(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: example_2/Test
package example_2;


public class Test {

public static int divBy(int x){
int r = 0;
int y;
while (x > 0) {
y = 2;
x = x/y;
r = r + x;
}
return r;
}

public static void main(String[] args) {
if (args.length > 0) {
int x = args[0].length();
int r = divBy(x);
// System.out.println("Result: " + r);
}
// else System.out.println("Error: Incorrect call");
}
}


(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
example_2.Test.main([Ljava/lang/String;)V: Graph of 88 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: example_2.Test.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:
197_0_divBy_LE(EOS(STATIC_197), i37, i37) → 202_0_divBy_LE(EOS(STATIC_202), i37, i37)
202_0_divBy_LE(EOS(STATIC_202), i37, i37) → 206_0_divBy_ConstantStackPush(EOS(STATIC_206), i37) | >(i37, 0)
206_0_divBy_ConstantStackPush(EOS(STATIC_206), i37) → 211_0_divBy_Store(EOS(STATIC_211), i37, 2)
211_0_divBy_Store(EOS(STATIC_211), i37, matching1) → 215_0_divBy_Load(EOS(STATIC_215), i37, 2) | =(matching1, 2)
215_0_divBy_Load(EOS(STATIC_215), i37, matching1) → 220_0_divBy_Load(EOS(STATIC_220), 2, i37) | =(matching1, 2)
220_0_divBy_Load(EOS(STATIC_220), matching1, i37) → 224_0_divBy_IntArithmetic(EOS(STATIC_224), i37, 2) | =(matching1, 2)
224_0_divBy_IntArithmetic(EOS(STATIC_224), i37, matching1) → 226_0_divBy_Store(EOS(STATIC_226), /(i37, 2)) | &&(>=(i37, 1), =(matching1, 2))
226_0_divBy_Store(EOS(STATIC_226), i39) → 228_0_divBy_Load(EOS(STATIC_228), i39)
228_0_divBy_Load(EOS(STATIC_228), i39) → 231_0_divBy_Load(EOS(STATIC_231), i39)
231_0_divBy_Load(EOS(STATIC_231), i39) → 233_0_divBy_IntArithmetic(EOS(STATIC_233), i39, i39)
233_0_divBy_IntArithmetic(EOS(STATIC_233), i39, i39) → 234_0_divBy_Store(EOS(STATIC_234), i39) | >=(i39, 0)
234_0_divBy_Store(EOS(STATIC_234), i39) → 237_0_divBy_JMP(EOS(STATIC_237), i39)
237_0_divBy_JMP(EOS(STATIC_237), i39) → 242_0_divBy_Load(EOS(STATIC_242), i39)
242_0_divBy_Load(EOS(STATIC_242), i39) → 192_0_divBy_Load(EOS(STATIC_192), i39)
192_0_divBy_Load(EOS(STATIC_192), i26) → 197_0_divBy_LE(EOS(STATIC_197), i26, i26)
R rules:

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


P rules:
197_0_divBy_LE(EOS(STATIC_197), x0, x0) → 197_0_divBy_LE(EOS(STATIC_197), /(x0, 2), /(x0, 2)) | &&(>(+(x0, 1), 1), <=(0, /(x0, 2)))
R rules:

Filtered ground terms:



197_0_divBy_LE(x1, x2, x3) → 197_0_divBy_LE(x2, x3)
EOS(x1) → EOS
Cond_197_0_divBy_LE(x1, x2, x3, x4) → Cond_197_0_divBy_LE(x1, x3, x4)

Filtered duplicate args:



197_0_divBy_LE(x1, x2) → 197_0_divBy_LE(x2)
Cond_197_0_divBy_LE(x1, x2, x3) → Cond_197_0_divBy_LE(x1, x3)

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


P rules:
197_0_divBy_LE(x0) → 197_0_divBy_LE(/(x0, 2)) | &&(>(x0, 0), <=(0, /(x0, 2)))
R rules:

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


P rules:
197_0_DIVBY_LE(x0) → COND_197_0_DIVBY_LE(&&(>(x0, 0), <=(0, /(x0, 2))), x0)
COND_197_0_DIVBY_LE(TRUE, x0) → 197_0_DIVBY_LE(/(x0, 2))
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): 197_0_DIVBY_LE(x0[0]) → COND_197_0_DIVBY_LE(x0[0] > 0 && 0 <= x0[0] / 2, x0[0])
(1): COND_197_0_DIVBY_LE(TRUE, x0[1]) → 197_0_DIVBY_LE(x0[1] / 2)

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


(1) -> (0), if (x0[1] / 2* 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@ebe5687 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 197_0_DIVBY_LE(x0) → COND_197_0_DIVBY_LE(&&(>(x0, 0), <=(0, /(x0, 2))), x0) the following chains were created:
  • We consider the chain 197_0_DIVBY_LE(x0[0]) → COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0]), COND_197_0_DIVBY_LE(TRUE, x0[1]) → 197_0_DIVBY_LE(/(x0[1], 2)) which results in the following constraint:

    (1)    (&&(>(x0[0], 0), <=(0, /(x0[0], 2)))=TRUEx0[0]=x0[1]197_0_DIVBY_LE(x0[0])≥NonInfC∧197_0_DIVBY_LE(x0[0])≥COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])∧(UIncreasing(COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])), ≥))



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

    (2)    (>(x0[0], 0)=TRUE<=(0, /(x0[0], 2))=TRUE197_0_DIVBY_LE(x0[0])≥NonInfC∧197_0_DIVBY_LE(x0[0])≥COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])∧(UIncreasing(COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])), ≥))



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

    (3)    (x0[0] + [-1] ≥ 0∧max{x0[0], [-1]x0[0]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [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)    (x0[0] + [-1] ≥ 0∧max{x0[0], [-1]x0[0]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [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)    (x0[0] + [-1] ≥ 0∧[2]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (6)    (x0[0] ≥ 0∧[2] + [2]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (7)    (x0[0] ≥ 0∧x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)







For Pair COND_197_0_DIVBY_LE(TRUE, x0) → 197_0_DIVBY_LE(/(x0, 2)) the following chains were created:
  • We consider the chain 197_0_DIVBY_LE(x0[0]) → COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0]), COND_197_0_DIVBY_LE(TRUE, x0[1]) → 197_0_DIVBY_LE(/(x0[1], 2)) which results in the following constraint:

    (8)    (&&(>(x0[0], 0), <=(0, /(x0[0], 2)))=TRUEx0[0]=x0[1]COND_197_0_DIVBY_LE(TRUE, x0[1])≥NonInfC∧COND_197_0_DIVBY_LE(TRUE, x0[1])≥197_0_DIVBY_LE(/(x0[1], 2))∧(UIncreasing(197_0_DIVBY_LE(/(x0[1], 2))), ≥))



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

    (9)    (>(x0[0], 0)=TRUE<=(0, /(x0[0], 2))=TRUECOND_197_0_DIVBY_LE(TRUE, x0[0])≥NonInfC∧COND_197_0_DIVBY_LE(TRUE, x0[0])≥197_0_DIVBY_LE(/(x0[0], 2))∧(UIncreasing(197_0_DIVBY_LE(/(x0[1], 2))), ≥))



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

    (10)    (x0[0] + [-1] ≥ 0∧max{x0[0], [-1]x0[0]} + [-1] ≥ 0 ⇒ (UIncreasing(197_0_DIVBY_LE(/(x0[1], 2))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_17] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)



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

    (11)    (x0[0] + [-1] ≥ 0∧max{x0[0], [-1]x0[0]} + [-1] ≥ 0 ⇒ (UIncreasing(197_0_DIVBY_LE(/(x0[1], 2))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_17] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)



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

    (12)    (x0[0] + [-1] ≥ 0∧[2]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(197_0_DIVBY_LE(/(x0[1], 2))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



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

    (13)    (x0[0] ≥ 0∧[2] + [2]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(197_0_DIVBY_LE(/(x0[1], 2))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



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

    (14)    (x0[0] ≥ 0∧x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(197_0_DIVBY_LE(/(x0[1], 2))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 197_0_DIVBY_LE(x0) → COND_197_0_DIVBY_LE(&&(>(x0, 0), <=(0, /(x0, 2))), x0)
    • (x0[0] ≥ 0∧x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

  • COND_197_0_DIVBY_LE(TRUE, x0) → 197_0_DIVBY_LE(/(x0, 2))
    • (x0[0] ≥ 0∧x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(197_0_DIVBY_LE(/(x0[1], 2))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_17] ≥ 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) = [1]   
POL(197_0_DIVBY_LE(x1)) = [-1] + x1   
POL(COND_197_0_DIVBY_LE(x1, x2)) = [-1] + x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(<=(x1, x2)) = [-1]   
POL(2) = [2]   

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, 2)1 @ {}) = max{x1, [-1]x1} + [-1]   
POL(/(x1, 2)1 @ {197_0_DIVBY_LE_1/0}) = max{x1, [-1]x1} + [-1]   

The following pairs are in P>:

COND_197_0_DIVBY_LE(TRUE, x0[1]) → 197_0_DIVBY_LE(/(x0[1], 2))

The following pairs are in Pbound:

197_0_DIVBY_LE(x0[0]) → COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])
COND_197_0_DIVBY_LE(TRUE, x0[1]) → 197_0_DIVBY_LE(/(x0[1], 2))

The following pairs are in P:

197_0_DIVBY_LE(x0[0]) → COND_197_0_DIVBY_LE(&&(>(x0[0], 0), <=(0, /(x0[0], 2))), x0[0])

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

&&(TRUE, TRUE)1TRUE1
FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)1
FALSE1&&(FALSE, FALSE)1
/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): 197_0_DIVBY_LE(x0[0]) → COND_197_0_DIVBY_LE(x0[0] > 0 && 0 <= x0[0] / 2, 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