Termination proof of /tmp/tmpoClNHB/PastaB10.jar

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 IDP

        ↳14 IDPNonInfProof (⇐)

        ↳15 IDP

        ↳16 IDependencyGraphProof (⇔)

        ↳17 TRUE

    ↳18 IDP

        ↳19 IDependencyGraphProof (⇔)

        ↳20 IDP

        ↳21 IDPNonInfProof (⇐)

        ↳22 AND

            ↳23 IDP

                ↳24 IDependencyGraphProof (⇔)

                ↳25 TRUE

            ↳26 IDP

                ↳27 IDependencyGraphProof (⇔)

                ↳28 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB10

/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaB10 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x + y > 0) {
            if (x > 0) {
                x--;
            } else if (y > 0) {
                y--;
            } else {
                continue;
            }            
        }
    }
}


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 197 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:
Load974(i153, i120) → Cond_Load974(i153 > 0 && i153 + i120 > 0, i153, i120)
Cond_Load974(TRUE, i153, i120) → Load974(i153 + -1, i120)
Load974(i154, i161) → Cond_Load9741(i161 > 0 && i154 <= 0 && i154 + i161 > 0, i154, i161)
Cond_Load9741(TRUE, i154, i161) → Load974(i154, i161 + -1)
Load974(i154, i162) → Cond_Load9742(i162 <= 0 && i154 <= 0 && i154 + i162 > 0, i154, i162)
Cond_Load9742(TRUE, i154, i162) → Load974(i154, i162)
The set Q consists of the following terms:
Load974(x0, x1)
Cond_Load974(TRUE, x0, x1)
Cond_Load9741(TRUE, x0, x1)
Cond_Load9742(TRUE, x0, x1)

(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:
Load974(i153, i120) → Cond_Load974(i153 > 0 && i153 + i120 > 0, i153, i120)
Cond_Load974(TRUE, i153, i120) → Load974(i153 + -1, i120)
Load974(i154, i161) → Cond_Load9741(i161 > 0 && i154 <= 0 && i154 + i161 > 0, i154, i161)
Cond_Load9741(TRUE, i154, i161) → Load974(i154, i161 + -1)
Load974(i154, i162) → Cond_Load9742(i162 <= 0 && i154 <= 0 && i154 + i162 > 0, i154, i162)
Cond_Load9742(TRUE, i154, i162) → Load974(i154, i162)

The integer pair graph contains the following rules and edges:
(0): LOAD974(i153[0], i120[0]) → COND_LOAD974(i153[0] > 0 && i153[0] + i120[0] > 0, i153[0], i120[0])
(1): COND_LOAD974(TRUE, i153[1], i120[1]) → LOAD974(i153[1] + -1, i120[1])
(2): LOAD974(i154[2], i161[2]) → COND_LOAD9741(i161[2] > 0 && i154[2] <= 0 && i154[2] + i161[2] > 0, i154[2], i161[2])
(3): COND_LOAD9741(TRUE, i154[3], i161[3]) → LOAD974(i154[3], i161[3] + -1)
(4): LOAD974(i154[4], i162[4]) → COND_LOAD9742(i162[4] <= 0 && i154[4] <= 0 && i154[4] + i162[4] > 0, i154[4], i162[4])
(5): COND_LOAD9742(TRUE, i154[5], i162[5]) → LOAD974(i154[5], i162[5])

(0) -> (1), if ((i120[0] →^* i120[1])∧(i153[0] > 0 && i153[0] + i120[0] > 0 →^* TRUE)∧(i153[0] →^* i153[1]))

(1) -> (0), if ((i120[1] →^* i120[0])∧(i153[1] + -1 →^* i153[0]))

(1) -> (2), if ((i120[1] →^* i161[2])∧(i153[1] + -1 →^* i154[2]))

(1) -> (4), if ((i153[1] + -1 →^* i154[4])∧(i120[1] →^* i162[4]))

(2) -> (3), if ((i161[2] > 0 && i154[2] <= 0 && i154[2] + i161[2] > 0 →^* TRUE)∧(i161[2] →^* i161[3])∧(i154[2] →^* i154[3]))

(3) -> (0), if ((i154[3] →^* i153[0])∧(i161[3] + -1 →^* i120[0]))

(3) -> (2), if ((i154[3] →^* i154[2])∧(i161[3] + -1 →^* i161[2]))

(3) -> (4), if ((i161[3] + -1 →^* i162[4])∧(i154[3] →^* i154[4]))

(4) -> (5), if ((i162[4] <= 0 && i154[4] <= 0 && i154[4] + i162[4] > 0 →^* TRUE)∧(i154[4] →^* i154[5])∧(i162[4] →^* i162[5]))

(5) -> (0), if ((i154[5] →^* i153[0])∧(i162[5] →^* i120[0]))

(5) -> (2), if ((i162[5] →^* i161[2])∧(i154[5] →^* i154[2]))

(5) -> (4), if ((i162[5] →^* i162[4])∧(i154[5] →^* i154[4]))

The set Q consists of the following terms:
Load974(x0, x1)
Cond_Load974(TRUE, x0, x1)
Cond_Load9741(TRUE, x0, x1)
Cond_Load9742(TRUE, x0, x1)

(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): LOAD974(i153[0], i120[0]) → COND_LOAD974(i153[0] > 0 && i153[0] + i120[0] > 0, i153[0], i120[0])
(1): COND_LOAD974(TRUE, i153[1], i120[1]) → LOAD974(i153[1] + -1, i120[1])
(2): LOAD974(i154[2], i161[2]) → COND_LOAD9741(i161[2] > 0 && i154[2] <= 0 && i154[2] + i161[2] > 0, i154[2], i161[2])
(3): COND_LOAD9741(TRUE, i154[3], i161[3]) → LOAD974(i154[3], i161[3] + -1)
(4): LOAD974(i154[4], i162[4]) → COND_LOAD9742(i162[4] <= 0 && i154[4] <= 0 && i154[4] + i162[4] > 0, i154[4], i162[4])
(5): COND_LOAD9742(TRUE, i154[5], i162[5]) → LOAD974(i154[5], i162[5])