Termination of the given JBC could be proven:

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_25 (Sun Microsystems Inc.) Main-Class: ClassAnalysisRec/ClassAnalysisRec

package ClassAnalysisRec;

public class ClassAnalysisRec {
	A field;

	public static void main(String[] args) {
		Random.args = args;
		ClassAnalysisRec t = new ClassAnalysisRec();
		t.field = new A();
		t.field = new B();
		t.eval();
	}

	public void eval() {
		int x = Random.random() % 100;
		this.field.test(x);
	}
}

class A {
	public boolean test(int x) {
		return this.test(x-1);
	}
}

class B extends A {
	public boolean test(int x) {
		if (x <= 0) return true;
		return test(x - 1);
	}
}


package ClassAnalysisRec;

public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    final String string = args[index];
    index++;
    return string.length();
  }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
ClassAnalysisRec.ClassAnalysisRec.main([Ljava/lang/String;)V: Graph of 71 nodes with 0 SCCs.

ClassAnalysisRec.ClassAnalysisRec.eval()V: Graph of 82 nodes with 0 SCCs.

ClassAnalysisRec.B.test(I)Z: Graph of 20 nodes with 0 SCCs.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: ClassAnalysisRec.B.test(I)Z
SCC calls the following helper methods: ClassAnalysisRec.B.test(I)Z
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 10 rules for P and 10 rules for R.

P rules:
254_0_test_GT(EOS(STATIC_254), i24, i24) → 266_0_test_GT(EOS(STATIC_266), i24, i24)
266_0_test_GT(EOS(STATIC_266), i24, i24) → 280_0_test_Load(EOS(STATIC_280), i24) | >(i24, 0)
280_0_test_Load(EOS(STATIC_280), i24) → 294_0_test_Load(EOS(STATIC_294), i24)
294_0_test_Load(EOS(STATIC_294), i24) → 314_0_test_ConstantStackPush(EOS(STATIC_314), i24)
314_0_test_ConstantStackPush(EOS(STATIC_314), i24) → 327_0_test_IntArithmetic(EOS(STATIC_327), i24, 1)
327_0_test_IntArithmetic(EOS(STATIC_327), i24, matching1) → 334_0_test_InvokeMethod(EOS(STATIC_334), -(i24, 1)) | &&(>(i24, 0), =(matching1, 1))
334_0_test_InvokeMethod(EOS(STATIC_334), i32) → 345_1_test_InvokeMethod(345_0_test_Load(EOS(STATIC_345), i32), i32)
345_0_test_Load(EOS(STATIC_345), i32) → 353_0_test_Load(EOS(STATIC_353), i32)
353_0_test_Load(EOS(STATIC_353), i32) → 247_0_test_Load(EOS(STATIC_247), i32)
247_0_test_Load(EOS(STATIC_247), i20) → 254_0_test_GT(EOS(STATIC_254), i20, i20)
R rules:
254_0_test_GT(EOS(STATIC_254), matching1, matching2) → 265_0_test_GT(EOS(STATIC_265), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
265_0_test_GT(EOS(STATIC_265), matching1, matching2) → 278_0_test_ConstantStackPush(EOS(STATIC_278), 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
278_0_test_ConstantStackPush(EOS(STATIC_278), matching1) → 291_0_test_Return(EOS(STATIC_291), 0, 1) | =(matching1, 0)
345_1_test_InvokeMethod(291_0_test_Return(EOS(STATIC_291), matching1, matching2), matching3) → 397_0_test_Return(EOS(STATIC_397), 0, 0, 1) | &&(&&(=(matching1, 0), =(matching2, 1)), =(matching3, 0))
345_1_test_InvokeMethod(404_0_test_Return(EOS(STATIC_404), matching1), i52) → 424_0_test_Return(EOS(STATIC_424), i52, 1) | =(matching1, 1)
345_1_test_InvokeMethod(431_0_test_Return(EOS(STATIC_431), matching1), i61) → 458_0_test_Return(EOS(STATIC_458), i61, 1) | =(matching1, 1)
397_0_test_Return(EOS(STATIC_397), matching1, matching2, matching3) → 404_0_test_Return(EOS(STATIC_404), 1) | &&(&&(=(matching1, 0), =(matching2, 0)), =(matching3, 1))
404_0_test_Return(EOS(STATIC_404), matching1) → 431_0_test_Return(EOS(STATIC_431), 1) | =(matching1, 1)
424_0_test_Return(EOS(STATIC_424), i52, matching1) → 431_0_test_Return(EOS(STATIC_431), 1) | =(matching1, 1)
458_0_test_Return(EOS(STATIC_458), i61, matching1) → 424_0_test_Return(EOS(STATIC_424), i61, 1) | =(matching1, 1)

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

P rules:
254_0_test_GT(EOS(STATIC_254), x0, x0) → 345_1_test_InvokeMethod(254_0_test_GT(EOS(STATIC_254), -(x0, 1), -(x0, 1)), -(x0, 1)) | >(x0, 0)
R rules:
254_0_test_GT(EOS(STATIC_254), 0, 0) → 291_0_test_Return(EOS(STATIC_291), 0, 1)
345_1_test_InvokeMethod(404_0_test_Return(EOS(STATIC_404), 1), x1) → 431_0_test_Return(EOS(STATIC_431), 1)
345_1_test_InvokeMethod(431_0_test_Return(EOS(STATIC_431), 1), x1) → 431_0_test_Return(EOS(STATIC_431), 1)
345_1_test_InvokeMethod(291_0_test_Return(EOS(STATIC_291), 0, 1), 0) → 431_0_test_Return(EOS(STATIC_431), 1)

Filtered ground terms:

254_0_test_GT(x1, x2, x3) → 254_0_test_GT(x2, x3)
Cond_254_0_test_GT(x1, x2, x3, x4) → Cond_254_0_test_GT(x1, x3, x4)
431_0_test_Return(x1, x2) → 431_0_test_Return
291_0_test_Return(x1, x2, x3) → 291_0_test_Return
404_0_test_Return(x1, x2) → 404_0_test_Return

Filtered duplicate args:

254_0_test_GT(x1, x2) → 254_0_test_GT(x2)
Cond_254_0_test_GT(x1, x2, x3) → Cond_254_0_test_GT(x1, x3)

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

P rules:
254_0_test_GT(x0) → 345_1_test_InvokeMethod(254_0_test_GT(-(x0, 1)), -(x0, 1)) | >(x0, 0)
R rules:
254_0_test_GT(0) → 291_0_test_Return
345_1_test_InvokeMethod(404_0_test_Return, x1) → 431_0_test_Return
345_1_test_InvokeMethod(431_0_test_Return, x1) → 431_0_test_Return
345_1_test_InvokeMethod(291_0_test_Return, 0) → 431_0_test_Return

Performed bisimulation on rules. Used the following equivalence classes: {[291_0_test_Return, 404_0_test_Return, 431_0_test_Return]=291_0_test_Return}

Finished conversion. Obtained 2 rules for P and 3 rules for R. System has predefined symbols.

P rules:
254_0_TEST_GT(x0) → COND_254_0_TEST_GT(>(x0, 0), x0)
COND_254_0_TEST_GT(TRUE, x0) → 254_0_TEST_GT(-(x0, 1))
R rules:
254_0_test_GT(0) → 291_0_test_Return
345_1_test_InvokeMethod(291_0_test_Return, x1) → 291_0_test_Return
345_1_test_InvokeMethod(291_0_test_Return, 0) → 291_0_test_Return

(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:

Integer

The ITRS R consists of the following rules:
254_0_test_GT(0) → 291_0_test_Return
345_1_test_InvokeMethod(291_0_test_Return, x1) → 291_0_test_Return
345_1_test_InvokeMethod(291_0_test_Return, 0) → 291_0_test_Return

The integer pair graph contains the following rules and edges:
(0): 254_0_TEST_GT(x0[0]) → COND_254_0_TEST_GT(x0[0] > 0, x0[0])
(1): COND_254_0_TEST_GT(TRUE, x0[1]) → 254_0_TEST_GT(x0[1] - 1)

(0) -> (1), if (x0[0] > 0 ∧x0[0] →^* x0[1])

(1) -> (0), if (x0[1] - 1 →^* x0[0])

The set Q consists of the following terms:
254_0_test_GT(0)
345_1_test_InvokeMethod(291_0_test_Return, x0)

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@29086036 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 254_0_TEST_GT(x0) → COND_254_0_TEST_GT(>(x0, 0), x0) the following chains were created:

• We consider the chain 254_0_TEST_GT(x0[0]) → COND_254_0_TEST_GT(>(x0[0], 0), x0[0]), COND_254_0_TEST_GT(TRUE, x0[1]) → 254_0_TEST_GT(-(x0[1], 1)) which results in the following constraint:
  

  (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 254_0_TEST_GT(x0[0])≥NonInfC∧254_0_TEST_GT(x0[0])≥COND_254_0_TEST_GT(>(x0[0], 0), x0[0])∧(U^Increasing(COND_254_0_TEST_GT(>(x0[0], 0), x0[0])), ≥))

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

  (2)    (>(x0[0], 0)=TRUE ⇒ 254_0_TEST_GT(x0[0])≥NonInfC∧254_0_TEST_GT(x0[0])≥COND_254_0_TEST_GT(>(x0[0], 0), x0[0])∧(U^Increasing(COND_254_0_TEST_GT(>(x0[0], 0), x0[0])), ≥))

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

  (3)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_254_0_TEST_GT(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

  (4)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_254_0_TEST_GT(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

  (5)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_254_0_TEST_GT(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

  
  
  We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_254_0_TEST_GT(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11 + (2)bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

  
  
  

For Pair COND_254_0_TEST_GT(TRUE, x0) → 254_0_TEST_GT(-(x0, 1)) the following chains were created:

• We consider the chain COND_254_0_TEST_GT(TRUE, x0[1]) → 254_0_TEST_GT(-(x0[1], 1)) which results in the following constraint:
  

  (7)    (COND_254_0_TEST_GT(TRUE, x0[1])≥NonInfC∧COND_254_0_TEST_GT(TRUE, x0[1])≥254_0_TEST_GT(-(x0[1], 1))∧(U^Increasing(254_0_TEST_GT(-(x0[1], 1))), ≥))

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

  (8)    ((U^Increasing(254_0_TEST_GT(-(x0[1], 1))), ≥)∧[bni_13] = 0∧[2 + (-1)bso_14] ≥ 0)

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

  (9)    ((U^Increasing(254_0_TEST_GT(-(x0[1], 1))), ≥)∧[bni_13] = 0∧[2 + (-1)bso_14] ≥ 0)

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

  (10)    ((U^Increasing(254_0_TEST_GT(-(x0[1], 1))), ≥)∧[bni_13] = 0∧[2 + (-1)bso_14] ≥ 0)

  
  
  We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (11)    ((U^Increasing(254_0_TEST_GT(-(x0[1], 1))), ≥)∧[bni_13] = 0∧0 = 0∧[2 + (-1)bso_14] ≥ 0)

  
  
  

To summarize, we get the following constraints P_≥ for the following pairs.

• 254_0_TEST_GT(x0) → COND_254_0_TEST_GT(>(x0, 0), x0)
  
  • (x0[0] ≥ 0 ⇒ (U^Increasing(COND_254_0_TEST_GT(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11 + (2)bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
    
  
  
• COND_254_0_TEST_GT(TRUE, x0) → 254_0_TEST_GT(-(x0, 1))
  
  • ((U^Increasing(254_0_TEST_GT(-(x0[1], 1))), ≥)∧[bni_13] = 0∧0 = 0∧[2 + (-1)bso_14] ≥ 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(254_0_test_GT(x₁)) = [-1]   
POL(0) = 0   
POL(291_0_test_Return) = [-1]   
POL(345_1_test_InvokeMethod(x₁, x₂)) = [-1]   
POL(254_0_TEST_GT(x₁)) = [2]x₁   
POL(COND_254_0_TEST_GT(x₁, x₂)) = [2]x₂   
POL(>(x₁, x₂)) = [-1]   
POL(-(x₁, x₂)) = x₁ + [-1]x₂   
POL(1) = [1]   

The following pairs are in P_>:

COND_254_0_TEST_GT(TRUE, x0[1]) → 254_0_TEST_GT(-(x0[1], 1))

The following pairs are in P_bound:

254_0_TEST_GT(x0[0]) → COND_254_0_TEST_GT(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

254_0_TEST_GT(x0[0]) → COND_254_0_TEST_GT(>(x0[0], 0), x0[0])

There are no usable rules.

(8) Complex Obligation (AND)

(9) 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:
254_0_test_GT(0) → 291_0_test_Return
345_1_test_InvokeMethod(291_0_test_Return, x1) → 291_0_test_Return
345_1_test_InvokeMethod(291_0_test_Return, 0) → 291_0_test_Return

The integer pair graph contains the following rules and edges:
(0): 254_0_TEST_GT(x0[0]) → COND_254_0_TEST_GT(x0[0] > 0, x0[0])


The set Q consists of the following terms:
254_0_test_GT(0)
345_1_test_InvokeMethod(291_0_test_Return, x0)

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) 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:
254_0_test_GT(0) → 291_0_test_Return
345_1_test_InvokeMethod(291_0_test_Return, x1) → 291_0_test_Return
345_1_test_InvokeMethod(291_0_test_Return, 0) → 291_0_test_Return

The integer pair graph contains the following rules and edges:
(1): COND_254_0_TEST_GT(TRUE, x0[1]) → 254_0_TEST_GT(x0[1] - 1)


The set Q consists of the following terms:
254_0_test_GT(0)
345_1_test_InvokeMethod(291_0_test_Return, x0)

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE