(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA6
/**
* 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 PastaA6 {
public static void main(String[] args) {
Random.args = args;
int x = Random.random();
int y = Random.random();
int z = Random.random();

while (x > y + z) {
y++;
z++;
}
}
}


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:
PastaA6.main([Ljava/lang/String;)V: Graph of 244 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: PastaA6.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 10 rules for P and 0 rules for R.


P rules:
506_0_main_Load(EOS(STATIC_506), i18, i47, i95, i18) → 512_0_main_Load(EOS(STATIC_512), i18, i47, i95, i18, i47)
512_0_main_Load(EOS(STATIC_512), i18, i47, i95, i18, i47) → 526_0_main_IntArithmetic(EOS(STATIC_526), i18, i47, i95, i18, i47, i95)
526_0_main_IntArithmetic(EOS(STATIC_526), i18, i47, i95, i18, i47, i95) → 535_0_main_LE(EOS(STATIC_535), i18, i47, i95, i18, +(i47, i95)) | &&(>=(i47, 0), >=(i95, 0))
535_0_main_LE(EOS(STATIC_535), i18, i47, i95, i18, i109) → 542_0_main_LE(EOS(STATIC_542), i18, i47, i95, i18, i109)
542_0_main_LE(EOS(STATIC_542), i18, i47, i95, i18, i109) → 554_0_main_Inc(EOS(STATIC_554), i18, i47, i95) | >(i18, i109)
554_0_main_Inc(EOS(STATIC_554), i18, i47, i95) → 565_0_main_Inc(EOS(STATIC_565), i18, +(i47, 1), i95) | >=(i47, 0)
565_0_main_Inc(EOS(STATIC_565), i18, i115, i95) → 574_0_main_JMP(EOS(STATIC_574), i18, i115, +(i95, 1)) | >=(i95, 0)
574_0_main_JMP(EOS(STATIC_574), i18, i115, i117) → 601_0_main_Load(EOS(STATIC_601), i18, i115, i117)
601_0_main_Load(EOS(STATIC_601), i18, i115, i117) → 498_0_main_Load(EOS(STATIC_498), i18, i115, i117)
498_0_main_Load(EOS(STATIC_498), i18, i47, i95) → 506_0_main_Load(EOS(STATIC_506), i18, i47, i95, i18)
R rules:

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


P rules:
506_0_main_Load(EOS(STATIC_506), x0, x1, x2, x0) → 506_0_main_Load(EOS(STATIC_506), x0, +(x1, 1), +(x2, 1), x0) | &&(&&(>(+(x2, 1), 0), >(+(x1, 1), 0)), >(x0, +(x1, x2)))
R rules:

Filtered ground terms:



506_0_main_Load(x1, x2, x3, x4, x5) → 506_0_main_Load(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_506_0_main_Load(x1, x2, x3, x4, x5, x6) → Cond_506_0_main_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:



506_0_main_Load(x1, x2, x3, x4) → 506_0_main_Load(x2, x3, x4)
Cond_506_0_main_Load(x1, x2, x3, x4, x5) → Cond_506_0_main_Load(x1, x3, x4, x5)

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


P rules:
506_0_main_Load(x1, x2, x0) → 506_0_main_Load(+(x1, 1), +(x2, 1), x0) | &&(&&(>(x2, -1), >(x1, -1)), >(x0, +(x1, x2)))
R rules:

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


P rules:
506_0_MAIN_LOAD(x1, x2, x0) → COND_506_0_MAIN_LOAD(&&(&&(>(x2, -1), >(x1, -1)), >(x0, +(x1, x2))), x1, x2, x0)
COND_506_0_MAIN_LOAD(TRUE, x1, x2, x0) → 506_0_MAIN_LOAD(+(x1, 1), +(x2, 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): 506_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_506_0_MAIN_LOAD(x2[0] > -1 && x1[0] > -1 && x0[0] > x1[0] + x2[0], x1[0], x2[0], x0[0])
(1): COND_506_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 506_0_MAIN_LOAD(x1[1] + 1, x2[1] + 1, x0[1])

(0) -> (1), if (x2[0] > -1 && x1[0] > -1 && x0[0] > x1[0] + x2[0]x1[0]* x1[1]x2[0]* x2[1]x0[0]* x0[1])


(1) -> (0), if (x1[1] + 1* x1[0]x2[1] + 1* x2[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@18b1625c 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 506_0_MAIN_LOAD(x1, x2, x0) → COND_506_0_MAIN_LOAD(&&(&&(>(x2, -1), >(x1, -1)), >(x0, +(x1, x2))), x1, x2, x0) the following chains were created:
  • We consider the chain 506_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0]), COND_506_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 506_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1]) which results in the following constraint:

    (1)    (&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0])))=TRUEx1[0]=x1[1]x2[0]=x2[1]x0[0]=x0[1]506_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧506_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])∧(UIncreasing(COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥))



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

    (2)    (>(x0[0], +(x1[0], x2[0]))=TRUE>(x2[0], -1)=TRUE>(x1[0], -1)=TRUE506_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧506_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])∧(UIncreasing(COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥))



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

    (3)    (x0[0] + [-1] + [-1]x1[0] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x2[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)    (x0[0] + [-1] + [-1]x1[0] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x2[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)    (x0[0] + [-1] + [-1]x1[0] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x2[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)    (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)







For Pair COND_506_0_MAIN_LOAD(TRUE, x1, x2, x0) → 506_0_MAIN_LOAD(+(x1, 1), +(x2, 1), x0) the following chains were created:
  • We consider the chain COND_506_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 506_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1]) which results in the following constraint:

    (7)    (COND_506_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_506_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥506_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])∧(UIncreasing(506_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥))



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

    (8)    ((UIncreasing(506_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)



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

    (9)    ((UIncreasing(506_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)



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

    (10)    ((UIncreasing(506_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)



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

    (11)    ((UIncreasing(506_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_13] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 506_0_MAIN_LOAD(x1, x2, x0) → COND_506_0_MAIN_LOAD(&&(&&(>(x2, -1), >(x1, -1)), >(x0, +(x1, x2))), x1, x2, x0)
    • (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

  • COND_506_0_MAIN_LOAD(TRUE, x1, x2, x0) → 506_0_MAIN_LOAD(+(x1, 1), +(x2, 1), x0)
    • ((UIncreasing(506_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-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(506_0_MAIN_LOAD(x1, x2, x3)) = [1] + x3 + [-1]x2 + [-1]x1   
POL(COND_506_0_MAIN_LOAD(x1, x2, x3, x4)) = [1] + x4 + [-1]x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(-1) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   

The following pairs are in P>:

COND_506_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 506_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])

The following pairs are in Pbound:

506_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])

The following pairs are in P:

506_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_506_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[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): 506_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_506_0_MAIN_LOAD(x2[0] > -1 && x1[0] > -1 && x0[0] > x1[0] + x2[0], x1[0], x2[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_506_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 506_0_MAIN_LOAD(x1[1] + 1, x2[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