Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 110 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 80 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 160 ms)
↳8 AND
    ↳9 IDP
        ↳10 IDependencyGraphProof (⇔, 0 ms)
        ↳11 TRUE
    ↳12 IDP
        ↳13 IDependencyGraphProof (⇔, 0 ms)
        ↳14 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Sun Microsystems Inc.) Main-Class: AG313 
public class AG313 {
  public static void main(String[] args) {
    int x, y;
    x = args[0].length();
    y = args[1].length() + 1;
    quot(x,y);

  }


  public static int quot(int x, int y) {
    int i = 0;
    if(x==0) return 0;
    while (x > 0 && y > 0) {
      i += 1;
      x = (x - 1)- (y - 1);

    }
    return i;
  }
}


(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
AG313.main([Ljava/lang/String;)V: Graph of 180 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: AG313.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 16 rules for P and 0 rules for R.


P rules:
568_0_quot_LE(EOS(STATIC_568), i40, i138, i40, i138) → 572_0_quot_LE(EOS(STATIC_572), i40, i138, i40, i138)
572_0_quot_LE(EOS(STATIC_572), i40, i138, i40, i138) → 576_0_quot_Load(EOS(STATIC_576), i40, i138, i40) | >(i138, 0)
576_0_quot_Load(EOS(STATIC_576), i40, i138, i40) → 581_0_quot_LE(EOS(STATIC_581), i40, i138, i40, i40)
581_0_quot_LE(EOS(STATIC_581), i40, i138, i40, i40) → 587_0_quot_Inc(EOS(STATIC_587), i40, i138, i40) | >(i40, 0)
587_0_quot_Inc(EOS(STATIC_587), i40, i138, i40) → 591_0_quot_Load(EOS(STATIC_591), i40, i138, i40)
591_0_quot_Load(EOS(STATIC_591), i40, i138, i40) → 595_0_quot_ConstantStackPush(EOS(STATIC_595), i40, i40, i138)
595_0_quot_ConstantStackPush(EOS(STATIC_595), i40, i40, i138) → 597_0_quot_IntArithmetic(EOS(STATIC_597), i40, i40, i138, 1)
597_0_quot_IntArithmetic(EOS(STATIC_597), i40, i40, i138, matching1) → 599_0_quot_Load(EOS(STATIC_599), i40, i40, -(i138, 1)) | &&(>(i138, 0), =(matching1, 1))
599_0_quot_Load(EOS(STATIC_599), i40, i40, i144) → 601_0_quot_ConstantStackPush(EOS(STATIC_601), i40, i40, i144, i40)
601_0_quot_ConstantStackPush(EOS(STATIC_601), i40, i40, i144, i40) → 603_0_quot_IntArithmetic(EOS(STATIC_603), i40, i40, i144, i40, 1)
603_0_quot_IntArithmetic(EOS(STATIC_603), i40, i40, i144, i40, matching1) → 606_0_quot_IntArithmetic(EOS(STATIC_606), i40, i40, i144, -(i40, 1)) | &&(>(i40, 0), =(matching1, 1))
606_0_quot_IntArithmetic(EOS(STATIC_606), i40, i40, i144, i146) → 608_0_quot_Store(EOS(STATIC_608), i40, i40, -(i144, i146)) | &&(>=(i144, 0), >=(i146, 0))
608_0_quot_Store(EOS(STATIC_608), i40, i40, i147) → 610_0_quot_JMP(EOS(STATIC_610), i40, i147, i40)
610_0_quot_JMP(EOS(STATIC_610), i40, i147, i40) → 613_0_quot_Load(EOS(STATIC_613), i40, i147, i40)
613_0_quot_Load(EOS(STATIC_613), i40, i147, i40) → 564_0_quot_Load(EOS(STATIC_564), i40, i147, i40)
564_0_quot_Load(EOS(STATIC_564), i40, i128, i40) → 568_0_quot_LE(EOS(STATIC_568), i40, i128, i40, i128)
R rules:

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


P rules:
568_0_quot_LE(EOS(STATIC_568), x0, x1, x0, x1) → 568_0_quot_LE(EOS(STATIC_568), x0, -(-(x1, 1), -(x0, 1)), x0, -(-(x1, 1), -(x0, 1))) | &&(>(+(x1, 1), 1), >(+(x0, 1), 1))
R rules:

Filtered ground terms:



568_0_quot_LE(x1, x2, x3, x4, x5) → 568_0_quot_LE(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_568_0_quot_LE(x1, x2, x3, x4, x5, x6) → Cond_568_0_quot_LE(x1, x3, x4, x5, x6)

Filtered duplicate args:



568_0_quot_LE(x1, x2, x3, x4) → 568_0_quot_LE(x3, x4)
Cond_568_0_quot_LE(x1, x2, x3, x4, x5) → Cond_568_0_quot_LE(x1, x4, x5)

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


P rules:
568_0_quot_LE(x0, x1) → 568_0_quot_LE(x0, -(-(x1, 1), -(x0, 1))) | &&(>(x1, 0), >(x0, 0))
R rules:

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


P rules:
568_0_QUOT_LE(x0, x1) → COND_568_0_QUOT_LE(&&(>(x1, 0), >(x0, 0)), x0, x1)
COND_568_0_QUOT_LE(TRUE, x0, x1) → 568_0_QUOT_LE(x0, -(-(x1, 1), -(x0, 1)))
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): 568_0_QUOT_LE(x0[0], x1[0]) → COND_568_0_QUOT_LE(x1[0] > 0 && x0[0] > 0, x0[0], x1[0])
(1): COND_568_0_QUOT_LE(TRUE, x0[1], x1[1]) → 568_0_QUOT_LE(x0[1], x1[1] - 1 - x0[1] - 1)

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


(1) -> (0), if (x0[1] →* x0[0]∧x1[1] - 1 - x0[1] - 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@133321d6 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 568_0_QUOT_LE(x0, x1) → COND_568_0_QUOT_LE(&&(>(x1, 0), >(x0, 0)), x0, x1) the following chains were created:
  • We consider the chain 568_0_QUOT_LE(x0[0], x1[0]) → COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]), COND_568_0_QUOT_LE(TRUE, x0[1], x1[1]) → 568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1))) which results in the following constraint:

    (1)    (&&(>(x1[0], 0), >(x0[0], 0))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 568_0_QUOT_LE(x0[0], x1[0])≥NonInfC∧568_0_QUOT_LE(x0[0], x1[0])≥COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_568_0_QUOT_LE(&&(>(x1[0], 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], 0)=TRUE∧>(x0[0], 0)=TRUE ⇒ 568_0_QUOT_LE(x0[0], x1[0])≥NonInfC∧568_0_QUOT_LE(x0[0], x1[0])≥COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_568_0_QUOT_LE(&&(>(x1[0], 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] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[-1 + (-1)bso_13] + x0[0] ≥ 0)



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

    (4)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[-1 + (-1)bso_13] + x0[0] ≥ 0)



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

    (5)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[-1 + (-1)bso_13] + x0[0] ≥ 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_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[-1 + (-1)bso_13] + x0[0] ≥ 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_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] + x0[0] ≥ 0)







For Pair COND_568_0_QUOT_LE(TRUE, x0, x1) → 568_0_QUOT_LE(x0, -(-(x1, 1), -(x0, 1))) the following chains were created:
  • We consider the chain 568_0_QUOT_LE(x0[0], x1[0]) → COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]), COND_568_0_QUOT_LE(TRUE, x0[1], x1[1]) → 568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1))), 568_0_QUOT_LE(x0[0], x1[0]) → COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]) which results in the following constraint:

    (8)    (&&(>(x1[0], 0), >(x0[0], 0))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x0[1]=x0[0]1∧-(-(x1[1], 1), -(x0[1], 1))=x1[0]1 ⇒ COND_568_0_QUOT_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_568_0_QUOT_LE(TRUE, x0[1], x1[1])≥568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))∧(UIncreasing(568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥))



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

    (9)    (>(x1[0], 0)=TRUE∧>(x0[0], 0)=TRUE ⇒ COND_568_0_QUOT_LE(TRUE, x0[0], x1[0])≥NonInfC∧COND_568_0_QUOT_LE(TRUE, x0[0], x1[0])≥568_0_QUOT_LE(x0[0], -(-(x1[0], 1), -(x0[0], 1)))∧(UIncreasing(568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥))



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

    (10)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)Bound*bni_14] + [bni_14]x1[0] + [(-1)bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (11)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)Bound*bni_14] + [bni_14]x1[0] + [(-1)bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (12)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)Bound*bni_14] + [bni_14]x1[0] + [(-1)bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 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(568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]x1[0] + [(-1)bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 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(568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)Bound*bni_14] + [bni_14]x1[0] + [(-1)bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 568_0_QUOT_LE(x0, x1) → COND_568_0_QUOT_LE(&&(>(x1, 0), >(x0, 0)), x0, x1)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] + x0[0] ≥ 0)

  • COND_568_0_QUOT_LE(TRUE, x0, x1) → 568_0_QUOT_LE(x0, -(-(x1, 1), -(x0, 1)))
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)Bound*bni_14] + [bni_14]x1[0] + [(-1)bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 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(568_0_QUOT_LE(x1, x2)) = [-1] + x2   
POL(COND_568_0_QUOT_LE(x1, x2, x3)) = x3 + [-1]x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

COND_568_0_QUOT_LE(TRUE, x0[1], x1[1]) → 568_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))

The following pairs are in Pbound:

568_0_QUOT_LE(x0[0], x1[0]) → COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])

The following pairs are in P≥:

568_0_QUOT_LE(x0[0], x1[0]) → COND_568_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])

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

TRUE1 → &&(TRUE, TRUE)1
&&(TRUE, FALSE)1 ↔ FALSE1
&&(FALSE, TRUE)1 ↔ FALSE1
&&(FALSE, FALSE)1 ↔ FALSE1

(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): 568_0_QUOT_LE(x0[0], x1[0]) → COND_568_0_QUOT_LE(x1[0] > 0 && x0[0] > 0, x0[0], x1[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_568_0_QUOT_LE(TRUE, x0[1], x1[1]) → 568_0_QUOT_LE(x0[1], x1[1] - 1 - x0[1] - 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