### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Round3
`public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}public class Round3{  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    while (x % 3 != 0) {      x++;    }  }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 124 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:
The set Q consists of the following terms:

### (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:

The integer pair graph contains the following rules and edges:

(0) -> (1), if ((i17[0]* i17[1])∧(!(i17[0] % 3 = 0) →* TRUE))

(1) -> (0), if ((i17[1] + 1* i17[0]))

The set Q consists of the following terms:

### (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) -> (1), if ((i17[0]* i17[1])∧(!(i17[0] % 3 = 0) →* TRUE))

(1) -> (0), if ((i17[1] + 1* i17[0]))

The set Q consists of the following terms:

### (9) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

### (10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

The TRS R consists of the following rules:

not(true) → false
not(false) → true
equal_int(pos(01), pos(01)) → true
equal_int(neg(01), pos(01)) → true
equal_int(neg(01), neg(01)) → true
equal_int(pos(01), neg(01)) → true
equal_int(pos(01), pos(s(y))) → false
equal_int(neg(01), pos(s(y))) → false
equal_int(pos(01), neg(s(y))) → false
equal_int(neg(01), neg(s(y))) → false
equal_int(pos(s(x)), pos(01)) → false
equal_int(pos(s(x)), neg(01)) → false
equal_int(neg(s(x)), pos(01)) → false
equal_int(neg(s(x)), neg(01)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(01, s(x)) → 01
mod_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(true, x, y) → x
if(false, x, y) → y
minus_nat_s(x, 01) → x
minus_nat_s(01, s(y)) → 01
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(01, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(01, 01) → pos(01)
minus_nat(01, s(y)) → neg(s(y))
minus_nat(s(x), 01) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

not(true)
not(false)
equal_int(pos(01), pos(01))
equal_int(neg(01), pos(01))
equal_int(neg(01), neg(01))
equal_int(pos(01), neg(01))
equal_int(pos(01), pos(s(x0)))
equal_int(neg(01), pos(s(x0)))
equal_int(pos(01), neg(s(x0)))
equal_int(neg(01), neg(s(x0)))
equal_int(pos(s(x0)), pos(01))
equal_int(pos(s(x0)), neg(01))
equal_int(neg(s(x0)), pos(01))
equal_int(neg(s(x0)), neg(01))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(01, s(x0))
mod_nat(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 01)
minus_nat_s(01, s(x0))
minus_nat_s(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(01, x0)
plus_nat(s(x0), x1)
minus_nat(01, 01)
minus_nat(01, s(x0))
minus_nat(s(x0), 01)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

### (11) 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.

### (12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(01, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(01, 01) → pos(01)
minus_nat(01, s(y)) → neg(s(y))
minus_nat(s(x), 01) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(01), pos(01)) → true
equal_int(neg(01), pos(01)) → true
equal_int(pos(s(x)), pos(01)) → false
equal_int(neg(s(x)), pos(01)) → false
not(true) → false
not(false) → true
mod_nat(01, s(x)) → 01
mod_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
minus_nat_s(x, 01) → x
minus_nat_s(01, s(y)) → 01
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)

The set Q consists of the following terms:

not(true)
not(false)
equal_int(pos(01), pos(01))
equal_int(neg(01), pos(01))
equal_int(neg(01), neg(01))
equal_int(pos(01), neg(01))
equal_int(pos(01), pos(s(x0)))
equal_int(neg(01), pos(s(x0)))
equal_int(pos(01), neg(s(x0)))
equal_int(neg(01), neg(s(x0)))
equal_int(pos(s(x0)), pos(01))
equal_int(pos(s(x0)), neg(01))
equal_int(neg(s(x0)), pos(01))
equal_int(neg(s(x0)), neg(01))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(01, s(x0))
mod_nat(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 01)
minus_nat_s(01, s(x0))
minus_nat_s(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(01, x0)
plus_nat(s(x0), x1)
minus_nat(01, 01)
minus_nat(01, s(x0))
minus_nat(s(x0), 01)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

### (13) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

### (14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(01, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(01, 01) → pos(01)
minus_nat(01, s(y)) → neg(s(y))
minus_nat(s(x), 01) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(01), pos(01)) → true
equal_int(neg(01), pos(01)) → true
equal_int(pos(s(x)), pos(01)) → false
equal_int(neg(s(x)), pos(01)) → false
not(true) → false
not(false) → true
mod_nat(01, s(x)) → 01
mod_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
minus_nat_s(x, 01) → x
minus_nat_s(01, s(y)) → 01
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)

The set Q consists of the following terms:

not(true)
not(false)
equal_int(pos(01), pos(01))
equal_int(neg(01), pos(01))
equal_int(neg(01), neg(01))
equal_int(pos(01), neg(01))
equal_int(pos(01), pos(s(x0)))
equal_int(neg(01), pos(s(x0)))
equal_int(pos(01), neg(s(x0)))
equal_int(neg(01), neg(s(x0)))
equal_int(pos(s(x0)), pos(01))
equal_int(pos(s(x0)), neg(01))
equal_int(neg(s(x0)), pos(01))
equal_int(neg(s(x0)), neg(01))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(01, s(x0))
mod_nat(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 01)
minus_nat_s(01, s(x0))
minus_nat_s(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(01, x0)
plus_nat(s(x0), x1)
minus_nat(01, 01)
minus_nat(01, s(x0))
minus_nat(s(x0), 01)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

### (15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

 [ 0, 0 ]
·x1

 [ 0, 1 ]
·x1 +
 [ 0, 0 ]
·x2

POL(not(x1)) =
 / 0 \ \ 0 /
+
 / 0 1 \ \ 1 0 /
·x1

POL(equal_int(x1, x2)) =
 / 0 \ \ 0 /
+
 / 0 1 \ \ 1 0 /
·x1 +
 / 0 0 \ \ 0 0 /
·x2

POL(mod_int(x1, x2)) =
 / 0 \ \ 0 /
+
 / 1 0 \ \ 0 0 /
·x1 +
 / 0 0 \ \ 1 0 /
·x2

POL(pos(x1)) =
 / 0 \ \ 0 /
+
 / 0 1 \ \ 1 0 /
·x1

POL(s(x1)) =
 / 1 \ \ 0 /
+
 / 1 1 \ \ 0 0 /
·x1

POL(01) =
 / 0 \ \ 1 /

POL(true) =
 / 0 \ \ 1 /

POL(plus_int(x1, x2)) =
 / 0 \ \ 0 /
+
 / 0 0 \ \ 0 0 /
·x1 +
 / 0 0 \ \ 1 0 /
·x2

POL(mod_nat(x1, x2)) =
 / 0 \ \ 0 /
+
 / 0 0 \ \ 0 1 /
·x1 +
 / 0 0 \ \ 0 0 /
·x2

POL(neg(x1)) =
 / 0 \ \ 0 /
+
 / 0 1 \ \ 1 0 /
·x1

POL(false) =
 / 1 \ \ 0 /

POL(minus_nat(x1, x2)) =
 / 0 \ \ 0 /
+
 / 0 0 \ \ 0 0 /
·x1 +
 / 0 1 \ \ 0 0 /
·x2

POL(plus_nat(x1, x2)) =
 / 0 \ \ 0 /
+
 / 1 1 \ \ 0 1 /
·x1 +
 / 0 0 \ \ 1 1 /
·x2

POL(if(x1, x2, x3)) =
 / 0 \ \ 0 /
+
 / 0 0 \ \ 0 0 /
·x1 +
 / 0 0 \ \ 0 0 /
·x2 +
 / 0 0 \ \ 0 0 /
·x3

POL(greatereq_int(x1, x2)) =
 / 0 \ \ 0 /
+
 / 0 0 \ \ 0 0 /
·x1 +
 / 0 0 \ \ 0 0 /
·x2

POL(minus_nat_s(x1, x2)) =
 / 0 \ \ 0 /
+
 / 0 0 \ \ 0 0 /
·x1 +
 / 1 0 \ \ 0 0 /
·x2

The following usable rules [FROCOS05] were oriented:

mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(01), pos(01)) → true
equal_int(neg(01), pos(01)) → true
equal_int(pos(s(x)), pos(01)) → false
equal_int(neg(s(x)), pos(01)) → false
not(true) → false
not(false) → true
mod_nat(01, s(x)) → 01
mod_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))

### (16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(01, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(01, 01) → pos(01)
minus_nat(01, s(y)) → neg(s(y))
minus_nat(s(x), 01) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(01), pos(01)) → true
equal_int(neg(01), pos(01)) → true
equal_int(pos(s(x)), pos(01)) → false
equal_int(neg(s(x)), pos(01)) → false
not(true) → false
not(false) → true
mod_nat(01, s(x)) → 01
mod_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
minus_nat_s(x, 01) → x
minus_nat_s(01, s(y)) → 01
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)

The set Q consists of the following terms:

not(true)
not(false)
equal_int(pos(01), pos(01))
equal_int(neg(01), pos(01))
equal_int(neg(01), neg(01))
equal_int(pos(01), neg(01))
equal_int(pos(01), pos(s(x0)))
equal_int(neg(01), pos(s(x0)))
equal_int(pos(01), neg(s(x0)))
equal_int(neg(01), neg(s(x0)))
equal_int(pos(s(x0)), pos(01))
equal_int(pos(s(x0)), neg(01))
equal_int(neg(s(x0)), pos(01))
equal_int(neg(s(x0)), neg(01))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(01, s(x0))
mod_nat(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 01)
minus_nat_s(01, s(x0))
minus_nat_s(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(01, x0)
plus_nat(s(x0), x1)
minus_nat(01, 01)
minus_nat(01, s(x0))
minus_nat(s(x0), 01)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

### (17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.