(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA4
/**
* 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 PastaA4 {
public static void main(String[] args) {
Random.args = args;
int x = Random.random();
int y = Random.random();

while (x > y) {
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:
PastaA4.main([Ljava/lang/String;)V: Graph of 174 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: PastaA4.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:
313_0_main_Load(EOS(STATIC_313), i18, i47, i18) → 319_0_main_LE(EOS(STATIC_319), i18, i47, i18, i47)
319_0_main_LE(EOS(STATIC_319), i18, i47, i18, i47) → 334_0_main_LE(EOS(STATIC_334), i18, i47, i18, i47)
334_0_main_LE(EOS(STATIC_334), i18, i47, i18, i47) → 344_0_main_Inc(EOS(STATIC_344), i18, i47) | >(i18, i47)
344_0_main_Inc(EOS(STATIC_344), i18, i47) → 352_0_main_JMP(EOS(STATIC_352), i18, +(i47, 1)) | >=(i47, 0)
352_0_main_JMP(EOS(STATIC_352), i18, i52) → 366_0_main_Load(EOS(STATIC_366), i18, i52)
366_0_main_Load(EOS(STATIC_366), i18, i52) → 307_0_main_Load(EOS(STATIC_307), i18, i52)
307_0_main_Load(EOS(STATIC_307), i18, i47) → 313_0_main_Load(EOS(STATIC_313), i18, i47, i18)
R rules:

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


P rules:
313_0_main_Load(EOS(STATIC_313), x0, x1, x0) → 313_0_main_Load(EOS(STATIC_313), x0, +(x1, 1), x0) | &&(>(+(x1, 1), 0), <(x1, x0))
R rules:

Filtered ground terms:



313_0_main_Load(x1, x2, x3, x4) → 313_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_313_0_main_Load(x1, x2, x3, x4, x5) → Cond_313_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:



313_0_main_Load(x1, x2, x3) → 313_0_main_Load(x2, x3)
Cond_313_0_main_Load(x1, x2, x3, x4) → Cond_313_0_main_Load(x1, x3, x4)

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


P rules:
313_0_main_Load(x1, x0) → 313_0_main_Load(+(x1, 1), 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:
313_0_MAIN_LOAD(x1, x0) → COND_313_0_MAIN_LOAD(&&(>(x1, -1), <(x1, x0)), x1, x0)
COND_313_0_MAIN_LOAD(TRUE, x1, x0) → 313_0_MAIN_LOAD(+(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 313_0_MAIN_LOAD(x1[0], x0[0]) → COND_313_0_MAIN_LOAD(x1[0] > -1 && x1[0] < x0[0], x1[0], x0[0])
(1): COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 313_0_MAIN_LOAD(x1[1] + 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] + 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.IdpCand1ShapeHeuristic@7814d044 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 313_0_MAIN_LOAD(x1, x0) → COND_313_0_MAIN_LOAD(&&(>(x1, -1), <(x1, x0)), x1, x0) the following chains were created:
  • We consider the chain 313_0_MAIN_LOAD(x1[0], x0[0]) → COND_313_0_MAIN_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), x1[0], x0[0]), COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 313_0_MAIN_LOAD(+(x1[1], 1), x0[1]) which results in the following constraint:

    (1)    (&&(>(x1[0], -1), <(x1[0], x0[0]))=TRUEx1[0]=x1[1]x0[0]=x0[1]313_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧313_0_MAIN_LOAD(x1[0], x0[0])≥COND_313_0_MAIN_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), x1[0], x0[0])∧(UIncreasing(COND_313_0_MAIN_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])=TRUE313_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧313_0_MAIN_LOAD(x1[0], x0[0])≥COND_313_0_MAIN_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), x1[0], x0[0])∧(UIncreasing(COND_313_0_MAIN_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] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_313_0_MAIN_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (4)    (x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_313_0_MAIN_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (5)    (x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_313_0_MAIN_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (6)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_313_0_MAIN_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)







For Pair COND_313_0_MAIN_LOAD(TRUE, x1, x0) → 313_0_MAIN_LOAD(+(x1, 1), x0) the following chains were created:
  • We consider the chain COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 313_0_MAIN_LOAD(+(x1[1], 1), x0[1]) which results in the following constraint:

    (7)    (COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥313_0_MAIN_LOAD(+(x1[1], 1), x0[1])∧(UIncreasing(313_0_MAIN_LOAD(+(x1[1], 1), x0[1])), ≥))



    We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (8)    ((UIncreasing(313_0_MAIN_LOAD(+(x1[1], 1), x0[1])), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)



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

    (9)    ((UIncreasing(313_0_MAIN_LOAD(+(x1[1], 1), x0[1])), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)



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

    (10)    ((UIncreasing(313_0_MAIN_LOAD(+(x1[1], 1), x0[1])), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)



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

    (11)    ((UIncreasing(313_0_MAIN_LOAD(+(x1[1], 1), x0[1])), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 313_0_MAIN_LOAD(x1, x0) → COND_313_0_MAIN_LOAD(&&(>(x1, -1), <(x1, x0)), x1, x0)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_313_0_MAIN_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

  • COND_313_0_MAIN_LOAD(TRUE, x1, x0) → 313_0_MAIN_LOAD(+(x1, 1), x0)
    • ((UIncreasing(313_0_MAIN_LOAD(+(x1[1], 1), x0[1])), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 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(313_0_MAIN_LOAD(x1, x2)) = [-1] + [2]x2 + [-1]x1   
POL(COND_313_0_MAIN_LOAD(x1, x2, x3)) = [-1] + [2]x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(-1) = [-1]   
POL(<(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   

The following pairs are in P>:

COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 313_0_MAIN_LOAD(+(x1[1], 1), x0[1])

The following pairs are in Pbound:

313_0_MAIN_LOAD(x1[0], x0[0]) → COND_313_0_MAIN_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), x1[0], x0[0])

The following pairs are in P:

313_0_MAIN_LOAD(x1[0], x0[0]) → COND_313_0_MAIN_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), x1[0], x0[0])

There are no usable rules.

(8) Complex Obligation (AND)

(9) 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): 313_0_MAIN_LOAD(x1[0], x0[0]) → COND_313_0_MAIN_LOAD(x1[0] > -1 && x1[0] < x0[0], x1[0], x0[0])


The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) 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_313_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 313_0_MAIN_LOAD(x1[1] + 1, x0[1])


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE