Termination proof of /tmp/tmpuMCTDK/PastaB15.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 TRUE

    ↳14 IDP

        ↳15 IDependencyGraphProof (⇔)

        ↳16 IDP

        ↳17 IDPNonInfProof (⇐)

        ↳18 IDP

        ↳19 IDependencyGraphProof (⇔)

        ↳20 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB15

/**
 * 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 PastaB15 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();
        int z = Random.random();

        while (x == y && x > z) {
            while (y > z) {
                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 265 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:
Load848(i20, i37) → Cond_Load848(i20 <= i37 && i20 > i37, i20, i37)
Cond_Load848(TRUE, i20, i37) → Load848(i20, i37)
Inc1074(i55, i20, i37) → Cond_Inc1074(i20 + -1 <= i37, i55, i20, i37)
Cond_Inc1074(TRUE, i55, i20, i37) → Load848(i55, i37)
Load848(i20, i37) → Cond_Load8481(i20 + -1 <= i37 && i20 > i37, i20, i37)
Cond_Load8481(TRUE, i20, i37) → Load848(i20 + -1, i37)
Inc1074(i55, i20, i37) → Cond_Inc10741(i20 + -1 > i37, i55, i20, i37)
Cond_Inc10741(TRUE, i55, i20, i37) → Inc1074(i55 + -1, i20 + -1, i37)
Load848(i20, i37) → Cond_Load8482(i20 + -1 > i37 && i20 > i37, i20, i37)
Cond_Load8482(TRUE, i20, i37) → Inc1074(i20 + -1 + -1, i20 + -1, i37)
The set Q consists of the following terms:
Load848(x0, x1)
Cond_Load848(TRUE, x0, x1)
Inc1074(x0, x1, x2)
Cond_Inc1074(TRUE, x0, x1, x2)
Cond_Load8481(TRUE, x0, x1)
Cond_Inc10741(TRUE, x0, x1, x2)
Cond_Load8482(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:
Load848(i20, i37) → Cond_Load848(i20 <= i37 && i20 > i37, i20, i37)
Cond_Load848(TRUE, i20, i37) → Load848(i20, i37)
Inc1074(i55, i20, i37) → Cond_Inc1074(i20 + -1 <= i37, i55, i20, i37)
Cond_Inc1074(TRUE, i55, i20, i37) → Load848(i55, i37)
Load848(i20, i37) → Cond_Load8481(i20 + -1 <= i37 && i20 > i37, i20, i37)
Cond_Load8481(TRUE, i20, i37) → Load848(i20 + -1, i37)
Inc1074(i55, i20, i37) → Cond_Inc10741(i20 + -1 > i37, i55, i20, i37)
Cond_Inc10741(TRUE, i55, i20, i37) → Inc1074(i55 + -1, i20 + -1, i37)
Load848(i20, i37) → Cond_Load8482(i20 + -1 > i37 && i20 > i37, i20, i37)
Cond_Load8482(TRUE, i20, i37) → Inc1074(i20 + -1 + -1, i20 + -1, i37)

The integer pair graph contains the following rules and edges:
(0): LOAD848(i20[0], i37[0]) → COND_LOAD848(i20[0] <= i37[0] && i20[0] > i37[0], i20[0], i37[0])
(1): COND_LOAD848(TRUE, i20[1], i37[1]) → LOAD848(i20[1], i37[1])
(2): INC1074(i55[2], i20[2], i37[2]) → COND_INC1074(i20[2] + -1 <= i37[2], i55[2], i20[2], i37[2])
(3): COND_INC1074(TRUE, i55[3], i20[3], i37[3]) → LOAD848(i55[3], i37[3])
(4): LOAD848(i20[4], i37[4]) → COND_LOAD8481(i20[4] + -1 <= i37[4] && i20[4] > i37[4], i20[4], i37[4])
(5): COND_LOAD8481(TRUE, i20[5], i37[5]) → LOAD848(i20[5] + -1, i37[5])
(6): INC1074(i55[6], i20[6], i37[6]) → COND_INC10741(i20[6] + -1 > i37[6], i55[6], i20[6], i37[6])
(7): COND_INC10741(TRUE, i55[7], i20[7], i37[7]) → INC1074(i55[7] + -1, i20[7] + -1, i37[7])
(8): LOAD848(i20[8], i37[8]) → COND_LOAD8482(i20[8] + -1 > i37[8] && i20[8] > i37[8], i20[8], i37[8])
(9): COND_LOAD8482(TRUE, i20[9], i37[9]) → INC1074(i20[9] + -1 + -1, i20[9] + -1, i37[9])

(0) -> (1), if ((i20[0] <= i37[0] && i20[0] > i37[0] →^* TRUE)∧(i20[0] →^* i20[1])∧(i37[0] →^* i37[1]))

(1) -> (0), if ((i37[1] →^* i37[0])∧(i20[1] →^* i20[0]))

(1) -> (4), if ((i37[1] →^* i37[4])∧(i20[1] →^* i20[4]))

(1) -> (8), if ((i37[1] →^* i37[8])∧(i20[1] →^* i20[8]))

(2) -> (3), if ((i37[2] →^* i37[3])∧(i55[2] →^* i55[3])∧(i20[2] →^* i20[3])∧(i20[2] + -1 <= i37[2] →^* TRUE))

(3) -> (0), if ((i37[3] →^* i37[0])∧(i55[3] →^* i20[0]))

(3) -> (4), if ((i37[3] →^* i37[4])∧(i55[3] →^* i20[4]))

(3) -> (8), if ((i55[3] →^* i20[8])∧(i37[3] →^* i37[8]))

(4) -> (5), if ((i20[4] →^* i20[5])∧(i20[4] + -1 <= i37[4] && i20[4] > i37[4] →^* TRUE)∧(i37[4] →^* i37[5]))

(5) -> (0), if ((i37[5] →^* i37[0])∧(i20[5] + -1 →^* i20[0]))

(5) -> (4), if ((i20[5] + -1 →^* i20[4])∧(i37[5] →^* i37[4]))

(5) -> (8), if ((i37[5] →^* i37[8])∧(i20[5] + -1 →^* i20[8]))

(6) -> (7), if ((i55[6] →^* i55[7])∧(i20[6] + -1 > i37[6] →^* TRUE)∧(i37[6] →^* i37[7])∧(i20[6] →^* i20[7]))

(7) -> (2), if ((i55[7] + -1 →^* i55[2])∧(i37[7] →^* i37[2])∧(i20[7] + -1 →^* i20[2]))

(7) -> (6), if ((i55[7] + -1 →^* i55[6])∧(i20[7] + -1 →^* i20[6])∧(i37[7] →^* i37[6]))

(8) -> (9), if ((i20[8] + -1 > i37[8] && i20[8] > i37[8] →^* TRUE)∧(i37[8] →^* i37[9])∧(i20[8] →^* i20[9]))

(9) -> (2), if ((i20[9] + -1 + -1 →^* i55[2])∧(i37[9] →^* i37[2])∧(i20[9] + -1 →^* i20[2]))

(9) -> (6), if ((i20[9] + -1 + -1 →^* i55[6])∧(i37[9] →^* i37[6])∧(i20[9] + -1 →^* i20[6]))

The set Q consists of the following terms:
Load848(x0, x1)
Cond_Load848(TRUE, x0, x1)
Inc1074(x0, x1, x2)
Cond_Inc1074(TRUE, x0, x1, x2)
Cond_Load8481(TRUE, x0, x1)
Cond_Inc10741(TRUE, x0, x1, x2)
Cond_Load8482(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): LOAD848(i20[0], i37[0]) → COND_LOAD848(i20[0] <= i37[0] && i20[0] > i37[0], i20[0], i37[0])
(1): COND_LOAD848(TRUE, i20[1], i37[1]) → LOAD848(i20[1], i37[1])
(2): INC1074(i55[2], i20[2], i37[2]) → COND_INC1074(i20[2] + -1 <= i37[2], i55[2], i20[2], i37[2])
(3): COND_INC1074(TRUE, i55[3], i20[3], i37[3]) → LOAD848(i55[3], i37[3])
(4): LOAD848(i20[4], i37[4]) → COND_LOAD8481(i20[4] + -1 <= i37[4] && i20[4] > i37[4], i20[4], i37[4])
(5): COND_LOAD8481(TRUE, i20[5], i37[5]) → LOAD848(i20[5] + -1, i37[5])
(6): INC1074(i55[6], i20[6], i37[6]) → COND_INC10741(i20[6] + -1 > i37[6], i55[6], i20[6], i37[6])
(7): COND_INC10741(TRUE, i55[7], i20[7], i37[7]) → INC1074(i55[7] + -1, i20[7] + -1, i37[7])
(8): LOAD848(i20[8], i37[8]) → COND_LOAD8482(i20[8] + -1 > i37[8] && i20[8] > i37[8], i20[8], i37[8])
(9): COND_LOAD8482(TRUE, i20[9], i37[9]) → INC1074(i20[9] + -1 + -1, i20[9] + -1, i37[9])