(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: CAppE
`public class CAppE {	CAppE next;	public static void main(String[] args) {		Random.args = args;		CAppE list = createList();		cappE(Random.random());	}	public static void cappE(int j) {		CAppE a = new CAppE();		if (j > 0) {			a.appE(j);			while (a.next == null) {}		}	}	public void appE(int i) {		if (next == null) {			if (i <= 0) {				return;			} else {				next = new CAppE();			}			i--;	 	}		next.appE(i);	}	public static CAppE createList() {		CAppE result = null;		int length = Random.random();		while (length > 0) {			result = new CAppE(result);			length--;		}		return result;	}	public CAppE() {		this.next = null;	}	public CAppE(CAppE n) {		this.next = n;	}}class Random {	static String[] args;	static int index = 0;	public static int random() {		String string = args[index];		index++;		return string.length();	}}`

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
CAppE.main([Ljava/lang/String;)V: Graph of 138 nodes with 0 SCCs.

CAppE.createList()LCAppE;: Graph of 91 nodes with 1 SCC.

CAppE.appE(I)V: Graph of 35 nodes with 0 SCCs.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 2 SCCss.

(5) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: CAppE.appE(I)V
SCC calls the following helper methods: CAppE.appE(I)V
Performed SCC analyses: UsedFieldsAnalysis

(6) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 24 rules for P and 9 rules for R.

P rules:
668_0_appE_FieldAccess(EOS(STATIC_668), i76) → 670_0_appE_NONNULL(EOS(STATIC_670), i76)
670_0_appE_NONNULL(EOS(STATIC_670), i76) → 671_0_appE_Load(EOS(STATIC_671), i76)
671_0_appE_Load(EOS(STATIC_671), i76) → 672_0_appE_GT(EOS(STATIC_672), i76, i76)
672_0_appE_GT(EOS(STATIC_672), i83, i83) → 675_0_appE_GT(EOS(STATIC_675), i83, i83)
675_0_appE_GT(EOS(STATIC_675), i83, i83) → 678_0_appE_Load(EOS(STATIC_678), i83) | >(i83, 0)
678_0_appE_Load(EOS(STATIC_678), i83) → 682_0_appE_New(EOS(STATIC_682), i83)
682_0_appE_New(EOS(STATIC_682), i83) → 687_0_appE_Duplicate(EOS(STATIC_687), i83)
687_0_appE_Duplicate(EOS(STATIC_687), i83) → 691_0_appE_InvokeMethod(EOS(STATIC_691), i83)
691_0_appE_InvokeMethod(EOS(STATIC_691), i83) → 692_0_<init>_Load(EOS(STATIC_692), i83)
692_0_<init>_Load(EOS(STATIC_692), i83) → 695_0_<init>_InvokeMethod(EOS(STATIC_695), i83)
695_0_<init>_InvokeMethod(EOS(STATIC_695), i83) → 697_0_<init>_Load(EOS(STATIC_697), i83)
697_0_<init>_Load(EOS(STATIC_697), i83) → 699_0_<init>_ConstantStackPush(EOS(STATIC_699), i83)
699_0_<init>_ConstantStackPush(EOS(STATIC_699), i83) → 701_0_<init>_FieldAccess(EOS(STATIC_701), i83)
701_0_<init>_FieldAccess(EOS(STATIC_701), i83) → 702_0_<init>_Return(EOS(STATIC_702), i83)
702_0_<init>_Return(EOS(STATIC_702), i83) → 705_0_appE_FieldAccess(EOS(STATIC_705), i83)
705_0_appE_FieldAccess(EOS(STATIC_705), i83) → 708_0_appE_Inc(EOS(STATIC_708), i83)
708_0_appE_Inc(EOS(STATIC_708), i83) → 710_0_appE_Load(EOS(STATIC_710), +(i83, -1)) | >(i83, 0)
710_0_appE_Load(EOS(STATIC_710), i86) → 712_0_appE_FieldAccess(EOS(STATIC_712), i86)
712_0_appE_FieldAccess(EOS(STATIC_712), i86) → 714_0_appE_Load(EOS(STATIC_714), i86)
714_0_appE_Load(EOS(STATIC_714), i86) → 716_0_appE_InvokeMethod(EOS(STATIC_716), i86)
716_0_appE_InvokeMethod(EOS(STATIC_716), i86) → 718_1_appE_InvokeMethod(718_0_appE_Load(EOS(STATIC_718), i86), i86)
665_0_appE_Load(EOS(STATIC_665), i76) → 668_0_appE_FieldAccess(EOS(STATIC_668), i76)
R rules:
672_0_appE_GT(EOS(STATIC_672), matching1, matching2) → 674_0_appE_GT(EOS(STATIC_674), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
674_0_appE_GT(EOS(STATIC_674), matching1, matching2) → 677_0_appE_Return(EOS(STATIC_677), 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
718_1_appE_InvokeMethod(677_0_appE_Return(EOS(STATIC_677), matching1), matching2) → 732_0_appE_Return(EOS(STATIC_732), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
718_1_appE_InvokeMethod(734_0_appE_Return(EOS(STATIC_734)), i94) → 749_0_appE_Return(EOS(STATIC_749), i94)
718_1_appE_InvokeMethod(787_0_appE_Return(EOS(STATIC_787)), i113) → 815_0_appE_Return(EOS(STATIC_815), i113)
732_0_appE_Return(EOS(STATIC_732), matching1, matching2) → 734_0_appE_Return(EOS(STATIC_734)) | &&(=(matching1, 0), =(matching2, 0))
749_0_appE_Return(EOS(STATIC_749), i94) → 782_0_appE_Return(EOS(STATIC_782), i94)
782_0_appE_Return(EOS(STATIC_782), i107) → 787_0_appE_Return(EOS(STATIC_787))
815_0_appE_Return(EOS(STATIC_815), i113) → 782_0_appE_Return(EOS(STATIC_782), i113)

Combined rules. Obtained 1 conditional rules for P and 3 conditional rules for R.

P rules:
668_0_appE_FieldAccess(EOS(STATIC_668), x0) → 718_1_appE_InvokeMethod(668_0_appE_FieldAccess(EOS(STATIC_668), +(x0, -1)), +(x0, -1)) | >(x0, 0)
R rules:
718_1_appE_InvokeMethod(677_0_appE_Return(EOS(STATIC_677), 0), 0) → 734_0_appE_Return(EOS(STATIC_734))
718_1_appE_InvokeMethod(734_0_appE_Return(EOS(STATIC_734)), x0) → 787_0_appE_Return(EOS(STATIC_787))
718_1_appE_InvokeMethod(787_0_appE_Return(EOS(STATIC_787)), x0) → 787_0_appE_Return(EOS(STATIC_787))

Filtered ground terms:

668_0_appE_FieldAccess(x1, x2) → 668_0_appE_FieldAccess(x2)
Cond_668_0_appE_FieldAccess(x1, x2, x3) → Cond_668_0_appE_FieldAccess(x1, x3)
787_0_appE_Return(x1) → 787_0_appE_Return
734_0_appE_Return(x1) → 734_0_appE_Return
677_0_appE_Return(x1, x2) → 677_0_appE_Return

Combined rules. Obtained 1 conditional rules for P and 3 conditional rules for R.

P rules:
668_0_appE_FieldAccess(x0) → 718_1_appE_InvokeMethod(668_0_appE_FieldAccess(+(x0, -1)), +(x0, -1)) | >(x0, 0)
R rules:
718_1_appE_InvokeMethod(677_0_appE_Return, 0) → 734_0_appE_Return
718_1_appE_InvokeMethod(734_0_appE_Return, x0) → 787_0_appE_Return
718_1_appE_InvokeMethod(787_0_appE_Return, x0) → 787_0_appE_Return

Performed bisimulation on rules. Used the following equivalence classes: {[677_0_appE_Return, 734_0_appE_Return, 787_0_appE_Return]=677_0_appE_Return}

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

P rules:
668_0_APPE_FIELDACCESS(x0) → COND_668_0_APPE_FIELDACCESS(>(x0, 0), x0)
COND_668_0_APPE_FIELDACCESS(TRUE, x0) → 668_0_APPE_FIELDACCESS(+(x0, -1))
R rules:
718_1_appE_InvokeMethod(677_0_appE_Return, 0) → 677_0_appE_Return
718_1_appE_InvokeMethod(677_0_appE_Return, x0) → 677_0_appE_Return

(7) 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:
718_1_appE_InvokeMethod(677_0_appE_Return, 0) → 677_0_appE_Return
718_1_appE_InvokeMethod(677_0_appE_Return, x0) → 677_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 668_0_APPE_FIELDACCESS(x0[0]) → COND_668_0_APPE_FIELDACCESS(x0[0] > 0, x0[0])
(1): COND_668_0_APPE_FIELDACCESS(TRUE, x0[1]) → 668_0_APPE_FIELDACCESS(x0[1] + -1)

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

(1) -> (0), if (x0[1] + -1* x0[0])

The set Q consists of the following terms:
718_1_appE_InvokeMethod(677_0_appE_Return, x0)

(8) 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@42fcb4f 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 668_0_APPE_FIELDACCESS(x0) → COND_668_0_APPE_FIELDACCESS(>(x0, 0), x0) the following chains were created:
• We consider the chain 668_0_APPE_FIELDACCESS(x0[0]) → COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0]), COND_668_0_APPE_FIELDACCESS(TRUE, x0[1]) → 668_0_APPE_FIELDACCESS(+(x0[1], -1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]668_0_APPE_FIELDACCESS(x0[0])≥NonInfC∧668_0_APPE_FIELDACCESS(x0[0])≥COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])∧(UIncreasing(COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE668_0_APPE_FIELDACCESS(x0[0])≥NonInfC∧668_0_APPE_FIELDACCESS(x0[0])≥COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])∧(UIncreasing(COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_668_0_APPE_FIELDACCESS(TRUE, x0) → 668_0_APPE_FIELDACCESS(+(x0, -1)) the following chains were created:
• We consider the chain COND_668_0_APPE_FIELDACCESS(TRUE, x0[1]) → 668_0_APPE_FIELDACCESS(+(x0[1], -1)) which results in the following constraint:

(7)    (COND_668_0_APPE_FIELDACCESS(TRUE, x0[1])≥NonInfC∧COND_668_0_APPE_FIELDACCESS(TRUE, x0[1])≥668_0_APPE_FIELDACCESS(+(x0[1], -1))∧(UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥))

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

(8)    ((UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)

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

(9)    ((UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[bni_12] = 0∧0 = 0∧[2 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 668_0_APPE_FIELDACCESS(x0) → COND_668_0_APPE_FIELDACCESS(>(x0, 0), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

• COND_668_0_APPE_FIELDACCESS(TRUE, x0) → 668_0_APPE_FIELDACCESS(+(x0, -1))
• ((UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[bni_12] = 0∧0 = 0∧[2 + (-1)bso_13] ≥ 0)

The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(718_1_appE_InvokeMethod(x1, x2)) = [-1]
POL(677_0_appE_Return) = [-1]
POL(0) = 0
POL(668_0_APPE_FIELDACCESS(x1)) = [2]x1
POL(COND_668_0_APPE_FIELDACCESS(x1, x2)) = [2]x2
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

COND_668_0_APPE_FIELDACCESS(TRUE, x0[1]) → 668_0_APPE_FIELDACCESS(+(x0[1], -1))

The following pairs are in Pbound:

668_0_APPE_FIELDACCESS(x0[0]) → COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])

The following pairs are in P:

668_0_APPE_FIELDACCESS(x0[0]) → COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])

There are no usable rules.

(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

The ITRS R consists of the following rules:
718_1_appE_InvokeMethod(677_0_appE_Return, 0) → 677_0_appE_Return
718_1_appE_InvokeMethod(677_0_appE_Return, x0) → 677_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 668_0_APPE_FIELDACCESS(x0[0]) → COND_668_0_APPE_FIELDACCESS(x0[0] > 0, x0[0])

The set Q consists of the following terms:
718_1_appE_InvokeMethod(677_0_appE_Return, x0)

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(13) 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:
718_1_appE_InvokeMethod(677_0_appE_Return, 0) → 677_0_appE_Return
718_1_appE_InvokeMethod(677_0_appE_Return, x0) → 677_0_appE_Return

The integer pair graph contains the following rules and edges:
(1): COND_668_0_APPE_FIELDACCESS(TRUE, x0[1]) → 668_0_APPE_FIELDACCESS(x0[1] + -1)

The set Q consists of the following terms:
718_1_appE_InvokeMethod(677_0_appE_Return, x0)

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(16) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: CAppE.createList()LCAppE;
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(17) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 17 rules for P and 0 rules for R.

P rules:
331_0_createList_LE(EOS(STATIC_331), i39, i39) → 335_0_createList_LE(EOS(STATIC_335), i39, i39)
335_0_createList_LE(EOS(STATIC_335), i39, i39) → 346_0_createList_New(EOS(STATIC_346), i39) | >(i39, 0)
346_0_createList_New(EOS(STATIC_346), i39) → 366_0_createList_Duplicate(EOS(STATIC_366), i39)
366_0_createList_Duplicate(EOS(STATIC_366), i39) → 377_0_createList_Load(EOS(STATIC_377), i39)
377_0_createList_Load(EOS(STATIC_377), i39) → 389_0_createList_InvokeMethod(EOS(STATIC_389), i39)
389_0_createList_InvokeMethod(EOS(STATIC_389), i39) → 403_0_<init>_Load(EOS(STATIC_403), i39)
403_0_<init>_Load(EOS(STATIC_403), i39) → 408_0_<init>_InvokeMethod(EOS(STATIC_408), i39)
408_0_<init>_InvokeMethod(EOS(STATIC_408), i39) → 412_0_<init>_Load(EOS(STATIC_412), i39)
414_0_<init>_Load(EOS(STATIC_414), i39) → 417_0_<init>_FieldAccess(EOS(STATIC_417), i39)
417_0_<init>_FieldAccess(EOS(STATIC_417), i39) → 423_0_<init>_Return(EOS(STATIC_423), i39)
423_0_<init>_Return(EOS(STATIC_423), i39) → 428_0_createList_Store(EOS(STATIC_428), i39)
428_0_createList_Store(EOS(STATIC_428), i39) → 434_0_createList_Inc(EOS(STATIC_434), i39)
434_0_createList_Inc(EOS(STATIC_434), i39) → 438_0_createList_JMP(EOS(STATIC_438), +(i39, -1)) | >(i39, 0)
438_0_createList_JMP(EOS(STATIC_438), i52) → 445_0_createList_Load(EOS(STATIC_445), i52)
321_0_createList_Load(EOS(STATIC_321), i35) → 331_0_createList_LE(EOS(STATIC_331), i35, i35)
R rules:

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
331_0_createList_LE(EOS(STATIC_331), x0, x0) → 331_0_createList_LE(EOS(STATIC_331), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

331_0_createList_LE(x1, x2, x3) → 331_0_createList_LE(x2, x3)
EOS(x1) → EOS
Cond_331_0_createList_LE(x1, x2, x3, x4) → Cond_331_0_createList_LE(x1, x3, x4)

Filtered duplicate args:

331_0_createList_LE(x1, x2) → 331_0_createList_LE(x2)
Cond_331_0_createList_LE(x1, x2, x3) → Cond_331_0_createList_LE(x1, x3)

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
331_0_createList_LE(x0) → 331_0_createList_LE(+(x0, -1)) | >(x0, 0)
R rules:

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

P rules:
331_0_CREATELIST_LE(x0) → COND_331_0_CREATELIST_LE(>(x0, 0), x0)
COND_331_0_CREATELIST_LE(TRUE, x0) → 331_0_CREATELIST_LE(+(x0, -1))
R rules:

(18) 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): 331_0_CREATELIST_LE(x0[0]) → COND_331_0_CREATELIST_LE(x0[0] > 0, x0[0])
(1): COND_331_0_CREATELIST_LE(TRUE, x0[1]) → 331_0_CREATELIST_LE(x0[1] + -1)

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

(1) -> (0), if (x0[1] + -1* x0[0])

The set Q is empty.

(19) 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@42fcb4f 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 331_0_CREATELIST_LE(x0) → COND_331_0_CREATELIST_LE(>(x0, 0), x0) the following chains were created:
• We consider the chain 331_0_CREATELIST_LE(x0[0]) → COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0]), COND_331_0_CREATELIST_LE(TRUE, x0[1]) → 331_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]331_0_CREATELIST_LE(x0[0])≥NonInfC∧331_0_CREATELIST_LE(x0[0])≥COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE331_0_CREATELIST_LE(x0[0])≥NonInfC∧331_0_CREATELIST_LE(x0[0])≥COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_331_0_CREATELIST_LE(TRUE, x0) → 331_0_CREATELIST_LE(+(x0, -1)) the following chains were created:
• We consider the chain COND_331_0_CREATELIST_LE(TRUE, x0[1]) → 331_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

(7)    (COND_331_0_CREATELIST_LE(TRUE, x0[1])≥NonInfC∧COND_331_0_CREATELIST_LE(TRUE, x0[1])≥331_0_CREATELIST_LE(+(x0[1], -1))∧(UIncreasing(331_0_CREATELIST_LE(+(x0[1], -1))), ≥))

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

(8)    ((UIncreasing(331_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(9)    ((UIncreasing(331_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(10)    ((UIncreasing(331_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(11)    ((UIncreasing(331_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 331_0_CREATELIST_LE(x0) → COND_331_0_CREATELIST_LE(>(x0, 0), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

• COND_331_0_CREATELIST_LE(TRUE, x0) → 331_0_CREATELIST_LE(+(x0, -1))
• ((UIncreasing(331_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(331_0_CREATELIST_LE(x1)) = [2]x1
POL(COND_331_0_CREATELIST_LE(x1, x2)) = [2]x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

COND_331_0_CREATELIST_LE(TRUE, x0[1]) → 331_0_CREATELIST_LE(+(x0[1], -1))

The following pairs are in Pbound:

331_0_CREATELIST_LE(x0[0]) → COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P:

331_0_CREATELIST_LE(x0[0]) → COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])

There are no usable rules.

(21) 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): 331_0_CREATELIST_LE(x0[0]) → COND_331_0_CREATELIST_LE(x0[0] > 0, x0[0])

The set Q is empty.

(22) IDependencyGraphProof (EQUIVALENT transformation)

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

(24) 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:
(1): COND_331_0_CREATELIST_LE(TRUE, x0[1]) → 331_0_CREATELIST_LE(x0[1] + -1)

The set Q is empty.

(25) IDependencyGraphProof (EQUIVALENT transformation)

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