Termination proof of /tmp/tmpzPW

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 120 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 200 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 150 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_22 (Sun Microsystems Inc.) Main-Class: EqUserDefRec

public class EqUserDefRec {
	public static void main(String[] args) {
		int x = args[0].length();
		int y = args[1].length();
		eq(x, y);
	}

	public static boolean eq(int x, int y) {
		if (x > 0 && y > 0) {
			return eq(x-1, y-1);
		} else {
			return (x == 0 && y == 0);
		}
	}
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
EqUserDefRec.main([Ljava/lang/String;)V: Graph of 153 nodes with 0 SCCs.

EqUserDefRec.eq(II)Z: Graph of 45 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: EqUserDefRec.eq(II)Z
SCC calls the following helper methods: EqUserDefRec.eq(II)Z
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 15 rules for P and 30 rules for R.

P rules:
263_0_eq_LE(EOS(STATIC_263), i43, i37, i43) → 272_0_eq_LE(EOS(STATIC_272), i43, i37, i43)
272_0_eq_LE(EOS(STATIC_272), i43, i37, i43) → 281_0_eq_Load(EOS(STATIC_281), i43, i37) | >(i43, 0)
281_0_eq_Load(EOS(STATIC_281), i43, i37) → 292_0_eq_LE(EOS(STATIC_292), i43, i37, i37)
292_0_eq_LE(EOS(STATIC_292), i43, i46, i46) → 307_0_eq_LE(EOS(STATIC_307), i43, i46, i46)
307_0_eq_LE(EOS(STATIC_307), i43, i46, i46) → 321_0_eq_Load(EOS(STATIC_321), i43, i46) | >(i46, 0)
321_0_eq_Load(EOS(STATIC_321), i43, i46) → 336_0_eq_ConstantStackPush(EOS(STATIC_336), i43, i46, i43)
336_0_eq_ConstantStackPush(EOS(STATIC_336), i43, i46, i43) → 351_0_eq_IntArithmetic(EOS(STATIC_351), i43, i46, i43, 1)
351_0_eq_IntArithmetic(EOS(STATIC_351), i43, i46, i43, matching1) → 365_0_eq_Load(EOS(STATIC_365), i43, i46, -(i43, 1)) | &&(>(i43, 0), =(matching1, 1))
365_0_eq_Load(EOS(STATIC_365), i43, i46, i53) → 380_0_eq_ConstantStackPush(EOS(STATIC_380), i43, i46, i53, i46)
380_0_eq_ConstantStackPush(EOS(STATIC_380), i43, i46, i53, i46) → 405_0_eq_IntArithmetic(EOS(STATIC_405), i43, i46, i53, i46, 1)
405_0_eq_IntArithmetic(EOS(STATIC_405), i43, i46, i53, i46, matching1) → 434_0_eq_InvokeMethod(EOS(STATIC_434), i43, i46, i53, -(i46, 1)) | &&(>(i46, 0), =(matching1, 1))
434_0_eq_InvokeMethod(EOS(STATIC_434), i43, i46, i53, i64) → 456_1_eq_InvokeMethod(456_0_eq_Load(EOS(STATIC_456), i53, i64), i43, i46, i53, i64)
456_0_eq_Load(EOS(STATIC_456), i53, i64) → 461_0_eq_Load(EOS(STATIC_461), i53, i64)
461_0_eq_Load(EOS(STATIC_461), i53, i64) → 251_0_eq_Load(EOS(STATIC_251), i53, i64)
251_0_eq_Load(EOS(STATIC_251), i18, i37) → 263_0_eq_LE(EOS(STATIC_263), i18, i37, i18)
R rules:
263_0_eq_LE(EOS(STATIC_263), matching1, i37, matching2) → 271_0_eq_LE(EOS(STATIC_271), 0, i37, 0) | &&(=(matching1, 0), =(matching2, 0))
271_0_eq_LE(EOS(STATIC_271), matching1, i37, matching2) → 280_0_eq_Load(EOS(STATIC_280), 0, i37) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
280_0_eq_Load(EOS(STATIC_280), matching1, i37) → 289_0_eq_NE(EOS(STATIC_289), i37, 0) | =(matching1, 0)
289_0_eq_NE(EOS(STATIC_289), i37, matching1) → 304_0_eq_Load(EOS(STATIC_304), i37) | =(matching1, 0)
292_0_eq_LE(EOS(STATIC_292), i43, matching1, matching2) → 306_0_eq_LE(EOS(STATIC_306), i43, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
304_0_eq_Load(EOS(STATIC_304), i37) → 317_0_eq_NE(EOS(STATIC_317), i37)
306_0_eq_LE(EOS(STATIC_306), i43, matching1, matching2) → 319_0_eq_Load(EOS(STATIC_319), i43, 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
317_0_eq_NE(EOS(STATIC_317), i50) → 331_0_eq_NE(EOS(STATIC_331), i50)
317_0_eq_NE(EOS(STATIC_317), matching1) → 332_0_eq_NE(EOS(STATIC_332), 0) | =(matching1, 0)
319_0_eq_Load(EOS(STATIC_319), i43, matching1) → 334_0_eq_NE(EOS(STATIC_334), 0, i43) | =(matching1, 0)
331_0_eq_NE(EOS(STATIC_331), i50) → 344_0_eq_ConstantStackPush(EOS(STATIC_344)) | >(i50, 0)
332_0_eq_NE(EOS(STATIC_332), matching1) → 346_0_eq_ConstantStackPush(EOS(STATIC_346)) | =(matching1, 0)
334_0_eq_NE(EOS(STATIC_334), matching1, i43) → 349_0_eq_ConstantStackPush(EOS(STATIC_349)) | &&(>(i43, 0), =(matching1, 0))
344_0_eq_ConstantStackPush(EOS(STATIC_344)) → 359_0_eq_Return(EOS(STATIC_359), 0)
346_0_eq_ConstantStackPush(EOS(STATIC_346)) → 360_0_eq_JMP(EOS(STATIC_360), 1)
349_0_eq_ConstantStackPush(EOS(STATIC_349)) → 363_0_eq_Return(EOS(STATIC_363), 0)
360_0_eq_JMP(EOS(STATIC_360), matching1) → 376_0_eq_Return(EOS(STATIC_376), 1) | =(matching1, 1)
456_1_eq_InvokeMethod(359_0_eq_Return(EOS(STATIC_359), matching1), i43, i46, matching2, i77) → 487_0_eq_Return(EOS(STATIC_487), i43, i46, 0, i77, 0) | &&(=(matching1, 0), =(matching2, 0))
456_1_eq_InvokeMethod(363_0_eq_Return(EOS(STATIC_363), matching1), i43, i46, i80, matching2) → 494_0_eq_Return(EOS(STATIC_494), i43, i46, i80, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
456_1_eq_InvokeMethod(376_0_eq_Return(EOS(STATIC_376), matching1), i43, i46, matching2, matching3) → 502_0_eq_Return(EOS(STATIC_502), i43, i46, 0, 0, 1) | &&(&&(=(matching1, 1), =(matching2, 0)), =(matching3, 0))
456_1_eq_InvokeMethod(506_0_eq_Return(EOS(STATIC_506), i95, i96, i84), i43, i46, i95, i96) → 516_0_eq_Return(EOS(STATIC_516), i43, i46, i95, i96, i95, i96, i84)
456_1_eq_InvokeMethod(522_0_eq_Return(EOS(STATIC_522), i104, i105, i84), i43, i46, i104, i105) → 540_0_eq_Return(EOS(STATIC_540), i43, i46, i104, i105, i104, i105, i84)
487_0_eq_Return(EOS(STATIC_487), i43, i46, matching1, i77, matching2) → 495_0_eq_Return(EOS(STATIC_495), i43, i46, 0, i77, 0) | &&(=(matching1, 0), =(matching2, 0))
494_0_eq_Return(EOS(STATIC_494), i43, i46, i80, matching1, matching2) → 495_0_eq_Return(EOS(STATIC_495), i43, i46, i80, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
495_0_eq_Return(EOS(STATIC_495), i43, i46, i82, i81, matching1) → 503_0_eq_Return(EOS(STATIC_503), i43, i46, i82, i81, 0) | =(matching1, 0)
502_0_eq_Return(EOS(STATIC_502), i43, i46, matching1, matching2, matching3) → 503_0_eq_Return(EOS(STATIC_503), i43, i46, 0, 0, 1) | &&(&&(=(matching1, 0), =(matching2, 0)), =(matching3, 1))
503_0_eq_Return(EOS(STATIC_503), i43, i46, i86, i85, i84) → 506_0_eq_Return(EOS(STATIC_506), i43, i46, i84)
506_0_eq_Return(EOS(STATIC_506), i43, i46, i84) → 522_0_eq_Return(EOS(STATIC_522), i43, i46, i84)
516_0_eq_Return(EOS(STATIC_516), i43, i46, i95, i96, i95, i96, i84) → 522_0_eq_Return(EOS(STATIC_522), i43, i46, i84)
540_0_eq_Return(EOS(STATIC_540), i43, i46, i104, i105, i104, i105, i84) → 516_0_eq_Return(EOS(STATIC_516), i43, i46, i104, i105, i104, i105, i84)

Combined rules. Obtained 1 conditional rules for P and 7 conditional rules for R.

P rules:
263_0_eq_LE(EOS(STATIC_263), x0, x1, x0) → 456_1_eq_InvokeMethod(263_0_eq_LE(EOS(STATIC_263), -(x0, 1), -(x1, 1), -(x0, 1)), x0, x1, -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
R rules:
263_0_eq_LE(EOS(STATIC_263), 0, x1, 0) → 359_0_eq_Return(EOS(STATIC_359), 0) | >(x1, 0)
263_0_eq_LE(EOS(STATIC_263), 0, 0, 0) → 376_0_eq_Return(EOS(STATIC_376), 1)
456_1_eq_InvokeMethod(506_0_eq_Return(EOS(STATIC_506), x0, x1, x2), x3, x4, x0, x1) → 522_0_eq_Return(EOS(STATIC_522), x3, x4, x2)
456_1_eq_InvokeMethod(522_0_eq_Return(EOS(STATIC_522), x0, x1, x2), x3, x4, x0, x1) → 522_0_eq_Return(EOS(STATIC_522), x3, x4, x2)
456_1_eq_InvokeMethod(376_0_eq_Return(EOS(STATIC_376), 1), x1, x2, 0, 0) → 522_0_eq_Return(EOS(STATIC_522), x1, x2, 1)
456_1_eq_InvokeMethod(359_0_eq_Return(EOS(STATIC_359), 0), x1, x2, 0, x4) → 522_0_eq_Return(EOS(STATIC_522), x1, x2, 0)
456_1_eq_InvokeMethod(363_0_eq_Return(EOS(STATIC_363), 0), x1, x2, x3, 0) → 522_0_eq_Return(EOS(STATIC_522), x1, x2, 0)

Filtered ground terms:

263_0_eq_LE(x1, x2, x3, x4) → 263_0_eq_LE(x2, x3, x4)
Cond_263_0_eq_LE(x1, x2, x3, x4, x5) → Cond_263_0_eq_LE(x1, x3, x4, x5)
522_0_eq_Return(x1, x2, x3, x4) → 522_0_eq_Return(x2, x3, x4)
363_0_eq_Return(x1, x2) → 363_0_eq_Return
359_0_eq_Return(x1, x2) → 359_0_eq_Return
376_0_eq_Return(x1, x2) → 376_0_eq_Return
506_0_eq_Return(x1, x2, x3, x4) → 506_0_eq_Return(x2, x3, x4)

Filtered duplicate args:

263_0_eq_LE(x1, x2, x3) → 263_0_eq_LE(x2, x3)
Cond_263_0_eq_LE(x1, x2, x3, x4) → Cond_263_0_eq_LE(x1, x3, x4)

Filtered unneeded arguments:

456_1_eq_InvokeMethod(x1, x2, x3, x4, x5) → 456_1_eq_InvokeMethod(x1, x4, x5)

Combined rules. Obtained 1 conditional rules for P and 7 conditional rules for R.

P rules:
263_0_eq_LE(x1, x0) → 456_1_eq_InvokeMethod(263_0_eq_LE(-(x1, 1), -(x0, 1)), -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
R rules:
263_0_eq_LE(x1, 0) → 359_0_eq_Return | >(x1, 0)
263_0_eq_LE(0, 0) → 376_0_eq_Return
456_1_eq_InvokeMethod(506_0_eq_Return(x0, x1, x2), x0, x1) → 522_0_eq_Return(x3, x4, x2)
456_1_eq_InvokeMethod(522_0_eq_Return(x0, x1, x2), x0, x1) → 522_0_eq_Return(x3, x4, x2)
456_1_eq_InvokeMethod(376_0_eq_Return, 0, 0) → 522_0_eq_Return(x1, x2, 1)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, x4) → 522_0_eq_Return(x1, x2, 0)
456_1_eq_InvokeMethod(363_0_eq_Return, x3, 0) → 522_0_eq_Return(x1, x2, 0)

Performed bisimulation on rules. Used the following equivalence classes: {[359_0_eq_Return, 376_0_eq_Return, 363_0_eq_Return]=359_0_eq_Return}

Finished conversion. Obtained 2 rules for P and 8 rules for R. System has predefined symbols.

P rules:
263_0_EQ_LE(x1, x0) → COND_263_0_EQ_LE(&&(>(x1, 0), >(x0, 0)), x1, x0)
COND_263_0_EQ_LE(TRUE, x1, x0) → 263_0_EQ_LE(-(x1, 1), -(x0, 1))
R rules:
263_0_eq_LE(x1, 0) → Cond_263_0_eq_LE(>(x1, 0), x1, 0)
Cond_263_0_eq_LE(TRUE, x1, 0) → 359_0_eq_Return
263_0_eq_LE(0, 0) → 359_0_eq_Return
456_1_eq_InvokeMethod(506_0_eq_Return(x0, x1, x2), x0, x1) → 522_0_eq_Return(x3, x4, x2)
456_1_eq_InvokeMethod(522_0_eq_Return(x0, x1, x2), x0, x1) → 522_0_eq_Return(x3, x4, x2)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, 0) → 522_0_eq_Return(x1, x2, 1)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, x4) → 522_0_eq_Return(x1, x2, 0)
456_1_eq_InvokeMethod(359_0_eq_Return, x3, 0) → 522_0_eq_Return(x1, x2, 0)

(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:
263_0_eq_LE(x1, 0) → Cond_263_0_eq_LE(x1 > 0, x1, 0)
Cond_263_0_eq_LE(TRUE, x1, 0) → 359_0_eq_Return
263_0_eq_LE(0, 0) → 359_0_eq_Return
456_1_eq_InvokeMethod(506_0_eq_Return(x0, x1, x2), x0, x1) → 522_0_eq_Return(x3, x4, x2)
456_1_eq_InvokeMethod(522_0_eq_Return(x0, x1, x2), x0, x1) → 522_0_eq_Return(x3, x4, x2)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, 0) → 522_0_eq_Return(x1, x2, 1)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, x4) → 522_0_eq_Return(x1, x2, 0)
456_1_eq_InvokeMethod(359_0_eq_Return, x3, 0) → 522_0_eq_Return(x1, x2, 0)

The integer pair graph contains the following rules and edges:
(0): 263_0_EQ_LE(x1[0], x0[0]) → COND_263_0_EQ_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_263_0_EQ_LE(TRUE, x1[1], x0[1]) → 263_0_EQ_LE(x1[1] - 1, x0[1] - 1)

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

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

The set Q consists of the following terms:
263_0_eq_LE(x0, 0)
Cond_263_0_eq_LE(TRUE, x0, 0)
456_1_eq_InvokeMethod(506_0_eq_Return(x0, x1, x2), x0, x1)
456_1_eq_InvokeMethod(522_0_eq_Return(x0, x1, x2), x0, x1)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, x0)
456_1_eq_InvokeMethod(359_0_eq_Return, x0, 0)

(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@12bc8f01 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 263_0_EQ_LE(x1, x0) → COND_263_0_EQ_LE(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:

We consider the chain 263_0_EQ_LE(x1[0], x0[0]) → COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_263_0_EQ_LE(TRUE, x1[1], x0[1]) → 263_0_EQ_LE(-(x1[1], 1), -(x0[1], 1)) which results in the following constraint:
(1)    (&&(>(x1[0], 0), >(x0[0], 0))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 263_0_EQ_LE(x1[0], x0[0])≥NonInfC∧263_0_EQ_LE(x1[0], x0[0])≥COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], 0)=TRUE∧>(x0[0], 0)=TRUE ⇒ 263_0_EQ_LE(x1[0], x0[0])≥NonInfC∧263_0_EQ_LE(x1[0], x0[0])≥COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x0[0] + [(2)bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x0[0] + [(2)bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x0[0] + [(2)bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x0[0] + [(2)bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 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_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(5)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x0[0] + [(2)bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)

For Pair COND_263_0_EQ_LE(TRUE, x1, x0) → 263_0_EQ_LE(-(x1, 1), -(x0, 1)) the following chains were created:

We consider the chain COND_263_0_EQ_LE(TRUE, x1[1], x0[1]) → 263_0_EQ_LE(-(x1[1], 1), -(x0[1], 1)) which results in the following constraint:
(8)    (COND_263_0_EQ_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_263_0_EQ_LE(TRUE, x1[1], x0[1])≥263_0_EQ_LE(-(x1[1], 1), -(x0[1], 1))∧(U^Increasing(263_0_EQ_LE(-(x1[1], 1), -(x0[1], 1))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(263_0_EQ_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[bni_19] = 0∧[4 + (-1)bso_20] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(263_0_EQ_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[bni_19] = 0∧[4 + (-1)bso_20] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(263_0_EQ_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[bni_19] = 0∧[4 + (-1)bso_20] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(263_0_EQ_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[bni_19] = 0∧0 = 0∧0 = 0∧[4 + (-1)bso_20] ≥ 0)

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

263_0_EQ_LE(x1, x0) → COND_263_0_EQ_LE(&&(>(x1, 0), >(x0, 0)), x1, x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(5)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x0[0] + [(2)bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)
COND_263_0_EQ_LE(TRUE, x1, x0) → 263_0_EQ_LE(-(x1, 1), -(x0, 1))
- ((U^Increasing(263_0_EQ_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[bni_19] = 0∧0 = 0∧0 = 0∧[4 + (-1)bso_20] ≥ 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(263_0_eq_LE(x₁, x₂)) = [-1] + [-1]x₁
POL(0) = 0
POL(Cond_263_0_eq_LE(x₁, x₂, x₃)) = [-1] + [-1]x₂
POL(>(x₁, x₂)) = [-1]
POL(359_0_eq_Return) = [-1]
POL(456_1_eq_InvokeMethod(x₁, x₂, x₃)) = [-1]
POL(506_0_eq_Return(x₁, x₂, x₃)) = [-1]
POL(522_0_eq_Return(x₁, x₂, x₃)) = [-1]
POL(1) = [1]
POL(263_0_EQ_LE(x₁, x₂)) = [1] + [2]x₂ + [2]x₁
POL(COND_263_0_EQ_LE(x₁, x₂, x₃)) = [1] + [2]x₃ + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂

The following pairs are in P_>:

COND_263_0_EQ_LE(TRUE, x1[1], x0[1]) → 263_0_EQ_LE(-(x1[1], 1), -(x0[1], 1))

The following pairs are in P_bound:

263_0_EQ_LE(x1[0], x0[0]) → COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])

The following pairs are in P_≥:

263_0_EQ_LE(x1[0], x0[0]) → COND_263_0_EQ_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[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:

Integer, Boolean

The ITRS R consists of the following rules:
263_0_eq_LE(x1, 0) → Cond_263_0_eq_LE(x1 > 0, x1, 0)
Cond_263_0_eq_LE(TRUE, x1, 0) → 359_0_eq_Return
263_0_eq_LE(0, 0) → 359_0_eq_Return
456_1_eq_InvokeMethod(506_0_eq_Return(x0, x1, x2), x0, x1) → 522_0_eq_Return(x3, x4, x2)
456_1_eq_InvokeMethod(522_0_eq_Return(x0, x1, x2), x0, x1) → 522_0_eq_Return(x3, x4, x2)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, 0) → 522_0_eq_Return(x1, x2, 1)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, x4) → 522_0_eq_Return(x1, x2, 0)
456_1_eq_InvokeMethod(359_0_eq_Return, x3, 0) → 522_0_eq_Return(x1, x2, 0)

The integer pair graph contains the following rules and edges:
(0): 263_0_EQ_LE(x1[0], x0[0]) → COND_263_0_EQ_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])

The set Q consists of the following terms:
263_0_eq_LE(x0, 0)
Cond_263_0_eq_LE(TRUE, x0, 0)
456_1_eq_InvokeMethod(506_0_eq_Return(x0, x1, x2), x0, x1)
456_1_eq_InvokeMethod(522_0_eq_Return(x0, x1, x2), x0, x1)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, x0)
456_1_eq_InvokeMethod(359_0_eq_Return, x0, 0)

(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

The ITRS R consists of the following rules:
263_0_eq_LE(x1, 0) → Cond_263_0_eq_LE(x1 > 0, x1, 0)
Cond_263_0_eq_LE(TRUE, x1, 0) → 359_0_eq_Return
263_0_eq_LE(0, 0) → 359_0_eq_Return
456_1_eq_InvokeMethod(506_0_eq_Return(x0, x1, x2), x0, x1) → 522_0_eq_Return(x3, x4, x2)
456_1_eq_InvokeMethod(522_0_eq_Return(x0, x1, x2), x0, x1) → 522_0_eq_Return(x3, x4, x2)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, 0) → 522_0_eq_Return(x1, x2, 1)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, x4) → 522_0_eq_Return(x1, x2, 0)
456_1_eq_InvokeMethod(359_0_eq_Return, x3, 0) → 522_0_eq_Return(x1, x2, 0)

The integer pair graph contains the following rules and edges:
(1): COND_263_0_EQ_LE(TRUE, x1[1], x0[1]) → 263_0_EQ_LE(x1[1] - 1, x0[1] - 1)

The set Q consists of the following terms:
263_0_eq_LE(x0, 0)
Cond_263_0_eq_LE(TRUE, x0, 0)
456_1_eq_InvokeMethod(506_0_eq_Return(x0, x1, x2), x0, x1)
456_1_eq_InvokeMethod(522_0_eq_Return(x0, x1, x2), x0, x1)
456_1_eq_InvokeMethod(359_0_eq_Return, 0, x0)
456_1_eq_InvokeMethod(359_0_eq_Return, x0, 0)

(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