### (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:
676_0_appE_FieldAccess(EOS(STATIC_676), i82) → 678_0_appE_NONNULL(EOS(STATIC_678), i82)
680_0_appE_Load(EOS(STATIC_680), i82) → 683_0_appE_GT(EOS(STATIC_683), i82, i82)
683_0_appE_GT(EOS(STATIC_683), i84, i84) → 686_0_appE_GT(EOS(STATIC_686), i84, i84)
686_0_appE_GT(EOS(STATIC_686), i84, i84) → 690_0_appE_Load(EOS(STATIC_690), i84) | >(i84, 0)
693_0_appE_New(EOS(STATIC_693), i84) → 697_0_appE_Duplicate(EOS(STATIC_697), i84)
697_0_appE_Duplicate(EOS(STATIC_697), i84) → 700_0_appE_InvokeMethod(EOS(STATIC_700), i84)
708_0_<init>_ConstantStackPush(EOS(STATIC_708), i84) → 710_0_<init>_FieldAccess(EOS(STATIC_710), i84)
710_0_<init>_FieldAccess(EOS(STATIC_710), i84) → 712_0_<init>_Return(EOS(STATIC_712), i84)
712_0_<init>_Return(EOS(STATIC_712), i84) → 714_0_appE_FieldAccess(EOS(STATIC_714), i84)
714_0_appE_FieldAccess(EOS(STATIC_714), i84) → 717_0_appE_Inc(EOS(STATIC_717), i84)
717_0_appE_Inc(EOS(STATIC_717), i84) → 719_0_appE_Load(EOS(STATIC_719), +(i84, -1)) | >(i84, 0)
725_0_appE_InvokeMethod(EOS(STATIC_725), i86) → 727_1_appE_InvokeMethod(727_0_appE_Load(EOS(STATIC_727), i86), i86)
R rules:
683_0_appE_GT(EOS(STATIC_683), matching1, matching2) → 685_0_appE_GT(EOS(STATIC_685), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
685_0_appE_GT(EOS(STATIC_685), matching1, matching2) → 688_0_appE_Return(EOS(STATIC_688), 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
727_1_appE_InvokeMethod(688_0_appE_Return(EOS(STATIC_688), matching1), matching2) → 737_0_appE_Return(EOS(STATIC_737), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
727_1_appE_InvokeMethod(739_0_appE_Return(EOS(STATIC_739)), i95) → 749_0_appE_Return(EOS(STATIC_749), i95)
727_1_appE_InvokeMethod(794_0_appE_Return(EOS(STATIC_794)), i113) → 823_0_appE_Return(EOS(STATIC_823), i113)
737_0_appE_Return(EOS(STATIC_737), matching1, matching2) → 739_0_appE_Return(EOS(STATIC_739)) | &&(=(matching1, 0), =(matching2, 0))
749_0_appE_Return(EOS(STATIC_749), i95) → 788_0_appE_Return(EOS(STATIC_788), i95)
788_0_appE_Return(EOS(STATIC_788), i107) → 794_0_appE_Return(EOS(STATIC_794))
823_0_appE_Return(EOS(STATIC_823), i113) → 788_0_appE_Return(EOS(STATIC_788), i113)

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

P rules:
676_0_appE_FieldAccess(EOS(STATIC_676), x0) → 727_1_appE_InvokeMethod(676_0_appE_FieldAccess(EOS(STATIC_676), +(x0, -1)), +(x0, -1)) | >(x0, 0)
R rules:
727_1_appE_InvokeMethod(688_0_appE_Return(EOS(STATIC_688), 0), 0) → 739_0_appE_Return(EOS(STATIC_739))
727_1_appE_InvokeMethod(739_0_appE_Return(EOS(STATIC_739)), x0) → 794_0_appE_Return(EOS(STATIC_794))
727_1_appE_InvokeMethod(794_0_appE_Return(EOS(STATIC_794)), x0) → 794_0_appE_Return(EOS(STATIC_794))

Filtered ground terms:

676_0_appE_FieldAccess(x1, x2) → 676_0_appE_FieldAccess(x2)
Cond_676_0_appE_FieldAccess(x1, x2, x3) → Cond_676_0_appE_FieldAccess(x1, x3)
794_0_appE_Return(x1) → 794_0_appE_Return
739_0_appE_Return(x1) → 739_0_appE_Return
688_0_appE_Return(x1, x2) → 688_0_appE_Return

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

P rules:
676_0_appE_FieldAccess(x0) → 727_1_appE_InvokeMethod(676_0_appE_FieldAccess(+(x0, -1)), +(x0, -1)) | >(x0, 0)
R rules:
727_1_appE_InvokeMethod(688_0_appE_Return, 0) → 739_0_appE_Return
727_1_appE_InvokeMethod(739_0_appE_Return, x0) → 794_0_appE_Return
727_1_appE_InvokeMethod(794_0_appE_Return, x0) → 794_0_appE_Return

Performed bisimulation on rules. Used the following equivalence classes: {[688_0_appE_Return, 739_0_appE_Return, 794_0_appE_Return]=688_0_appE_Return}

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

P rules:
676_0_APPE_FIELDACCESS(x0) → COND_676_0_APPE_FIELDACCESS(>(x0, 0), x0)
COND_676_0_APPE_FIELDACCESS(TRUE, x0) → 676_0_APPE_FIELDACCESS(+(x0, -1))
R rules:
727_1_appE_InvokeMethod(688_0_appE_Return, 0) → 688_0_appE_Return
727_1_appE_InvokeMethod(688_0_appE_Return, x0) → 688_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:
727_1_appE_InvokeMethod(688_0_appE_Return, 0) → 688_0_appE_Return
727_1_appE_InvokeMethod(688_0_appE_Return, x0) → 688_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 676_0_APPE_FIELDACCESS(x0[0]) → COND_676_0_APPE_FIELDACCESS(x0[0] > 0, x0[0])
(1): COND_676_0_APPE_FIELDACCESS(TRUE, x0[1]) → 676_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:
727_1_appE_InvokeMethod(688_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@1a543937 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 676_0_APPE_FIELDACCESS(x0) → COND_676_0_APPE_FIELDACCESS(>(x0, 0), x0) the following chains were created:
• We consider the chain 676_0_APPE_FIELDACCESS(x0[0]) → COND_676_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0]), COND_676_0_APPE_FIELDACCESS(TRUE, x0[1]) → 676_0_APPE_FIELDACCESS(+(x0[1], -1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]676_0_APPE_FIELDACCESS(x0[0])≥NonInfC∧676_0_APPE_FIELDACCESS(x0[0])≥COND_676_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])∧(UIncreasing(COND_676_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)=TRUE676_0_APPE_FIELDACCESS(x0[0])≥NonInfC∧676_0_APPE_FIELDACCESS(x0[0])≥COND_676_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])∧(UIncreasing(COND_676_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_676_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_676_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_676_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_676_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_676_0_APPE_FIELDACCESS(TRUE, x0) → 676_0_APPE_FIELDACCESS(+(x0, -1)) the following chains were created:
• We consider the chain COND_676_0_APPE_FIELDACCESS(TRUE, x0[1]) → 676_0_APPE_FIELDACCESS(+(x0[1], -1)) which results in the following constraint:

(7)    (COND_676_0_APPE_FIELDACCESS(TRUE, x0[1])≥NonInfC∧COND_676_0_APPE_FIELDACCESS(TRUE, x0[1])≥676_0_APPE_FIELDACCESS(+(x0[1], -1))∧(UIncreasing(676_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥))

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

(8)    ((UIncreasing(676_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(676_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(676_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(676_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.
• 676_0_APPE_FIELDACCESS(x0) → COND_676_0_APPE_FIELDACCESS(>(x0, 0), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_676_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_676_0_APPE_FIELDACCESS(TRUE, x0) → 676_0_APPE_FIELDACCESS(+(x0, -1))
• ((UIncreasing(676_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(727_1_appE_InvokeMethod(x1, x2)) = [-1]
POL(688_0_appE_Return) = [-1]
POL(0) = 0
POL(676_0_APPE_FIELDACCESS(x1)) = [2]x1
POL(COND_676_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_676_0_APPE_FIELDACCESS(TRUE, x0[1]) → 676_0_APPE_FIELDACCESS(+(x0[1], -1))

The following pairs are in Pbound:

676_0_APPE_FIELDACCESS(x0[0]) → COND_676_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])

The following pairs are in P:

676_0_APPE_FIELDACCESS(x0[0]) → COND_676_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:
727_1_appE_InvokeMethod(688_0_appE_Return, 0) → 688_0_appE_Return
727_1_appE_InvokeMethod(688_0_appE_Return, x0) → 688_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 676_0_APPE_FIELDACCESS(x0[0]) → COND_676_0_APPE_FIELDACCESS(x0[0] > 0, x0[0])

The set Q consists of the following terms:
727_1_appE_InvokeMethod(688_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:
727_1_appE_InvokeMethod(688_0_appE_Return, 0) → 688_0_appE_Return
727_1_appE_InvokeMethod(688_0_appE_Return, x0) → 688_0_appE_Return

The integer pair graph contains the following rules and edges:
(1): COND_676_0_APPE_FIELDACCESS(TRUE, x0[1]) → 676_0_APPE_FIELDACCESS(x0[1] + -1)

The set Q consists of the following terms:
727_1_appE_InvokeMethod(688_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:
324_0_createList_LE(EOS(STATIC_324), i39, i39) → 331_0_createList_LE(EOS(STATIC_331), i39, i39)
331_0_createList_LE(EOS(STATIC_331), i39, i39) → 340_0_createList_New(EOS(STATIC_340), i39) | >(i39, 0)
340_0_createList_New(EOS(STATIC_340), i39) → 359_0_createList_Duplicate(EOS(STATIC_359), i39)
410_0_<init>_FieldAccess(EOS(STATIC_410), i39) → 415_0_<init>_Return(EOS(STATIC_415), i39)
415_0_<init>_Return(EOS(STATIC_415), i39) → 420_0_createList_Store(EOS(STATIC_420), i39)
420_0_createList_Store(EOS(STATIC_420), i39) → 425_0_createList_Inc(EOS(STATIC_425), i39)
425_0_createList_Inc(EOS(STATIC_425), i39) → 429_0_createList_JMP(EOS(STATIC_429), +(i39, -1)) | >(i39, 0)
313_0_createList_Load(EOS(STATIC_313), i35) → 324_0_createList_LE(EOS(STATIC_324), i35, i35)
R rules:

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

P rules:
324_0_createList_LE(EOS(STATIC_324), x0, x0) → 324_0_createList_LE(EOS(STATIC_324), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

324_0_createList_LE(x1, x2, x3) → 324_0_createList_LE(x2, x3)
EOS(x1) → EOS
Cond_324_0_createList_LE(x1, x2, x3, x4) → Cond_324_0_createList_LE(x1, x3, x4)

Filtered duplicate args:

324_0_createList_LE(x1, x2) → 324_0_createList_LE(x2)
Cond_324_0_createList_LE(x1, x2, x3) → Cond_324_0_createList_LE(x1, x3)

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

P rules:
324_0_createList_LE(x0) → 324_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:
324_0_CREATELIST_LE(x0) → COND_324_0_CREATELIST_LE(>(x0, 0), x0)
COND_324_0_CREATELIST_LE(TRUE, x0) → 324_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): 324_0_CREATELIST_LE(x0[0]) → COND_324_0_CREATELIST_LE(x0[0] > 0, x0[0])
(1): COND_324_0_CREATELIST_LE(TRUE, x0[1]) → 324_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@1a543937 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 324_0_CREATELIST_LE(x0) → COND_324_0_CREATELIST_LE(>(x0, 0), x0) the following chains were created:
• We consider the chain 324_0_CREATELIST_LE(x0[0]) → COND_324_0_CREATELIST_LE(>(x0[0], 0), x0[0]), COND_324_0_CREATELIST_LE(TRUE, x0[1]) → 324_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]324_0_CREATELIST_LE(x0[0])≥NonInfC∧324_0_CREATELIST_LE(x0[0])≥COND_324_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_324_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)=TRUE324_0_CREATELIST_LE(x0[0])≥NonInfC∧324_0_CREATELIST_LE(x0[0])≥COND_324_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_324_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_324_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_324_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_324_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_324_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_324_0_CREATELIST_LE(TRUE, x0) → 324_0_CREATELIST_LE(+(x0, -1)) the following chains were created:
• We consider the chain COND_324_0_CREATELIST_LE(TRUE, x0[1]) → 324_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

(7)    (COND_324_0_CREATELIST_LE(TRUE, x0[1])≥NonInfC∧COND_324_0_CREATELIST_LE(TRUE, x0[1])≥324_0_CREATELIST_LE(+(x0[1], -1))∧(UIncreasing(324_0_CREATELIST_LE(+(x0[1], -1))), ≥))

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

(8)    ((UIncreasing(324_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(324_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(324_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(324_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.
• 324_0_CREATELIST_LE(x0) → COND_324_0_CREATELIST_LE(>(x0, 0), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_324_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_324_0_CREATELIST_LE(TRUE, x0) → 324_0_CREATELIST_LE(+(x0, -1))
• ((UIncreasing(324_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(324_0_CREATELIST_LE(x1)) = [2]x1
POL(COND_324_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_324_0_CREATELIST_LE(TRUE, x0[1]) → 324_0_CREATELIST_LE(+(x0[1], -1))

The following pairs are in Pbound:

324_0_CREATELIST_LE(x0[0]) → COND_324_0_CREATELIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P:

324_0_CREATELIST_LE(x0[0]) → COND_324_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): 324_0_CREATELIST_LE(x0[0]) → COND_324_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_324_0_CREATELIST_LE(TRUE, x0[1]) → 324_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.