Termination proof of /tmp/tmpc5klbm/LeUserDefRec.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 120 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 120 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 120 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: LeUserDefRec

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

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

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
LeUserDefRec.main([Ljava/lang/String;)V: Graph of 151 nodes with 0 SCCs.

LeUserDefRec.le(II)Z: Graph of 36 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: LeUserDefRec.le(II)Z
SCC calls the following helper methods: LeUserDefRec.le(II)Z
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 15 rules for P and 21 rules for R.

P rules:
262_0_le_LE(EOS(STATIC_262), i42, i37, i42) → 271_0_le_LE(EOS(STATIC_271), i42, i37, i42)
271_0_le_LE(EOS(STATIC_271), i42, i37, i42) → 278_0_le_Load(EOS(STATIC_278), i42, i37) | >(i42, 0)
278_0_le_Load(EOS(STATIC_278), i42, i37) → 290_0_le_LE(EOS(STATIC_290), i42, i37, i37)
290_0_le_LE(EOS(STATIC_290), i42, i46, i46) → 306_0_le_LE(EOS(STATIC_306), i42, i46, i46)
306_0_le_LE(EOS(STATIC_306), i42, i46, i46) → 319_0_le_Load(EOS(STATIC_319), i42, i46) | >(i46, 0)
319_0_le_Load(EOS(STATIC_319), i42, i46) → 332_0_le_ConstantStackPush(EOS(STATIC_332), i42, i46, i42)
332_0_le_ConstantStackPush(EOS(STATIC_332), i42, i46, i42) → 346_0_le_IntArithmetic(EOS(STATIC_346), i42, i46, i42, 1)
346_0_le_IntArithmetic(EOS(STATIC_346), i42, i46, i42, matching1) → 365_0_le_Load(EOS(STATIC_365), i42, i46, -(i42, 1)) | &&(>(i42, 0), =(matching1, 1))
365_0_le_Load(EOS(STATIC_365), i42, i46, i54) → 375_0_le_ConstantStackPush(EOS(STATIC_375), i42, i54, i46)
375_0_le_ConstantStackPush(EOS(STATIC_375), i42, i54, i46) → 385_0_le_IntArithmetic(EOS(STATIC_385), i42, i54, i46, 1)
385_0_le_IntArithmetic(EOS(STATIC_385), i42, i54, i46, matching1) → 403_0_le_InvokeMethod(EOS(STATIC_403), i42, i54, -(i46, 1)) | &&(>(i46, 0), =(matching1, 1))
403_0_le_InvokeMethod(EOS(STATIC_403), i42, i54, i61) → 409_1_le_InvokeMethod(409_0_le_Load(EOS(STATIC_409), i54, i61), i42, i54, i61)
409_0_le_Load(EOS(STATIC_409), i54, i61) → 413_0_le_Load(EOS(STATIC_413), i54, i61)
413_0_le_Load(EOS(STATIC_413), i54, i61) → 251_0_le_Load(EOS(STATIC_251), i54, i61)
251_0_le_Load(EOS(STATIC_251), i18, i37) → 262_0_le_LE(EOS(STATIC_262), i18, i37, i18)
R rules:
262_0_le_LE(EOS(STATIC_262), matching1, i37, matching2) → 270_0_le_LE(EOS(STATIC_270), 0, i37, 0) | &&(=(matching1, 0), =(matching2, 0))
270_0_le_LE(EOS(STATIC_270), matching1, i37, matching2) → 277_0_le_Load(EOS(STATIC_277), 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
277_0_le_Load(EOS(STATIC_277), matching1) → 287_0_le_NE(EOS(STATIC_287), 0) | =(matching1, 0)
287_0_le_NE(EOS(STATIC_287), matching1) → 302_0_le_ConstantStackPush(EOS(STATIC_302)) | =(matching1, 0)
290_0_le_LE(EOS(STATIC_290), i42, matching1, matching2) → 305_0_le_LE(EOS(STATIC_305), i42, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
302_0_le_ConstantStackPush(EOS(STATIC_302)) → 315_0_le_JMP(EOS(STATIC_315), 1)
305_0_le_LE(EOS(STATIC_305), i42, matching1, matching2) → 317_0_le_Load(EOS(STATIC_317), i42) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
315_0_le_JMP(EOS(STATIC_315), matching1) → 327_0_le_Return(EOS(STATIC_327), 1) | =(matching1, 1)
317_0_le_Load(EOS(STATIC_317), i42) → 330_0_le_NE(EOS(STATIC_330), i42)
330_0_le_NE(EOS(STATIC_330), i42) → 344_0_le_ConstantStackPush(EOS(STATIC_344)) | >(i42, 0)
344_0_le_ConstantStackPush(EOS(STATIC_344)) → 364_0_le_Return(EOS(STATIC_364), 0)
409_1_le_InvokeMethod(327_0_le_Return(EOS(STATIC_327), matching1), i42, matching2, i65) → 420_0_le_Return(EOS(STATIC_420), i42, 0, i65, 1) | &&(=(matching1, 1), =(matching2, 0))
409_1_le_InvokeMethod(364_0_le_Return(EOS(STATIC_364), matching1), i42, i68, matching2) → 424_0_le_Return(EOS(STATIC_424), i42, i68, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
409_1_le_InvokeMethod(427_0_le_Return(EOS(STATIC_427), i77, i69), i42, i77, i78) → 438_0_le_Return(EOS(STATIC_438), i42, i77, i78, i77, i69)
409_1_le_InvokeMethod(444_0_le_Return(EOS(STATIC_444), i85, i69), i42, i85, i86) → 461_0_le_Return(EOS(STATIC_461), i42, i85, i86, i85, i69)
420_0_le_Return(EOS(STATIC_420), i42, matching1, i65, matching2) → 425_0_le_Return(EOS(STATIC_425), i42, 0, i65, 1) | &&(=(matching1, 0), =(matching2, 1))
424_0_le_Return(EOS(STATIC_424), i42, i68, matching1, matching2) → 425_0_le_Return(EOS(STATIC_425), i42, i68, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
425_0_le_Return(EOS(STATIC_425), i42, i71, i70, i69) → 427_0_le_Return(EOS(STATIC_427), i42, i69)
427_0_le_Return(EOS(STATIC_427), i42, i69) → 444_0_le_Return(EOS(STATIC_444), i42, i69)
438_0_le_Return(EOS(STATIC_438), i42, i77, i78, i77, i69) → 444_0_le_Return(EOS(STATIC_444), i42, i69)
461_0_le_Return(EOS(STATIC_461), i42, i85, i86, i85, i69) → 438_0_le_Return(EOS(STATIC_438), i42, i85, i86, i85, i69)

Combined rules. Obtained 1 conditional rules for P and 5 conditional rules for R.

P rules:
262_0_le_LE(EOS(STATIC_262), x0, x1, x0) → 409_1_le_InvokeMethod(262_0_le_LE(EOS(STATIC_262), -(x0, 1), -(x1, 1), -(x0, 1)), x0, -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
R rules:
262_0_le_LE(EOS(STATIC_262), 0, x1, 0) → 327_0_le_Return(EOS(STATIC_327), 1)
409_1_le_InvokeMethod(427_0_le_Return(EOS(STATIC_427), x0, x1), x2, x0, x3) → 444_0_le_Return(EOS(STATIC_444), x2, x1)
409_1_le_InvokeMethod(444_0_le_Return(EOS(STATIC_444), x0, x1), x2, x0, x3) → 444_0_le_Return(EOS(STATIC_444), x2, x1)
409_1_le_InvokeMethod(327_0_le_Return(EOS(STATIC_327), 1), x1, 0, x3) → 444_0_le_Return(EOS(STATIC_444), x1, 1)
409_1_le_InvokeMethod(364_0_le_Return(EOS(STATIC_364), 0), x1, x2, 0) → 444_0_le_Return(EOS(STATIC_444), x1, 0)

Filtered ground terms:

262_0_le_LE(x1, x2, x3, x4) → 262_0_le_LE(x2, x3, x4)
Cond_262_0_le_LE(x1, x2, x3, x4, x5) → Cond_262_0_le_LE(x1, x3, x4, x5)
444_0_le_Return(x1, x2, x3) → 444_0_le_Return(x2, x3)
364_0_le_Return(x1, x2) → 364_0_le_Return
327_0_le_Return(x1, x2) → 327_0_le_Return
427_0_le_Return(x1, x2, x3) → 427_0_le_Return(x2, x3)

Filtered duplicate args:

262_0_le_LE(x1, x2, x3) → 262_0_le_LE(x2, x3)
Cond_262_0_le_LE(x1, x2, x3, x4) → Cond_262_0_le_LE(x1, x3, x4)

Filtered unneeded arguments:

409_1_le_InvokeMethod(x1, x2, x3, x4) → 409_1_le_InvokeMethod(x1, x3, x4)

Combined rules. Obtained 1 conditional rules for P and 5 conditional rules for R.

P rules:
262_0_le_LE(x1, x0) → 409_1_le_InvokeMethod(262_0_le_LE(-(x1, 1), -(x0, 1)), -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
R rules:
262_0_le_LE(x1, 0) → 327_0_le_Return
409_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x3) → 444_0_le_Return(x2, x1)
409_1_le_InvokeMethod(444_0_le_Return(x0, x1), x0, x3) → 444_0_le_Return(x2, x1)
409_1_le_InvokeMethod(327_0_le_Return, 0, x3) → 444_0_le_Return(x1, 1)
409_1_le_InvokeMethod(364_0_le_Return, x2, 0) → 444_0_le_Return(x1, 0)

Performed bisimulation on rules. Used the following equivalence classes: {[327_0_le_Return, 364_0_le_Return]=327_0_le_Return}

Finished conversion. Obtained 2 rules for P and 5 rules for R. System has predefined symbols.

P rules:
262_0_LE_LE(x1, x0) → COND_262_0_LE_LE(&&(>(x1, 0), >(x0, 0)), x1, x0)
COND_262_0_LE_LE(TRUE, x1, x0) → 262_0_LE_LE(-(x1, 1), -(x0, 1))
R rules:
262_0_le_LE(x1, 0) → 327_0_le_Return
409_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x3) → 444_0_le_Return(x2, x1)
409_1_le_InvokeMethod(444_0_le_Return(x0, x1), x0, x3) → 444_0_le_Return(x2, x1)
409_1_le_InvokeMethod(327_0_le_Return, 0, x3) → 444_0_le_Return(x1, 1)
409_1_le_InvokeMethod(327_0_le_Return, x2, 0) → 444_0_le_Return(x1, 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:

Boolean, Integer

The ITRS R consists of the following rules:
262_0_le_LE(x1, 0) → 327_0_le_Return
409_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x3) → 444_0_le_Return(x2, x1)
409_1_le_InvokeMethod(444_0_le_Return(x0, x1), x0, x3) → 444_0_le_Return(x2, x1)
409_1_le_InvokeMethod(327_0_le_Return, 0, x3) → 444_0_le_Return(x1, 1)
409_1_le_InvokeMethod(327_0_le_Return, x2, 0) → 444_0_le_Return(x1, 0)

The integer pair graph contains the following rules and edges:
(0): 262_0_LE_LE(x1[0], x0[0]) → COND_262_0_LE_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_262_0_LE_LE(TRUE, x1[1], x0[1]) → 262_0_LE_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:
262_0_le_LE(x0, 0)
409_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x2)
409_1_le_InvokeMethod(444_0_le_Return(x0, x1), x0, x2)
409_1_le_InvokeMethod(327_0_le_Return, 0, x0)
409_1_le_InvokeMethod(327_0_le_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@461fbe88 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 262_0_LE_LE(x1, x0) → COND_262_0_LE_LE(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:

We consider the chain 262_0_LE_LE(x1[0], x0[0]) → COND_262_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_262_0_LE_LE(TRUE, x1[1], x0[1]) → 262_0_LE_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] ⇒ 262_0_LE_LE(x1[0], x0[0])≥NonInfC∧262_0_LE_LE(x1[0], x0[0])≥COND_262_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_262_0_LE_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 ⇒ 262_0_LE_LE(x1[0], x0[0])≥NonInfC∧262_0_LE_LE(x1[0], x0[0])≥COND_262_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_262_0_LE_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_262_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 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_262_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 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_262_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 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_262_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 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_262_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(5)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_262_0_LE_LE(TRUE, x1, x0) → 262_0_LE_LE(-(x1, 1), -(x0, 1)) the following chains were created:

We consider the chain COND_262_0_LE_LE(TRUE, x1[1], x0[1]) → 262_0_LE_LE(-(x1[1], 1), -(x0[1], 1)) which results in the following constraint:
(8)    (COND_262_0_LE_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_262_0_LE_LE(TRUE, x1[1], x0[1])≥262_0_LE_LE(-(x1[1], 1), -(x0[1], 1))∧(U^Increasing(262_0_LE_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(262_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[bni_16] = 0∧[4 + (-1)bso_17] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(262_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[bni_16] = 0∧[4 + (-1)bso_17] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(262_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[bni_16] = 0∧[4 + (-1)bso_17] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(262_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[bni_16] = 0∧0 = 0∧0 = 0∧[4 + (-1)bso_17] ≥ 0)

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

262_0_LE_LE(x1, x0) → COND_262_0_LE_LE(&&(>(x1, 0), >(x0, 0)), x1, x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_262_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(5)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)
COND_262_0_LE_LE(TRUE, x1, x0) → 262_0_LE_LE(-(x1, 1), -(x0, 1))
- ((U^Increasing(262_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[bni_16] = 0∧0 = 0∧0 = 0∧[4 + (-1)bso_17] ≥ 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(262_0_le_LE(x₁, x₂)) = [-1]
POL(0) = 0
POL(327_0_le_Return) = [-1]
POL(409_1_le_InvokeMethod(x₁, x₂, x₃)) = [-1]
POL(427_0_le_Return(x₁, x₂)) = [-1]
POL(444_0_le_Return(x₁, x₂)) = [-1]
POL(1) = [1]
POL(262_0_LE_LE(x₁, x₂)) = [1] + [2]x₂ + [2]x₁
POL(COND_262_0_LE_LE(x₁, x₂, x₃)) = [1] + [2]x₃ + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂

The following pairs are in P_>:

COND_262_0_LE_LE(TRUE, x1[1], x0[1]) → 262_0_LE_LE(-(x1[1], 1), -(x0[1], 1))

The following pairs are in P_bound:

262_0_LE_LE(x1[0], x0[0]) → COND_262_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])

The following pairs are in P_≥:

262_0_LE_LE(x1[0], x0[0]) → COND_262_0_LE_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:

Boolean, Integer

The ITRS R consists of the following rules:
262_0_le_LE(x1, 0) → 327_0_le_Return
409_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x3) → 444_0_le_Return(x2, x1)
409_1_le_InvokeMethod(444_0_le_Return(x0, x1), x0, x3) → 444_0_le_Return(x2, x1)
409_1_le_InvokeMethod(327_0_le_Return, 0, x3) → 444_0_le_Return(x1, 1)
409_1_le_InvokeMethod(327_0_le_Return, x2, 0) → 444_0_le_Return(x1, 0)

The integer pair graph contains the following rules and edges:
(0): 262_0_LE_LE(x1[0], x0[0]) → COND_262_0_LE_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])

The set Q consists of the following terms:
262_0_le_LE(x0, 0)
409_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x2)
409_1_le_InvokeMethod(444_0_le_Return(x0, x1), x0, x2)
409_1_le_InvokeMethod(327_0_le_Return, 0, x0)
409_1_le_InvokeMethod(327_0_le_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:
262_0_le_LE(x1, 0) → 327_0_le_Return
409_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x3) → 444_0_le_Return(x2, x1)
409_1_le_InvokeMethod(444_0_le_Return(x0, x1), x0, x3) → 444_0_le_Return(x2, x1)
409_1_le_InvokeMethod(327_0_le_Return, 0, x3) → 444_0_le_Return(x1, 1)
409_1_le_InvokeMethod(327_0_le_Return, x2, 0) → 444_0_le_Return(x1, 0)

The integer pair graph contains the following rules and edges:
(1): COND_262_0_LE_LE(TRUE, x1[1], x0[1]) → 262_0_LE_LE(x1[1] - 1, x0[1] - 1)

The set Q consists of the following terms:
262_0_le_LE(x0, 0)
409_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x2)
409_1_le_InvokeMethod(444_0_le_Return(x0, x1), x0, x2)
409_1_le_InvokeMethod(327_0_le_Return, 0, x0)
409_1_le_InvokeMethod(327_0_le_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