Termination proof of /tmp/tmpE1Ra

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 150 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 70 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: PastaA6

/**
 * 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 PastaA6 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();
        int z = Random.random();

        while (x > y + z) {
            y++;
            z++;
        }
    }
}


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:
PastaA6.main([Ljava/lang/String;)V: Graph of 244 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: PastaA6.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:
499_0_main_Load(EOS(STATIC_499), i18, i47, i93, i18) → 506_0_main_Load(EOS(STATIC_506), i18, i47, i93, i18, i47)
506_0_main_Load(EOS(STATIC_506), i18, i47, i93, i18, i47) → 520_0_main_IntArithmetic(EOS(STATIC_520), i18, i47, i93, i18, i47, i93)
520_0_main_IntArithmetic(EOS(STATIC_520), i18, i47, i93, i18, i47, i93) → 529_0_main_LE(EOS(STATIC_529), i18, i47, i93, i18, +(i47, i93)) | &&(>=(i47, 0), >=(i93, 0))
529_0_main_LE(EOS(STATIC_529), i18, i47, i93, i18, i108) → 536_0_main_LE(EOS(STATIC_536), i18, i47, i93, i18, i108)
536_0_main_LE(EOS(STATIC_536), i18, i47, i93, i18, i108) → 549_0_main_Inc(EOS(STATIC_549), i18, i47, i93) | >(i18, i108)
549_0_main_Inc(EOS(STATIC_549), i18, i47, i93) → 561_0_main_Inc(EOS(STATIC_561), i18, +(i47, 1), i93) | >=(i47, 0)
561_0_main_Inc(EOS(STATIC_561), i18, i113, i93) → 570_0_main_JMP(EOS(STATIC_570), i18, i113, +(i93, 1)) | >=(i93, 0)
570_0_main_JMP(EOS(STATIC_570), i18, i113, i116) → 584_0_main_Load(EOS(STATIC_584), i18, i113, i116)
584_0_main_Load(EOS(STATIC_584), i18, i113, i116) → 491_0_main_Load(EOS(STATIC_491), i18, i113, i116)
491_0_main_Load(EOS(STATIC_491), i18, i47, i93) → 499_0_main_Load(EOS(STATIC_499), i18, i47, i93, i18)
R rules:

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

P rules:
499_0_main_Load(EOS(STATIC_499), x0, x1, x2, x0) → 499_0_main_Load(EOS(STATIC_499), x0, +(x1, 1), +(x2, 1), x0) | &&(&&(>(+(x2, 1), 0), >(+(x1, 1), 0)), >(x0, +(x1, x2)))
R rules:

Filtered ground terms:

499_0_main_Load(x1, x2, x3, x4, x5) → 499_0_main_Load(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_499_0_main_Load(x1, x2, x3, x4, x5, x6) → Cond_499_0_main_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:

499_0_main_Load(x1, x2, x3, x4) → 499_0_main_Load(x2, x3, x4)
Cond_499_0_main_Load(x1, x2, x3, x4, x5) → Cond_499_0_main_Load(x1, x3, x4, x5)

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

P rules:
499_0_main_Load(x1, x2, x0) → 499_0_main_Load(+(x1, 1), +(x2, 1), x0) | &&(&&(>(x2, -1), >(x1, -1)), >(x0, +(x1, x2)))
R rules:

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

P rules:
499_0_MAIN_LOAD(x1, x2, x0) → COND_499_0_MAIN_LOAD(&&(&&(>(x2, -1), >(x1, -1)), >(x0, +(x1, x2))), x1, x2, x0)
COND_499_0_MAIN_LOAD(TRUE, x1, x2, x0) → 499_0_MAIN_LOAD(+(x1, 1), +(x2, 1), 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): 499_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_499_0_MAIN_LOAD(x2[0] > -1 && x1[0] > -1 && x0[0] > x1[0] + x2[0], x1[0], x2[0], x0[0])
(1): COND_499_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 499_0_MAIN_LOAD(x1[1] + 1, x2[1] + 1, x0[1])

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

(1) -> (0), if (x1[1] + 1 →^* x1[0]∧x2[1] + 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.IdpCand1ShapeHeuristic@548ea21d 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 499_0_MAIN_LOAD(x1, x2, x0) → COND_499_0_MAIN_LOAD(&&(&&(>(x2, -1), >(x1, -1)), >(x0, +(x1, x2))), x1, x2, x0) the following chains were created:

We consider the chain 499_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0]), COND_499_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 499_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1]) which results in the following constraint:
(1)    (&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0])))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1] ⇒ 499_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧499_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])∧(U^Increasing(COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x0[0], +(x1[0], x2[0]))=TRUE∧>(x2[0], -1)=TRUE∧>(x1[0], -1)=TRUE ⇒ 499_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧499_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])∧(U^Increasing(COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[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] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x2[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] + [-1]x1[0] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x2[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 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] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x2[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_499_0_MAIN_LOAD(TRUE, x1, x2, x0) → 499_0_MAIN_LOAD(+(x1, 1), +(x2, 1), x0) the following chains were created:

We consider the chain COND_499_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 499_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1]) which results in the following constraint:
(7)    (COND_499_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_499_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥499_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])∧(U^Increasing(499_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(499_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(499_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(499_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(499_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_13] ≥ 0)

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

499_0_MAIN_LOAD(x1, x2, x0) → COND_499_0_MAIN_LOAD(&&(&&(>(x2, -1), >(x1, -1)), >(x0, +(x1, x2))), x1, x2, x0)
- (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)
COND_499_0_MAIN_LOAD(TRUE, x1, x2, x0) → 499_0_MAIN_LOAD(+(x1, 1), +(x2, 1), x0)
- ((U^Increasing(499_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_13] ≥ 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(499_0_MAIN_LOAD(x₁, x₂, x₃)) = [1] + x₃ + [-1]x₂ + [-1]x₁
POL(COND_499_0_MAIN_LOAD(x₁, x₂, x₃, x₄)) = [1] + x₄ + [-1]x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(-1) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]

The following pairs are in P_>:

COND_499_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 499_0_MAIN_LOAD(+(x1[1], 1), +(x2[1], 1), x0[1])

The following pairs are in P_bound:

499_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[0], x0[0])

The following pairs are in P_≥:

499_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_499_0_MAIN_LOAD(&&(&&(>(x2[0], -1), >(x1[0], -1)), >(x0[0], +(x1[0], x2[0]))), x1[0], x2[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

(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_499_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 499_0_MAIN_LOAD(x1[1] + 1, x2[1] + 1, x0[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