Termination proof of /tmp/tmp64

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 250 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 80 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 80 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: PastaB4

/**
 * 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 PastaB4 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x > y) {
            int t = x;
            x = y;
            y = t;
        }
    }
}


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:
PastaB4.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: PastaB4.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 12 rules for P and 0 rules for R.

P rules:
292_0_main_Load(EOS(STATIC_292), i18, i46, i18) → 300_0_main_LE(EOS(STATIC_300), i18, i46, i18, i46)
300_0_main_LE(EOS(STATIC_300), i18, i46, i18, i46) → 314_0_main_LE(EOS(STATIC_314), i18, i46, i18, i46)
314_0_main_LE(EOS(STATIC_314), i18, i46, i18, i46) → 327_0_main_Load(EOS(STATIC_327), i18, i46) | >(i18, i46)
327_0_main_Load(EOS(STATIC_327), i18, i46) → 337_0_main_Store(EOS(STATIC_337), i46, i18)
337_0_main_Store(EOS(STATIC_337), i46, i18) → 345_0_main_Load(EOS(STATIC_345), i46, i18)
345_0_main_Load(EOS(STATIC_345), i46, i18) → 352_0_main_Store(EOS(STATIC_352), i18, i46)
352_0_main_Store(EOS(STATIC_352), i18, i46) → 359_0_main_Load(EOS(STATIC_359), i46, i18)
359_0_main_Load(EOS(STATIC_359), i46, i18) → 366_0_main_Store(EOS(STATIC_366), i46, i18)
366_0_main_Store(EOS(STATIC_366), i46, i18) → 375_0_main_JMP(EOS(STATIC_375), i46, i18)
375_0_main_JMP(EOS(STATIC_375), i46, i18) → 397_0_main_Load(EOS(STATIC_397), i46, i18)
397_0_main_Load(EOS(STATIC_397), i46, i18) → 286_0_main_Load(EOS(STATIC_286), i46, i18)
286_0_main_Load(EOS(STATIC_286), i18, i46) → 292_0_main_Load(EOS(STATIC_292), i18, i46, i18)
R rules:

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

P rules:
292_0_main_Load(EOS(STATIC_292), x0, x1, x0) → 292_0_main_Load(EOS(STATIC_292), x1, x0, x1) | <(x1, x0)
R rules:

Filtered ground terms:

292_0_main_Load(x1, x2, x3, x4) → 292_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_292_0_main_Load(x1, x2, x3, x4, x5) → Cond_292_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

292_0_main_Load(x1, x2, x3) → 292_0_main_Load(x2, x3)
Cond_292_0_main_Load(x1, x2, x3, x4) → Cond_292_0_main_Load(x1, x3, x4)

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

P rules:
292_0_main_Load(x1, x0) → 292_0_main_Load(x0, x1) | <(x1, x0)
R rules:

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

P rules:
292_0_MAIN_LOAD(x1, x0) → COND_292_0_MAIN_LOAD(<(x1, x0), x1, x0)
COND_292_0_MAIN_LOAD(TRUE, x1, x0) → 292_0_MAIN_LOAD(x0, x1)
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): 292_0_MAIN_LOAD(x1[0], x0[0]) → COND_292_0_MAIN_LOAD(x1[0] < x0[0], x1[0], x0[0])
(1): COND_292_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 292_0_MAIN_LOAD(x0[1], x1[1])

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

(1) -> (0), if (x0[1] →^* x1[0]∧x1[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.IdpDefaultShapeHeuristic@1b710c15 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 292_0_MAIN_LOAD(x1, x0) → COND_292_0_MAIN_LOAD(<(x1, x0), x1, x0) the following chains were created:

We consider the chain COND_292_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 292_0_MAIN_LOAD(x0[1], x1[1]), 292_0_MAIN_LOAD(x1[0], x0[0]) → COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0]), COND_292_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 292_0_MAIN_LOAD(x0[1], x1[1]) which results in the following constraint:
(1)    (x0[1]=x1[0]∧x1[1]=x0[0]∧<(x1[0], x0[0])=TRUE∧x1[0]=x1[1]1∧x0[0]=x0[1]1 ⇒ 292_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧292_0_MAIN_LOAD(x1[0], x0[0])≥COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])∧(U^Increasing(COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (<(x1[0], x0[0])=TRUE ⇒ 292_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧292_0_MAIN_LOAD(x1[0], x0[0])≥COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])∧(U^Increasing(COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] + [(-1)bni_11]x1[0] ≥ 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] + [(-1)bni_11]x1[0] ≥ 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] + [(-1)bni_11]x1[0] ≥ 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

(8)    (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

For Pair COND_292_0_MAIN_LOAD(TRUE, x1, x0) → 292_0_MAIN_LOAD(x0, x1) the following chains were created:

We consider the chain 292_0_MAIN_LOAD(x1[0], x0[0]) → COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0]), COND_292_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 292_0_MAIN_LOAD(x0[1], x1[1]), 292_0_MAIN_LOAD(x1[0], x0[0]) → COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0]) which results in the following constraint:
(9)    (<(x1[0], x0[0])=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1]∧x0[1]=x1[0]1∧x1[1]=x0[0]1 ⇒ COND_292_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_292_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥292_0_MAIN_LOAD(x0[1], x1[1])∧(U^Increasing(292_0_MAIN_LOAD(x0[1], x1[1])), ≥))

We simplified constraint (9) using rules (III), (IV) which results in the following new constraint:
(10)    (<(x1[0], x0[0])=TRUE ⇒ COND_292_0_MAIN_LOAD(TRUE, x1[0], x0[0])≥NonInfC∧COND_292_0_MAIN_LOAD(TRUE, x1[0], x0[0])≥292_0_MAIN_LOAD(x0[0], x1[0])∧(U^Increasing(292_0_MAIN_LOAD(x0[1], x1[1])), ≥))

We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11)    (x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(292_0_MAIN_LOAD(x0[1], x1[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[(-1)bso_14] + [2]x0[0] + [-2]x1[0] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    (x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(292_0_MAIN_LOAD(x0[1], x1[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[(-1)bso_14] + [2]x0[0] + [-2]x1[0] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    (x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(292_0_MAIN_LOAD(x0[1], x1[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[(-1)bso_14] + [2]x0[0] + [-2]x1[0] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x0[0] ≥ 0 ⇒ (U^Increasing(292_0_MAIN_LOAD(x0[1], x1[1])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] + [2]x0[0] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(15)    (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(292_0_MAIN_LOAD(x0[1], x1[1])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] + [2]x0[0] ≥ 0)

(16)    (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(292_0_MAIN_LOAD(x0[1], x1[1])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] + [2]x0[0] ≥ 0)

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

292_0_MAIN_LOAD(x1, x0) → COND_292_0_MAIN_LOAD(<(x1, x0), x1, x0)
- (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
- (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
COND_292_0_MAIN_LOAD(TRUE, x1, x0) → 292_0_MAIN_LOAD(x0, x1)
- (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(292_0_MAIN_LOAD(x0[1], x1[1])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] + [2]x0[0] ≥ 0)
- (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(292_0_MAIN_LOAD(x0[1], x1[1])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] + [2]x0[0] ≥ 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) = [1]
POL(FALSE) = 0
POL(292_0_MAIN_LOAD(x₁, x₂)) = [-1] + x₂ + [-1]x₁
POL(COND_292_0_MAIN_LOAD(x₁, x₂, x₃)) = [-1] + x₃ + [-1]x₂
POL(<(x₁, x₂)) = [-1]

The following pairs are in P_>:

COND_292_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 292_0_MAIN_LOAD(x0[1], x1[1])

The following pairs are in P_bound:

292_0_MAIN_LOAD(x1[0], x0[0]) → COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])
COND_292_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 292_0_MAIN_LOAD(x0[1], x1[1])

The following pairs are in P_≥:

292_0_MAIN_LOAD(x1[0], x0[0]) → COND_292_0_MAIN_LOAD(<(x1[0], x0[0]), x1[0], x0[0])

There are no usable rules.

(8) Obligation: