### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: TwoWay/TwoWay
`package TwoWay;public class Random {  static String[] args;  static int index = 0;  public static int random() {    final String string = args[index];    index++;    return string.length();  }}package TwoWay;public class TwoWay {	public static void main(String[] args) {		Random.args = args;		twoWay(true, Random.random());	}	public static int twoWay(boolean terminate, int n) {		if (n < 0) {			return 1;		} else {			int m = n;			if (terminate) {				m--;			} else {				m++;			}			return m*twoWay(terminate, m);		}	}}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
TwoWay.TwoWay.main([Ljava/lang/String;)V: Graph of 86 nodes with 0 SCCs.

TwoWay.TwoWay.twoWay(ZI)I: Graph of 29 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 15 rules for P and 14 rules for R.

Combined rules. Obtained 1 rules for P and 4 rules for R.

Filtered ground terms:

329_1_twoWay_InvokeMethod(x1, x2, x3, x4) → 329_1_twoWay_InvokeMethod(x1, x2, x4)
275_0_twoWay_GE(x1, x2, x3, x4) → 275_0_twoWay_GE(x3, x4)
Cond_275_0_twoWay_GE(x1, x2, x3, x4, x5) → Cond_275_0_twoWay_GE(x1, x4, x5)
357_0_twoWay_Return(x1, x2) → 357_0_twoWay_Return(x2)
Cond_329_1_twoWay_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_329_1_twoWay_InvokeMethod1(x1, x2, x3, x5)
Cond_329_1_twoWay_InvokeMethod(x1, x2, x3, x4, x5) → Cond_329_1_twoWay_InvokeMethod(x1, x3, x5)
340_0_twoWay_Return(x1, x2) → 340_0_twoWay_Return
301_0_twoWay_Return(x1, x2, x3, x4) → 301_0_twoWay_Return

Filtered duplicate args:

329_1_twoWay_InvokeMethod(x1, x2, x3) → 329_1_twoWay_InvokeMethod(x1, x3)
275_0_twoWay_GE(x1, x2) → 275_0_twoWay_GE(x2)
Cond_275_0_twoWay_GE(x1, x2, x3) → Cond_275_0_twoWay_GE(x1, x3)
Cond_329_1_twoWay_InvokeMethod1(x1, x2, x3, x4) → Cond_329_1_twoWay_InvokeMethod1(x1, x2, x4)
Cond_329_1_twoWay_InvokeMethod(x1, x2, x3) → Cond_329_1_twoWay_InvokeMethod(x1, x3)

Combined rules. Obtained 1 rules for P and 4 rules for R.

Finished conversion. Obtained 1 rules for P and 4 rules for R. System has predefined symbols.

### (4) 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:

Integer

The ITRS R consists of the following rules:
275_0_twoWay_GE(-1) → Cond_275_0_twoWay_GE(0 > -1, -1)
Cond_275_0_twoWay_GE(TRUE, -1) → 301_0_twoWay_Return
329_1_twoWay_InvokeMethod(301_0_twoWay_Return, -1) → 357_0_twoWay_Return(-1)
329_1_twoWay_InvokeMethod(340_0_twoWay_Return, x1) → Cond_329_1_twoWay_InvokeMethod(1 > -1, 340_0_twoWay_Return, x1)
Cond_329_1_twoWay_InvokeMethod(TRUE, 340_0_twoWay_Return, x1) → 357_0_twoWay_Return(x1 * -1)
329_1_twoWay_InvokeMethod(357_0_twoWay_Return(x0), x1) → Cond_329_1_twoWay_InvokeMethod1(x0 < 1, 357_0_twoWay_Return(x0), x1)
Cond_329_1_twoWay_InvokeMethod1(TRUE, 357_0_twoWay_Return(x0), x1) → 357_0_twoWay_Return(x1 * x0)

The integer pair graph contains the following rules and edges:
(0): 275_0_TWOWAY_GE(x1[0]) → COND_275_0_TWOWAY_GE(x1[0] >= 0, x1[0])
(1): COND_275_0_TWOWAY_GE(TRUE, x1[1]) → 275_0_TWOWAY_GE(x1[1] + -1)

(0) -> (1), if ((x1[0] >= 0* TRUE)∧(x1[0]* x1[1]))

(1) -> (0), if ((x1[1] + -1* x1[0]))

The set Q consists of the following terms:
275_0_twoWay_GE(-1)
Cond_275_0_twoWay_GE(TRUE, -1)
329_1_twoWay_InvokeMethod(301_0_twoWay_Return, -1)
329_1_twoWay_InvokeMethod(340_0_twoWay_Return, x0)
Cond_329_1_twoWay_InvokeMethod(TRUE, 340_0_twoWay_Return, x0)
329_1_twoWay_InvokeMethod(357_0_twoWay_Return(x0), x1)
Cond_329_1_twoWay_InvokeMethod1(TRUE, 357_0_twoWay_Return(x0), x1)

### (5) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair 275_0_TWOWAY_GE(x1) → COND_275_0_TWOWAY_GE(>=(x1, 0), x1) the following chains were created:
• We consider the chain 275_0_TWOWAY_GE(x1[0]) → COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0]), COND_275_0_TWOWAY_GE(TRUE, x1[1]) → 275_0_TWOWAY_GE(+(x1[1], -1)) which results in the following constraint:

(1)    (>=(x1[0], 0)=TRUEx1[0]=x1[1]275_0_TWOWAY_GE(x1[0])≥NonInfC∧275_0_TWOWAY_GE(x1[0])≥COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])∧(UIncreasing(COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:

(2)    (>=(x1[0], 0)=TRUE275_0_TWOWAY_GE(x1[0])≥NonInfC∧275_0_TWOWAY_GE(x1[0])≥COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])∧(UIncreasing(COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(3)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_21] + [(2)bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(4)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_21] + [(2)bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(5)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_21] + [(2)bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

For Pair COND_275_0_TWOWAY_GE(TRUE, x1) → 275_0_TWOWAY_GE(+(x1, -1)) the following chains were created:
• We consider the chain COND_275_0_TWOWAY_GE(TRUE, x1[1]) → 275_0_TWOWAY_GE(+(x1[1], -1)) which results in the following constraint:

(6)    (COND_275_0_TWOWAY_GE(TRUE, x1[1])≥NonInfC∧COND_275_0_TWOWAY_GE(TRUE, x1[1])≥275_0_TWOWAY_GE(+(x1[1], -1))∧(UIncreasing(275_0_TWOWAY_GE(+(x1[1], -1))), ≥))

We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(7)    ((UIncreasing(275_0_TWOWAY_GE(+(x1[1], -1))), ≥)∧[2 + (-1)bso_24] ≥ 0)

We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(8)    ((UIncreasing(275_0_TWOWAY_GE(+(x1[1], -1))), ≥)∧[2 + (-1)bso_24] ≥ 0)

We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(9)    ((UIncreasing(275_0_TWOWAY_GE(+(x1[1], -1))), ≥)∧[2 + (-1)bso_24] ≥ 0)

We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(10)    ((UIncreasing(275_0_TWOWAY_GE(+(x1[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_24] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 275_0_TWOWAY_GE(x1) → COND_275_0_TWOWAY_GE(>=(x1, 0), x1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_21] + [(2)bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

• COND_275_0_TWOWAY_GE(TRUE, x1) → 275_0_TWOWAY_GE(+(x1, -1))
• ((UIncreasing(275_0_TWOWAY_GE(+(x1[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_24] ≥ 0)

The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(275_0_twoWay_GE(x1)) = [-1]
POL(-1) = [-1]
POL(Cond_275_0_twoWay_GE(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(301_0_twoWay_Return) = [-1]
POL(329_1_twoWay_InvokeMethod(x1, x2)) = [-1] + [-1]x1
POL(357_0_twoWay_Return(x1)) = x1
POL(340_0_twoWay_Return) = [-1]
POL(Cond_329_1_twoWay_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3
POL(1) = [1]
POL(*(x1, x2)) = x1·x2
POL(Cond_329_1_twoWay_InvokeMethod1(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2
POL(<(x1, x2)) = [-1]
POL(275_0_TWOWAY_GE(x1)) = [2]x1
POL(COND_275_0_TWOWAY_GE(x1, x2)) = [2]x2
POL(>=(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2

The following pairs are in P>:

COND_275_0_TWOWAY_GE(TRUE, x1[1]) → 275_0_TWOWAY_GE(+(x1[1], -1))

The following pairs are in Pbound:

275_0_TWOWAY_GE(x1[0]) → COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])

The following pairs are in P:

275_0_TWOWAY_GE(x1[0]) → COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])

There are no usable rules.

### (7) 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:

Integer

The ITRS R consists of the following rules:
275_0_twoWay_GE(-1) → Cond_275_0_twoWay_GE(0 > -1, -1)
Cond_275_0_twoWay_GE(TRUE, -1) → 301_0_twoWay_Return
329_1_twoWay_InvokeMethod(301_0_twoWay_Return, -1) → 357_0_twoWay_Return(-1)
329_1_twoWay_InvokeMethod(340_0_twoWay_Return, x1) → Cond_329_1_twoWay_InvokeMethod(1 > -1, 340_0_twoWay_Return, x1)
Cond_329_1_twoWay_InvokeMethod(TRUE, 340_0_twoWay_Return, x1) → 357_0_twoWay_Return(x1 * -1)
329_1_twoWay_InvokeMethod(357_0_twoWay_Return(x0), x1) → Cond_329_1_twoWay_InvokeMethod1(x0 < 1, 357_0_twoWay_Return(x0), x1)
Cond_329_1_twoWay_InvokeMethod1(TRUE, 357_0_twoWay_Return(x0), x1) → 357_0_twoWay_Return(x1 * x0)

The integer pair graph contains the following rules and edges:
(0): 275_0_TWOWAY_GE(x1[0]) → COND_275_0_TWOWAY_GE(x1[0] >= 0, x1[0])

The set Q consists of the following terms:
275_0_twoWay_GE(-1)
Cond_275_0_twoWay_GE(TRUE, -1)
329_1_twoWay_InvokeMethod(301_0_twoWay_Return, -1)
329_1_twoWay_InvokeMethod(340_0_twoWay_Return, x0)
Cond_329_1_twoWay_InvokeMethod(TRUE, 340_0_twoWay_Return, x0)
329_1_twoWay_InvokeMethod(357_0_twoWay_Return(x0), x1)
Cond_329_1_twoWay_InvokeMethod1(TRUE, 357_0_twoWay_Return(x0), x1)

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:

Integer

The ITRS R consists of the following rules:
275_0_twoWay_GE(-1) → Cond_275_0_twoWay_GE(0 > -1, -1)
Cond_275_0_twoWay_GE(TRUE, -1) → 301_0_twoWay_Return
329_1_twoWay_InvokeMethod(301_0_twoWay_Return, -1) → 357_0_twoWay_Return(-1)
329_1_twoWay_InvokeMethod(340_0_twoWay_Return, x1) → Cond_329_1_twoWay_InvokeMethod(1 > -1, 340_0_twoWay_Return, x1)
Cond_329_1_twoWay_InvokeMethod(TRUE, 340_0_twoWay_Return, x1) → 357_0_twoWay_Return(x1 * -1)
329_1_twoWay_InvokeMethod(357_0_twoWay_Return(x0), x1) → Cond_329_1_twoWay_InvokeMethod1(x0 < 1, 357_0_twoWay_Return(x0), x1)
Cond_329_1_twoWay_InvokeMethod1(TRUE, 357_0_twoWay_Return(x0), x1) → 357_0_twoWay_Return(x1 * x0)

The integer pair graph contains the following rules and edges:
(1): COND_275_0_TWOWAY_GE(TRUE, x1[1]) → 275_0_TWOWAY_GE(x1[1] + -1)

The set Q consists of the following terms:
275_0_twoWay_GE(-1)
Cond_275_0_twoWay_GE(TRUE, -1)
329_1_twoWay_InvokeMethod(301_0_twoWay_Return, -1)
329_1_twoWay_InvokeMethod(340_0_twoWay_Return, x0)
Cond_329_1_twoWay_InvokeMethod(TRUE, 340_0_twoWay_Return, x0)
329_1_twoWay_InvokeMethod(357_0_twoWay_Return(x0), x1)
Cond_329_1_twoWay_InvokeMethod1(TRUE, 357_0_twoWay_Return(x0), x1)

### (11) IDependencyGraphProof (EQUIVALENT transformation)

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