Termination proof of /tmp/tmpIQrvMO/LogRecursive.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 230 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 100 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 230 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_14 (Sun Microsystems Inc.) Main-Class: LogRecursive

public class LogRecursive {
  public static void main(String[] args) {
    Random.args = args;
    log(Random.random(), Random.random());
  }

  public static int log(int x, int y) {
    if (x >= y && y > 1) {
      return 1 + log(x/y, y);
    }
    return 0;
  }
}


public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    if (args.length <= index) {
      return 0;
    }
    String string = args[index];
    index++;
    if (string == null) {
      return 0;
    }
    return string.length();
  }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
LogRecursive.main([Ljava/lang/String;)V: Graph of 137 nodes with 0 SCCs.

LogRecursive.log(II)I: Graph of 37 nodes with 0 SCCs.

(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: LogRecursive.log(II)I
SCC calls the following helper methods: LogRecursive.log(II)I
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 16 rules for P and 18 rules for R.

P rules:
409_0_log_Load(EOS(STATIC_409), i68, i69, i68) → 412_0_log_LT(EOS(STATIC_412), i68, i69, i68, i69)
412_0_log_LT(EOS(STATIC_412), i68, i69, i68, i69) → 415_0_log_LT(EOS(STATIC_415), i68, i69, i68, i69)
415_0_log_LT(EOS(STATIC_415), i68, i69, i68, i69) → 419_0_log_Load(EOS(STATIC_419), i68, i69) | >=(i68, i69)
419_0_log_Load(EOS(STATIC_419), i68, i69) → 422_0_log_ConstantStackPush(EOS(STATIC_422), i68, i69, i69)
422_0_log_ConstantStackPush(EOS(STATIC_422), i68, i69, i69) → 427_0_log_LE(EOS(STATIC_427), i68, i69, i69, 1)
427_0_log_LE(EOS(STATIC_427), i68, i80, i80, matching1) → 447_0_log_LE(EOS(STATIC_447), i68, i80, i80, 1) | =(matching1, 1)
447_0_log_LE(EOS(STATIC_447), i68, i80, i80, matching1) → 460_0_log_ConstantStackPush(EOS(STATIC_460), i68, i80) | &&(>(i80, 1), =(matching1, 1))
460_0_log_ConstantStackPush(EOS(STATIC_460), i68, i80) → 467_0_log_Load(EOS(STATIC_467), i68, i80, 1)
467_0_log_Load(EOS(STATIC_467), i68, i80, matching1) → 474_0_log_Load(EOS(STATIC_474), i80, 1, i68) | =(matching1, 1)
474_0_log_Load(EOS(STATIC_474), i80, matching1, i68) → 493_0_log_IntArithmetic(EOS(STATIC_493), i80, 1, i68, i80) | =(matching1, 1)
493_0_log_IntArithmetic(EOS(STATIC_493), i80, matching1, i68, i80) → 505_0_log_Load(EOS(STATIC_505), i80, 1, /(i68, i80)) | &&(>(i80, 1), =(matching1, 1))
505_0_log_Load(EOS(STATIC_505), i80, matching1, i92) → 507_0_log_InvokeMethod(EOS(STATIC_507), 1, i92, i80) | =(matching1, 1)
507_0_log_InvokeMethod(EOS(STATIC_507), matching1, i92, i80) → 509_1_log_InvokeMethod(509_0_log_Load(EOS(STATIC_509), i92, i80), 1, i92, i80) | =(matching1, 1)
509_0_log_Load(EOS(STATIC_509), i92, i80) → 511_0_log_Load(EOS(STATIC_511), i92, i80)
511_0_log_Load(EOS(STATIC_511), i92, i80) → 407_0_log_Load(EOS(STATIC_407), i92, i80)
407_0_log_Load(EOS(STATIC_407), i68, i69) → 409_0_log_Load(EOS(STATIC_409), i68, i69, i68)
R rules:
412_0_log_LT(EOS(STATIC_412), i68, i69, i68, i69) → 414_0_log_LT(EOS(STATIC_414), i68, i69, i68, i69)
414_0_log_LT(EOS(STATIC_414), i68, i69, i68, i69) → 417_0_log_ConstantStackPush(EOS(STATIC_417)) | <(i68, i69)
417_0_log_ConstantStackPush(EOS(STATIC_417)) → 421_0_log_Return(EOS(STATIC_421))
427_0_log_LE(EOS(STATIC_427), i68, i79, i79, matching1) → 446_0_log_LE(EOS(STATIC_446), i68, i79, i79, 1) | =(matching1, 1)
446_0_log_LE(EOS(STATIC_446), i68, i79, i79, matching1) → 458_0_log_ConstantStackPush(EOS(STATIC_458)) | &&(<=(i79, 1), =(matching1, 1))
458_0_log_ConstantStackPush(EOS(STATIC_458)) → 465_0_log_Return(EOS(STATIC_465))
509_1_log_InvokeMethod(421_0_log_Return(EOS(STATIC_421)), matching1, i98, i99) → 523_0_log_Return(EOS(STATIC_523), 1, i98, i99) | =(matching1, 1)
509_1_log_InvokeMethod(528_0_log_Return(EOS(STATIC_528), matching1), matching2, i112, i113) → 549_0_log_Return(EOS(STATIC_549), 1, i112, i113, 1) | &&(=(matching1, 1), =(matching2, 1))
509_1_log_InvokeMethod(742_0_log_Return(EOS(STATIC_742), i299), matching1, i308, i309) → 760_0_log_Return(EOS(STATIC_760), 1, i308, i309, i299) | =(matching1, 1)
523_0_log_Return(EOS(STATIC_523), matching1, i98, i99) → 526_0_log_IntArithmetic(EOS(STATIC_526), 1) | =(matching1, 1)
526_0_log_IntArithmetic(EOS(STATIC_526), matching1) → 528_0_log_Return(EOS(STATIC_528), 1) | =(matching1, 1)
549_0_log_Return(EOS(STATIC_549), matching1, i112, i113, matching2) → 594_0_log_Return(EOS(STATIC_594), 1, i112, i113, 1) | &&(=(matching1, 1), =(matching2, 1))
594_0_log_Return(EOS(STATIC_594), matching1, i143, i139, i142) → 645_0_log_Return(EOS(STATIC_645), 1, i143, i139, i142) | =(matching1, 1)
645_0_log_Return(EOS(STATIC_645), matching1, i185, i181, i184) → 692_0_log_Return(EOS(STATIC_692), 1, i185, i181, i184) | =(matching1, 1)
692_0_log_Return(EOS(STATIC_692), matching1, i232, i228, i231) → 731_0_log_Return(EOS(STATIC_731), 1, i232, i228, i231) | =(matching1, 1)
731_0_log_Return(EOS(STATIC_731), matching1, i279, i275, i278) → 736_0_log_IntArithmetic(EOS(STATIC_736), 1, i278) | =(matching1, 1)
736_0_log_IntArithmetic(EOS(STATIC_736), matching1, i278) → 742_0_log_Return(EOS(STATIC_742), +(1, i278)) | &&(>(i278, 0), =(matching1, 1))
760_0_log_Return(EOS(STATIC_760), matching1, i308, i309, i299) → 731_0_log_Return(EOS(STATIC_731), 1, i308, i309, i299) | =(matching1, 1)

Combined rules. Obtained 1 conditional rules for P and 3 conditional rules for R.

P rules:
409_0_log_Load(EOS(STATIC_409), x0, x1, x0) → 509_1_log_InvokeMethod(409_0_log_Load(EOS(STATIC_409), /(x0, x1), x1, /(x0, x1)), 1, /(x0, x1), x1) | &&(>(x1, 1), <=(x1, x0))
R rules:
509_1_log_InvokeMethod(421_0_log_Return(EOS(STATIC_421)), 1, x1, x2) → 528_0_log_Return(EOS(STATIC_528), 1)
509_1_log_InvokeMethod(742_0_log_Return(EOS(STATIC_742), x0), 1, x2, x3) → 742_0_log_Return(EOS(STATIC_742), +(1, x0)) | >(x0, 0)
509_1_log_InvokeMethod(528_0_log_Return(EOS(STATIC_528), 1), 1, x2, x3) → 742_0_log_Return(EOS(STATIC_742), 2)

Filtered ground terms:

509_1_log_InvokeMethod(x1, x2, x3, x4) → 509_1_log_InvokeMethod(x1, x3, x4)
409_0_log_Load(x1, x2, x3, x4) → 409_0_log_Load(x2, x3, x4)
Cond_409_0_log_Load(x1, x2, x3, x4, x5) → Cond_409_0_log_Load(x1, x3, x4, x5)
742_0_log_Return(x1, x2) → 742_0_log_Return(x2)
528_0_log_Return(x1, x2) → 528_0_log_Return
Cond_509_1_log_InvokeMethod(x1, x2, x3, x4, x5) → Cond_509_1_log_InvokeMethod(x1, x2, x4, x5)
421_0_log_Return(x1) → 421_0_log_Return

Filtered duplicate args:

409_0_log_Load(x1, x2, x3) → 409_0_log_Load(x2, x3)
Cond_409_0_log_Load(x1, x2, x3, x4) → Cond_409_0_log_Load(x1, x3, x4)

Filtered unneeded arguments:

509_1_log_InvokeMethod(x1, x2, x3) → 509_1_log_InvokeMethod(x1)
Cond_509_1_log_InvokeMethod(x1, x2, x3, x4) → Cond_509_1_log_InvokeMethod(x1, x2)

Combined rules. Obtained 1 conditional rules for P and 3 conditional rules for R.

P rules:
409_0_log_Load(x1, x0) → 509_1_log_InvokeMethod(409_0_log_Load(x1, /(x0, x1))) | &&(>(x1, 1), <=(x1, x0))
R rules:
509_1_log_InvokeMethod(421_0_log_Return) → 528_0_log_Return
509_1_log_InvokeMethod(742_0_log_Return(x0)) → 742_0_log_Return(+(1, x0)) | >(x0, 0)
509_1_log_InvokeMethod(528_0_log_Return) → 742_0_log_Return(2)

Performed bisimulation on rules. Used the following equivalence classes: {[421_0_log_Return, 528_0_log_Return]=421_0_log_Return}

Finished conversion. Obtained 2 rules for P and 4 rules for R. System has predefined symbols.

P rules:
409_0_LOG_LOAD(x1, x0) → COND_409_0_LOG_LOAD(&&(>(x1, 1), <=(x1, x0)), x1, x0)
COND_409_0_LOG_LOAD(TRUE, x1, x0) → 409_0_LOG_LOAD(x1, /(x0, x1))
R rules:
509_1_log_InvokeMethod(421_0_log_Return) → 421_0_log_Return
509_1_log_InvokeMethod(742_0_log_Return(x0)) → Cond_509_1_log_InvokeMethod(>(x0, 0), 742_0_log_Return(x0))
Cond_509_1_log_InvokeMethod(TRUE, 742_0_log_Return(x0)) → 742_0_log_Return(+(1, x0))
509_1_log_InvokeMethod(421_0_log_Return) → 742_0_log_Return(2)

(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, Boolean

The ITRS R consists of the following rules:
509_1_log_InvokeMethod(421_0_log_Return) → 421_0_log_Return
509_1_log_InvokeMethod(742_0_log_Return(x0)) → Cond_509_1_log_InvokeMethod(x0 > 0, 742_0_log_Return(x0))
Cond_509_1_log_InvokeMethod(TRUE, 742_0_log_Return(x0)) → 742_0_log_Return(1 + x0)
509_1_log_InvokeMethod(421_0_log_Return) → 742_0_log_Return(2)

The integer pair graph contains the following rules and edges:
(0): 409_0_LOG_LOAD(x1[0], x0[0]) → COND_409_0_LOG_LOAD(x1[0] > 1 && x1[0] <= x0[0], x1[0], x0[0])
(1): COND_409_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 409_0_LOG_LOAD(x1[1], x0[1] / x1[1])

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

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

The set Q consists of the following terms:
509_1_log_InvokeMethod(421_0_log_Return)
509_1_log_InvokeMethod(742_0_log_Return(x0))
Cond_509_1_log_InvokeMethod(TRUE, 742_0_log_Return(x0))

(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@57f428b 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 409_0_LOG_LOAD(x1, x0) → COND_409_0_LOG_LOAD(&&(>(x1, 1), <=(x1, x0)), x1, x0) the following chains were created:

We consider the chain 409_0_LOG_LOAD(x1[0], x0[0]) → COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0]), COND_409_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 409_0_LOG_LOAD(x1[1], /(x0[1], x1[1])) which results in the following constraint:
(1)    (&&(>(x1[0], 1), <=(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 409_0_LOG_LOAD(x1[0], x0[0])≥NonInfC∧409_0_LOG_LOAD(x1[0], x0[0])≥COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])∧(U^Increasing(COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], 1)=TRUE∧<=(x1[0], x0[0])=TRUE ⇒ 409_0_LOG_LOAD(x1[0], x0[0])≥NonInfC∧409_0_LOG_LOAD(x1[0], x0[0])≥COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])∧(U^Increasing(COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] + [(-1)bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] + [(-1)bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] + [(-1)bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-3)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] + [(-1)bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] ≥ 0∧[(-1)bso_22] ≥ 0)

For Pair COND_409_0_LOG_LOAD(TRUE, x1, x0) → 409_0_LOG_LOAD(x1, /(x0, x1)) the following chains were created:

We consider the chain 409_0_LOG_LOAD(x1[0], x0[0]) → COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0]), COND_409_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 409_0_LOG_LOAD(x1[1], /(x0[1], x1[1])), 409_0_LOG_LOAD(x1[0], x0[0]) → COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0]) which results in the following constraint:
(8)    (&&(>(x1[0], 1), <=(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1]∧x1[1]=x1[0]1∧/(x0[1], x1[1])=x0[0]1 ⇒ COND_409_0_LOG_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_409_0_LOG_LOAD(TRUE, x1[1], x0[1])≥409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))∧(U^Increasing(409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(x1[0], 1)=TRUE∧<=(x1[0], x0[0])=TRUE ⇒ COND_409_0_LOG_LOAD(TRUE, x1[0], x0[0])≥NonInfC∧COND_409_0_LOG_LOAD(TRUE, x1[0], x0[0])≥409_0_LOG_LOAD(x1[0], /(x0[0], x1[0]))∧(U^Increasing(409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] + [(-1)bni_23]x1[0] ≥ 0∧[(-1)bso_27] + x0[0] + [-1]max{x0[0], [-1]x0[0]} + min{max{x1[0], [-1]x1[0]} + [-1], max{x0[0], [-1]x0[0]}} ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] + [(-1)bni_23]x1[0] ≥ 0∧[(-1)bso_27] + x0[0] + [-1]max{x0[0], [-1]x0[0]} + min{max{x1[0], [-1]x1[0]} + [-1], max{x0[0], [-1]x0[0]}} ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0∧[2]x0[0] ≥ 0∧[2]x1[0] ≥ 0 ⇒ (U^Increasing(409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] + [(-1)bni_23]x1[0] ≥ 0∧[-1 + (-1)bso_27] + x1[0] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x1[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0∧[2]x0[0] ≥ 0∧[4] + [2]x1[0] ≥ 0 ⇒ (U^Increasing(409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-3)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] + [(-1)bni_23]x1[0] ≥ 0∧[1 + (-1)bso_27] + x1[0] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[4] + [2]x1[0] + [2]x0[0] ≥ 0∧[4] + [2]x1[0] ≥ 0 ⇒ (U^Increasing(409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[1 + (-1)bso_27] + x1[0] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_GCD) which results in the following new constraint:
(15)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (U^Increasing(409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[1 + (-1)bso_27] + x1[0] ≥ 0)

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

409_0_LOG_LOAD(x1, x0) → COND_409_0_LOG_LOAD(&&(>(x1, 1), <=(x1, x0)), x1, x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] ≥ 0∧[(-1)bso_22] ≥ 0)
COND_409_0_LOG_LOAD(TRUE, x1, x0) → 409_0_LOG_LOAD(x1, /(x0, x1))
- (x1[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (U^Increasing(409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[1 + (-1)bso_27] + x1[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) = 0
POL(FALSE) = 0
POL(509_1_log_InvokeMethod(x₁)) = [-1] + [-1]x₁
POL(421_0_log_Return) = [-1]
POL(742_0_log_Return(x₁)) = [-1] + [-1]x₁
POL(Cond_509_1_log_InvokeMethod(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(2) = [2]
POL(409_0_LOG_LOAD(x₁, x₂)) = [-1] + x₂ + [-1]x₁
POL(COND_409_0_LOG_LOAD(x₁, x₂, x₃)) = [-1] + x₃ + [-1]x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(<=(x₁, x₂)) = [-1]

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

POL(/(x₁, x1[0])¹ @ {409_0_LOG_LOAD_2/1}) = max{x₁, [-1]x₁} + [-1]min{max{x₂, [-1]x₂} + [-1], max{x₁, [-1]x₁}}

The following pairs are in P_>:

COND_409_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))

The following pairs are in P_bound:

409_0_LOG_LOAD(x1[0], x0[0]) → COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])
COND_409_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 409_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))

The following pairs are in P_≥:

409_0_LOG_LOAD(x1[0], x0[0]) → COND_409_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])

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

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

(8) 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, Boolean

The ITRS R consists of the following rules:
509_1_log_InvokeMethod(421_0_log_Return) → 421_0_log_Return
509_1_log_InvokeMethod(742_0_log_Return(x0)) → Cond_509_1_log_InvokeMethod(x0 > 0, 742_0_log_Return(x0))
Cond_509_1_log_InvokeMethod(TRUE, 742_0_log_Return(x0)) → 742_0_log_Return(1 + x0)
509_1_log_InvokeMethod(421_0_log_Return) → 742_0_log_Return(2)

The integer pair graph contains the following rules and edges:
(0): 409_0_LOG_LOAD(x1[0], x0[0]) → COND_409_0_LOG_LOAD(x1[0] > 1 && x1[0] <= x0[0], x1[0], x0[0])

The set Q consists of the following terms:
509_1_log_InvokeMethod(421_0_log_Return)
509_1_log_InvokeMethod(742_0_log_Return(x0))
Cond_509_1_log_InvokeMethod(TRUE, 742_0_log_Return(x0))

(9) 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) Obligation:

(9) IDependencyGraphProof (EQUIVALENT transformation)

(10) TRUE