Termination proof of /tmp/tmpKpUh1P/LogRecursive.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 140 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 140 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 220 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:
373_0_log_Load(EOS(STATIC_373), i66, i67, i66) → 375_0_log_LT(EOS(STATIC_375), i66, i67, i66, i67)
375_0_log_LT(EOS(STATIC_375), i66, i67, i66, i67) → 378_0_log_LT(EOS(STATIC_378), i66, i67, i66, i67)
378_0_log_LT(EOS(STATIC_378), i66, i67, i66, i67) → 382_0_log_Load(EOS(STATIC_382), i66, i67) | >=(i66, i67)
382_0_log_Load(EOS(STATIC_382), i66, i67) → 385_0_log_ConstantStackPush(EOS(STATIC_385), i66, i67, i67)
385_0_log_ConstantStackPush(EOS(STATIC_385), i66, i67, i67) → 389_0_log_LE(EOS(STATIC_389), i66, i67, i67, 1)
389_0_log_LE(EOS(STATIC_389), i66, i78, i78, matching1) → 396_0_log_LE(EOS(STATIC_396), i66, i78, i78, 1) | =(matching1, 1)
396_0_log_LE(EOS(STATIC_396), i66, i78, i78, matching1) → 407_0_log_ConstantStackPush(EOS(STATIC_407), i66, i78) | &&(>(i78, 1), =(matching1, 1))
407_0_log_ConstantStackPush(EOS(STATIC_407), i66, i78) → 412_0_log_Load(EOS(STATIC_412), i66, i78, 1)
412_0_log_Load(EOS(STATIC_412), i66, i78, matching1) → 417_0_log_Load(EOS(STATIC_417), i78, 1, i66) | =(matching1, 1)
417_0_log_Load(EOS(STATIC_417), i78, matching1, i66) → 437_0_log_IntArithmetic(EOS(STATIC_437), i78, 1, i66, i78) | =(matching1, 1)
437_0_log_IntArithmetic(EOS(STATIC_437), i78, matching1, i66, i78) → 449_0_log_Load(EOS(STATIC_449), i78, 1, /(i66, i78)) | &&(>(i78, 1), =(matching1, 1))
449_0_log_Load(EOS(STATIC_449), i78, matching1, i89) → 452_0_log_InvokeMethod(EOS(STATIC_452), 1, i89, i78) | =(matching1, 1)
452_0_log_InvokeMethod(EOS(STATIC_452), matching1, i89, i78) → 453_1_log_InvokeMethod(453_0_log_Load(EOS(STATIC_453), i89, i78), 1, i89, i78) | =(matching1, 1)
453_0_log_Load(EOS(STATIC_453), i89, i78) → 456_0_log_Load(EOS(STATIC_456), i89, i78)
456_0_log_Load(EOS(STATIC_456), i89, i78) → 372_0_log_Load(EOS(STATIC_372), i89, i78)
372_0_log_Load(EOS(STATIC_372), i66, i67) → 373_0_log_Load(EOS(STATIC_373), i66, i67, i66)
R rules:
375_0_log_LT(EOS(STATIC_375), i66, i67, i66, i67) → 377_0_log_LT(EOS(STATIC_377), i66, i67, i66, i67)
377_0_log_LT(EOS(STATIC_377), i66, i67, i66, i67) → 380_0_log_ConstantStackPush(EOS(STATIC_380)) | <(i66, i67)
380_0_log_ConstantStackPush(EOS(STATIC_380)) → 383_0_log_Return(EOS(STATIC_383))
389_0_log_LE(EOS(STATIC_389), i66, i77, i77, matching1) → 395_0_log_LE(EOS(STATIC_395), i66, i77, i77, 1) | =(matching1, 1)
395_0_log_LE(EOS(STATIC_395), i66, i77, i77, matching1) → 405_0_log_ConstantStackPush(EOS(STATIC_405)) | &&(<=(i77, 1), =(matching1, 1))
405_0_log_ConstantStackPush(EOS(STATIC_405)) → 410_0_log_Return(EOS(STATIC_410))
453_1_log_InvokeMethod(383_0_log_Return(EOS(STATIC_383)), matching1, i96, i97) → 467_0_log_Return(EOS(STATIC_467), 1, i96, i97) | =(matching1, 1)
453_1_log_InvokeMethod(472_0_log_Return(EOS(STATIC_472), matching1), matching2, i110, i111) → 494_0_log_Return(EOS(STATIC_494), 1, i110, i111, 1) | &&(=(matching1, 1), =(matching2, 1))
453_1_log_InvokeMethod(715_0_log_Return(EOS(STATIC_715), i298), matching1, i308, i309) → 736_0_log_Return(EOS(STATIC_736), 1, i308, i309, i298) | =(matching1, 1)
467_0_log_Return(EOS(STATIC_467), matching1, i96, i97) → 470_0_log_IntArithmetic(EOS(STATIC_470), 1) | =(matching1, 1)
470_0_log_IntArithmetic(EOS(STATIC_470), matching1) → 472_0_log_Return(EOS(STATIC_472), 1) | =(matching1, 1)
494_0_log_Return(EOS(STATIC_494), matching1, i110, i111, matching2) → 543_0_log_Return(EOS(STATIC_543), 1, i110, i111, 1) | &&(=(matching1, 1), =(matching2, 1))
543_0_log_Return(EOS(STATIC_543), matching1, i141, i137, i140) → 595_0_log_Return(EOS(STATIC_595), 1, i141, i137, i140) | =(matching1, 1)
595_0_log_Return(EOS(STATIC_595), matching1, i183, i179, i182) → 649_0_log_Return(EOS(STATIC_649), 1, i183, i179, i182) | =(matching1, 1)
649_0_log_Return(EOS(STATIC_649), matching1, i231, i227, i230) → 700_0_log_Return(EOS(STATIC_700), 1, i231, i227, i230) | =(matching1, 1)
700_0_log_Return(EOS(STATIC_700), matching1, i278, i274, i277) → 707_0_log_IntArithmetic(EOS(STATIC_707), 1, i277) | =(matching1, 1)
707_0_log_IntArithmetic(EOS(STATIC_707), matching1, i277) → 715_0_log_Return(EOS(STATIC_715), +(1, i277)) | &&(>(i277, 0), =(matching1, 1))
736_0_log_Return(EOS(STATIC_736), matching1, i308, i309, i298) → 700_0_log_Return(EOS(STATIC_700), 1, i308, i309, i298) | =(matching1, 1)

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

P rules:
373_0_log_Load(EOS(STATIC_373), x0, x1, x0) → 453_1_log_InvokeMethod(373_0_log_Load(EOS(STATIC_373), /(x0, x1), x1, /(x0, x1)), 1, /(x0, x1), x1) | &&(>(x1, 1), <=(x1, x0))
R rules:
453_1_log_InvokeMethod(383_0_log_Return(EOS(STATIC_383)), 1, x1, x2) → 472_0_log_Return(EOS(STATIC_472), 1)
453_1_log_InvokeMethod(715_0_log_Return(EOS(STATIC_715), x0), 1, x2, x3) → 715_0_log_Return(EOS(STATIC_715), +(1, x0)) | >(x0, 0)
453_1_log_InvokeMethod(472_0_log_Return(EOS(STATIC_472), 1), 1, x2, x3) → 715_0_log_Return(EOS(STATIC_715), 2)

Filtered ground terms:

453_1_log_InvokeMethod(x1, x2, x3, x4) → 453_1_log_InvokeMethod(x1, x3, x4)
373_0_log_Load(x1, x2, x3, x4) → 373_0_log_Load(x2, x3, x4)
Cond_373_0_log_Load(x1, x2, x3, x4, x5) → Cond_373_0_log_Load(x1, x3, x4, x5)
715_0_log_Return(x1, x2) → 715_0_log_Return(x2)
472_0_log_Return(x1, x2) → 472_0_log_Return
Cond_453_1_log_InvokeMethod(x1, x2, x3, x4, x5) → Cond_453_1_log_InvokeMethod(x1, x2, x4, x5)
383_0_log_Return(x1) → 383_0_log_Return

Filtered duplicate args:

373_0_log_Load(x1, x2, x3) → 373_0_log_Load(x2, x3)
Cond_373_0_log_Load(x1, x2, x3, x4) → Cond_373_0_log_Load(x1, x3, x4)

Filtered unneeded arguments:

453_1_log_InvokeMethod(x1, x2, x3) → 453_1_log_InvokeMethod(x1)
Cond_453_1_log_InvokeMethod(x1, x2, x3, x4) → Cond_453_1_log_InvokeMethod(x1, x2)

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

P rules:
373_0_log_Load(x1, x0) → 453_1_log_InvokeMethod(373_0_log_Load(x1, /(x0, x1))) | &&(>(x1, 1), <=(x1, x0))
R rules:
453_1_log_InvokeMethod(383_0_log_Return) → 472_0_log_Return
453_1_log_InvokeMethod(715_0_log_Return(x0)) → 715_0_log_Return(+(1, x0)) | >(x0, 0)
453_1_log_InvokeMethod(472_0_log_Return) → 715_0_log_Return(2)

Performed bisimulation on rules. Used the following equivalence classes: {[383_0_log_Return, 472_0_log_Return]=383_0_log_Return}

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

P rules:
373_0_LOG_LOAD(x1, x0) → COND_373_0_LOG_LOAD(&&(>(x1, 1), <=(x1, x0)), x1, x0)
COND_373_0_LOG_LOAD(TRUE, x1, x0) → 373_0_LOG_LOAD(x1, /(x0, x1))
R rules:
453_1_log_InvokeMethod(383_0_log_Return) → 383_0_log_Return
453_1_log_InvokeMethod(715_0_log_Return(x0)) → Cond_453_1_log_InvokeMethod(>(x0, 0), 715_0_log_Return(x0))
Cond_453_1_log_InvokeMethod(TRUE, 715_0_log_Return(x0)) → 715_0_log_Return(+(1, x0))
453_1_log_InvokeMethod(383_0_log_Return) → 715_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:
453_1_log_InvokeMethod(383_0_log_Return) → 383_0_log_Return
453_1_log_InvokeMethod(715_0_log_Return(x0)) → Cond_453_1_log_InvokeMethod(x0 > 0, 715_0_log_Return(x0))
Cond_453_1_log_InvokeMethod(TRUE, 715_0_log_Return(x0)) → 715_0_log_Return(1 + x0)
453_1_log_InvokeMethod(383_0_log_Return) → 715_0_log_Return(2)

The integer pair graph contains the following rules and edges:
(0): 373_0_LOG_LOAD(x1[0], x0[0]) → COND_373_0_LOG_LOAD(x1[0] > 1 && x1[0] <= x0[0], x1[0], x0[0])
(1): COND_373_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 373_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:
453_1_log_InvokeMethod(383_0_log_Return)
453_1_log_InvokeMethod(715_0_log_Return(x0))
Cond_453_1_log_InvokeMethod(TRUE, 715_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@7186fe17 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 373_0_LOG_LOAD(x1, x0) → COND_373_0_LOG_LOAD(&&(>(x1, 1), <=(x1, x0)), x1, x0) the following chains were created:

We consider the chain 373_0_LOG_LOAD(x1[0], x0[0]) → COND_373_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0]), COND_373_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 373_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] ⇒ 373_0_LOG_LOAD(x1[0], x0[0])≥NonInfC∧373_0_LOG_LOAD(x1[0], x0[0])≥COND_373_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])∧(U^Increasing(COND_373_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 ⇒ 373_0_LOG_LOAD(x1[0], x0[0])≥NonInfC∧373_0_LOG_LOAD(x1[0], x0[0])≥COND_373_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])∧(U^Increasing(COND_373_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_373_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_373_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_373_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_373_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_373_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_373_0_LOG_LOAD(TRUE, x1, x0) → 373_0_LOG_LOAD(x1, /(x0, x1)) the following chains were created:

We consider the chain 373_0_LOG_LOAD(x1[0], x0[0]) → COND_373_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0]), COND_373_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 373_0_LOG_LOAD(x1[1], /(x0[1], x1[1])), 373_0_LOG_LOAD(x1[0], x0[0]) → COND_373_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_373_0_LOG_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_373_0_LOG_LOAD(TRUE, x1[1], x0[1])≥373_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))∧(U^Increasing(373_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_373_0_LOG_LOAD(TRUE, x1[0], x0[0])≥NonInfC∧COND_373_0_LOG_LOAD(TRUE, x1[0], x0[0])≥373_0_LOG_LOAD(x1[0], /(x0[0], x1[0]))∧(U^Increasing(373_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(373_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(373_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(373_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(373_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(373_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(373_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.

373_0_LOG_LOAD(x1, x0) → COND_373_0_LOG_LOAD(&&(>(x1, 1), <=(x1, x0)), x1, x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_373_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_373_0_LOG_LOAD(TRUE, x1, x0) → 373_0_LOG_LOAD(x1, /(x0, x1))
- (x1[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (U^Increasing(373_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) = [1]
POL(FALSE) = [2]
POL(453_1_log_InvokeMethod(x₁)) = [-1] + [-1]x₁
POL(383_0_log_Return) = [-1]
POL(715_0_log_Return(x₁)) = [-1] + [-1]x₁
POL(Cond_453_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(373_0_LOG_LOAD(x₁, x₂)) = [-1] + x₂ + [-1]x₁
POL(COND_373_0_LOG_LOAD(x₁, x₂, x₃)) = x₃ + [-1]x₂ + [-1]x₁
POL(&&(x₁, x₂)) = [1]
POL(<=(x₁, x₂)) = [-1]

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

POL(/(x₁, x1[0])¹ @ {373_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_373_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 373_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))

The following pairs are in P_bound:

373_0_LOG_LOAD(x1[0], x0[0]) → COND_373_0_LOG_LOAD(&&(>(x1[0], 1), <=(x1[0], x0[0])), x1[0], x0[0])
COND_373_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 373_0_LOG_LOAD(x1[1], /(x0[1], x1[1]))

The following pairs are in P_≥:

373_0_LOG_LOAD(x1[0], x0[0]) → COND_373_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:

TRUE¹ → &&(TRUE, TRUE)¹
FALSE¹ → &&(TRUE, FALSE)¹
FALSE¹ → &&(FALSE, TRUE)¹
FALSE¹ → &&(FALSE, FALSE)¹
/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:
453_1_log_InvokeMethod(383_0_log_Return) → 383_0_log_Return
453_1_log_InvokeMethod(715_0_log_Return(x0)) → Cond_453_1_log_InvokeMethod(x0 > 0, 715_0_log_Return(x0))
Cond_453_1_log_InvokeMethod(TRUE, 715_0_log_Return(x0)) → 715_0_log_Return(1 + x0)
453_1_log_InvokeMethod(383_0_log_Return) → 715_0_log_Return(2)

The integer pair graph contains the following rules and edges:
(0): 373_0_LOG_LOAD(x1[0], x0[0]) → COND_373_0_LOG_LOAD(x1[0] > 1 && x1[0] <= x0[0], x1[0], x0[0])

The set Q consists of the following terms:
453_1_log_InvokeMethod(383_0_log_Return)
453_1_log_InvokeMethod(715_0_log_Return(x0))
Cond_453_1_log_InvokeMethod(TRUE, 715_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