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: TerminatorRec01

public class TerminatorRec01 {
	static int z = 0;

	public static void main(String[] args) {
		z = args.length;
		f(z);
	}

	public static void f(int x) {
		int y = 0;
		if (x > 0) {
			y = 2;
			while (y > 0) {
				z = z - 1;
				f(x - y);
				y = y - 1;
			}
		}
	}
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
TerminatorRec01.main([Ljava/lang/String;)V: Graph of 39 nodes with 0 SCCs.

TerminatorRec01.f(I)V: Graph of 37 nodes with 1 SCC.

(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: TerminatorRec01.f(I)V
SCC calls the following helper methods: TerminatorRec01.f(I)V
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 33 rules for P and 4 rules for R.

P rules:
372_0_f_Store(EOS(STATIC_372), i62, matching1) → 374_0_f_Load(EOS(STATIC_374), i62) | =(matching1, 0)
374_0_f_Load(EOS(STATIC_374), i62) → 377_0_f_LE(EOS(STATIC_377), i62, i62)
377_0_f_LE(EOS(STATIC_377), i71, i71) → 381_0_f_LE(EOS(STATIC_381), i71, i71)
381_0_f_LE(EOS(STATIC_381), i71, i71) → 385_0_f_ConstantStackPush(EOS(STATIC_385), i71) | >(i71, 0)
385_0_f_ConstantStackPush(EOS(STATIC_385), i71) → 389_0_f_Store(EOS(STATIC_389), i71, 2)
389_0_f_Store(EOS(STATIC_389), i71, matching1) → 397_0_f_Load(EOS(STATIC_397), i71, 2) | =(matching1, 2)
397_0_f_Load(EOS(STATIC_397), i71, matching1) → 451_0_f_Load(EOS(STATIC_451), i71, 2) | =(matching1, 2)
451_0_f_Load(EOS(STATIC_451), i71, i85) → 499_0_f_Load(EOS(STATIC_499), i71, i85)
499_0_f_Load(EOS(STATIC_499), i71, i102) → 502_0_f_LE(EOS(STATIC_502), i71, i102, i102)
502_0_f_LE(EOS(STATIC_502), i71, i108, i108) → 505_0_f_LE(EOS(STATIC_505), i71, i108, i108)
505_0_f_LE(EOS(STATIC_505), i71, i108, i108) → 509_0_f_FieldAccess(EOS(STATIC_509), i71, i108) | >(i108, 0)
509_0_f_FieldAccess(EOS(STATIC_509), i71, i108) → 513_0_f_ConstantStackPush(EOS(STATIC_513), i71, i108)
513_0_f_ConstantStackPush(EOS(STATIC_513), i71, i108) → 519_0_f_IntArithmetic(EOS(STATIC_519), i71, i108)
519_0_f_IntArithmetic(EOS(STATIC_519), i71, i108) → 530_0_f_FieldAccess(EOS(STATIC_530), i71, i108)
530_0_f_FieldAccess(EOS(STATIC_530), i71, i108) → 534_0_f_Load(EOS(STATIC_534), i71, i108)
534_0_f_Load(EOS(STATIC_534), i71, i108) → 537_0_f_Load(EOS(STATIC_537), i71, i108, i71)
537_0_f_Load(EOS(STATIC_537), i71, i108, i71) → 539_0_f_IntArithmetic(EOS(STATIC_539), i71, i108, i71, i108)
539_0_f_IntArithmetic(EOS(STATIC_539), i71, i108, i71, i108) → 541_0_f_InvokeMethod(EOS(STATIC_541), i71, i108, -(i71, i108)) | &&(>(i71, 0), >(i108, 0))
541_0_f_InvokeMethod(EOS(STATIC_541), i71, i108, i116) → 543_1_f_InvokeMethod(543_0_f_ConstantStackPush(EOS(STATIC_543), i116), i71, i108, i116)
543_0_f_ConstantStackPush(EOS(STATIC_543), i116) → 545_0_f_ConstantStackPush(EOS(STATIC_545), i116)
543_1_f_InvokeMethod(383_0_f_Return(EOS(STATIC_383)), i71, i108, i120) → 554_0_f_Return(EOS(STATIC_554), i71, i108, i120)
543_1_f_InvokeMethod(508_0_f_Return(EOS(STATIC_508)), i71, i108, i123) → 559_0_f_Return(EOS(STATIC_559), i71, i108, i123)
545_0_f_ConstantStackPush(EOS(STATIC_545), i116) → 370_0_f_ConstantStackPush(EOS(STATIC_370), i116)
370_0_f_ConstantStackPush(EOS(STATIC_370), i62) → 372_0_f_Store(EOS(STATIC_372), i62, 0)
554_0_f_Return(EOS(STATIC_554), i71, i108, i120) → 560_0_f_Return(EOS(STATIC_560), i71, i108, i120)
560_0_f_Return(EOS(STATIC_560), i71, i108, i125) → 563_0_f_Load(EOS(STATIC_563), i71, i108)
563_0_f_Load(EOS(STATIC_563), i71, i108) → 565_0_f_ConstantStackPush(EOS(STATIC_565), i71, i108)
565_0_f_ConstantStackPush(EOS(STATIC_565), i71, i108) → 567_0_f_IntArithmetic(EOS(STATIC_567), i71, i108, 1)
567_0_f_IntArithmetic(EOS(STATIC_567), i71, i108, matching1) → 569_0_f_Store(EOS(STATIC_569), i71, -(i108, 1)) | &&(>(i108, 0), =(matching1, 1))
569_0_f_Store(EOS(STATIC_569), i71, i129) → 570_0_f_JMP(EOS(STATIC_570), i71, i129)
570_0_f_JMP(EOS(STATIC_570), i71, i129) → 573_0_f_Load(EOS(STATIC_573), i71, i129)
573_0_f_Load(EOS(STATIC_573), i71, i129) → 499_0_f_Load(EOS(STATIC_499), i71, i129)
559_0_f_Return(EOS(STATIC_559), i71, i108, i123) → 560_0_f_Return(EOS(STATIC_560), i71, i108, i123)
R rules:
377_0_f_LE(EOS(STATIC_377), i70, i70) → 380_0_f_LE(EOS(STATIC_380), i70, i70)
380_0_f_LE(EOS(STATIC_380), i70, i70) → 383_0_f_Return(EOS(STATIC_383)) | <=(i70, 0)
502_0_f_LE(EOS(STATIC_502), i71, matching1, matching2) → 504_0_f_LE(EOS(STATIC_504), i71, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
504_0_f_LE(EOS(STATIC_504), i71, matching1, matching2) → 508_0_f_Return(EOS(STATIC_508)) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))

Combined rules. Obtained 3 conditional rules for P and 0 conditional rules for R.

P rules:
372_0_f_Store(EOS(STATIC_372), x0, 0) → 543_1_f_InvokeMethod(372_0_f_Store(EOS(STATIC_372), -(x0, 2), 0), x0, 2, -(x0, 2)) | >(x0, 0)
543_1_f_InvokeMethod(383_0_f_Return(EOS(STATIC_383)), x0, x1, x2) → 543_1_f_InvokeMethod(372_0_f_Store(EOS(STATIC_372), -(x0, -(x1, 1)), 0), x0, -(x1, 1), -(x0, -(x1, 1))) | &&(>(x1, 1), >(x0, 0))
543_1_f_InvokeMethod(508_0_f_Return(EOS(STATIC_508)), x0, x1, x2) → 543_1_f_InvokeMethod(372_0_f_Store(EOS(STATIC_372), -(x0, -(x1, 1)), 0), x0, -(x1, 1), -(x0, -(x1, 1))) | &&(>(x1, 1), >(x0, 0))
R rules:

Filtered ground terms:

372_0_f_Store(x1, x2, x3) → 372_0_f_Store(x2)
Cond_543_1_f_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_543_1_f_InvokeMethod1(x1, x3, x4, x5)
508_0_f_Return(x1) → 508_0_f_Return
Cond_543_1_f_InvokeMethod(x1, x2, x3, x4, x5) → Cond_543_1_f_InvokeMethod(x1, x3, x4, x5)
383_0_f_Return(x1) → 383_0_f_Return
Cond_372_0_f_Store(x1, x2, x3, x4) → Cond_372_0_f_Store(x1, x3)

Filtered unneeded arguments:

543_1_f_InvokeMethod(x1, x2, x3, x4) → 543_1_f_InvokeMethod(x1, x2, x3)
Cond_543_1_f_InvokeMethod(x1, x2, x3, x4) → Cond_543_1_f_InvokeMethod(x1, x2, x3)
Cond_543_1_f_InvokeMethod1(x1, x2, x3, x4) → Cond_543_1_f_InvokeMethod1(x1, x2, x3)

Combined rules. Obtained 3 conditional rules for P and 0 conditional rules for R.

P rules:
372_0_f_Store(x0) → 543_1_f_InvokeMethod(372_0_f_Store(-(x0, 2)), x0, 2) | >(x0, 0)
543_1_f_InvokeMethod(383_0_f_Return, x0, x1) → 543_1_f_InvokeMethod(372_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1)) | &&(>(x1, 1), >(x0, 0))
543_1_f_InvokeMethod(508_0_f_Return, x0, x1) → 543_1_f_InvokeMethod(372_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1)) | &&(>(x1, 1), >(x0, 0))
R rules:

Performed bisimulation on rules. Used the following equivalence classes: {[383_0_f_Return, 508_0_f_Return]=383_0_f_Return, [Cond_543_1_f_InvokeMethod_4, Cond_543_1_f_InvokeMethod1_4]=Cond_543_1_f_InvokeMethod_4}

Finished conversion. Obtained 6 rules for P and 0 rules for R. System has predefined symbols.

P rules:
372_0_F_STORE(x0) → COND_372_0_F_STORE(>(x0, 0), x0)
COND_372_0_F_STORE(TRUE, x0) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0, 2)), x0, 2)
COND_372_0_F_STORE(TRUE, x0) → 372_0_F_STORE(-(x0, 2))
543_1_F_INVOKEMETHOD(383_0_f_Return, x0, x1) → COND_543_1_F_INVOKEMETHOD(&&(>(x1, 1), >(x0, 0)), 383_0_f_Return, x0, x1)
COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0, x1) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1))
COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0, x1) → 372_0_F_STORE(-(x0, -(x1, 1)))
R rules:

(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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(x0[0] > 0, x0[0])
(1): COND_372_0_F_STORE(TRUE, x0[1]) → 543_1_F_INVOKEMETHOD(372_0_f_Store(x0[1] - 2), x0[1], 2)
(2): COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(x0[2] - 2)
(3): 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3]) → COND_543_1_F_INVOKEMETHOD(x1[3] > 1 && x0[3] > 0, 383_0_f_Return, x0[3], x1[3])
(4): COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[4], x1[4]) → 543_1_F_INVOKEMETHOD(372_0_f_Store(x0[4] - x1[4] - 1), x0[4], x1[4] - 1)
(5): COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[5], x1[5]) → 372_0_F_STORE(x0[5] - x1[5] - 1)

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

(0) -> (2), if (x0[0] > 0 ∧x0[0] →^* x0[2])

(1) -> (3), if (372_0_f_Store(x0[1] - 2) →^* 383_0_f_Return∧x0[1] →^* x0[3]∧2 →^* x1[3])

(2) -> (0), if (x0[2] - 2 →^* x0[0])

(3) -> (4), if (x1[3] > 1 && x0[3] > 0 ∧x0[3] →^* x0[4]∧x1[3] →^* x1[4])

(3) -> (5), if (x1[3] > 1 && x0[3] > 0 ∧x0[3] →^* x0[5]∧x1[3] →^* x1[5])

(4) -> (3), if (372_0_f_Store(x0[4] - x1[4] - 1) →^* 383_0_f_Return∧x0[4] →^* x0[3]∧x1[4] - 1 →^* x1[3])

(5) -> (0), if (x0[5] - x1[5] - 1 →^* x0[0])

The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: true Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@491cb6f7 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 372_0_F_STORE(x0) → COND_372_0_F_STORE(>(x0, 0), x0) the following chains were created:

• We consider the chain 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0]), COND_372_0_F_STORE(TRUE, x0[1]) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[1], 2)), x0[1], 2) which results in the following constraint:
  

  (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 372_0_F_STORE(x0[0])≥NonInfC∧372_0_F_STORE(x0[0])≥COND_372_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥))

  
  
  We simplified constraint (1) using rule (IV) which results in the following new constraint:
  

  (2)    (>(x0[0], 0)=TRUE ⇒ 372_0_F_STORE(x0[0])≥NonInfC∧372_0_F_STORE(x0[0])≥COND_372_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥))

  
  
  We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (3)    (0 ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)

  
  
  We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (4)    (0 ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)

  
  
  We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (5)    (0 ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)

  
  
  We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (6)    (0 ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)

  
  
  
• We consider the chain 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0]), COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(-(x0[2], 2)) which results in the following constraint:
  

  (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 372_0_F_STORE(x0[0])≥NonInfC∧372_0_F_STORE(x0[0])≥COND_372_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥))

  
  
  We simplified constraint (7) using rule (IV) which results in the following new constraint:
  

  (8)    (>(x0[0], 0)=TRUE ⇒ 372_0_F_STORE(x0[0])≥NonInfC∧372_0_F_STORE(x0[0])≥COND_372_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥))

  
  
  We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (9)    (0 ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)

  
  
  We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (10)    (0 ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)

  
  
  We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (11)    (0 ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)

  
  
  We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (12)    (0 ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)

  
  
  

For Pair COND_372_0_F_STORE(TRUE, x0) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0, 2)), x0, 2) the following chains were created:

• We consider the chain 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0]), COND_372_0_F_STORE(TRUE, x0[1]) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[1], 2)), x0[1], 2), 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3]) → COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3]) which results in the following constraint:
  

  (13)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1]∧372_0_f_Store(-(x0[1], 2))=383_0_f_Return∧x0[1]=x0[3]∧2=x1[3] ⇒ COND_372_0_F_STORE(TRUE, x0[1])≥NonInfC∧COND_372_0_F_STORE(TRUE, x0[1])≥543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[1], 2)), x0[1], 2)∧(U^Increasing(543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[1], 2)), x0[1], 2)), ≥))

  
  
  We solved constraint (13) using rules (I), (II).

For Pair COND_372_0_F_STORE(TRUE, x0) → 372_0_F_STORE(-(x0, 2)) the following chains were created:

• We consider the chain 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0]), COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(-(x0[2], 2)), 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:
  

  (14)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2]∧-(x0[2], 2)=x0[0]1 ⇒ COND_372_0_F_STORE(TRUE, x0[2])≥NonInfC∧COND_372_0_F_STORE(TRUE, x0[2])≥372_0_F_STORE(-(x0[2], 2))∧(U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥))

  
  
  We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:
  

  (15)    (>(x0[0], 0)=TRUE ⇒ COND_372_0_F_STORE(TRUE, x0[0])≥NonInfC∧COND_372_0_F_STORE(TRUE, x0[0])≥372_0_F_STORE(-(x0[0], 2))∧(U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥))

  
  
  We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (16)    (0 ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

  
  
  We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (17)    (0 ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

  
  
  We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (18)    (0 ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

  
  
  We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (19)    (0 ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)

  
  
  

For Pair 543_1_F_INVOKEMETHOD(383_0_f_Return, x0, x1) → COND_543_1_F_INVOKEMETHOD(&&(>(x1, 1), >(x0, 0)), 383_0_f_Return, x0, x1) the following chains were created:

• We consider the chain 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3]) → COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3]), COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[4], x1[4]) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1)) which results in the following constraint:
  

  (20)    (&&(>(x1[3], 1), >(x0[3], 0))=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4] ⇒ 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3])≥NonInfC∧543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3])≥COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥))

  
  
  We simplified constraint (20) using rule (IV) which results in the following new constraint:
  

  (21)    (&&(>(x1[3], 1), >(x0[3], 0))=TRUE ⇒ 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3])≥NonInfC∧543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3])≥COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥))

  
  
  We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (22)    (0 ≥ 0 ⇒ (U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)

  
  
  We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (23)    (0 ≥ 0 ⇒ (U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)

  
  
  We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (24)    (0 ≥ 0 ⇒ (U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)

  
  
  We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (25)    (0 ≥ 0 ⇒ (U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)

  
  
  
• We consider the chain 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3]) → COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3]), COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[5], x1[5]) → 372_0_F_STORE(-(x0[5], -(x1[5], 1))) which results in the following constraint:
  

  (26)    (&&(>(x1[3], 1), >(x0[3], 0))=TRUE∧x0[3]=x0[5]∧x1[3]=x1[5] ⇒ 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3])≥NonInfC∧543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3])≥COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥))

  
  
  We simplified constraint (26) using rule (IV) which results in the following new constraint:
  

  (27)    (&&(>(x1[3], 1), >(x0[3], 0))=TRUE ⇒ 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3])≥NonInfC∧543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3])≥COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥))

  
  
  We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (28)    (0 ≥ 0 ⇒ (U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)

  
  
  We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (29)    (0 ≥ 0 ⇒ (U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)

  
  
  We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (30)    (0 ≥ 0 ⇒ (U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)

  
  
  We simplified constraint (30) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (31)    (0 ≥ 0 ⇒ (U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)

  
  
  

For Pair COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0, x1) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1)) the following chains were created:

• We consider the chain 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3]) → COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3]), COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[4], x1[4]) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1)), 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3]) → COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3]) which results in the following constraint:
  

  (32)    (&&(>(x1[3], 1), >(x0[3], 0))=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧372_0_f_Store(-(x0[4], -(x1[4], 1)))=383_0_f_Return∧x0[4]=x0[3]1∧-(x1[4], 1)=x1[3]1 ⇒ COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[4], x1[4])≥NonInfC∧COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[4], x1[4])≥543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))∧(U^Increasing(543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))), ≥))

  
  
  We solved constraint (32) using rules (I), (II).

For Pair COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0, x1) → 372_0_F_STORE(-(x0, -(x1, 1))) the following chains were created:

• We consider the chain 543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3]) → COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3]), COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[5], x1[5]) → 372_0_F_STORE(-(x0[5], -(x1[5], 1))), 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:
  

  (33)    (&&(>(x1[3], 1), >(x0[3], 0))=TRUE∧x0[3]=x0[5]∧x1[3]=x1[5]∧-(x0[5], -(x1[5], 1))=x0[0] ⇒ COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[5], x1[5])≥NonInfC∧COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[5], x1[5])≥372_0_F_STORE(-(x0[5], -(x1[5], 1)))∧(U^Increasing(372_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥))

  
  
  We simplified constraint (33) using rules (III), (IV) which results in the following new constraint:
  

  (34)    (&&(>(x1[3], 1), >(x0[3], 0))=TRUE ⇒ COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[3], x1[3])≥NonInfC∧COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[3], x1[3])≥372_0_F_STORE(-(x0[3], -(x1[3], 1)))∧(U^Increasing(372_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥))

  
  
  We simplified constraint (34) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (35)    (0 ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)

  
  
  We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (36)    (0 ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)

  
  
  We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (37)    (0 ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)

  
  
  We simplified constraint (37) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (38)    (0 ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0)

  
  
  

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

• 372_0_F_STORE(x0) → COND_372_0_F_STORE(>(x0, 0), x0)
  
  • (0 ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)
    
  • (0 ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)
    
  
  
• COND_372_0_F_STORE(TRUE, x0) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0, 2)), x0, 2)
  
  
• COND_372_0_F_STORE(TRUE, x0) → 372_0_F_STORE(-(x0, 2))
  
  • (0 ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)
    
  
  
• 543_1_F_INVOKEMETHOD(383_0_f_Return, x0, x1) → COND_543_1_F_INVOKEMETHOD(&&(>(x1, 1), >(x0, 0)), 383_0_f_Return, x0, x1)
  
  • (0 ≥ 0 ⇒ (U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)
    
  • (0 ≥ 0 ⇒ (U^Increasing(COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)
    
  
  
• COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0, x1) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1))
  
  
• COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0, x1) → 372_0_F_STORE(-(x0, -(x1, 1)))
  
  • (0 ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(372_0_F_STORE(x₁)) = [-1] + x₁   
POL(COND_372_0_F_STORE(x₁, x₂)) = [-1]   
POL(>(x₁, x₂)) = 0   
POL(0) = 0   
POL(543_1_F_INVOKEMETHOD(x₁, x₂, x₃)) = [1] + x₃ + x₂ + [-1]x₁   
POL(372_0_f_Store(x₁)) = 0   
POL(-(x₁, x₂)) = 0   
POL(2) = 0   
POL(383_0_f_Return) = 0   
POL(COND_543_1_F_INVOKEMETHOD(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₂   
POL(&&(x₁, x₂)) = 0   
POL(1) = 0   

The following pairs are in P_>:

COND_372_0_F_STORE(TRUE, x0[1]) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[1], 2)), x0[1], 2)
543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3]) → COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])
COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[4], x1[4]) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))

The following pairs are in P_bound:

372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0])
COND_372_0_F_STORE(TRUE, x0[1]) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[1], 2)), x0[1], 2)
COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(-(x0[2], 2))
543_1_F_INVOKEMETHOD(383_0_f_Return, x0[3], x1[3]) → COND_543_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 383_0_f_Return, x0[3], x1[3])
COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[4], x1[4]) → 543_1_F_INVOKEMETHOD(372_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))
COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[5], x1[5]) → 372_0_F_STORE(-(x0[5], -(x1[5], 1)))

The following pairs are in P_≥:

372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0])
COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(-(x0[2], 2))
COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[5], x1[5]) → 372_0_F_STORE(-(x0[5], -(x1[5], 1)))

There are no usable rules.

(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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(x0[0] > 0, x0[0])
(2): COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(x0[2] - 2)
(5): COND_543_1_F_INVOKEMETHOD(TRUE, 383_0_f_Return, x0[5], x1[5]) → 372_0_F_STORE(x0[5] - x1[5] - 1)

(2) -> (0), if (x0[2] - 2 →^* x0[0])

(5) -> (0), if (x0[5] - x1[5] - 1 →^* x0[0])

(0) -> (2), if (x0[0] > 0 ∧x0[0] →^* x0[2])

The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(10) 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

R is empty.

The integer pair graph contains the following rules and edges:
(2): COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(x0[2] - 2)
(0): 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(x0[0] > 0, x0[0])

(2) -> (0), if (x0[2] - 2 →^* x0[0])

(0) -> (2), if (x0[0] > 0 ∧x0[0] →^* x0[2])

The set Q is empty.

(11) 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@6f9a2170 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 COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(-(x0[2], 2)) the following chains were created:

• We consider the chain 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0]), COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(-(x0[2], 2)), 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:
  

  (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2]∧-(x0[2], 2)=x0[0]1 ⇒ COND_372_0_F_STORE(TRUE, x0[2])≥NonInfC∧COND_372_0_F_STORE(TRUE, x0[2])≥372_0_F_STORE(-(x0[2], 2))∧(U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥))

  
  
  We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
  

  (2)    (>(x0[0], 0)=TRUE ⇒ COND_372_0_F_STORE(TRUE, x0[0])≥NonInfC∧COND_372_0_F_STORE(TRUE, x0[0])≥372_0_F_STORE(-(x0[0], 2))∧(U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥))

  
  
  We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (3)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

  
  
  We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (4)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

  
  
  We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (5)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

  
  
  We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (6)    (x0[0] ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

  
  
  

For Pair 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0]) the following chains were created:

• We consider the chain 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0]), COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(-(x0[2], 2)) which results in the following constraint:
  

  (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 372_0_F_STORE(x0[0])≥NonInfC∧372_0_F_STORE(x0[0])≥COND_372_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥))

  
  
  We simplified constraint (7) using rule (IV) which results in the following new constraint:
  

  (8)    (>(x0[0], 0)=TRUE ⇒ 372_0_F_STORE(x0[0])≥NonInfC∧372_0_F_STORE(x0[0])≥COND_372_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥))

  
  
  We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (9)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

  
  
  We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (10)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

  
  
  We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (11)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

  
  
  We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (12)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

  
  
  

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

• COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(-(x0[2], 2))
  
  • (x0[0] ≥ 0 ⇒ (U^Increasing(372_0_F_STORE(-(x0[2], 2))), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)
    
  
  
• 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0])
  
  • (x0[0] ≥ 0 ⇒ (U^Increasing(COND_372_0_F_STORE(>(x0[0], 0), x0[0])), ≥)∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 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) = [3]   
POL(FALSE) = 0   
POL(COND_372_0_F_STORE(x₁, x₂)) = [2] + x₂   
POL(372_0_F_STORE(x₁)) = [2] + x₁   
POL(-(x₁, x₂)) = x₁ + [-1]x₂   
POL(2) = [2]   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   

The following pairs are in P_>:

COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(-(x0[2], 2))

The following pairs are in P_bound:

COND_372_0_F_STORE(TRUE, x0[2]) → 372_0_F_STORE(-(x0[2], 2))
372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(>(x0[0], 0), x0[0])

There are no usable rules.

(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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 372_0_F_STORE(x0[0]) → COND_372_0_F_STORE(x0[0] > 0, x0[0])


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(14) TRUE