Termination proof of /tmp/tmp8hznw9/TwoWay.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 IDP

↳5 IDPNonInfProof (⇒)

↳6 AND

    ↳7 IDP

        ↳8 IDependencyGraphProof (⇔)

        ↳9 TRUE

    ↳10 IDP

        ↳11 IDependencyGraphProof (⇔)

        ↳12 TRUE

(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)=TRUE∧x1[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])∧(U^Increasing(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)=TRUE ⇒ 275_0_TWOWAY_GE(x1[0])≥NonInfC∧275_0_TWOWAY_GE(x1[0])≥COND_275_0_TWOWAY_GE(>=(x1[0], 0), x1[0])∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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))∧(U^Increasing(275_0_TWOWAY_GE(+(x1[1], -1))), ≥))

We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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 ⇒ (U^Increasing(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))
- ((U^Increasing(275_0_TWOWAY_GE(+(x1[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_24] ≥ 0)

The constraints for P_> respective P_bound 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 P_bound.
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 P_bound:

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.

(6) Complex Obligation (AND)

(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.

(9) TRUE

(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.

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Obligation:

(5) IDPNonInfProof (SOUND transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) IDependencyGraphProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE