Termination proof of /tmp/tmpg2NG

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 250 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 90 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 110 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: LogBuiltIn

public class LogBuiltIn{
  public static int log(int x) {

    int res = 0;

    while (x > 1) {

      x = x/2;
      res++;

    }

    return res;

  }


  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    log(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:
LogBuiltIn.main([Ljava/lang/String;)V: Graph of 122 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: LogBuiltIn.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 11 rules for P and 0 rules for R.

P rules:
432_0_log_ConstantStackPush(EOS(STATIC_432), i79, i79) → 436_0_log_LE(EOS(STATIC_436), i79, i79, 1)
436_0_log_LE(EOS(STATIC_436), i89, i89, matching1) → 441_0_log_LE(EOS(STATIC_441), i89, i89, 1) | =(matching1, 1)
441_0_log_LE(EOS(STATIC_441), i89, i89, matching1) → 446_0_log_Load(EOS(STATIC_446), i89) | &&(>(i89, 1), =(matching1, 1))
446_0_log_Load(EOS(STATIC_446), i89) → 451_0_log_ConstantStackPush(EOS(STATIC_451), i89)
451_0_log_ConstantStackPush(EOS(STATIC_451), i89) → 456_0_log_IntArithmetic(EOS(STATIC_456), i89, 2)
456_0_log_IntArithmetic(EOS(STATIC_456), i89, matching1) → 461_0_log_Store(EOS(STATIC_461), /(i89, 2)) | &&(>(i89, 1), =(matching1, 2))
461_0_log_Store(EOS(STATIC_461), i91) → 464_0_log_Inc(EOS(STATIC_464), i91)
464_0_log_Inc(EOS(STATIC_464), i91) → 467_0_log_JMP(EOS(STATIC_467), i91)
467_0_log_JMP(EOS(STATIC_467), i91) → 490_0_log_Load(EOS(STATIC_490), i91)
490_0_log_Load(EOS(STATIC_490), i91) → 429_0_log_Load(EOS(STATIC_429), i91)
429_0_log_Load(EOS(STATIC_429), i79) → 432_0_log_ConstantStackPush(EOS(STATIC_432), i79, i79)
R rules:

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

P rules:
432_0_log_ConstantStackPush(EOS(STATIC_432), x0, x0) → 432_0_log_ConstantStackPush(EOS(STATIC_432), /(x0, 2), /(x0, 2)) | >(x0, 1)
R rules:

Filtered ground terms:

432_0_log_ConstantStackPush(x1, x2, x3) → 432_0_log_ConstantStackPush(x2, x3)
EOS(x1) → EOS
Cond_432_0_log_ConstantStackPush(x1, x2, x3, x4) → Cond_432_0_log_ConstantStackPush(x1, x3, x4)

Filtered duplicate args:

432_0_log_ConstantStackPush(x1, x2) → 432_0_log_ConstantStackPush(x2)
Cond_432_0_log_ConstantStackPush(x1, x2, x3) → Cond_432_0_log_ConstantStackPush(x1, x3)

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

P rules:
432_0_log_ConstantStackPush(x0) → 432_0_log_ConstantStackPush(/(x0, 2)) | >(x0, 1)
R rules:

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

P rules:
432_0_LOG_CONSTANTSTACKPUSH(x0) → COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0, 1), x0)
COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0) → 432_0_LOG_CONSTANTSTACKPUSH(/(x0, 2))
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): 432_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_432_0_LOG_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1]) → 432_0_LOG_CONSTANTSTACKPUSH(x0[1] / 2)

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

(1) -> (0), if (x0[1] / 2 →^* 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@3513126e 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 432_0_LOG_CONSTANTSTACKPUSH(x0) → COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:

We consider the chain 432_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1]) → 432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2)) which results in the following constraint:
(1)    (>(x0[0], 1)=TRUE∧x0[0]=x0[1] ⇒ 432_0_LOG_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧432_0_LOG_CONSTANTSTACKPUSH(x0[0])≥COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x0[0], 1)=TRUE ⇒ 432_0_LOG_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧432_0_LOG_CONSTANTSTACKPUSH(x0[0])≥COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(U^Increasing(COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)

For Pair COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0) → 432_0_LOG_CONSTANTSTACKPUSH(/(x0, 2)) the following chains were created:

We consider the chain 432_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1]) → 432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2)), 432_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]) which results in the following constraint:
(7)    (>(x0[0], 1)=TRUE∧x0[0]=x0[1]∧/(x0[1], 2)=x0[0]1 ⇒ COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1])≥432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))∧(U^Increasing(432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥))

We simplified constraint (7) using rules (III), (IV) which results in the following new constraint:
(8)    (>(x0[0], 1)=TRUE ⇒ COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[0])≥NonInfC∧COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[0])≥432_0_LOG_CONSTANTSTACKPUSH(/(x0[0], 2))∧(U^Increasing(432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (x0[0] + [-2] ≥ 0∧[2]x0[0] ≥ 0 ⇒ (U^Increasing(432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12)    (x0[0] ≥ 0∧[4] + [2]x0[0] ≥ 0 ⇒ (U^Increasing(432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (12) using rule (IDP_POLY_GCD) which results in the following new constraint:
(13)    (x0[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (U^Increasing(432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

432_0_LOG_CONSTANTSTACKPUSH(x0) → COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0, 1), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)
COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0) → 432_0_LOG_CONSTANTSTACKPUSH(/(x0, 2))
- (x0[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (U^Increasing(432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 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(432_0_LOG_CONSTANTSTACKPUSH(x₁)) = [-1] + x₁
POL(COND_432_0_LOG_CONSTANTSTACKPUSH(x₁, x₂)) = [-1] + x₂
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(2) = [2]

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(Term^CSAR-Mode @ Context)

POL(/(x₁, 2)¹ @ {432_0_LOG_CONSTANTSTACKPUSH_1/0}) = max{x₁, [-1]x₁} + [-1]

The following pairs are in P_>:

COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1]) → 432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))

The following pairs are in P_bound:

432_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])
COND_432_0_LOG_CONSTANTSTACKPUSH(TRUE, x0[1]) → 432_0_LOG_CONSTANTSTACKPUSH(/(x0[1], 2))

The following pairs are in P_≥:

432_0_LOG_CONSTANTSTACKPUSH(x0[0]) → COND_432_0_LOG_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

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

/¹ →

(8) Obligation: