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: LogMult

public class LogMult{

  public static int log(int x, int y) {

    int res = 1;

    if (x < 0 || y < 1) return 0;
    else {
      while (x > y) { 
        y = y*y;
        res = 2*res;
      }
    }
    return res;

  }





  public static void main(String[] args) {
    Random.args = args;

    int x = Random.random();
    log(x,2);
  }
}


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:
LogMult.main([Ljava/lang/String;)V: Graph of 131 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: LogMult.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 14 rules for P and 0 rules for R.

P rules:
491_0_log_Load(EOS(STATIC_491), i32, i32, i65, i32) → 494_0_log_LE(EOS(STATIC_494), i32, i32, i65, i32, i65)
494_0_log_LE(EOS(STATIC_494), i32, i32, i65, i32, i65) → 498_0_log_LE(EOS(STATIC_498), i32, i32, i65, i32, i65)
498_0_log_LE(EOS(STATIC_498), i32, i32, i65, i32, i65) → 502_0_log_Load(EOS(STATIC_502), i32, i32, i65) | >(i32, i65)
502_0_log_Load(EOS(STATIC_502), i32, i32, i65) → 505_0_log_Load(EOS(STATIC_505), i32, i32, i65, i65)
505_0_log_Load(EOS(STATIC_505), i32, i32, i65, i65) → 507_0_log_IntArithmetic(EOS(STATIC_507), i32, i32, i65, i65)
507_0_log_IntArithmetic(EOS(STATIC_507), i32, i32, i65, i65) → 509_0_log_Store(EOS(STATIC_509), i32, i32, *(i65, i65)) | &&(>(i65, 1), >(i65, 1))
509_0_log_Store(EOS(STATIC_509), i32, i32, i71) → 513_0_log_ConstantStackPush(EOS(STATIC_513), i32, i32, i71)
513_0_log_ConstantStackPush(EOS(STATIC_513), i32, i32, i71) → 515_0_log_Load(EOS(STATIC_515), i32, i32, i71)
515_0_log_Load(EOS(STATIC_515), i32, i32, i71) → 517_0_log_IntArithmetic(EOS(STATIC_517), i32, i32, i71)
517_0_log_IntArithmetic(EOS(STATIC_517), i32, i32, i71) → 519_0_log_Store(EOS(STATIC_519), i32, i32, i71)
519_0_log_Store(EOS(STATIC_519), i32, i32, i71) → 521_0_log_JMP(EOS(STATIC_521), i32, i32, i71)
521_0_log_JMP(EOS(STATIC_521), i32, i32, i71) → 526_0_log_Load(EOS(STATIC_526), i32, i32, i71)
526_0_log_Load(EOS(STATIC_526), i32, i32, i71) → 488_0_log_Load(EOS(STATIC_488), i32, i32, i71)
488_0_log_Load(EOS(STATIC_488), i32, i32, i65) → 491_0_log_Load(EOS(STATIC_491), i32, i32, i65, i32)
R rules:

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

P rules:
491_0_log_Load(EOS(STATIC_491), x0, x0, x1, x0) → 491_0_log_Load(EOS(STATIC_491), x0, x0, *(x1, x1), x0) | &&(>(x1, 1), <(x1, x0))
R rules:

Filtered ground terms:

491_0_log_Load(x1, x2, x3, x4, x5) → 491_0_log_Load(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_491_0_log_Load(x1, x2, x3, x4, x5, x6) → Cond_491_0_log_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:

491_0_log_Load(x1, x2, x3, x4) → 491_0_log_Load(x3, x4)
Cond_491_0_log_Load(x1, x2, x3, x4, x5) → Cond_491_0_log_Load(x1, x4, x5)

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

P rules:
491_0_log_Load(x1, x0) → 491_0_log_Load(*(x1, x1), x0) | &&(>(x1, 1), <(x1, x0))
R rules:

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

P rules:
491_0_LOG_LOAD(x1, x0) → COND_491_0_LOG_LOAD(&&(>(x1, 1), <(x1, x0)), x1, x0)
COND_491_0_LOG_LOAD(TRUE, x1, x0) → 491_0_LOG_LOAD(*(x1, x1), x0)
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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 491_0_LOG_LOAD(x1[0], x0[0]) → COND_491_0_LOG_LOAD(x1[0] > 1 && x1[0] < x0[0], x1[0], x0[0])
(1): COND_491_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 491_0_LOG_LOAD(x1[1] * x1[1], x0[1])

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

(1) -> (0), if (x1[1] * x1[1] →^* x1[0]∧x0[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.IdpDefaultShapeHeuristic@318430a5 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 491_0_LOG_LOAD(x1, x0) → COND_491_0_LOG_LOAD(&&(>(x1, 1), <(x1, x0)), x1, x0) the following chains were created:

• We consider the chain 491_0_LOG_LOAD(x1[0], x0[0]) → COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0]), COND_491_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1]) which results in the following constraint:
  

  (1)    (&&(>(x1[0], 1), <(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 491_0_LOG_LOAD(x1[0], x0[0])≥NonInfC∧491_0_LOG_LOAD(x1[0], x0[0])≥COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])∧(U^Increasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥))

  
  
  We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (2)    (>(x1[0], 1)=TRUE∧<(x1[0], x0[0])=TRUE ⇒ 491_0_LOG_LOAD(x1[0], x0[0])≥NonInfC∧491_0_LOG_LOAD(x1[0], x0[0])≥COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])∧(U^Increasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥))

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

  (3)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

  (4)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

  (5)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

  (6)    (x1[0] ≥ 0∧x0[0] + [-3] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

  (7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

  
  
  

For Pair COND_491_0_LOG_LOAD(TRUE, x1, x0) → 491_0_LOG_LOAD(*(x1, x1), x0) the following chains were created:

• We consider the chain 491_0_LOG_LOAD(x1[0], x0[0]) → COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0]), COND_491_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1]), 491_0_LOG_LOAD(x1[0], x0[0]) → COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0]) which results in the following constraint:
  

  (8)    (&&(>(x1[0], 1), <(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1]∧*(x1[1], x1[1])=x1[0]1∧x0[1]=x0[0]1 ⇒ COND_491_0_LOG_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_491_0_LOG_LOAD(TRUE, x1[1], x0[1])≥491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])∧(U^Increasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥))

  
  
  We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (9)    (>(x1[0], 1)=TRUE∧<(x1[0], x0[0])=TRUE ⇒ COND_491_0_LOG_LOAD(TRUE, x1[0], x0[0])≥NonInfC∧COND_491_0_LOG_LOAD(TRUE, x1[0], x0[0])≥491_0_LOG_LOAD(*(x1[0], x1[0]), x0[0])∧(U^Increasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥))

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

  (10)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[-2 + (-1)bso_16] + [-1]x1[0] + x1[0]² ≥ 0)

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

  (11)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[-2 + (-1)bso_16] + [-1]x1[0] + x1[0]² ≥ 0)

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

  (12)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[-2 + (-1)bso_16] + [-1]x1[0] + x1[0]² ≥ 0)

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

  (13)    (x1[0] ≥ 0∧x0[0] + [-3] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-3)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[(-1)bso_16] + [3]x1[0] + x1[0]² ≥ 0)

  
  
  We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (14)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + [3]x1[0] + x1[0]² ≥ 0)

  
  
  

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

• 491_0_LOG_LOAD(x1, x0) → COND_491_0_LOG_LOAD(&&(>(x1, 1), <(x1, x0)), x1, x0)
  
  • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)
    
  
  
• COND_491_0_LOG_LOAD(TRUE, x1, x0) → 491_0_LOG_LOAD(*(x1, x1), x0)
  
  • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + [3]x1[0] + x1[0]² ≥ 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(491_0_LOG_LOAD(x₁, x₂)) = [1] + x₂ + [-1]x₁   
POL(COND_491_0_LOG_LOAD(x₁, x₂, x₃)) = [-1] + x₃ + [-1]x₂ + [-1]x₁   
POL(&&(x₁, x₂)) = 0   
POL(>(x₁, x₂)) = [-1]   
POL(1) = [1]   
POL(<(x₁, x₂)) = [-1]   
POL(*(x₁, x₂)) = x₁·x₂   

The following pairs are in P_>:

491_0_LOG_LOAD(x1[0], x0[0]) → COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])

The following pairs are in P_bound:

491_0_LOG_LOAD(x1[0], x0[0]) → COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])
COND_491_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])

The following pairs are in P_≥:

COND_491_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(TRUE, TRUE)¹ ↔ TRUE¹
&&(TRUE, FALSE)¹ ↔ FALSE¹
&&(FALSE, TRUE)¹ ↔ FALSE¹
&&(FALSE, FALSE)¹ ↔ FALSE¹

(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_491_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 491_0_LOG_LOAD(x1[1] * x1[1], x0[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