Termination proof of /tmp/tmpt

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 320 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 30 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: PlusSwap

public class PlusSwap{
  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    int z;
    int res = 0;

    while (y > 0) {

      z = x;
      x = y-1;
      y = z;
      res++;

    }

    res = res + x;
  }
}


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:
PlusSwap.main([Ljava/lang/String;)V: Graph of 191 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: PlusSwap.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 14 rules for P and 0 rules for R.

P rules:
622_0_main_LE(EOS(STATIC_622), i118, i129, i129) → 625_0_main_LE(EOS(STATIC_625), i118, i129, i129)
625_0_main_LE(EOS(STATIC_625), i118, i129, i129) → 629_0_main_Load(EOS(STATIC_629), i118, i129) | >(i129, 0)
629_0_main_Load(EOS(STATIC_629), i118, i129) → 633_0_main_Store(EOS(STATIC_633), i129, i118)
633_0_main_Store(EOS(STATIC_633), i129, i118) → 637_0_main_Load(EOS(STATIC_637), i129, i118)
637_0_main_Load(EOS(STATIC_637), i129, i118) → 642_0_main_ConstantStackPush(EOS(STATIC_642), i118, i129)
642_0_main_ConstantStackPush(EOS(STATIC_642), i118, i129) → 646_0_main_IntArithmetic(EOS(STATIC_646), i118, i129, 1)
646_0_main_IntArithmetic(EOS(STATIC_646), i118, i129, matching1) → 650_0_main_Store(EOS(STATIC_650), i118, -(i129, 1)) | &&(>(i129, 0), =(matching1, 1))
650_0_main_Store(EOS(STATIC_650), i118, i133) → 652_0_main_Load(EOS(STATIC_652), i133, i118)
652_0_main_Load(EOS(STATIC_652), i133, i118) → 654_0_main_Store(EOS(STATIC_654), i133, i118)
654_0_main_Store(EOS(STATIC_654), i133, i118) → 656_0_main_Inc(EOS(STATIC_656), i133, i118)
656_0_main_Inc(EOS(STATIC_656), i133, i118) → 658_0_main_JMP(EOS(STATIC_658), i133, i118)
658_0_main_JMP(EOS(STATIC_658), i133, i118) → 675_0_main_Load(EOS(STATIC_675), i133, i118)
675_0_main_Load(EOS(STATIC_675), i133, i118) → 618_0_main_Load(EOS(STATIC_618), i133, i118)
618_0_main_Load(EOS(STATIC_618), i118, i119) → 622_0_main_LE(EOS(STATIC_622), i118, i119, i119)
R rules:

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

P rules:
622_0_main_LE(EOS(STATIC_622), x0, x1, x1) → 622_0_main_LE(EOS(STATIC_622), -(x1, 1), x0, x0) | >(x1, 0)
R rules:

Filtered ground terms:

622_0_main_LE(x1, x2, x3, x4) → 622_0_main_LE(x2, x3, x4)
EOS(x1) → EOS
Cond_622_0_main_LE(x1, x2, x3, x4, x5) → Cond_622_0_main_LE(x1, x3, x4, x5)

Filtered duplicate args:

622_0_main_LE(x1, x2, x3) → 622_0_main_LE(x1, x3)
Cond_622_0_main_LE(x1, x2, x3, x4) → Cond_622_0_main_LE(x1, x2, x4)

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

P rules:
622_0_main_LE(x0, x1) → 622_0_main_LE(-(x1, 1), x0) | >(x1, 0)
R rules:

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

P rules:
622_0_MAIN_LE(x0, x1) → COND_622_0_MAIN_LE(>(x1, 0), x0, x1)
COND_622_0_MAIN_LE(TRUE, x0, x1) → 622_0_MAIN_LE(-(x1, 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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(x1[0] > 0, x0[0], x1[0])
(1): COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(x1[1] - 1, x0[1])

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

(1) -> (0), if (x1[1] - 1 →^* x0[0]∧x0[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.IdpDefaultShapeHeuristic@5a55a0da Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 3 Max Right Steps: 2

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 622_0_MAIN_LE(x0, x1) → COND_622_0_MAIN_LE(>(x1, 0), x0, x1) the following chains were created:

We consider the chain 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1]) which results in the following constraint:
(1)    (>(x1[0], 0)=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 622_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧622_0_MAIN_LE(x0[0], x1[0])≥COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])∧(U^Increasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x1[0], 0)=TRUE ⇒ 622_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧622_0_MAIN_LE(x0[0], x1[0])≥COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])∧(U^Increasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[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_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[0] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 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_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[0] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 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_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[0] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_11] = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_11] = 0∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

For Pair COND_622_0_MAIN_LE(TRUE, x0, x1) → 622_0_MAIN_LE(-(x1, 1), x0) the following chains were created:

We consider the chain 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1]), 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1]), 622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1]) which results in the following constraint:
(8)    (>(x1[0], 0)=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧-(x1[1], 1)=x0[0]1∧x0[1]=x1[0]1∧>(x1[0]1, 0)=TRUE∧x0[0]1=x0[1]1∧x1[0]1=x1[1]1∧-(x1[1]1, 1)=x0[0]2∧x0[1]1=x1[0]2∧>(x1[0]2, 0)=TRUE∧x0[0]2=x0[1]2∧x1[0]2=x1[1]2 ⇒ COND_622_0_MAIN_LE(TRUE, x0[1]1, x1[1]1)≥NonInfC∧COND_622_0_MAIN_LE(TRUE, x0[1]1, x1[1]1)≥622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)∧(U^Increasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥))

We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9)    (>(x1[0], 0)=TRUE∧>(x1[0]1, 0)=TRUE∧>(-(x1[0], 1), 0)=TRUE ⇒ COND_622_0_MAIN_LE(TRUE, -(x1[0], 1), x1[0]1)≥NonInfC∧COND_622_0_MAIN_LE(TRUE, -(x1[0], 1), x1[0]1)≥622_0_MAIN_LE(-(x1[0]1, 1), -(x1[0], 1))∧(U^Increasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (U^Increasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (U^Increasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (U^Increasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x1[0] ≥ 0∧x1[0]1 + [-1] ≥ 0∧[-1] + x1[0] ≥ 0 ⇒ (U^Increasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    ([1] + x1[0] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(15)    ([1] + x1[0] ≥ 0∧x1[0]1 ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

622_0_MAIN_LE(x0, x1) → COND_622_0_MAIN_LE(>(x1, 0), x0, x1)
- (x1[0] ≥ 0 ⇒ (U^Increasing(COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_11] = 0∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)
COND_622_0_MAIN_LE(TRUE, x0, x1) → 622_0_MAIN_LE(-(x1, 1), x0)
- ([1] + x1[0] ≥ 0∧x1[0]1 ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(622_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0]1 + [bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 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(622_0_MAIN_LE(x₁, x₂)) = [-1] + x₂ + x₁
POL(COND_622_0_MAIN_LE(x₁, x₂, x₃)) = [-1] + x₃ + x₂
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]

The following pairs are in P_>:

COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1])

The following pairs are in P_bound:

COND_622_0_MAIN_LE(TRUE, x0[1], x1[1]) → 622_0_MAIN_LE(-(x1[1], 1), x0[1])

The following pairs are in P_≥:

622_0_MAIN_LE(x0[0], x1[0]) → COND_622_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])

There are no usable rules.

(8) Obligation: