Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 320 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 90 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 210 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: DivMinus 
public class DivMinus {
  public static int div(int x, int y) {
    int res = 0;
    while (x >= y && y > 0) {
      x = x-y;
      res = res + 1;
    }
    return res;
  }

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    div(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:
DivMinus.main([Ljava/lang/String;)V: Graph of 200 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: DivMinus.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 17 rules for P and 0 rules for R.


P rules:
701_0_div_Load(EOS(STATIC_701), i98, i97, i98, i97) → 703_0_div_LT(EOS(STATIC_703), i98, i97, i98, i97, i98)
703_0_div_LT(EOS(STATIC_703), i98, i97, i98, i97, i98) → 707_0_div_LT(EOS(STATIC_707), i98, i97, i98, i97, i98)
707_0_div_LT(EOS(STATIC_707), i98, i97, i98, i97, i98) → 711_0_div_Load(EOS(STATIC_711), i98, i97, i98) | >=(i97, i98)
711_0_div_Load(EOS(STATIC_711), i98, i97, i98) → 716_0_div_LE(EOS(STATIC_716), i98, i97, i98, i98)
716_0_div_LE(EOS(STATIC_716), i108, i97, i108, i108) → 721_0_div_LE(EOS(STATIC_721), i108, i97, i108, i108)
721_0_div_LE(EOS(STATIC_721), i108, i97, i108, i108) → 728_0_div_Load(EOS(STATIC_728), i108, i97, i108) | >(i108, 0)
728_0_div_Load(EOS(STATIC_728), i108, i97, i108) → 736_0_div_Load(EOS(STATIC_736), i108, i108, i97)
736_0_div_Load(EOS(STATIC_736), i108, i108, i97) → 743_0_div_IntArithmetic(EOS(STATIC_743), i108, i108, i97, i108)
743_0_div_IntArithmetic(EOS(STATIC_743), i108, i108, i97, i108) → 747_0_div_Store(EOS(STATIC_747), i108, i108, -(i97, i108)) | >(i108, 0)
747_0_div_Store(EOS(STATIC_747), i108, i108, i109) → 750_0_div_Load(EOS(STATIC_750), i108, i109, i108)
750_0_div_Load(EOS(STATIC_750), i108, i109, i108) → 753_0_div_ConstantStackPush(EOS(STATIC_753), i108, i109, i108)
753_0_div_ConstantStackPush(EOS(STATIC_753), i108, i109, i108) → 755_0_div_IntArithmetic(EOS(STATIC_755), i108, i109, i108)
755_0_div_IntArithmetic(EOS(STATIC_755), i108, i109, i108) → 757_0_div_Store(EOS(STATIC_757), i108, i109, i108)
757_0_div_Store(EOS(STATIC_757), i108, i109, i108) → 759_0_div_JMP(EOS(STATIC_759), i108, i109, i108)
759_0_div_JMP(EOS(STATIC_759), i108, i109, i108) → 778_0_div_Load(EOS(STATIC_778), i108, i109, i108)
778_0_div_Load(EOS(STATIC_778), i108, i109, i108) → 697_0_div_Load(EOS(STATIC_697), i108, i109, i108)
697_0_div_Load(EOS(STATIC_697), i98, i97, i98) → 701_0_div_Load(EOS(STATIC_701), i98, i97, i98, i97)
R rules:

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


P rules:
701_0_div_Load(EOS(STATIC_701), x0, x1, x0, x1) → 701_0_div_Load(EOS(STATIC_701), x0, -(x1, x0), x0, -(x1, x0)) | &&(>=(x1, x0), >(x0, 0))
R rules:

Filtered ground terms:



701_0_div_Load(x1, x2, x3, x4, x5) → 701_0_div_Load(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_701_0_div_Load(x1, x2, x3, x4, x5, x6) → Cond_701_0_div_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:



701_0_div_Load(x1, x2, x3, x4) → 701_0_div_Load(x3, x4)
Cond_701_0_div_Load(x1, x2, x3, x4, x5) → Cond_701_0_div_Load(x1, x4, x5)

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


P rules:
701_0_div_Load(x0, x1) → 701_0_div_Load(x0, -(x1, x0)) | &&(>=(x1, x0), >(x0, 0))
R rules:

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


P rules:
701_0_DIV_LOAD(x0, x1) → COND_701_0_DIV_LOAD(&&(>=(x1, x0), >(x0, 0)), x0, x1)
COND_701_0_DIV_LOAD(TRUE, x0, x1) → 701_0_DIV_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): 701_0_DIV_LOAD(x0[0], x1[0]) → COND_701_0_DIV_LOAD(x1[0] >= x0[0] && x0[0] > 0, x0[0], x1[0])
(1): COND_701_0_DIV_LOAD(TRUE, x0[1], x1[1]) → 701_0_DIV_LOAD(x0[1], x1[1] - x0[1])

(0) -> (1), if (x1[0] >= x0[0] && x0[0] > 0 ∧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@551da1f8 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 701_0_DIV_LOAD(x0, x1) → COND_701_0_DIV_LOAD(&&(>=(x1, x0), >(x0, 0)), x0, x1) the following chains were created:
  • We consider the chain 701_0_DIV_LOAD(x0[0], x1[0]) → COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]), COND_701_0_DIV_LOAD(TRUE, x0[1], x1[1]) → 701_0_DIV_LOAD(x0[1], -(x1[1], x0[1])) which results in the following constraint:

    (1)    (&&(>=(x1[0], x0[0]), >(x0[0], 0))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 701_0_DIV_LOAD(x0[0], x1[0])≥NonInfC∧701_0_DIV_LOAD(x0[0], x1[0])≥COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), 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], 0)=TRUE ⇒ 701_0_DIV_LOAD(x0[0], x1[0])≥NonInfC∧701_0_DIV_LOAD(x0[0], x1[0])≥COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), 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] + [-1] ≥ 0 ⇒ (UIncreasing(COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-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] + [-1] ≥ 0 ⇒ (UIncreasing(COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-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] + [-1] ≥ 0 ⇒ (UIncreasing(COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-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] + [-1] ≥ 0 ⇒ (UIncreasing(COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-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_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_13 + (-1)bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)







For Pair COND_701_0_DIV_LOAD(TRUE, x0, x1) → 701_0_DIV_LOAD(x0, -(x1, x0)) the following chains were created:
  • We consider the chain 701_0_DIV_LOAD(x0[0], x1[0]) → COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]), COND_701_0_DIV_LOAD(TRUE, x0[1], x1[1]) → 701_0_DIV_LOAD(x0[1], -(x1[1], x0[1])), 701_0_DIV_LOAD(x0[0], x1[0]) → COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]) which results in the following constraint:

    (8)    (&&(>=(x1[0], x0[0]), >(x0[0], 0))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x0[1]=x0[0]1∧-(x1[1], x0[1])=x1[0]1 ⇒ COND_701_0_DIV_LOAD(TRUE, x0[1], x1[1])≥NonInfC∧COND_701_0_DIV_LOAD(TRUE, x0[1], x1[1])≥701_0_DIV_LOAD(x0[1], -(x1[1], x0[1]))∧(UIncreasing(701_0_DIV_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], 0)=TRUE ⇒ COND_701_0_DIV_LOAD(TRUE, x0[0], x1[0])≥NonInfC∧COND_701_0_DIV_LOAD(TRUE, x0[0], x1[0])≥701_0_DIV_LOAD(x0[0], -(x1[0], x0[0]))∧(UIncreasing(701_0_DIV_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] + [-1] ≥ 0 ⇒ (UIncreasing(701_0_DIV_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)bso_16] + x0[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] + [-1] ≥ 0 ⇒ (UIncreasing(701_0_DIV_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)bso_16] + x0[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] + [-1] ≥ 0 ⇒ (UIncreasing(701_0_DIV_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)bso_16] + 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] + [-1] ≥ 0 ⇒ (UIncreasing(701_0_DIV_LOAD(x0[1], -(x1[1], x0[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] + 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 ⇒ (UIncreasing(701_0_DIV_LOAD(x0[1], -(x1[1], x0[1]))), ≥)∧[(-1)Bound*bni_15 + (-1)bni_15] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_16] + x0[0] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 701_0_DIV_LOAD(x0, x1) → COND_701_0_DIV_LOAD(&&(>=(x1, x0), >(x0, 0)), x0, x1)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_13 + (-1)bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)

  • COND_701_0_DIV_LOAD(TRUE, x0, x1) → 701_0_DIV_LOAD(x0, -(x1, x0))
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(701_0_DIV_LOAD(x0[1], -(x1[1], x0[1]))), ≥)∧[(-1)Bound*bni_15 + (-1)bni_15] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_16] + 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) = [2]   
POL(701_0_DIV_LOAD(x1, x2)) = [-1] + x2 + [-1]x1   
POL(COND_701_0_DIV_LOAD(x1, x2, x3)) = [-1] + x3 + [-1]x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>=(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(-(x1, x2)) = x1 + [-1]x2   

The following pairs are in P>:

COND_701_0_DIV_LOAD(TRUE, x0[1], x1[1]) → 701_0_DIV_LOAD(x0[1], -(x1[1], x0[1]))

The following pairs are in Pbound:

701_0_DIV_LOAD(x0[0], x1[0]) → COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])
COND_701_0_DIV_LOAD(TRUE, x0[1], x1[1]) → 701_0_DIV_LOAD(x0[1], -(x1[1], x0[1]))

The following pairs are in P≥:

701_0_DIV_LOAD(x0[0], x1[0]) → COND_701_0_DIV_LOAD(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])

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

&&(TRUE, TRUE)1 ↔ TRUE1
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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 701_0_DIV_LOAD(x0[0], x1[0]) → COND_701_0_DIV_LOAD(x1[0] >= x0[0] && x0[0] > 0, x0[0], x1[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