(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PlusSwap
public class PlusSwap{
public static void main(String[] args) {
Random.args = args;
int x = Random.random();
int y = Random.random();
int z;
int res = 0;

while (y > 0) {

z = x;
x = y-1;
y = z;
res++;

}

res = res + 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:
PlusSwap.main([Ljava/lang/String;)V: Graph of 191 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: PlusSwap.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:
622_0_main_LE(EOS(STATIC_622), i118, i129, i129) → 625_0_main_LE(EOS(STATIC_625), i118, i129, i129)
625_0_main_LE(EOS(STATIC_625), i118, i129, i129) → 629_0_main_Load(EOS(STATIC_629), i118, i129) | >(i129, 0)
629_0_main_Load(EOS(STATIC_629), i118, i129) → 633_0_main_Store(EOS(STATIC_633), i129, i118)
633_0_main_Store(EOS(STATIC_633), i129, i118) → 637_0_main_Load(EOS(STATIC_637), i129, i118)
637_0_main_Load(EOS(STATIC_637), i129, i118) → 642_0_main_ConstantStackPush(EOS(STATIC_642), i118, i129)
642_0_main_ConstantStackPush(EOS(STATIC_642), i118, i129) → 646_0_main_IntArithmetic(EOS(STATIC_646), i118, i129, 1)
646_0_main_IntArithmetic(EOS(STATIC_646), i118, i129, matching1) → 650_0_main_Store(EOS(STATIC_650), i118, -(i129, 1)) | &&(>(i129, 0), =(matching1, 1))
650_0_main_Store(EOS(STATIC_650), i118, i133) → 652_0_main_Load(EOS(STATIC_652), i133, i118)
652_0_main_Load(EOS(STATIC_652), i133, i118) → 654_0_main_Store(EOS(STATIC_654), i133, i118)
654_0_main_Store(EOS(STATIC_654), i133, i118) → 656_0_main_Inc(EOS(STATIC_656), i133, i118)
656_0_main_Inc(EOS(STATIC_656), i133, i118) → 658_0_main_JMP(EOS(STATIC_658), i133, i118)
658_0_main_JMP(EOS(STATIC_658), i133, i118) → 675_0_main_Load(EOS(STATIC_675), i133, i118)
675_0_main_Load(EOS(STATIC_675), i133, i118) → 618_0_main_Load(EOS(STATIC_618), i133, i118)
618_0_main_Load(EOS(STATIC_618), i118, i119) → 622_0_main_LE(EOS(STATIC_622), i118, i119, i119)
R rules:

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


P rules:
622_0_main_LE(EOS(STATIC_622), x0, x1, x1) → 622_0_main_LE(EOS(STATIC_622), -(x1, 1), x0, x0) | >(x1, 0)
R rules:

Filtered ground terms:



622_0_main_LE(x1, x2, x3, x4) → 622_0_main_LE(x2, x3, x4)
EOS(x1) → EOS
Cond_622_0_main_LE(x1, x2, x3, x4, x5) → Cond_622_0_main_LE(x1, x3, x4, x5)

Filtered duplicate args:



622_0_main_LE(x1, x2, x3) → 622_0_main_LE(x1, x3)
Cond_622_0_main_LE(x1, x2, x3, x4) → Cond_622_0_main_LE(x1, x2, x4)

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


P rules:
622_0_main_LE(x0, x1) → 622_0_main_LE(-(x1, 1), x0) | >(x1, 0)
R rules:

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


P rules:
622_0_MAIN_LE(x0, x1) → COND_622_0_MAIN_LE(>(x1, 0), x0, x1)
COND_622_0_MAIN_LE(TRUE, x0, x1) → 622_0_MAIN_LE(-(x1, 1), 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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(x1[0] > 0, x0[0], x1[0])
(1): COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(x1[1] - 1, x0[1])

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


(1) -> (0), if (x1[1] - 1* x0[0]x0[1]* x1[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@5a55a0da Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 3 Max Right Steps: 2

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 622_0_MAIN_LE(x0, x1) → COND_622_0_MAIN_LE(>(x1, 0), x0, x1) the following chains were created:
  • We consider the chain 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1]) which results in the following constraint:

    (1)    (>(x1[0], 0)=TRUEx0[0]=x0[1]x1[0]=x1[1]622_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧622_0_MAIN_LE(x0[0], x1[0])≥COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])∧(UIncreasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥))



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

    (2)    (>(x1[0], 0)=TRUE622_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧622_0_MAIN_LE(x0[0], x1[0])≥COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])∧(UIncreasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥))



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

    (3)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[0] + [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)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[0] + [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)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[0] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (6)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_11] = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)



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

    (7)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_11] = 0∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)







For Pair COND_622_0_MAIN_LE(TRUE, x0, x1) → 622_0_MAIN_LE(-(x1, 1), x0) the following chains were created:
  • We consider the chain 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1]), 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1]), 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1]) which results in the following constraint:

    (8)    (>(x1[0], 0)=TRUEx0[0]=x0[1]x1[0]=x1[1]-(x1[1], 1)=x0[0]1x0[1]=x1[0]1>(x1[0]1, 0)=TRUEx0[0]1=x0[1]1x1[0]1=x1[1]1-(x1[1]1, 1)=x0[0]2x0[1]1=x1[0]2>(x1[0]2, 0)=TRUEx0[0]2=x0[1]2x1[0]2=x1[1]2COND_622_0_MAIN_LE(TRUE, x0[1]1, x1[1]1)≥NonInfC∧COND_622_0_MAIN_LE(TRUE, x0[1]1, x1[1]1)≥622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)∧(UIncreasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥))



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

    (9)    (>(x1[0], 0)=TRUE>(x1[0]1, 0)=TRUE>(-(x1[0], 1), 0)=TRUECOND_622_0_MAIN_LE(TRUE, -(x1[0], 1), x1[0]1)≥NonInfC∧COND_622_0_MAIN_LE(TRUE, -(x1[0], 1), x1[0]1)≥622_0_MAIN_LE(-(x1[0]1, 1), -(x1[0], 1))∧(UIncreasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥))



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

    (10)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (UIncreasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (11)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (UIncreasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (12)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (UIncreasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (13)    (x1[0] ≥ 0∧x1[0]1 + [-1] ≥ 0∧[-1] + x1[0] ≥ 0 ⇒ (UIncreasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (14)    ([1] + x1[0] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



    We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (15)    ([1] + x1[0] ≥ 0∧x1[0]1 ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 622_0_MAIN_LE(x0, x1) → COND_622_0_MAIN_LE(>(x1, 0), x0, x1)
    • (x1[0] ≥ 0 ⇒ (UIncreasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_11] = 0∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

  • COND_622_0_MAIN_LE(TRUE, x0, x1) → 622_0_MAIN_LE(-(x1, 1), x0)
    • ([1] + x1[0] ≥ 0∧x1[0]1 ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 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) = 0   
POL(622_0_MAIN_LE(x1, x2)) = [-1] + x2 + x1   
POL(COND_622_0_MAIN_LE(x1, x2, x3)) = [-1] + x3 + x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1])

The following pairs are in Pbound:

COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1])

The following pairs are in P:

622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])

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:
(0): 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(x1[0] > 0, x0[0], x1[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