Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 370 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 70 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 300 ms)
↳8 IDP
↳9 IDependencyGraphProof (⇔, 0 ms)
↳10 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: LogIterative 
public class LogIterative {
  public static int log(int x, int y) {
    int res = 0;
    while (x >= y && y > 1) {
      res++;
      x = x/y;
    }
    return res;
  } 

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    log(x, 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:
LogIterative.main([Ljava/lang/String;)V: Graph of 198 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: LogIterative.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:
711_0_log_Load(EOS(STATIC_711), i110, i109, i110, i109) → 713_0_log_LT(EOS(STATIC_713), i110, i109, i110, i109, i110)
713_0_log_LT(EOS(STATIC_713), i110, i109, i110, i109, i110) → 717_0_log_LT(EOS(STATIC_717), i110, i109, i110, i109, i110)
717_0_log_LT(EOS(STATIC_717), i110, i109, i110, i109, i110) → 722_0_log_Load(EOS(STATIC_722), i110, i109, i110) | >=(i109, i110)
722_0_log_Load(EOS(STATIC_722), i110, i109, i110) → 726_0_log_ConstantStackPush(EOS(STATIC_726), i110, i109, i110, i110)
726_0_log_ConstantStackPush(EOS(STATIC_726), i110, i109, i110, i110) → 732_0_log_LE(EOS(STATIC_732), i110, i109, i110, i110, 1)
732_0_log_LE(EOS(STATIC_732), i121, i109, i121, i121, matching1) → 738_0_log_LE(EOS(STATIC_738), i121, i109, i121, i121, 1) | =(matching1, 1)
738_0_log_LE(EOS(STATIC_738), i121, i109, i121, i121, matching1) → 746_0_log_Inc(EOS(STATIC_746), i121, i109, i121) | &&(>(i121, 1), =(matching1, 1))
746_0_log_Inc(EOS(STATIC_746), i121, i109, i121) → 752_0_log_Load(EOS(STATIC_752), i121, i109, i121)
752_0_log_Load(EOS(STATIC_752), i121, i109, i121) → 758_0_log_Load(EOS(STATIC_758), i121, i121, i109)
758_0_log_Load(EOS(STATIC_758), i121, i121, i109) → 762_0_log_IntArithmetic(EOS(STATIC_762), i121, i121, i109, i121)
762_0_log_IntArithmetic(EOS(STATIC_762), i121, i121, i109, i121) → 766_0_log_Store(EOS(STATIC_766), i121, i121, /(i109, i121)) | >(i121, 1)
766_0_log_Store(EOS(STATIC_766), i121, i121, i124) → 767_0_log_JMP(EOS(STATIC_767), i121, i124, i121)
767_0_log_JMP(EOS(STATIC_767), i121, i124, i121) → 785_0_log_Load(EOS(STATIC_785), i121, i124, i121)
785_0_log_Load(EOS(STATIC_785), i121, i124, i121) → 707_0_log_Load(EOS(STATIC_707), i121, i124, i121)
707_0_log_Load(EOS(STATIC_707), i110, i109, i110) → 711_0_log_Load(EOS(STATIC_711), i110, i109, i110, i109)
R rules:

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


P rules:
711_0_log_Load(EOS(STATIC_711), x0, x1, x0, x1) → 711_0_log_Load(EOS(STATIC_711), x0, /(x1, x0), x0, /(x1, x0)) | &&(>=(x1, x0), >(x0, 1))
R rules:

Filtered ground terms:



711_0_log_Load(x1, x2, x3, x4, x5) → 711_0_log_Load(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_711_0_log_Load(x1, x2, x3, x4, x5, x6) → Cond_711_0_log_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:



711_0_log_Load(x1, x2, x3, x4) → 711_0_log_Load(x3, x4)
Cond_711_0_log_Load(x1, x2, x3, x4, x5) → Cond_711_0_log_Load(x1, x4, x5)

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


P rules:
711_0_log_Load(x0, x1) → 711_0_log_Load(x0, /(x1, x0)) | &&(>=(x1, x0), >(x0, 1))
R rules:

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


P rules:
711_0_LOG_LOAD(x0, x1) → COND_711_0_LOG_LOAD(&&(>=(x1, x0), >(x0, 1)), x0, x1)
COND_711_0_LOG_LOAD(TRUE, x0, x1) → 711_0_LOG_LOAD(x0, /(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): 711_0_LOG_LOAD(x0[0], x1[0]) → COND_711_0_LOG_LOAD(x1[0] >= x0[0] && x0[0] > 1, x0[0], x1[0])
(1): COND_711_0_LOG_LOAD(TRUE, x0[1], x1[1]) → 711_0_LOG_LOAD(x0[1], x1[1] / x0[1])

(0) -> (1), if (x1[0] >= x0[0] && x0[0] > 1 ∧x0[0] →* x0[1]∧x1[0] →* x1[1])


(1) -> (0), if (x0[1] →* x0[0]∧x1[1] / 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@53622a11 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 711_0_LOG_LOAD(x0, x1) → COND_711_0_LOG_LOAD(&&(>=(x1, x0), >(x0, 1)), x0, x1) the following chains were created:
  • We consider the chain 711_0_LOG_LOAD(x0[0], x1[0]) → COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]), COND_711_0_LOG_LOAD(TRUE, x0[1], x1[1]) → 711_0_LOG_LOAD(x0[1], /(x1[1], x0[1])) which results in the following constraint:

    (1)    (&&(>=(x1[0], x0[0]), >(x0[0], 1))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 711_0_LOG_LOAD(x0[0], x1[0])≥NonInfC∧711_0_LOG_LOAD(x0[0], x1[0])≥COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])∧(UIncreasing(COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥))



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

    (2)    (>=(x1[0], x0[0])=TRUE∧>(x0[0], 1)=TRUE ⇒ 711_0_LOG_LOAD(x0[0], x1[0])≥NonInfC∧711_0_LOG_LOAD(x0[0], x1[0])≥COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])∧(UIncreasing(COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥))



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

    (3)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (4)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (5)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (6)    (x1[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)







For Pair COND_711_0_LOG_LOAD(TRUE, x0, x1) → 711_0_LOG_LOAD(x0, /(x1, x0)) the following chains were created:
  • We consider the chain 711_0_LOG_LOAD(x0[0], x1[0]) → COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]), COND_711_0_LOG_LOAD(TRUE, x0[1], x1[1]) → 711_0_LOG_LOAD(x0[1], /(x1[1], x0[1])), 711_0_LOG_LOAD(x0[0], x1[0]) → COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]) which results in the following constraint:

    (8)    (&&(>=(x1[0], x0[0]), >(x0[0], 1))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x0[1]=x0[0]1∧/(x1[1], x0[1])=x1[0]1 ⇒ COND_711_0_LOG_LOAD(TRUE, x0[1], x1[1])≥NonInfC∧COND_711_0_LOG_LOAD(TRUE, x0[1], x1[1])≥711_0_LOG_LOAD(x0[1], /(x1[1], x0[1]))∧(UIncreasing(711_0_LOG_LOAD(x0[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], x0[0])=TRUE∧>(x0[0], 1)=TRUE ⇒ COND_711_0_LOG_LOAD(TRUE, x0[0], x1[0])≥NonInfC∧COND_711_0_LOG_LOAD(TRUE, x0[0], x1[0])≥711_0_LOG_LOAD(x0[0], /(x1[0], x0[0]))∧(UIncreasing(711_0_LOG_LOAD(x0[1], /(x1[1], x0[1]))), ≥))



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

    (10)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(711_0_LOG_LOAD(x0[1], /(x1[1], x0[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[-1 + (-1)bso_19] + x1[0] + [-1]max{x1[0], [-1]x1[0]} + min{max{x0[0], [-1]x0[0]} + [-1], max{x1[0], [-1]x1[0]}} ≥ 0)



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

    (11)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(711_0_LOG_LOAD(x0[1], /(x1[1], x0[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[-1 + (-1)bso_19] + x1[0] + [-1]max{x1[0], [-1]x1[0]} + min{max{x0[0], [-1]x0[0]} + [-1], max{x1[0], [-1]x1[0]}} ≥ 0)



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

    (12)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0∧[2]x1[0] ≥ 0∧[2]x0[0] ≥ 0 ⇒ (UIncreasing(711_0_LOG_LOAD(x0[1], /(x1[1], x0[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[-2 + (-1)bso_19] + x0[0] ≥ 0)



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

    (13)    (x1[0] ≥ 0∧x0[0] + [-2] ≥ 0∧[2]x0[0] + [2]x1[0] ≥ 0∧[2]x0[0] ≥ 0 ⇒ (UIncreasing(711_0_LOG_LOAD(x0[1], /(x1[1], x0[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[-2 + (-1)bso_19] + x0[0] ≥ 0)



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

    (14)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[4] + [2]x0[0] + [2]x1[0] ≥ 0∧[4] + [2]x0[0] ≥ 0 ⇒ (UIncreasing(711_0_LOG_LOAD(x0[1], /(x1[1], x0[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (15)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(711_0_LOG_LOAD(x0[1], /(x1[1], x0[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 711_0_LOG_LOAD(x0, x1) → COND_711_0_LOG_LOAD(&&(>=(x1, x0), >(x0, 1)), x0, x1)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

  • COND_711_0_LOG_LOAD(TRUE, x0, x1) → 711_0_LOG_LOAD(x0, /(x1, x0))
    • (x1[0] ≥ 0∧x0[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(711_0_LOG_LOAD(x0[1], /(x1[1], x0[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 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(711_0_LOG_LOAD(x1, x2)) = x2 + [-1]x1   
POL(COND_711_0_LOG_LOAD(x1, x2, x3)) = [-1] + x3 + [-1]x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>=(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(1) = [1]   

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

POL(/(x1, x0[0])1 @ {711_0_LOG_LOAD_2/1}) = max{x1, [-1]x1} + [-1]min{max{x2, [-1]x2} + [-1], max{x1, [-1]x1}}   

The following pairs are in P>:

711_0_LOG_LOAD(x0[0], x1[0]) → COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])

The following pairs are in Pbound:

711_0_LOG_LOAD(x0[0], x1[0]) → COND_711_0_LOG_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])
COND_711_0_LOG_LOAD(TRUE, x0[1], x1[1]) → 711_0_LOG_LOAD(x0[1], /(x1[1], x0[1]))

The following pairs are in P≥:

COND_711_0_LOG_LOAD(TRUE, x0[1], x1[1]) → 711_0_LOG_LOAD(x0[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
&&(FALSE, TRUE)1 ↔ FALSE1
&&(FALSE, FALSE)1 ↔ FALSE1
/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_711_0_LOG_LOAD(TRUE, x0[1], x1[1]) → 711_0_LOG_LOAD(x0[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