(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: LogBuiltIn
public class LogBuiltIn{
public static int log(int x) {

int res = 0;

while (x > 1) {

x = x/2;
res++;

}

return res;

}


public static void main(String[] args) {
Random.args = args;
int x = Random.random();
log(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:
LogBuiltIn.main([Ljava/lang/String;)V: Graph of 122 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: LogBuiltIn.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 11 rules for P and 0 rules for R.


P rules:
384_0_log_ConstantStackPush(EOS(STATIC_384), i74, i74) → 387_0_log_LE(EOS(STATIC_387), i74, i74, 1)
387_0_log_LE(EOS(STATIC_387), i82, i82, matching1) → 390_0_log_LE(EOS(STATIC_390), i82, i82, 1) | =(matching1, 1)
390_0_log_LE(EOS(STATIC_390), i82, i82, matching1) → 395_0_log_Load(EOS(STATIC_395), i82) | &&(>(i82, 1), =(matching1, 1))
395_0_log_Load(EOS(STATIC_395), i82) → 400_0_log_ConstantStackPush(EOS(STATIC_400), i82)
400_0_log_ConstantStackPush(EOS(STATIC_400), i82) → 404_0_log_IntArithmetic(EOS(STATIC_404), i82, 2)
404_0_log_IntArithmetic(EOS(STATIC_404), i82, matching1) → 408_0_log_Store(EOS(STATIC_408), /(i82, 2)) | &&(>(i82, 1), =(matching1, 2))
408_0_log_Store(EOS(STATIC_408), i85) → 412_0_log_Inc(EOS(STATIC_412), i85)
412_0_log_Inc(EOS(STATIC_412), i85) → 415_0_log_JMP(EOS(STATIC_415), i85)
415_0_log_JMP(EOS(STATIC_415), i85) → 418_0_log_Load(EOS(STATIC_418), i85)
418_0_log_Load(EOS(STATIC_418), i85) → 381_0_log_Load(EOS(STATIC_381), i85)
381_0_log_Load(EOS(STATIC_381), i74) → 384_0_log_ConstantStackPush(EOS(STATIC_384), i74, i74)
R rules:

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


P rules:
384_0_log_ConstantStackPush(EOS(STATIC_384), x0, x0) → 384_0_log_ConstantStackPush(EOS(STATIC_384), /(x0, 2), /(x0, 2)) | >(x0, 1)
R rules:

Filtered ground terms:



384_0_log_ConstantStackPush(x1, x2, x3) → 384_0_log_ConstantStackPush(x2, x3)
EOS(x1) → EOS
Cond_384_0_log_ConstantStackPush(x1, x2, x3, x4) → Cond_384_0_log_ConstantStackPush(x1, x3, x4)

Filtered duplicate args:



384_0_log_ConstantStackPush(x1, x2) → 384_0_log_ConstantStackPush(x2)
Cond_384_0_log_ConstantStackPush(x1, x2, x3) → Cond_384_0_log_ConstantStackPush(x1, x3)

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


P rules:
384_0_log_ConstantStackPush(x0) → 384_0_log_ConstantStackPush(/(x0, 2)) | >(x0, 1)
R rules:

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


P rules:
384_0_LOG_CONSTANTSTACKPUSH(x0) → COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0, 1), x0)
COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0) → 384_0_LOG_CONSTANTSTACKPUSH(/(x0, 2))
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): 384_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_384_0_LOG_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1]) → 384_0_LOG_CONSTANTSTACKPUSH(x0[1] / 2)

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


(1) -> (0), if (x0[1] / 2* 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@7d480d96 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 384_0_LOG_CONSTANTSTACKPUSH(x0) → COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:
  • We consider the chain 384_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1]) → 384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2)) which results in the following constraint:

    (1)    (>(x0[0], 1)=TRUEx0[0]=x0[1]384_0_LOG_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧384_0_LOG_CONSTANTSTACKPUSH(x0[0])≥COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))



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

    (2)    (>(x0[0], 1)=TRUE384_0_LOG_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧384_0_LOG_CONSTANTSTACKPUSH(x0[0])≥COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))



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

    (3)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (4)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (5)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)







For Pair COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0) → 384_0_LOG_CONSTANTSTACKPUSH(/(x0, 2)) the following chains were created:
  • We consider the chain 384_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1]) → 384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2)), 384_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]) which results in the following constraint:

    (7)    (>(x0[0], 1)=TRUEx0[0]=x0[1]/(x0[1], 2)=x0[0]1COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1])≥384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))∧(UIncreasing(384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥))



    We simplified constraint (7) using rules (III), (IV) which results in the following new constraint:

    (8)    (>(x0[0], 1)=TRUECOND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[0])≥NonInfC∧COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[0])≥384_0_LOG_CONSTANTSTACKPUSH(/(x0[0], 2))∧(UIncreasing(384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥))



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

    (9)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)



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

    (10)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)



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

    (11)    (x0[0] + [-2] ≥ 0∧[2]x0[0] ≥ 0 ⇒ (UIncreasing(384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (12)    (x0[0] ≥ 0∧[4] + [2]x0[0] ≥ 0 ⇒ (UIncreasing(384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (13)    (x0[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 384_0_LOG_CONSTANTSTACKPUSH(x0) → COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0, 1), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)

  • COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0) → 384_0_LOG_CONSTANTSTACKPUSH(/(x0, 2))
    • (x0[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 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) = [1]   
POL(FALSE) = 0   
POL(384_0_LOG_CONSTANTSTACKPUSH(x1)) = [-1] + x1   
POL(COND_384_0_LOG_CONSTANTSTACKPUSH(x1, x2)) = [-1] + x2   
POL(>(x1, x2)) = [-1]   
POL(1) = [1]   
POL(2) = [2]   

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, 2)1 @ {384_0_LOG_CONSTANTSTACKPUSH_1/0}) = max{x1, [-1]x1} + [-1]   

The following pairs are in P>:

COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1]) → 384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))

The following pairs are in Pbound:

384_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])
COND_384_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1]) → 384_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))

The following pairs are in P:

384_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_384_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

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

/1

(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:
(0): 384_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_384_0_LOG_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])


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