Termination proof of /tmp/tmp9ARXTN/PastaB6.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 230 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 60 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_16 (Sun Microsystems Inc.) Main-Class: PastaB6

/**
 * 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 PastaB6 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x > 0 && 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:
PastaB6.main([Ljava/lang/String;)V: Graph of 179 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: PastaB6.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 10 rules for P and 0 rules for R.

P rules:
345_0_main_LE(EOS(STATIC_345), i61, i51, i61) → 352_0_main_LE(EOS(STATIC_352), i61, i51, i61)
352_0_main_LE(EOS(STATIC_352), i61, i51, i61) → 369_0_main_Load(EOS(STATIC_369), i61, i51) | >(i61, 0)
369_0_main_Load(EOS(STATIC_369), i61, i51) → 379_0_main_LE(EOS(STATIC_379), i61, i51, i51)
379_0_main_LE(EOS(STATIC_379), i61, i69, i69) → 387_0_main_LE(EOS(STATIC_387), i61, i69, i69)
387_0_main_LE(EOS(STATIC_387), i61, i69, i69) → 405_0_main_Inc(EOS(STATIC_405), i61, i69) | >(i69, 0)
405_0_main_Inc(EOS(STATIC_405), i61, i69) → 416_0_main_Inc(EOS(STATIC_416), +(i61, -1), i69) | >(i61, 0)
416_0_main_Inc(EOS(STATIC_416), i76, i69) → 424_0_main_JMP(EOS(STATIC_424), i76, +(i69, -1)) | >(i69, 0)
424_0_main_JMP(EOS(STATIC_424), i76, i78) → 448_0_main_Load(EOS(STATIC_448), i76, i78)
448_0_main_Load(EOS(STATIC_448), i76, i78) → 336_0_main_Load(EOS(STATIC_336), i76, i78)
336_0_main_Load(EOS(STATIC_336), i18, i51) → 345_0_main_LE(EOS(STATIC_345), i18, i51, i18)
R rules:

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

P rules:
345_0_main_LE(EOS(STATIC_345), x0, x1, x0) → 345_0_main_LE(EOS(STATIC_345), +(x0, -1), +(x1, -1), +(x0, -1)) | &&(>(x1, 0), >(x0, 0))
R rules:

Filtered ground terms:

345_0_main_LE(x1, x2, x3, x4) → 345_0_main_LE(x2, x3, x4)
EOS(x1) → EOS
Cond_345_0_main_LE(x1, x2, x3, x4, x5) → Cond_345_0_main_LE(x1, x3, x4, x5)

Filtered duplicate args:

345_0_main_LE(x1, x2, x3) → 345_0_main_LE(x2, x3)
Cond_345_0_main_LE(x1, x2, x3, x4) → Cond_345_0_main_LE(x1, x3, x4)

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

P rules:
345_0_main_LE(x1, x0) → 345_0_main_LE(+(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:
345_0_MAIN_LE(x1, x0) → COND_345_0_MAIN_LE(&&(>(x1, 0), >(x0, 0)), x1, x0)
COND_345_0_MAIN_LE(TRUE, x1, x0) → 345_0_MAIN_LE(+(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): 345_0_MAIN_LE(x1[0], x0[0]) → COND_345_0_MAIN_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_345_0_MAIN_LE(TRUE, x1[1], x0[1]) → 345_0_MAIN_LE(x1[1] + -1, x0[1] + -1)

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

(1) -> (0), if (x1[1] + -1 →^* x1[0]∧x0[1] + -1 →^* 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.IdpCand1ShapeHeuristic@7c298508 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 345_0_MAIN_LE(x1, x0) → COND_345_0_MAIN_LE(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:

We consider the chain 345_0_MAIN_LE(x1[0], x0[0]) → COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_345_0_MAIN_LE(TRUE, x1[1], x0[1]) → 345_0_MAIN_LE(+(x1[1], -1), +(x0[1], -1)) which results in the following constraint:
(1)    (&&(>(x1[0], 0), >(x0[0], 0))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 345_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧345_0_MAIN_LE(x1[0], x0[0])≥COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[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 ⇒ 345_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧345_0_MAIN_LE(x1[0], x0[0])≥COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[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 ⇒ (U^Increasing(COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 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 ⇒ (U^Increasing(COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 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 ⇒ (U^Increasing(COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 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 ⇒ (U^Increasing(COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(5)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 0)

For Pair COND_345_0_MAIN_LE(TRUE, x1, x0) → 345_0_MAIN_LE(+(x1, -1), +(x0, -1)) the following chains were created:

We consider the chain COND_345_0_MAIN_LE(TRUE, x1[1], x0[1]) → 345_0_MAIN_LE(+(x1[1], -1), +(x0[1], -1)) which results in the following constraint:
(8)    (COND_345_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_345_0_MAIN_LE(TRUE, x1[1], x0[1])≥345_0_MAIN_LE(+(x1[1], -1), +(x0[1], -1))∧(U^Increasing(345_0_MAIN_LE(+(x1[1], -1), +(x0[1], -1))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(345_0_MAIN_LE(+(x1[1], -1), +(x0[1], -1))), ≥)∧[bni_11] = 0∧[4 + (-1)bso_12] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(345_0_MAIN_LE(+(x1[1], -1), +(x0[1], -1))), ≥)∧[bni_11] = 0∧[4 + (-1)bso_12] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(345_0_MAIN_LE(+(x1[1], -1), +(x0[1], -1))), ≥)∧[bni_11] = 0∧[4 + (-1)bso_12] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(345_0_MAIN_LE(+(x1[1], -1), +(x0[1], -1))), ≥)∧[bni_11] = 0∧0 = 0∧0 = 0∧[4 + (-1)bso_12] ≥ 0)

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

345_0_MAIN_LE(x1, x0) → COND_345_0_MAIN_LE(&&(>(x1, 0), >(x0, 0)), x1, x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(5)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x0[0] + [(2)bni_9]x1[0] ≥ 0∧[(-1)bso_10] ≥ 0)
COND_345_0_MAIN_LE(TRUE, x1, x0) → 345_0_MAIN_LE(+(x1, -1), +(x0, -1))
- ((U^Increasing(345_0_MAIN_LE(+(x1[1], -1), +(x0[1], -1))), ≥)∧[bni_11] = 0∧0 = 0∧0 = 0∧[4 + (-1)bso_12] ≥ 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(345_0_MAIN_LE(x₁, x₂)) = [1] + [2]x₂ + [2]x₁
POL(COND_345_0_MAIN_LE(x₁, x₂, x₃)) = [1] + [2]x₃ + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_345_0_MAIN_LE(TRUE, x1[1], x0[1]) → 345_0_MAIN_LE(+(x1[1], -1), +(x0[1], -1))

The following pairs are in P_bound:

345_0_MAIN_LE(x1[0], x0[0]) → COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])

The following pairs are in P_≥:

345_0_MAIN_LE(x1[0], x0[0]) → COND_345_0_MAIN_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 345_0_MAIN_LE(x1[0], x0[0]) → COND_345_0_MAIN_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[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_345_0_MAIN_LE(TRUE, x1[1], x0[1]) → 345_0_MAIN_LE(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.

(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