Termination of the given JBC could be proven:

(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:
249_0_le_LE(EOS(STATIC_249), i42, i37, i42) → 256_0_le_LE(EOS(STATIC_256), i42, i37, i42)
256_0_le_LE(EOS(STATIC_256), i42, i37, i42) → 262_0_le_Load(EOS(STATIC_262), i42, i37) | >(i42, 0)
262_0_le_Load(EOS(STATIC_262), i42, i37) → 269_0_le_LE(EOS(STATIC_269), i42, i37, i37)
269_0_le_LE(EOS(STATIC_269), i42, i44, i44) → 281_0_le_LE(EOS(STATIC_281), i42, i44, i44)
281_0_le_LE(EOS(STATIC_281), i42, i44, i44) → 291_0_le_Load(EOS(STATIC_291), i42, i44) | >(i44, 0)
291_0_le_Load(EOS(STATIC_291), i42, i44) → 297_0_le_ConstantStackPush(EOS(STATIC_297), i42, i44, i42)
297_0_le_ConstantStackPush(EOS(STATIC_297), i42, i44, i42) → 309_0_le_IntArithmetic(EOS(STATIC_309), i42, i44, i42, 1)
309_0_le_IntArithmetic(EOS(STATIC_309), i42, i44, i42, matching1) → 330_0_le_Load(EOS(STATIC_330), i42, i44, -(i42, 1)) | &&(>(i42, 0), =(matching1, 1))
330_0_le_Load(EOS(STATIC_330), i42, i44, i50) → 340_0_le_ConstantStackPush(EOS(STATIC_340), i42, i50, i44)
340_0_le_ConstantStackPush(EOS(STATIC_340), i42, i50, i44) → 355_0_le_IntArithmetic(EOS(STATIC_355), i42, i50, i44, 1)
355_0_le_IntArithmetic(EOS(STATIC_355), i42, i50, i44, matching1) → 377_0_le_InvokeMethod(EOS(STATIC_377), i42, i50, -(i44, 1)) | &&(>(i44, 0), =(matching1, 1))
377_0_le_InvokeMethod(EOS(STATIC_377), i42, i50, i59) → 385_1_le_InvokeMethod(385_0_le_Load(EOS(STATIC_385), i50, i59), i42, i50, i59)
385_0_le_Load(EOS(STATIC_385), i50, i59) → 389_0_le_Load(EOS(STATIC_389), i50, i59)
389_0_le_Load(EOS(STATIC_389), i50, i59) → 239_0_le_Load(EOS(STATIC_239), i50, i59)
239_0_le_Load(EOS(STATIC_239), i18, i37) → 249_0_le_LE(EOS(STATIC_249), i18, i37, i18)
R rules:
249_0_le_LE(EOS(STATIC_249), matching1, i37, matching2) → 255_0_le_LE(EOS(STATIC_255), 0, i37, 0) | &&(=(matching1, 0), =(matching2, 0))
255_0_le_LE(EOS(STATIC_255), matching1, i37, matching2) → 260_0_le_Load(EOS(STATIC_260), 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
260_0_le_Load(EOS(STATIC_260), matching1) → 267_0_le_NE(EOS(STATIC_267), 0) | =(matching1, 0)
267_0_le_NE(EOS(STATIC_267), matching1) → 278_0_le_ConstantStackPush(EOS(STATIC_278)) | =(matching1, 0)
269_0_le_LE(EOS(STATIC_269), i42, matching1, matching2) → 280_0_le_LE(EOS(STATIC_280), i42, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
278_0_le_ConstantStackPush(EOS(STATIC_278)) → 287_0_le_JMP(EOS(STATIC_287), 1)
280_0_le_LE(EOS(STATIC_280), i42, matching1, matching2) → 289_0_le_Load(EOS(STATIC_289), i42) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
287_0_le_JMP(EOS(STATIC_287), matching1) → 295_0_le_Return(EOS(STATIC_295), 1) | =(matching1, 1)
289_0_le_Load(EOS(STATIC_289), i42) → 296_0_le_NE(EOS(STATIC_296), i42)
296_0_le_NE(EOS(STATIC_296), i42) → 307_0_le_ConstantStackPush(EOS(STATIC_307)) | >(i42, 0)
307_0_le_ConstantStackPush(EOS(STATIC_307)) → 327_0_le_Return(EOS(STATIC_327), 0)
385_1_le_InvokeMethod(295_0_le_Return(EOS(STATIC_295), matching1), i42, matching2, i63) → 400_0_le_Return(EOS(STATIC_400), i42, 0, i63, 1) | &&(=(matching1, 1), =(matching2, 0))
385_1_le_InvokeMethod(327_0_le_Return(EOS(STATIC_327), matching1), i42, i65, matching2) → 404_0_le_Return(EOS(STATIC_404), i42, i65, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
385_1_le_InvokeMethod(408_0_le_Return(EOS(STATIC_408), i74, i66), i42, i74, i75) → 419_0_le_Return(EOS(STATIC_419), i42, i74, i75, i74, i66)
385_1_le_InvokeMethod(427_0_le_Return(EOS(STATIC_427), i84, i66), i42, i84, i85) → 447_0_le_Return(EOS(STATIC_447), i42, i84, i85, i84, i66)
400_0_le_Return(EOS(STATIC_400), i42, matching1, i63, matching2) → 405_0_le_Return(EOS(STATIC_405), i42, 0, i63, 1) | &&(=(matching1, 0), =(matching2, 1))
404_0_le_Return(EOS(STATIC_404), i42, i65, matching1, matching2) → 405_0_le_Return(EOS(STATIC_405), i42, i65, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
405_0_le_Return(EOS(STATIC_405), i42, i68, i67, i66) → 408_0_le_Return(EOS(STATIC_408), i42, i66)
408_0_le_Return(EOS(STATIC_408), i42, i66) → 427_0_le_Return(EOS(STATIC_427), i42, i66)
419_0_le_Return(EOS(STATIC_419), i42, i74, i75, i74, i66) → 427_0_le_Return(EOS(STATIC_427), i42, i66)
447_0_le_Return(EOS(STATIC_447), i42, i84, i85, i84, i66) → 419_0_le_Return(EOS(STATIC_419), i42, i84, i85, i84, i66)

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

P rules:
249_0_le_LE(EOS(STATIC_249), x0, x1, x0) → 385_1_le_InvokeMethod(249_0_le_LE(EOS(STATIC_249), -(x0, 1), -(x1, 1), -(x0, 1)), x0, -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
R rules:
249_0_le_LE(EOS(STATIC_249), 0, x1, 0) → 295_0_le_Return(EOS(STATIC_295), 1)
385_1_le_InvokeMethod(408_0_le_Return(EOS(STATIC_408), x0, x1), x2, x0, x3) → 427_0_le_Return(EOS(STATIC_427), x2, x1)
385_1_le_InvokeMethod(427_0_le_Return(EOS(STATIC_427), x0, x1), x2, x0, x3) → 427_0_le_Return(EOS(STATIC_427), x2, x1)
385_1_le_InvokeMethod(295_0_le_Return(EOS(STATIC_295), 1), x1, 0, x3) → 427_0_le_Return(EOS(STATIC_427), x1, 1)
385_1_le_InvokeMethod(327_0_le_Return(EOS(STATIC_327), 0), x1, x2, 0) → 427_0_le_Return(EOS(STATIC_427), x1, 0)

Filtered ground terms:

249_0_le_LE(x1, x2, x3, x4) → 249_0_le_LE(x2, x3, x4)
Cond_249_0_le_LE(x1, x2, x3, x4, x5) → Cond_249_0_le_LE(x1, x3, x4, x5)
427_0_le_Return(x1, x2, x3) → 427_0_le_Return(x2, x3)
327_0_le_Return(x1, x2) → 327_0_le_Return
295_0_le_Return(x1, x2) → 295_0_le_Return
408_0_le_Return(x1, x2, x3) → 408_0_le_Return(x2, x3)

Filtered duplicate args:

249_0_le_LE(x1, x2, x3) → 249_0_le_LE(x2, x3)
Cond_249_0_le_LE(x1, x2, x3, x4) → Cond_249_0_le_LE(x1, x3, x4)

Filtered unneeded arguments:

385_1_le_InvokeMethod(x1, x2, x3, x4) → 385_1_le_InvokeMethod(x1, x3, x4)

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

P rules:
249_0_le_LE(x1, x0) → 385_1_le_InvokeMethod(249_0_le_LE(-(x1, 1), -(x0, 1)), -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
R rules:
249_0_le_LE(x1, 0) → 295_0_le_Return
385_1_le_InvokeMethod(408_0_le_Return(x0, x1), x0, x3) → 427_0_le_Return(x2, x1)
385_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x3) → 427_0_le_Return(x2, x1)
385_1_le_InvokeMethod(295_0_le_Return, 0, x3) → 427_0_le_Return(x1, 1)
385_1_le_InvokeMethod(327_0_le_Return, x2, 0) → 427_0_le_Return(x1, 0)

Performed bisimulation on rules. Used the following equivalence classes: {[295_0_le_Return, 327_0_le_Return]=295_0_le_Return}

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

P rules:
249_0_LE_LE(x1, x0) → COND_249_0_LE_LE(&&(>(x1, 0), >(x0, 0)), x1, x0)
COND_249_0_LE_LE(TRUE, x1, x0) → 249_0_LE_LE(-(x1, 1), -(x0, 1))
R rules:
249_0_le_LE(x1, 0) → 295_0_le_Return
385_1_le_InvokeMethod(408_0_le_Return(x0, x1), x0, x3) → 427_0_le_Return(x2, x1)
385_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x3) → 427_0_le_Return(x2, x1)
385_1_le_InvokeMethod(295_0_le_Return, 0, x3) → 427_0_le_Return(x1, 1)
385_1_le_InvokeMethod(295_0_le_Return, x2, 0) → 427_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:
249_0_le_LE(x1, 0) → 295_0_le_Return
385_1_le_InvokeMethod(408_0_le_Return(x0, x1), x0, x3) → 427_0_le_Return(x2, x1)
385_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x3) → 427_0_le_Return(x2, x1)
385_1_le_InvokeMethod(295_0_le_Return, 0, x3) → 427_0_le_Return(x1, 1)
385_1_le_InvokeMethod(295_0_le_Return, x2, 0) → 427_0_le_Return(x1, 0)

The integer pair graph contains the following rules and edges:
(0): 249_0_LE_LE(x1[0], x0[0]) → COND_249_0_LE_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_249_0_LE_LE(TRUE, x1[1], x0[1]) → 249_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:
249_0_le_LE(x0, 0)
385_1_le_InvokeMethod(408_0_le_Return(x0, x1), x0, x2)
385_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x2)
385_1_le_InvokeMethod(295_0_le_Return, 0, x0)
385_1_le_InvokeMethod(295_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@6fd2300e 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 249_0_LE_LE(x1, x0) → COND_249_0_LE_LE(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:

• We consider the chain 249_0_LE_LE(x1[0], x0[0]) → COND_249_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_249_0_LE_LE(TRUE, x1[1], x0[1]) → 249_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] ⇒ 249_0_LE_LE(x1[0], x0[0])≥NonInfC∧249_0_LE_LE(x1[0], x0[0])≥COND_249_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_249_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 ⇒ 249_0_LE_LE(x1[0], x0[0])≥NonInfC∧249_0_LE_LE(x1[0], x0[0])≥COND_249_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_249_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_249_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_249_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_249_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_249_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_249_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_249_0_LE_LE(TRUE, x1, x0) → 249_0_LE_LE(-(x1, 1), -(x0, 1)) the following chains were created:

• We consider the chain COND_249_0_LE_LE(TRUE, x1[1], x0[1]) → 249_0_LE_LE(-(x1[1], 1), -(x0[1], 1)) which results in the following constraint:
  

  (8)    (COND_249_0_LE_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_249_0_LE_LE(TRUE, x1[1], x0[1])≥249_0_LE_LE(-(x1[1], 1), -(x0[1], 1))∧(U^Increasing(249_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(249_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(249_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(249_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(249_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.

• 249_0_LE_LE(x1, x0) → COND_249_0_LE_LE(&&(>(x1, 0), >(x0, 0)), x1, x0)
  
  • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_249_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_249_0_LE_LE(TRUE, x1, x0) → 249_0_LE_LE(-(x1, 1), -(x0, 1))
  
  • ((U^Increasing(249_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(249_0_le_LE(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(295_0_le_Return) = [-1]   
POL(385_1_le_InvokeMethod(x₁, x₂, x₃)) = [-1]   
POL(408_0_le_Return(x₁, x₂)) = [-1]   
POL(427_0_le_Return(x₁, x₂)) = [-1]   
POL(1) = [1]   
POL(249_0_LE_LE(x₁, x₂)) = [1] + [2]x₂ + [2]x₁   
POL(COND_249_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_249_0_LE_LE(TRUE, x1[1], x0[1]) → 249_0_LE_LE(-(x1[1], 1), -(x0[1], 1))

The following pairs are in P_bound:

249_0_LE_LE(x1[0], x0[0]) → COND_249_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])

The following pairs are in P_≥:

249_0_LE_LE(x1[0], x0[0]) → COND_249_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:
249_0_le_LE(x1, 0) → 295_0_le_Return
385_1_le_InvokeMethod(408_0_le_Return(x0, x1), x0, x3) → 427_0_le_Return(x2, x1)
385_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x3) → 427_0_le_Return(x2, x1)
385_1_le_InvokeMethod(295_0_le_Return, 0, x3) → 427_0_le_Return(x1, 1)
385_1_le_InvokeMethod(295_0_le_Return, x2, 0) → 427_0_le_Return(x1, 0)

The integer pair graph contains the following rules and edges:
(0): 249_0_LE_LE(x1[0], x0[0]) → COND_249_0_LE_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])


The set Q consists of the following terms:
249_0_le_LE(x0, 0)
385_1_le_InvokeMethod(408_0_le_Return(x0, x1), x0, x2)
385_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x2)
385_1_le_InvokeMethod(295_0_le_Return, 0, x0)
385_1_le_InvokeMethod(295_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:
249_0_le_LE(x1, 0) → 295_0_le_Return
385_1_le_InvokeMethod(408_0_le_Return(x0, x1), x0, x3) → 427_0_le_Return(x2, x1)
385_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x3) → 427_0_le_Return(x2, x1)
385_1_le_InvokeMethod(295_0_le_Return, 0, x3) → 427_0_le_Return(x1, 1)
385_1_le_InvokeMethod(295_0_le_Return, x2, 0) → 427_0_le_Return(x1, 0)

The integer pair graph contains the following rules and edges:
(1): COND_249_0_LE_LE(TRUE, x1[1], x0[1]) → 249_0_LE_LE(x1[1] - 1, x0[1] - 1)


The set Q consists of the following terms:
249_0_le_LE(x0, 0)
385_1_le_InvokeMethod(408_0_le_Return(x0, x1), x0, x2)
385_1_le_InvokeMethod(427_0_le_Return(x0, x1), x0, x2)
385_1_le_InvokeMethod(295_0_le_Return, 0, x0)
385_1_le_InvokeMethod(295_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.

(14) TRUE