(0) Obligation:

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

public static int round (int x) {

if (x % 2 == 0) return x;
else return x+1;
}


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

while ((x > y) && (y > 2)) {
x++;
y = 2*y;

}
}
}


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:
Duplicate.main([Ljava/lang/String;)V: Graph of 184 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: Duplicate.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 15 rules for P and 0 rules for R.


P rules:
279_0_main_Load(EOS(STATIC_279), i18, i46, i18) → 282_0_main_LE(EOS(STATIC_282), i18, i46, i18, i46)
282_0_main_LE(EOS(STATIC_282), i18, i46, i18, i46) → 296_0_main_LE(EOS(STATIC_296), i18, i46, i18, i46)
296_0_main_LE(EOS(STATIC_296), i18, i46, i18, i46) → 306_0_main_Load(EOS(STATIC_306), i18, i46) | >(i18, i46)
306_0_main_Load(EOS(STATIC_306), i18, i46) → 314_0_main_ConstantStackPush(EOS(STATIC_314), i18, i46, i46)
314_0_main_ConstantStackPush(EOS(STATIC_314), i18, i46, i46) → 319_0_main_LE(EOS(STATIC_319), i18, i46, i46, 2)
319_0_main_LE(EOS(STATIC_319), i18, i54, i54, matching1) → 324_0_main_LE(EOS(STATIC_324), i18, i54, i54, 2) | =(matching1, 2)
324_0_main_LE(EOS(STATIC_324), i18, i54, i54, matching1) → 344_0_main_Inc(EOS(STATIC_344), i18, i54) | &&(>(i54, 2), =(matching1, 2))
344_0_main_Inc(EOS(STATIC_344), i18, i54) → 358_0_main_ConstantStackPush(EOS(STATIC_358), +(i18, 1), i54) | >=(i18, 0)
358_0_main_ConstantStackPush(EOS(STATIC_358), i56, i54) → 375_0_main_Load(EOS(STATIC_375), i56, i54, 2)
375_0_main_Load(EOS(STATIC_375), i56, i54, matching1) → 387_0_main_IntArithmetic(EOS(STATIC_387), i56, 2, i54) | =(matching1, 2)
387_0_main_IntArithmetic(EOS(STATIC_387), i56, matching1, i54) → 400_0_main_Store(EOS(STATIC_400), i56, *(2, i54)) | &&(>(i54, 1), =(matching1, 2))
400_0_main_Store(EOS(STATIC_400), i56, i61) → 409_0_main_JMP(EOS(STATIC_409), i56, i61)
409_0_main_JMP(EOS(STATIC_409), i56, i61) → 440_0_main_Load(EOS(STATIC_440), i56, i61)
440_0_main_Load(EOS(STATIC_440), i56, i61) → 275_0_main_Load(EOS(STATIC_275), i56, i61)
275_0_main_Load(EOS(STATIC_275), i18, i46) → 279_0_main_Load(EOS(STATIC_279), i18, i46, i18)
R rules:

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


P rules:
279_0_main_Load(EOS(STATIC_279), x0, x1, x0) → 279_0_main_Load(EOS(STATIC_279), +(x0, 1), *(2, x1), +(x0, 1)) | &&(&&(>(x1, 2), <(x1, x0)), >(+(x0, 1), 0))
R rules:

Filtered ground terms:



279_0_main_Load(x1, x2, x3, x4) → 279_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_279_0_main_Load(x1, x2, x3, x4, x5) → Cond_279_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:



279_0_main_Load(x1, x2, x3) → 279_0_main_Load(x2, x3)
Cond_279_0_main_Load(x1, x2, x3, x4) → Cond_279_0_main_Load(x1, x3, x4)

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


P rules:
279_0_main_Load(x1, x0) → 279_0_main_Load(*(2, x1), +(x0, 1)) | &&(&&(>(x1, 2), <(x1, x0)), >(x0, -1))
R rules:

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


P rules:
279_0_MAIN_LOAD(x1, x0) → COND_279_0_MAIN_LOAD(&&(&&(>(x1, 2), <(x1, x0)), >(x0, -1)), x1, x0)
COND_279_0_MAIN_LOAD(TRUE, x1, x0) → 279_0_MAIN_LOAD(*(2, 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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 279_0_MAIN_LOAD(x1[0], x0[0]) → COND_279_0_MAIN_LOAD(x1[0] > 2 && x1[0] < x0[0] && x0[0] > -1, x1[0], x0[0])
(1): COND_279_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 279_0_MAIN_LOAD(2 * x1[1], x0[1] + 1)

(0) -> (1), if (x1[0] > 2 && x1[0] < x0[0] && x0[0] > -1x1[0]* x1[1]x0[0]* x0[1])


(1) -> (0), if (2 * 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.IdpDefaultShapeHeuristic@5fd9398d 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 279_0_MAIN_LOAD(x1, x0) → COND_279_0_MAIN_LOAD(&&(&&(>(x1, 2), <(x1, x0)), >(x0, -1)), x1, x0) the following chains were created:
  • We consider the chain 279_0_MAIN_LOAD(x1[0], x0[0]) → COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0]), COND_279_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1)) which results in the following constraint:

    (1)    (&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1))=TRUEx1[0]=x1[1]x0[0]=x0[1]279_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧279_0_MAIN_LOAD(x1[0], x0[0])≥COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])∧(UIncreasing(COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])), ≥))



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

    (2)    (>(x0[0], -1)=TRUE>(x1[0], 2)=TRUE<(x1[0], x0[0])=TRUE279_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧279_0_MAIN_LOAD(x1[0], x0[0])≥COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])∧(UIncreasing(COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])), ≥))



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

    (3)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (4)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (5)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (6)    ([1] + x1[0] + x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (7)    ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)







For Pair COND_279_0_MAIN_LOAD(TRUE, x1, x0) → 279_0_MAIN_LOAD(*(2, x1), +(x0, 1)) the following chains were created:
  • We consider the chain 279_0_MAIN_LOAD(x1[0], x0[0]) → COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0]), COND_279_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1)), 279_0_MAIN_LOAD(x1[0], x0[0]) → COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0]) which results in the following constraint:

    (8)    (&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1))=TRUEx1[0]=x1[1]x0[0]=x0[1]*(2, x1[1])=x1[0]1+(x0[1], 1)=x0[0]1COND_279_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_279_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))∧(UIncreasing(279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥))



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

    (9)    (>(x0[0], -1)=TRUE>(x1[0], 2)=TRUE<(x1[0], x0[0])=TRUECOND_279_0_MAIN_LOAD(TRUE, x1[0], x0[0])≥NonInfC∧COND_279_0_MAIN_LOAD(TRUE, x1[0], x0[0])≥279_0_MAIN_LOAD(*(2, x1[0]), +(x0[0], 1))∧(UIncreasing(279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥))



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

    (10)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[-1 + (-1)bso_16] + x1[0] ≥ 0)



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

    (11)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[-1 + (-1)bso_16] + x1[0] ≥ 0)



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

    (12)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[-1 + (-1)bso_16] + x1[0] ≥ 0)



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

    (13)    ([1] + x1[0] + x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[-1 + (-1)bso_16] + x1[0] ≥ 0)



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

    (14)    ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[2 + (-1)bso_16] + x1[0] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 279_0_MAIN_LOAD(x1, x0) → COND_279_0_MAIN_LOAD(&&(&&(>(x1, 2), <(x1, x0)), >(x0, -1)), x1, x0)
    • ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

  • COND_279_0_MAIN_LOAD(TRUE, x1, x0) → 279_0_MAIN_LOAD(*(2, x1), +(x0, 1))
    • ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[2 + (-1)bso_16] + x1[0] ≥ 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) = 0   
POL(FALSE) = [2]   
POL(279_0_MAIN_LOAD(x1, x2)) = [-1] + x2 + [-1]x1   
POL(COND_279_0_MAIN_LOAD(x1, x2, x3)) = [-1] + x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(2) = [2]   
POL(<(x1, x2)) = [-1]   
POL(-1) = [-1]   
POL(*(x1, x2)) = x1·x2   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   

The following pairs are in P>:

COND_279_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))

The following pairs are in Pbound:

279_0_MAIN_LOAD(x1[0], x0[0]) → COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])
COND_279_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 279_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))

The following pairs are in P:

279_0_MAIN_LOAD(x1[0], x0[0]) → COND_279_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >(x0[0], -1)), x1[0], x0[0])

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

FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 279_0_MAIN_LOAD(x1[0], x0[0]) → COND_279_0_MAIN_LOAD(x1[0] > 2 && x1[0] < x0[0] && x0[0] > -1, x1[0], 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