Termination proof of /tmp/tmpwQ98N9/DivMinus2.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 IDP

        ↳27 IDPNonInfProof (⇐)

        ↳28 AND

            ↳29 IDP

                ↳30 IDependencyGraphProof (⇔)

                ↳31 TRUE

            ↳32 IDP

                ↳33 IDependencyGraphProof (⇔)

                ↳34 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: DivMinus2

public class DivMinus2 {
  public static int div(int x, int y) {
    int res = 0;
    while (x >= y && y > 0) {
      x = minus(x,y);
      res = res + 1;
    }
    return res;
  }

  public static int minus(int x, int y) {
    while (y != 0) {
      if (y > 0)  {
        y--;
        x--;
      } else  {
        y++;
        x++;
      }
    }
    return x;
  }

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    div(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:
Load1381(i669, i614, i669, i616) → Cond_Load1381(i669 > 0 && i614 >= i669, i669, i614, i669, i616)
Cond_Load1381(TRUE, i669, i614, i669, i616) → Load1471(i669, i669, i616, i669, i614, i669)
Load1471(i669, i669, i616, i669, i730, i743) → Cond_Load1471(i743 > 0, i669, i669, i616, i669, i730, i743)
Cond_Load1471(TRUE, i669, i669, i616, i669, i730, i743) → Load1471(i669, i669, i616, i669, i730 + -1, i743 + -1)
Load1471(i669, i669, i616, i669, i730, 0) → Load1381(i669, i730, i669, i616 + 1)
The set Q consists of the following terms:
Load1381(x0, x1, x0, x2)
Cond_Load1381(TRUE, x0, x1, x0, x2)
Load1471(x0, x0, x1, x0, x2, x3)
Cond_Load1471(TRUE, x0, x0, x1, x0, x2, x3)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

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

Load1381(x1, x2, x3, x4) → Load1381(x2, x3, x4)
Load1471(x1, x2, x3, x4, x5, x6) → Load1471(x3, x4, x5, x6)
Cond_Load1471(x1, x2, x3, x4, x5, x6, x7) → Cond_Load1471(x1, x4, x5, x6, x7)
Cond_Load1381(x1, x2, x3, x4, x5) → Cond_Load1381(x1, x3, x4, x5)

(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:
Load1381(i614, i669, i616) → Cond_Load1381(i669 > 0 && i614 >= i669, i614, i669, i616)
Cond_Load1381(TRUE, i614, i669, i616) → Load1471(i616, i669, i614, i669)
Load1471(i616, i669, i730, i743) → Cond_Load1471(i743 > 0, i616, i669, i730, i743)
Cond_Load1471(TRUE, i616, i669, i730, i743) → Load1471(i616, i669, i730 + -1, i743 + -1)
Load1471(i616, i669, i730, 0) → Load1381(i730, i669, i616 + 1)
The set Q consists of the following terms:
Load1381(x0, x1, x2)
Cond_Load1381(TRUE, x0, x1, x2)
Load1471(x0, x1, x2, x3)
Cond_Load1471(TRUE, x0, x1, x2, x3)

(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:
Load1381(i614, i669, i616) → Cond_Load1381(i669 > 0 && i614 >= i669, i614, i669, i616)
Cond_Load1381(TRUE, i614, i669, i616) → Load1471(i616, i669, i614, i669)
Load1471(i616, i669, i730, i743) → Cond_Load1471(i743 > 0, i616, i669, i730, i743)
Cond_Load1471(TRUE, i616, i669, i730, i743) → Load1471(i616, i669, i730 + -1, i743 + -1)
Load1471(i616, i669, i730, 0) → Load1381(i730, i669, i616 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD1381(i614[0], i669[0], i616[0]) → COND_LOAD1381(i669[0] > 0 && i614[0] >= i669[0], i614[0], i669[0], i616[0])
(1): COND_LOAD1381(TRUE, i614[1], i669[1], i616[1]) → LOAD1471(i616[1], i669[1], i614[1], i669[1])
(2): LOAD1471(i616[2], i669[2], i730[2], i743[2]) → COND_LOAD1471(i743[2] > 0, i616[2], i669[2], i730[2], i743[2])
(3): COND_LOAD1471(TRUE, i616[3], i669[3], i730[3], i743[3]) → LOAD1471(i616[3], i669[3], i730[3] + -1, i743[3] + -1)
(4): LOAD1471(i616[4], i669[4], i730[4], 0) → LOAD1381(i730[4], i669[4], i616[4] + 1)

(0) -> (1), if ((i669[0] > 0 && i614[0] >= i669[0] →^* TRUE)∧(i616[0] →^* i616[1])∧(i614[0] →^* i614[1])∧(i669[0] →^* i669[1]))

(1) -> (2), if ((i614[1] →^* i730[2])∧(i669[1] →^* i743[2])∧(i669[1] →^* i669[2])∧(i616[1] →^* i616[2]))

(1) -> (4), if ((i614[1] →^* i730[4])∧(i669[1] →^* i669[4])∧(i616[1] →^* i616[4])∧(i669[1] →^* 0))

(2) -> (3), if ((i669[2] →^* i669[3])∧(i743[2] > 0 →^* TRUE)∧(i743[2] →^* i743[3])∧(i730[2] →^* i730[3])∧(i616[2] →^* i616[3]))

(3) -> (2), if ((i669[3] →^* i669[2])∧(i743[3] + -1 →^* i743[2])∧(i616[3] →^* i616[2])∧(i730[3] + -1 →^* i730[2]))

(3) -> (4), if ((i669[3] →^* i669[4])∧(i730[3] + -1 →^* i730[4])∧(i616[3] →^* i616[4])∧(i743[3] + -1 →^* 0))

(4) -> (0), if ((i669[4] →^* i669[0])∧(i616[4] + 1 →^* i616[0])∧(i730[4] →^* i614[0]))

The set Q consists of the following terms:
Load1381(x0, x1, x2)
Cond_Load1381(TRUE, x0, x1, x2)
Load1471(x0, x1, x2, x3)
Cond_Load1471(TRUE, x0, x1, x2, x3)

(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): LOAD1381(i614[0], i669[0], i616[0]) → COND_LOAD1381(i669[0] > 0 && i614[0] >= i669[0], i614[0], i669[0], i616[0])
(1): COND_LOAD1381(TRUE, i614[1], i669[1], i616[1]) → LOAD1471(i616[1], i669[1], i614[1], i669[1])
(2): LOAD1471(i616[2], i669[2], i730[2], i743[2]) → COND_LOAD1471(i743[2] > 0, i616[2], i669[2], i730[2], i743[2])
(3): COND_LOAD1471(TRUE, i616[3], i669[3], i730[3], i743[3]) → LOAD1471(i616[3], i669[3], i730[3] + -1, i743[3] + -1)
(4): LOAD1471(i616[4], i669[4], i730[4], 0) → LOAD1381(i730[4], i669[4], i616[4] + 1)