND_LOAD1303(TRUE, i427[1], i421[1]) → LOAD1438(i427[1], i421[1])
(2): LOAD1438(i507[2], i421[2]) → COND_LOAD1438(i421[2] > 0 && i507[2] >= i421[2], i507[2], i421[2])
(3): COND_LOAD1438(TRUE, i507[3], i421[3]) → LOAD1438(i507[3] - i421[3], i421[3])
(4): LOAD1438(i507[4], i421[4]) → COND_LOAD14381(i507[4] < i421[4], i507[4], i421[4])
(5): COND_LOAD14381(TRUE, i507[5], i421[5]) → LOAD1303(i421[5], i507[5])