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:
360_0_f_Store(EOS(STATIC_360), i62, matching1) → 362_0_f_Load(EOS(STATIC_362), i62) | =(matching1, 0)
362_0_f_Load(EOS(STATIC_362), i62) → 364_0_f_LE(EOS(STATIC_364), i62, i62)
364_0_f_LE(EOS(STATIC_364), i67, i67) → 367_0_f_LE(EOS(STATIC_367), i67, i67)
367_0_f_LE(EOS(STATIC_367), i67, i67) → 371_0_f_ConstantStackPush(EOS(STATIC_371), i67) | >(i67, 0)
371_0_f_ConstantStackPush(EOS(STATIC_371), i67) → 375_0_f_Store(EOS(STATIC_375), i67, 2)
375_0_f_Store(EOS(STATIC_375), i67, matching1) → 381_0_f_Load(EOS(STATIC_381), i67, 2) | =(matching1, 2)
381_0_f_Load(EOS(STATIC_381), i67, matching1) → 432_0_f_Load(EOS(STATIC_432), i67, 2) | =(matching1, 2)
432_0_f_Load(EOS(STATIC_432), i67, i82) → 479_0_f_Load(EOS(STATIC_479), i67, i82)
479_0_f_Load(EOS(STATIC_479), i67, i97) → 483_0_f_LE(EOS(STATIC_483), i67, i97, i97)
483_0_f_LE(EOS(STATIC_483), i67, i106, i106) → 485_0_f_LE(EOS(STATIC_485), i67, i106, i106)
485_0_f_LE(EOS(STATIC_485), i67, i106, i106) → 490_0_f_FieldAccess(EOS(STATIC_490), i67, i106) | >(i106, 0)
490_0_f_FieldAccess(EOS(STATIC_490), i67, i106) → 494_0_f_ConstantStackPush(EOS(STATIC_494), i67, i106)
494_0_f_ConstantStackPush(EOS(STATIC_494), i67, i106) → 499_0_f_IntArithmetic(EOS(STATIC_499), i67, i106)
499_0_f_IntArithmetic(EOS(STATIC_499), i67, i106) → 509_0_f_FieldAccess(EOS(STATIC_509), i67, i106)
509_0_f_FieldAccess(EOS(STATIC_509), i67, i106) → 513_0_f_Load(EOS(STATIC_513), i67, i106)
513_0_f_Load(EOS(STATIC_513), i67, i106) → 517_0_f_Load(EOS(STATIC_517), i67, i106, i67)
517_0_f_Load(EOS(STATIC_517), i67, i106, i67) → 518_0_f_IntArithmetic(EOS(STATIC_518), i67, i106, i67, i106)
518_0_f_IntArithmetic(EOS(STATIC_518), i67, i106, i67, i106) → 520_0_f_InvokeMethod(EOS(STATIC_520), i67, i106, -(i67, i106)) | &&(>(i67, 0), >(i106, 0))
520_0_f_InvokeMethod(EOS(STATIC_520), i67, i106, i115) → 522_1_f_InvokeMethod(522_0_f_ConstantStackPush(EOS(STATIC_522), i115), i67, i106, i115)
522_0_f_ConstantStackPush(EOS(STATIC_522), i115) → 524_0_f_ConstantStackPush(EOS(STATIC_524), i115)
522_1_f_InvokeMethod(369_0_f_Return(EOS(STATIC_369)), i67, i106, i118) → 535_0_f_Return(EOS(STATIC_535), i67, i106, i118)
522_1_f_InvokeMethod(488_0_f_Return(EOS(STATIC_488)), i67, i106, i121) → 542_0_f_Return(EOS(STATIC_542), i67, i106, i121)
524_0_f_ConstantStackPush(EOS(STATIC_524), i115) → 358_0_f_ConstantStackPush(EOS(STATIC_358), i115)
358_0_f_ConstantStackPush(EOS(STATIC_358), i62) → 360_0_f_Store(EOS(STATIC_360), i62, 0)
535_0_f_Return(EOS(STATIC_535), i67, i106, i118) → 543_0_f_Return(EOS(STATIC_543), i67, i106, i118)
543_0_f_Return(EOS(STATIC_543), i67, i106, i123) → 546_0_f_Load(EOS(STATIC_546), i67, i106)
546_0_f_Load(EOS(STATIC_546), i67, i106) → 548_0_f_ConstantStackPush(EOS(STATIC_548), i67, i106)
548_0_f_ConstantStackPush(EOS(STATIC_548), i67, i106) → 550_0_f_IntArithmetic(EOS(STATIC_550), i67, i106, 1)
550_0_f_IntArithmetic(EOS(STATIC_550), i67, i106, matching1) → 552_0_f_Store(EOS(STATIC_552), i67, -(i106, 1)) | &&(>(i106, 0), =(matching1, 1))
552_0_f_Store(EOS(STATIC_552), i67, i127) → 554_0_f_JMP(EOS(STATIC_554), i67, i127)
554_0_f_JMP(EOS(STATIC_554), i67, i127) → 557_0_f_Load(EOS(STATIC_557), i67, i127)
557_0_f_Load(EOS(STATIC_557), i67, i127) → 479_0_f_Load(EOS(STATIC_479), i67, i127)
542_0_f_Return(EOS(STATIC_542), i67, i106, i121) → 543_0_f_Return(EOS(STATIC_543), i67, i106, i121)
R rules:
364_0_f_LE(EOS(STATIC_364), i66, i66) → 366_0_f_LE(EOS(STATIC_366), i66, i66)
366_0_f_LE(EOS(STATIC_366), i66, i66) → 369_0_f_Return(EOS(STATIC_369)) | <=(i66, 0)
483_0_f_LE(EOS(STATIC_483), i67, matching1, matching2) → 484_0_f_LE(EOS(STATIC_484), i67, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
484_0_f_LE(EOS(STATIC_484), i67, matching1, matching2) → 488_0_f_Return(EOS(STATIC_488)) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))

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

P rules:
360_0_f_Store(EOS(STATIC_360), x0, 0) → 522_1_f_InvokeMethod(360_0_f_Store(EOS(STATIC_360), -(x0, 2), 0), x0, 2, -(x0, 2)) | >(x0, 0)
522_1_f_InvokeMethod(369_0_f_Return(EOS(STATIC_369)), x0, x1, x2) → 522_1_f_InvokeMethod(360_0_f_Store(EOS(STATIC_360), -(x0, -(x1, 1)), 0), x0, -(x1, 1), -(x0, -(x1, 1))) | &&(>(x1, 1), >(x0, 0))
522_1_f_InvokeMethod(488_0_f_Return(EOS(STATIC_488)), x0, x1, x2) → 522_1_f_InvokeMethod(360_0_f_Store(EOS(STATIC_360), -(x0, -(x1, 1)), 0), x0, -(x1, 1), -(x0, -(x1, 1))) | &&(>(x1, 1), >(x0, 0))
R rules:

Filtered ground terms:

360_0_f_Store(x1, x2, x3) → 360_0_f_Store(x2)
Cond_522_1_f_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_522_1_f_InvokeMethod1(x1, x3, x4, x5)
488_0_f_Return(x1) → 488_0_f_Return
Cond_522_1_f_InvokeMethod(x1, x2, x3, x4, x5) → Cond_522_1_f_InvokeMethod(x1, x3, x4, x5)
369_0_f_Return(x1) → 369_0_f_Return
Cond_360_0_f_Store(x1, x2, x3, x4) → Cond_360_0_f_Store(x1, x3)

Filtered unneeded arguments:

522_1_f_InvokeMethod(x1, x2, x3, x4) → 522_1_f_InvokeMethod(x1, x2, x3)
Cond_522_1_f_InvokeMethod(x1, x2, x3, x4) → Cond_522_1_f_InvokeMethod(x1, x2, x3)
Cond_522_1_f_InvokeMethod1(x1, x2, x3, x4) → Cond_522_1_f_InvokeMethod1(x1, x2, x3)

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

P rules:
360_0_f_Store(x0) → 522_1_f_InvokeMethod(360_0_f_Store(-(x0, 2)), x0, 2) | >(x0, 0)
522_1_f_InvokeMethod(369_0_f_Return, x0, x1) → 522_1_f_InvokeMethod(360_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1)) | &&(>(x1, 1), >(x0, 0))
522_1_f_InvokeMethod(488_0_f_Return, x0, x1) → 522_1_f_InvokeMethod(360_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1)) | &&(>(x1, 1), >(x0, 0))
R rules:

Performed bisimulation on rules. Used the following equivalence classes: {[369_0_f_Return, 488_0_f_Return]=369_0_f_Return, [Cond_522_1_f_InvokeMethod_4, Cond_522_1_f_InvokeMethod1_4]=Cond_522_1_f_InvokeMethod_4}

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

P rules:
360_0_F_STORE(x0) → COND_360_0_F_STORE(>(x0, 0), x0)
COND_360_0_F_STORE(TRUE, x0) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0, 2)), x0, 2)
COND_360_0_F_STORE(TRUE, x0) → 360_0_F_STORE(-(x0, 2))
522_1_F_INVOKEMETHOD(369_0_f_Return, x0, x1) → COND_522_1_F_INVOKEMETHOD(&&(>(x1, 1), >(x0, 0)), 369_0_f_Return, x0, x1)
COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0, x1) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1))
COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0, x1) → 360_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): 360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(x0[0] > 0, x0[0])
(1): COND_360_0_F_STORE(TRUE, x0[1]) → 522_1_F_INVOKEMETHOD(360_0_f_Store(x0[1] - 2), x0[1], 2)
(2): COND_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(x0[2] - 2)
(3): 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3]) → COND_522_1_F_INVOKEMETHOD(x1[3] > 1 && x0[3] > 0, 369_0_f_Return, x0[3], x1[3])
(4): COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[4], x1[4]) → 522_1_F_INVOKEMETHOD(360_0_f_Store(x0[4] - x1[4] - 1), x0[4], x1[4] - 1)
(5): COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[5], x1[5]) → 360_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 (360_0_f_Store(x0[1] - 2) →^* 369_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 (360_0_f_Store(x0[4] - x1[4] - 1) →^* 369_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@e381d2b 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 360_0_F_STORE(x0) → COND_360_0_F_STORE(>(x0, 0), x0) the following chains were created:

• We consider the chain 360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0]), COND_360_0_F_STORE(TRUE, x0[1]) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[1], 2)), x0[1], 2) which results in the following constraint:
  

  (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 360_0_F_STORE(x0[0])≥NonInfC∧360_0_F_STORE(x0[0])≥COND_360_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_360_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 ⇒ 360_0_F_STORE(x0[0])≥NonInfC∧360_0_F_STORE(x0[0])≥COND_360_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_360_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_360_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_360_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_360_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_360_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 360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0]), COND_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(-(x0[2], 2)) which results in the following constraint:
  

  (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 360_0_F_STORE(x0[0])≥NonInfC∧360_0_F_STORE(x0[0])≥COND_360_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_360_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 ⇒ 360_0_F_STORE(x0[0])≥NonInfC∧360_0_F_STORE(x0[0])≥COND_360_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_360_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_360_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_360_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_360_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_360_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_360_0_F_STORE(TRUE, x0) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0, 2)), x0, 2) the following chains were created:

• We consider the chain 360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0]), COND_360_0_F_STORE(TRUE, x0[1]) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[1], 2)), x0[1], 2), 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3]) → COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3]) which results in the following constraint:
  

  (13)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1]∧360_0_f_Store(-(x0[1], 2))=369_0_f_Return∧x0[1]=x0[3]∧2=x1[3] ⇒ COND_360_0_F_STORE(TRUE, x0[1])≥NonInfC∧COND_360_0_F_STORE(TRUE, x0[1])≥522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[1], 2)), x0[1], 2)∧(U^Increasing(522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[1], 2)), x0[1], 2)), ≥))

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

For Pair COND_360_0_F_STORE(TRUE, x0) → 360_0_F_STORE(-(x0, 2)) the following chains were created:

• We consider the chain 360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0]), COND_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(-(x0[2], 2)), 360_0_F_STORE(x0[0]) → COND_360_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_360_0_F_STORE(TRUE, x0[2])≥NonInfC∧COND_360_0_F_STORE(TRUE, x0[2])≥360_0_F_STORE(-(x0[2], 2))∧(U^Increasing(360_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_360_0_F_STORE(TRUE, x0[0])≥NonInfC∧COND_360_0_F_STORE(TRUE, x0[0])≥360_0_F_STORE(-(x0[0], 2))∧(U^Increasing(360_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(360_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(360_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(360_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(360_0_F_STORE(-(x0[2], 2))), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)

  
  
  

For Pair 522_1_F_INVOKEMETHOD(369_0_f_Return, x0, x1) → COND_522_1_F_INVOKEMETHOD(&&(>(x1, 1), >(x0, 0)), 369_0_f_Return, x0, x1) the following chains were created:

• We consider the chain 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3]) → COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3]), COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[4], x1[4]) → 522_1_F_INVOKEMETHOD(360_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] ⇒ 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3])≥NonInfC∧522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3])≥COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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 ⇒ 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3])≥NonInfC∧522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3])≥COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3]) → COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3]), COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[5], x1[5]) → 360_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] ⇒ 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3])≥NonInfC∧522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3])≥COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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 ⇒ 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3])≥NonInfC∧522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3])≥COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0, x1) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1)) the following chains were created:

• We consider the chain 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3]) → COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3]), COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[4], x1[4]) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1)), 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3]) → COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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]∧360_0_f_Store(-(x0[4], -(x1[4], 1)))=369_0_f_Return∧x0[4]=x0[3]1∧-(x1[4], 1)=x1[3]1 ⇒ COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[4], x1[4])≥NonInfC∧COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[4], x1[4])≥522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))∧(U^Increasing(522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))), ≥))

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

For Pair COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0, x1) → 360_0_F_STORE(-(x0, -(x1, 1))) the following chains were created:

• We consider the chain 522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3]) → COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3]), COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[5], x1[5]) → 360_0_F_STORE(-(x0[5], -(x1[5], 1))), 360_0_F_STORE(x0[0]) → COND_360_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_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[5], x1[5])≥NonInfC∧COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[5], x1[5])≥360_0_F_STORE(-(x0[5], -(x1[5], 1)))∧(U^Increasing(360_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_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[3], x1[3])≥NonInfC∧COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[3], x1[3])≥360_0_F_STORE(-(x0[3], -(x1[3], 1)))∧(U^Increasing(360_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(360_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(360_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(360_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(360_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.

• 360_0_F_STORE(x0) → COND_360_0_F_STORE(>(x0, 0), x0)
  
  • (0 ≥ 0 ⇒ (U^Increasing(COND_360_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_360_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_360_0_F_STORE(TRUE, x0) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0, 2)), x0, 2)
  
  
• COND_360_0_F_STORE(TRUE, x0) → 360_0_F_STORE(-(x0, 2))
  
  • (0 ≥ 0 ⇒ (U^Increasing(360_0_F_STORE(-(x0[2], 2))), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)
    
  
  
• 522_1_F_INVOKEMETHOD(369_0_f_Return, x0, x1) → COND_522_1_F_INVOKEMETHOD(&&(>(x1, 1), >(x0, 0)), 369_0_f_Return, x0, x1)
  
  • (0 ≥ 0 ⇒ (U^Increasing(COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_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_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0, x1) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1))
  
  
• COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0, x1) → 360_0_F_STORE(-(x0, -(x1, 1)))
  
  • (0 ≥ 0 ⇒ (U^Increasing(360_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(360_0_F_STORE(x₁)) = [-1] + x₁   
POL(COND_360_0_F_STORE(x₁, x₂)) = [-1]   
POL(>(x₁, x₂)) = 0   
POL(0) = 0   
POL(522_1_F_INVOKEMETHOD(x₁, x₂, x₃)) = [1] + x₃ + x₂ + [-1]x₁   
POL(360_0_f_Store(x₁)) = 0   
POL(-(x₁, x₂)) = 0   
POL(2) = 0   
POL(369_0_f_Return) = 0   
POL(COND_522_1_F_INVOKEMETHOD(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₂   
POL(&&(x₁, x₂)) = 0   
POL(1) = 0   

The following pairs are in P_>:

COND_360_0_F_STORE(TRUE, x0[1]) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[1], 2)), x0[1], 2)
522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3]) → COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3])
COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[4], x1[4]) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))

The following pairs are in P_bound:

360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0])
COND_360_0_F_STORE(TRUE, x0[1]) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[1], 2)), x0[1], 2)
COND_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(-(x0[2], 2))
522_1_F_INVOKEMETHOD(369_0_f_Return, x0[3], x1[3]) → COND_522_1_F_INVOKEMETHOD(&&(>(x1[3], 1), >(x0[3], 0)), 369_0_f_Return, x0[3], x1[3])
COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[4], x1[4]) → 522_1_F_INVOKEMETHOD(360_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))
COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[5], x1[5]) → 360_0_F_STORE(-(x0[5], -(x1[5], 1)))

The following pairs are in P_≥:

360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0])
COND_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(-(x0[2], 2))
COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[5], x1[5]) → 360_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): 360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(x0[0] > 0, x0[0])
(2): COND_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(x0[2] - 2)
(5): COND_522_1_F_INVOKEMETHOD(TRUE, 369_0_f_Return, x0[5], x1[5]) → 360_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_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(x0[2] - 2)
(0): 360_0_F_STORE(x0[0]) → COND_360_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@5e26e232 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_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(-(x0[2], 2)) the following chains were created:

• We consider the chain 360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0]), COND_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(-(x0[2], 2)), 360_0_F_STORE(x0[0]) → COND_360_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_360_0_F_STORE(TRUE, x0[2])≥NonInfC∧COND_360_0_F_STORE(TRUE, x0[2])≥360_0_F_STORE(-(x0[2], 2))∧(U^Increasing(360_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_360_0_F_STORE(TRUE, x0[0])≥NonInfC∧COND_360_0_F_STORE(TRUE, x0[0])≥360_0_F_STORE(-(x0[0], 2))∧(U^Increasing(360_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(360_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(360_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(360_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(360_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 360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0]) the following chains were created:

• We consider the chain 360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0]), COND_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(-(x0[2], 2)) which results in the following constraint:
  

  (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 360_0_F_STORE(x0[0])≥NonInfC∧360_0_F_STORE(x0[0])≥COND_360_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_360_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 ⇒ 360_0_F_STORE(x0[0])≥NonInfC∧360_0_F_STORE(x0[0])≥COND_360_0_F_STORE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_360_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_360_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_360_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_360_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_360_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_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(-(x0[2], 2))
  
  • (x0[0] ≥ 0 ⇒ (U^Increasing(360_0_F_STORE(-(x0[2], 2))), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)
    
  
  
• 360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0])
  
  • (x0[0] ≥ 0 ⇒ (U^Increasing(COND_360_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_360_0_F_STORE(x₁, x₂)) = [2] + x₂   
POL(360_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_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(-(x0[2], 2))

The following pairs are in P_bound:

COND_360_0_F_STORE(TRUE, x0[2]) → 360_0_F_STORE(-(x0[2], 2))
360_0_F_STORE(x0[0]) → COND_360_0_F_STORE(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

360_0_F_STORE(x0[0]) → COND_360_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): 360_0_F_STORE(x0[0]) → COND_360_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