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_16 (Sun Microsystems Inc.) Main-Class: PastaB1

/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaB1 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x > y) {
            x--;
        }
    }
}


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

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

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
PastaB1.main([Ljava/lang/String;)V: Graph of 178 nodes with 1 SCC.

(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: PastaB1.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 7 rules for P and 0 rules for R.

P rules:
532_0_main_Load(EOS(STATIC_532), i75, i46, i75) → 535_0_main_LE(EOS(STATIC_535), i75, i46, i75, i46)
535_0_main_LE(EOS(STATIC_535), i75, i46, i75, i46) → 538_0_main_LE(EOS(STATIC_538), i75, i46, i75, i46)
538_0_main_LE(EOS(STATIC_538), i75, i46, i75, i46) → 543_0_main_Inc(EOS(STATIC_543), i75, i46) | >(i75, i46)
543_0_main_Inc(EOS(STATIC_543), i75, i46) → 547_0_main_JMP(EOS(STATIC_547), +(i75, -1), i46)
547_0_main_JMP(EOS(STATIC_547), i78, i46) → 550_0_main_Load(EOS(STATIC_550), i78, i46)
550_0_main_Load(EOS(STATIC_550), i78, i46) → 530_0_main_Load(EOS(STATIC_530), i78, i46)
530_0_main_Load(EOS(STATIC_530), i75, i46) → 532_0_main_Load(EOS(STATIC_532), i75, i46, i75)
R rules:

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
532_0_main_Load(EOS(STATIC_532), x0, x1, x0) → 532_0_main_Load(EOS(STATIC_532), +(x0, -1), x1, +(x0, -1)) | <(x1, x0)
R rules:

Filtered ground terms:

532_0_main_Load(x1, x2, x3, x4) → 532_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_532_0_main_Load(x1, x2, x3, x4, x5) → Cond_532_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

532_0_main_Load(x1, x2, x3) → 532_0_main_Load(x2, x3)
Cond_532_0_main_Load(x1, x2, x3, x4) → Cond_532_0_main_Load(x1, x3, x4)

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
532_0_main_Load(x1, x0) → 532_0_main_Load(x1, +(x0, -1)) | <(x1, x0)
R rules:

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

P rules:
532_0_MAIN_LOAD(x1, x0) → COND_532_0_MAIN_LOAD(<(x1, x0), x1, x0)
COND_532_0_MAIN_LOAD(TRUE, x1, x0) → 532_0_MAIN_LOAD(x1, +(x0, -1))
R rules:

(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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 532_0_MAIN_LOAD(x1[0], x0[0]) → COND_532_0_MAIN_LOAD(x1[0] < x0[0], x1[0], x0[0])
(1): COND_532_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 532_0_MAIN_LOAD(x1[1], x0[1] + -1)

(0) -> (1), if (x1[0] < x0[0] ∧x1[0] →^* x1[1]∧x0[0] →^* x0[1])

(1) -> (0), if (x1[1] →^* x1[0]∧x0[1] + -1 →^* x0[0])

The set Q is empty.

(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@75e13ce3 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 532_0_MAIN_LOAD(x1, x0) → COND_532_0_MAIN_LOAD(<(x1, x0), x1, x0) the following chains were created:

• We consider the chain 532_0_MAIN_LOAD(x1[0], x0[0]) → COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0]), COND_532_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 532_0_MAIN_LOAD(x1[1], +(x0[1], -1)) which results in the following constraint:
  

  (1)    (<(x1[0], x0[0])=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 532_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧532_0_MAIN_LOAD(x1[0], x0[0])≥COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])∧(U^Increasing(COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥))

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

  (2)    (<(x1[0], x0[0])=TRUE ⇒ 532_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧532_0_MAIN_LOAD(x1[0], x0[0])≥COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])∧(U^Increasing(COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥))

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

  (3)    (x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

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

  (4)    (x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

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

  (5)    (x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

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

  (6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

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

  (7)    (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

  
  

  (8)    (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

  
  
  

For Pair COND_532_0_MAIN_LOAD(TRUE, x1, x0) → 532_0_MAIN_LOAD(x1, +(x0, -1)) the following chains were created:

• We consider the chain COND_532_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 532_0_MAIN_LOAD(x1[1], +(x0[1], -1)) which results in the following constraint:
  

  (9)    (COND_532_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_532_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥532_0_MAIN_LOAD(x1[1], +(x0[1], -1))∧(U^Increasing(532_0_MAIN_LOAD(x1[1], +(x0[1], -1))), ≥))

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

  (10)    ((U^Increasing(532_0_MAIN_LOAD(x1[1], +(x0[1], -1))), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

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

  (11)    ((U^Increasing(532_0_MAIN_LOAD(x1[1], +(x0[1], -1))), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

  
  
  We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (12)    ((U^Increasing(532_0_MAIN_LOAD(x1[1], +(x0[1], -1))), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

  
  
  We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (13)    ((U^Increasing(532_0_MAIN_LOAD(x1[1], +(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

  
  
  

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

• 532_0_MAIN_LOAD(x1, x0) → COND_532_0_MAIN_LOAD(<(x1, x0), x1, x0)
  
  • (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)
    
  • (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)
    
  
  
• COND_532_0_MAIN_LOAD(TRUE, x1, x0) → 532_0_MAIN_LOAD(x1, +(x0, -1))
  
  • ((U^Increasing(532_0_MAIN_LOAD(x1[1], +(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 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(532_0_MAIN_LOAD(x₁, x₂)) = [1] + x₂ + [-1]x₁   
POL(COND_532_0_MAIN_LOAD(x₁, x₂, x₃)) = x₃ + [-1]x₂   
POL(<(x₁, x₂)) = [-1]   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   

The following pairs are in P_>:

532_0_MAIN_LOAD(x1[0], x0[0]) → COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])

The following pairs are in P_bound:

532_0_MAIN_LOAD(x1[0], x0[0]) → COND_532_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])

The following pairs are in P_≥:

COND_532_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 532_0_MAIN_LOAD(x1[1], +(x0[1], -1))

There are no usable rules.

(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_532_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 532_0_MAIN_LOAD(x1[1], x0[1] + -1)


The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) TRUE