(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA9
/**
* 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 PastaA9 {
public static void main(String[] args) {
Random.args = args;
int x = Random.random();
int y = Random.random();
int z = Random.random();

if (y > 0) {
while (x >= z) {
z += 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:
PastaA9.main([Ljava/lang/String;)V: Graph of 249 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: PastaA9.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:
540_0_main_Load(EOS(STATIC_540), i18, i94, i88, i18) → 548_0_main_LT(EOS(STATIC_548), i18, i94, i88, i18, i88)
548_0_main_LT(EOS(STATIC_548), i18, i94, i88, i18, i88) → 559_0_main_LT(EOS(STATIC_559), i18, i94, i88, i18, i88)
559_0_main_LT(EOS(STATIC_559), i18, i94, i88, i18, i88) → 579_0_main_Load(EOS(STATIC_579), i18, i94, i88) | >=(i18, i88)
579_0_main_Load(EOS(STATIC_579), i18, i94, i88) → 590_0_main_Load(EOS(STATIC_590), i18, i94, i88)
590_0_main_Load(EOS(STATIC_590), i18, i94, i88) → 600_0_main_IntArithmetic(EOS(STATIC_600), i18, i94, i88, i94)
600_0_main_IntArithmetic(EOS(STATIC_600), i18, i94, i88, i94) → 613_0_main_Store(EOS(STATIC_613), i18, i94, +(i88, i94)) | &&(>=(i88, 0), >(i94, 0))
613_0_main_Store(EOS(STATIC_613), i18, i94, i101) → 623_0_main_JMP(EOS(STATIC_623), i18, i94, i101)
623_0_main_JMP(EOS(STATIC_623), i18, i94, i101) → 648_0_main_Load(EOS(STATIC_648), i18, i94, i101)
648_0_main_Load(EOS(STATIC_648), i18, i94, i101) → 531_0_main_Load(EOS(STATIC_531), i18, i94, i101)
531_0_main_Load(EOS(STATIC_531), i18, i94, i88) → 540_0_main_Load(EOS(STATIC_540), i18, i94, i88, i18)
R rules:

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


P rules:
540_0_main_Load(EOS(STATIC_540), x0, x1, x2, x0) → 540_0_main_Load(EOS(STATIC_540), x0, x1, +(x2, x1), x0) | &&(&&(>(+(x2, 1), 0), <=(x2, x0)), >(x1, 0))
R rules:

Filtered ground terms:



540_0_main_Load(x1, x2, x3, x4, x5) → 540_0_main_Load(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_540_0_main_Load(x1, x2, x3, x4, x5, x6) → Cond_540_0_main_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:



540_0_main_Load(x1, x2, x3, x4) → 540_0_main_Load(x2, x3, x4)
Cond_540_0_main_Load(x1, x2, x3, x4, x5) → Cond_540_0_main_Load(x1, x3, x4, x5)

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


P rules:
540_0_main_Load(x1, x2, x0) → 540_0_main_Load(x1, +(x2, x1), x0) | &&(&&(>(x2, -1), <=(x2, x0)), >(x1, 0))
R rules:

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


P rules:
540_0_MAIN_LOAD(x1, x2, x0) → COND_540_0_MAIN_LOAD(&&(&&(>(x2, -1), <=(x2, x0)), >(x1, 0)), x1, x2, x0)
COND_540_0_MAIN_LOAD(TRUE, x1, x2, x0) → 540_0_MAIN_LOAD(x1, +(x2, x1), 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): 540_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_540_0_MAIN_LOAD(x2[0] > -1 && x2[0] <= x0[0] && x1[0] > 0, x1[0], x2[0], x0[0])
(1): COND_540_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 540_0_MAIN_LOAD(x1[1], x2[1] + x1[1], x0[1])

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


(1) -> (0), if (x1[1]* x1[0]x2[1] + x1[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.IdpDefaultShapeHeuristic@67b99628 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 540_0_MAIN_LOAD(x1, x2, x0) → COND_540_0_MAIN_LOAD(&&(&&(>(x2, -1), <=(x2, x0)), >(x1, 0)), x1, x2, x0) the following chains were created:
  • We consider the chain 540_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0]), COND_540_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1]) which results in the following constraint:

    (1)    (&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0))=TRUEx1[0]=x1[1]x2[0]=x2[1]x0[0]=x0[1]540_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧540_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])∧(UIncreasing(COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥))



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

    (2)    (>(x1[0], 0)=TRUE>(x2[0], -1)=TRUE<=(x2[0], x0[0])=TRUE540_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧540_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])∧(UIncreasing(COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥))



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

    (3)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] + x1[0] ≥ 0)



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

    (4)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] + x1[0] ≥ 0)



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

    (5)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] + x1[0] ≥ 0)



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

    (6)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_16] + x1[0] ≥ 0)



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

    (7)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_16] + x1[0] ≥ 0)







For Pair COND_540_0_MAIN_LOAD(TRUE, x1, x2, x0) → 540_0_MAIN_LOAD(x1, +(x2, x1), x0) the following chains were created:
  • We consider the chain 540_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0]), COND_540_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1]), 540_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0]) which results in the following constraint:

    (8)    (&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0))=TRUEx1[0]=x1[1]x2[0]=x2[1]x0[0]=x0[1]x1[1]=x1[0]1+(x2[1], x1[1])=x2[0]1x0[1]=x0[0]1COND_540_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_540_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])∧(UIncreasing(540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥))



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

    (9)    (>(x1[0], 0)=TRUE>(x2[0], -1)=TRUE<=(x2[0], x0[0])=TRUECOND_540_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_540_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥540_0_MAIN_LOAD(x1[0], +(x2[0], x1[0]), x0[0])∧(UIncreasing(540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥))



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

    (10)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (11)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (12)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (13)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (14)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 540_0_MAIN_LOAD(x1, x2, x0) → COND_540_0_MAIN_LOAD(&&(&&(>(x2, -1), <=(x2, x0)), >(x1, 0)), x1, x2, x0)
    • (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_16] + x1[0] ≥ 0)

  • COND_540_0_MAIN_LOAD(TRUE, x1, x2, x0) → 540_0_MAIN_LOAD(x1, +(x2, x1), x0)
    • (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 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(540_0_MAIN_LOAD(x1, x2, x3)) = [-1] + x3 + [-1]x2 + x1   
POL(COND_540_0_MAIN_LOAD(x1, x2, x3, x4)) = [-1] + x4 + [-1]x3   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(-1) = [-1]   
POL(<=(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   

The following pairs are in P>:

540_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])

The following pairs are in Pbound:

540_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_540_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])
COND_540_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])

The following pairs are in P:

COND_540_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 540_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])

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

TRUE1&&(TRUE, TRUE)1
FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)1
FALSE1&&(FALSE, FALSE)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:
(1): COND_540_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 540_0_MAIN_LOAD(x1[1], x2[1] + x1[1], x0[1])


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