Termination proof of /tmp/tmp6bbfB5/PastaA9.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 130 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 70 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 180 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: PastaA9

/**
 * 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 PastaA9 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();
        int z = Random.random();

        if (y > 0) {
            while (x >= z) {
                z += 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:
PastaA9.main([Ljava/lang/String;)V: Graph of 249 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: PastaA9.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:
530_0_main_Load(EOS(STATIC_530), i18, i93, i88, i18) → 537_0_main_LT(EOS(STATIC_537), i18, i93, i88, i18, i88)
537_0_main_LT(EOS(STATIC_537), i18, i93, i88, i18, i88) → 548_0_main_LT(EOS(STATIC_548), i18, i93, i88, i18, i88)
548_0_main_LT(EOS(STATIC_548), i18, i93, i88, i18, i88) → 567_0_main_Load(EOS(STATIC_567), i18, i93, i88) | >=(i18, i88)
567_0_main_Load(EOS(STATIC_567), i18, i93, i88) → 578_0_main_Load(EOS(STATIC_578), i18, i93, i88)
578_0_main_Load(EOS(STATIC_578), i18, i93, i88) → 588_0_main_IntArithmetic(EOS(STATIC_588), i18, i93, i88, i93)
588_0_main_IntArithmetic(EOS(STATIC_588), i18, i93, i88, i93) → 601_0_main_Store(EOS(STATIC_601), i18, i93, +(i88, i93)) | &&(>=(i88, 0), >(i93, 0))
601_0_main_Store(EOS(STATIC_601), i18, i93, i100) → 611_0_main_JMP(EOS(STATIC_611), i18, i93, i100)
611_0_main_JMP(EOS(STATIC_611), i18, i93, i100) → 622_0_main_Load(EOS(STATIC_622), i18, i93, i100)
622_0_main_Load(EOS(STATIC_622), i18, i93, i100) → 519_0_main_Load(EOS(STATIC_519), i18, i93, i100)
519_0_main_Load(EOS(STATIC_519), i18, i93, i88) → 530_0_main_Load(EOS(STATIC_530), i18, i93, i88, i18)
R rules:

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

P rules:
530_0_main_Load(EOS(STATIC_530), x0, x1, x2, x0) → 530_0_main_Load(EOS(STATIC_530), x0, x1, +(x2, x1), x0) | &&(&&(>(+(x2, 1), 0), <=(x2, x0)), >(x1, 0))
R rules:

Filtered ground terms:

530_0_main_Load(x1, x2, x3, x4, x5) → 530_0_main_Load(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_530_0_main_Load(x1, x2, x3, x4, x5, x6) → Cond_530_0_main_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:

530_0_main_Load(x1, x2, x3, x4) → 530_0_main_Load(x2, x3, x4)
Cond_530_0_main_Load(x1, x2, x3, x4, x5) → Cond_530_0_main_Load(x1, x3, x4, x5)

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

P rules:
530_0_main_Load(x1, x2, x0) → 530_0_main_Load(x1, +(x2, x1), x0) | &&(&&(>(x2, -1), <=(x2, x0)), >(x1, 0))
R rules:

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

P rules:
530_0_MAIN_LOAD(x1, x2, x0) → COND_530_0_MAIN_LOAD(&&(&&(>(x2, -1), <=(x2, x0)), >(x1, 0)), x1, x2, x0)
COND_530_0_MAIN_LOAD(TRUE, x1, x2, x0) → 530_0_MAIN_LOAD(x1, +(x2, 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): 530_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_530_0_MAIN_LOAD(x2[0] > -1 && x2[0] <= x0[0] && x1[0] > 0, x1[0], x2[0], x0[0])
(1): COND_530_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 530_0_MAIN_LOAD(x1[1], x2[1] + x1[1], x0[1])

(0) -> (1), if (x2[0] > -1 && x2[0] <= x0[0] && x1[0] > 0 ∧x1[0] →^* x1[1]∧x2[0] →^* x2[1]∧x0[0] →^* x0[1])

(1) -> (0), if (x1[1] →^* x1[0]∧x2[1] + x1[1] →^* x2[0]∧x0[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@6bf491a6 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 530_0_MAIN_LOAD(x1, x2, x0) → COND_530_0_MAIN_LOAD(&&(&&(>(x2, -1), <=(x2, x0)), >(x1, 0)), x1, x2, x0) the following chains were created:

We consider the chain 530_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0]), COND_530_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1]) which results in the following constraint:
(1)    (&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1] ⇒ 530_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧530_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])∧(U^Increasing(COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], 0)=TRUE∧>(x2[0], -1)=TRUE∧<=(x2[0], x0[0])=TRUE ⇒ 530_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧530_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])∧(U^Increasing(COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] + [bni_15]x1[0] ≥ 0∧[-1 + (-1)bso_16] + x1[0] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] + [bni_15]x1[0] ≥ 0∧[-1 + (-1)bso_16] + x1[0] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] + [bni_15]x1[0] ≥ 0∧[-1 + (-1)bso_16] + x1[0] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] + x1[0] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] + x1[0] ≥ 0)

For Pair COND_530_0_MAIN_LOAD(TRUE, x1, x2, x0) → 530_0_MAIN_LOAD(x1, +(x2, x1), x0) the following chains were created:

We consider the chain 530_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0]), COND_530_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1]), 530_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0]) which results in the following constraint:
(8)    (&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1]∧x1[1]=x1[0]1∧+(x2[1], x1[1])=x2[0]1∧x0[1]=x0[0]1 ⇒ COND_530_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_530_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])∧(U^Increasing(530_0_MAIN_LOAD(x1[1], +(x2[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], 0)=TRUE∧>(x2[0], -1)=TRUE∧<=(x2[0], x0[0])=TRUE ⇒ COND_530_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_530_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥530_0_MAIN_LOAD(x1[0], +(x2[0], x1[0]), x0[0])∧(U^Increasing(530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

530_0_MAIN_LOAD(x1, x2, x0) → COND_530_0_MAIN_LOAD(&&(&&(>(x2, -1), <=(x2, x0)), >(x1, 0)), x1, x2, x0)
- (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] + x1[0] ≥ 0)
COND_530_0_MAIN_LOAD(TRUE, x1, x2, x0) → 530_0_MAIN_LOAD(x1, +(x2, x1), x0)
- (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 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(530_0_MAIN_LOAD(x₁, x₂, x₃)) = [-1] + x₃ + [-1]x₂ + x₁
POL(COND_530_0_MAIN_LOAD(x₁, x₂, x₃, x₄)) = x₄ + [-1]x₃ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(-1) = [-1]
POL(<=(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂

The following pairs are in P_>:

COND_530_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])

The following pairs are in P_bound:

530_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])
COND_530_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 530_0_MAIN_LOAD(x1[1], +(x2[1], x1[1]), x0[1])

The following pairs are in P_≥:

530_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_530_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x2[0], x0[0])

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

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

(8) Obligation: