Termination proof of /tmp/tmp7wmLib/PastaB14.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 180 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 50 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 130 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_16 (Sun Microsystems Inc.) Main-Class: PastaB14

/**
 * 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 PastaB14 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x == y && x > 0) {
            while (y > 0) {
                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:
PastaB14.main([Ljava/lang/String;)V: Graph of 189 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: PastaB14.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 19 rules for P and 0 rules for R.

P rules:
598_0_main_Load(EOS(STATIC_598), i143, i144, i143) → 601_0_main_NE(EOS(STATIC_601), i143, i144, i143, i144)
601_0_main_NE(EOS(STATIC_601), i144, i144, i144, i144) → 604_0_main_NE(EOS(STATIC_604), i144, i144, i144, i144)
604_0_main_NE(EOS(STATIC_604), i144, i144, i144, i144) → 608_0_main_Load(EOS(STATIC_608), i144, i144)
608_0_main_Load(EOS(STATIC_608), i144, i144) → 612_0_main_LE(EOS(STATIC_612), i144, i144, i144)
612_0_main_LE(EOS(STATIC_612), i151, i151, i151) → 615_0_main_LE(EOS(STATIC_615), i151, i151, i151)
615_0_main_LE(EOS(STATIC_615), i151, i151, i151) → 621_0_main_Load(EOS(STATIC_621), i151, i151) | >(i151, 0)
621_0_main_Load(EOS(STATIC_621), i151, i151) → 637_0_main_Load(EOS(STATIC_637), i151, i151)
637_0_main_Load(EOS(STATIC_637), i157, i158) → 659_0_main_Load(EOS(STATIC_659), i157, i158)
659_0_main_Load(EOS(STATIC_659), i169, i170) → 662_0_main_LE(EOS(STATIC_662), i169, i170, i170)
662_0_main_LE(EOS(STATIC_662), i169, matching1, matching2) → 664_0_main_LE(EOS(STATIC_664), i169, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
662_0_main_LE(EOS(STATIC_662), i169, i176, i176) → 665_0_main_LE(EOS(STATIC_665), i169, i176, i176)
664_0_main_LE(EOS(STATIC_664), i169, matching1, matching2) → 668_0_main_Load(EOS(STATIC_668), i169, 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
668_0_main_Load(EOS(STATIC_668), i169, matching1) → 595_0_main_Load(EOS(STATIC_595), i169, 0) | =(matching1, 0)
595_0_main_Load(EOS(STATIC_595), i143, i144) → 598_0_main_Load(EOS(STATIC_598), i143, i144, i143)
665_0_main_LE(EOS(STATIC_665), i169, i176, i176) → 671_0_main_Inc(EOS(STATIC_671), i169, i176) | >(i176, 0)
671_0_main_Inc(EOS(STATIC_671), i169, i176) → 672_0_main_Inc(EOS(STATIC_672), +(i169, -1), i176)
672_0_main_Inc(EOS(STATIC_672), i179, i176) → 675_0_main_JMP(EOS(STATIC_675), i179, +(i176, -1)) | >(i176, 0)
675_0_main_JMP(EOS(STATIC_675), i179, i181) → 678_0_main_Load(EOS(STATIC_678), i179, i181)
678_0_main_Load(EOS(STATIC_678), i179, i181) → 659_0_main_Load(EOS(STATIC_659), i179, i181)
R rules:

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
662_0_main_LE(EOS(STATIC_662), 0, x0, x1) → 662_0_main_LE(EOS(STATIC_662), 0, 0, 0) | FALSE
662_0_main_LE(EOS(STATIC_662), x0, x1, x1) → 662_0_main_LE(EOS(STATIC_662), +(x0, -1), +(x1, -1), +(x1, -1)) | >(x1, 0)
R rules:

Filtered ground terms:

662_0_main_LE(x1, x2, x3, x4) → 662_0_main_LE(x2, x3, x4)
EOS(x1) → EOS
Cond_662_0_main_LE(x1, x2, x3, x4, x5) → Cond_662_0_main_LE(x1, x3, x4, x5)

Filtered duplicate args:

662_0_main_LE(x1, x2, x3) → 662_0_main_LE(x1, x3)
Cond_662_0_main_LE(x1, x2, x3, x4) → Cond_662_0_main_LE(x1, x2, x4)

Filtered unneeded arguments:

Cond_662_0_main_LE(x1, x2, x3) → Cond_662_0_main_LE(x1, x3)
662_0_main_LE(x1, x2) → 662_0_main_LE(x2)

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

P rules:
662_0_main_LE(x1) → 662_0_main_LE(+(x1, -1)) | >(x1, 0)
R rules:

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

P rules:
662_0_MAIN_LE(x1) → COND_662_0_MAIN_LE(>(x1, 0), x1)
COND_662_0_MAIN_LE(TRUE, x1) → 662_0_MAIN_LE(+(x1, -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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 662_0_MAIN_LE(x1[0]) → COND_662_0_MAIN_LE(x1[0] > 0, x1[0])
(1): COND_662_0_MAIN_LE(TRUE, x1[1]) → 662_0_MAIN_LE(x1[1] + -1)

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

(1) -> (0), if (x1[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.IdpCand1ShapeHeuristic@165e6c89 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 662_0_MAIN_LE(x1) → COND_662_0_MAIN_LE(>(x1, 0), x1) the following chains were created:

We consider the chain 662_0_MAIN_LE(x1[0]) → COND_662_0_MAIN_LE(>(x1[0], 0), x1[0]), COND_662_0_MAIN_LE(TRUE, x1[1]) → 662_0_MAIN_LE(+(x1[1], -1)) which results in the following constraint:
(1)    (>(x1[0], 0)=TRUE∧x1[0]=x1[1] ⇒ 662_0_MAIN_LE(x1[0])≥NonInfC∧662_0_MAIN_LE(x1[0])≥COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])∧(U^Increasing(COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x1[0], 0)=TRUE ⇒ 662_0_MAIN_LE(x1[0])≥NonInfC∧662_0_MAIN_LE(x1[0])≥COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])∧(U^Increasing(COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_662_0_MAIN_LE(TRUE, x1) → 662_0_MAIN_LE(+(x1, -1)) the following chains were created:

We consider the chain COND_662_0_MAIN_LE(TRUE, x1[1]) → 662_0_MAIN_LE(+(x1[1], -1)) which results in the following constraint:
(7)    (COND_662_0_MAIN_LE(TRUE, x1[1])≥NonInfC∧COND_662_0_MAIN_LE(TRUE, x1[1])≥662_0_MAIN_LE(+(x1[1], -1))∧(U^Increasing(662_0_MAIN_LE(+(x1[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(662_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(662_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(662_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(662_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

662_0_MAIN_LE(x1) → COND_662_0_MAIN_LE(>(x1, 0), x1)
- (x1[0] ≥ 0 ⇒ (U^Increasing(COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)
COND_662_0_MAIN_LE(TRUE, x1) → 662_0_MAIN_LE(+(x1, -1))
- ((U^Increasing(662_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

The constraints for P_> respective P_bound 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 P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(662_0_MAIN_LE(x₁)) = [2]x₁
POL(COND_662_0_MAIN_LE(x₁, x₂)) = [2]x₂
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_662_0_MAIN_LE(TRUE, x1[1]) → 662_0_MAIN_LE(+(x1[1], -1))

The following pairs are in P_bound:

662_0_MAIN_LE(x1[0]) → COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])

The following pairs are in P_≥:

662_0_MAIN_LE(x1[0]) → COND_662_0_MAIN_LE(>(x1[0], 0), x1[0])

There are no usable rules.

(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 662_0_MAIN_LE(x1[0]) → COND_662_0_MAIN_LE(x1[0] > 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_662_0_MAIN_LE(TRUE, x1[1]) → 662_0_MAIN_LE(x1[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.

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Obligation:

(5) SCCToIDPv1Proof (SOUND transformation)

(6) Obligation:

(7) IDPNonInfProof (SOUND transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) IDependencyGraphProof (EQUIVALENT transformation)

(14) TRUE