### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: FibSLR
`public class FibSLR {    public static int fib(int n){	if (n  < 2) return 1;	else return fib(n-1) + fib(n-2);	      }    public static int doSum(List x){	if (x==null) return 1;	else return fib(x.head) + doSum(x.tail);	    }    public static void main(String [] args) {	Random.args = args;	List l = List.mk(Random.random()*Random.random());	//System.out.println(doSum(l));    }}public class List {    public int head;    public List tail;    public List(int head, List tail) {	this.head = head;	this.tail = tail;    }    public List getTail() {	return tail;    }    public static List mk(int len) {	List result = null;	while (len-- > 0)	    result = new List(Random.random(), result);	return result;    }}public class Random {  static String[] args;  static int index = 0;  public static int random() {      if (index >= args.length)	  return 0;      String string = args[index];      index++;      return string.length();  }}`

### (1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
FibSLR.main([Ljava/lang/String;)V: Graph of 183 nodes with 0 SCCs.

List.mk(I)LList;: Graph of 119 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: List.mk(I)LList;
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

### (5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 61 rules for P and 0 rules for R.

P rules:
1326_0_mk_Inc(EOS(STATIC_1326(i234)), i235, i235) → 1328_0_mk_LE(EOS(STATIC_1328(i234)), +(i235, -1), i235)
1328_0_mk_LE(EOS(STATIC_1328(i234)), i238, i242) → 1331_0_mk_LE(EOS(STATIC_1331(i234)), i238, i242)
1331_0_mk_LE(EOS(STATIC_1331(i234)), i238, i242) → 1334_0_mk_New(EOS(STATIC_1334(i234)), i238) | >(i242, 0)
1334_0_mk_New(EOS(STATIC_1334(i234)), i238) → 1337_0_mk_Duplicate(EOS(STATIC_1337(i234)), i238)
1337_0_mk_Duplicate(EOS(STATIC_1337(i234)), i238) → 1340_0_mk_InvokeMethod(EOS(STATIC_1340(i234)), i238)
1340_0_mk_InvokeMethod(EOS(STATIC_1340(i234)), i238) → 1347_0_random_FieldAccess(EOS(STATIC_1347(i234)), i238)
1347_0_random_FieldAccess(EOS(STATIC_1347(i234)), i238) → 1356_0_random_FieldAccess(EOS(STATIC_1356(i234)), i238, i234)
1356_0_random_FieldAccess(EOS(STATIC_1356(i234)), i238, i234) → 1357_0_random_ArrayLength(EOS(STATIC_1357(i234)), i238, i234, java.lang.Object(ARRAY(i66)))
1357_0_random_ArrayLength(EOS(STATIC_1357(i234)), i238, i234, java.lang.Object(ARRAY(i66))) → 1359_0_random_LT(EOS(STATIC_1359(i234)), i238, i234, i66) | >=(i66, 0)
1359_0_random_LT(EOS(STATIC_1359(i234)), i238, i234, i66) → 1361_0_random_LT(EOS(STATIC_1361(i234)), i238, i234, i66)
1359_0_random_LT(EOS(STATIC_1359(i234)), i238, i234, i66) → 1362_0_random_LT(EOS(STATIC_1362(i234)), i238, i234, i66)
1361_0_random_LT(EOS(STATIC_1361(i234)), i238, i234, i66) → 1363_0_random_FieldAccess(EOS(STATIC_1363(i234)), i238) | <(i234, i66)
1363_0_random_FieldAccess(EOS(STATIC_1363(i234)), i238) → 1366_0_random_FieldAccess(EOS(STATIC_1366(i234)), i238, java.lang.Object(ARRAY(i66)))
1366_0_random_FieldAccess(EOS(STATIC_1366(i234)), i238, java.lang.Object(ARRAY(i66))) → 1370_0_random_ArrayAccess(EOS(STATIC_1370(i234)), i238, java.lang.Object(ARRAY(i66)), i234)
1370_0_random_ArrayAccess(EOS(STATIC_1370(i234)), i238, java.lang.Object(ARRAY(i66)), i234) → 1373_0_random_ArrayAccess(EOS(STATIC_1373(i234)), i238, java.lang.Object(ARRAY(i66)), i234)
1373_0_random_ArrayAccess(EOS(STATIC_1373(i234)), i238, java.lang.Object(ARRAY(i66)), i234) → 1377_0_random_Store(EOS(STATIC_1377(i234)), i238, o275)
1377_0_random_Store(EOS(STATIC_1377(i234)), i238, o275) → 1381_0_random_FieldAccess(EOS(STATIC_1381(i234)), i238, o275)
1381_0_random_FieldAccess(EOS(STATIC_1381(i234)), i238, o275) → 1383_0_random_ConstantStackPush(EOS(STATIC_1383(i234)), i238, o275, i234)
1383_0_random_ConstantStackPush(EOS(STATIC_1383(i234)), i238, o275, i234) → 1388_0_random_IntArithmetic(EOS(STATIC_1388(i234)), i238, o275, i234, 1)
1388_0_random_IntArithmetic(EOS(STATIC_1388(i234)), i238, o275, i234, matching1) → 1393_0_random_FieldAccess(EOS(STATIC_1393(i234)), i238, o275, +(i234, 1)) | &&(>=(i234, 0), =(matching1, 1))
1393_0_random_FieldAccess(EOS(STATIC_1393(i234)), i238, o275, i253) → 1396_0_random_Load(EOS(STATIC_1396(i253)), i238, o275)
1396_0_random_Load(EOS(STATIC_1396(i253)), i238, o275) → 1401_0_random_InvokeMethod(EOS(STATIC_1401(i253)), i238, o275)
1401_0_random_InvokeMethod(EOS(STATIC_1401(i253)), i238, java.lang.Object(o278sub)) → 1406_0_random_InvokeMethod(EOS(STATIC_1406(i253)), i238, java.lang.Object(o278sub))
1406_0_random_InvokeMethod(EOS(STATIC_1406(i253)), i238, java.lang.Object(o278sub)) → 1411_0_length_Load(EOS(STATIC_1411(i253)), i238, java.lang.Object(o278sub), java.lang.Object(o278sub))
1411_0_length_Load(EOS(STATIC_1411(i253)), i238, java.lang.Object(o278sub), java.lang.Object(o278sub)) → 1423_0_length_FieldAccess(EOS(STATIC_1423(i253)), i238, java.lang.Object(o278sub), java.lang.Object(o278sub))
1423_0_length_FieldAccess(EOS(STATIC_1423(i253)), i238, java.lang.Object(java.lang.String(o282sub, i262)), java.lang.Object(java.lang.String(o282sub, i262))) → 1426_0_length_FieldAccess(EOS(STATIC_1426(i253)), i238, java.lang.Object(java.lang.String(o282sub, i262)), java.lang.Object(java.lang.String(o282sub, i262))) | &&(>=(i262, 0), >=(i263, 0))
1426_0_length_FieldAccess(EOS(STATIC_1426(i253)), i238, java.lang.Object(java.lang.String(o282sub, i262)), java.lang.Object(java.lang.String(o282sub, i262))) → 1431_0_length_Return(EOS(STATIC_1431(i253)), i238, java.lang.Object(java.lang.String(o282sub, i262)))
1431_0_length_Return(EOS(STATIC_1431(i253)), i238, java.lang.Object(java.lang.String(o282sub, i262))) → 1438_0_random_Return(EOS(STATIC_1438(i253)), i238)
1492_0_<init>_FieldAccess(EOS(STATIC_1492(i253)), i238) → 1501_0_<init>_Return(EOS(STATIC_1501(i253)), i238)
1501_0_<init>_Return(EOS(STATIC_1501(i253)), i238) → 1507_0_mk_Store(EOS(STATIC_1507(i253)), i238)
1507_0_mk_Store(EOS(STATIC_1507(i253)), i238) → 1514_0_mk_JMP(EOS(STATIC_1514(i253)), i238)
1323_0_mk_Load(EOS(STATIC_1323(i234)), i235) → 1326_0_mk_Inc(EOS(STATIC_1326(i234)), i235, i235)
1362_0_random_LT(EOS(STATIC_1362(i234)), i238, i234, i66) → 1365_0_random_ConstantStackPush(EOS(STATIC_1365(i234)), i238) | >=(i234, i66)
1365_0_random_ConstantStackPush(EOS(STATIC_1365(i234)), i238) → 1368_0_random_Return(EOS(STATIC_1368(i234)), i238)
1415_0_<init>_FieldAccess(EOS(STATIC_1415(i234)), i238) → 1421_0_<init>_Return(EOS(STATIC_1421(i234)), i238)
1421_0_<init>_Return(EOS(STATIC_1421(i234)), i238) → 1425_0_mk_Store(EOS(STATIC_1425(i234)), i238)
1425_0_mk_Store(EOS(STATIC_1425(i234)), i238) → 1429_0_mk_JMP(EOS(STATIC_1429(i234)), i238)
R rules:

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
1326_0_mk_Inc(EOS(STATIC_1326(x0)), x1, x1) → 1326_0_mk_Inc(EOS(STATIC_1326(+(x0, 1))), +(x1, -1), +(x1, -1)) | &&(>(x1, 0), >(+(x0, 1), 0))
1326_0_mk_Inc(EOS(STATIC_1326(x0)), x1, x1) → 1326_0_mk_Inc(EOS(STATIC_1326(x0)), +(x1, -1), +(x1, -1)) | >(x1, 0)
R rules:

Filtered duplicate args:

1326_0_mk_Inc(x1, x2, x3) → 1326_0_mk_Inc(x1, x3)
Cond_1326_0_mk_Inc(x1, x2, x3, x4) → Cond_1326_0_mk_Inc(x1, x2, x4)
Cond_1326_0_mk_Inc1(x1, x2, x3, x4) → Cond_1326_0_mk_Inc1(x1, x2, x4)

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
1326_0_mk_Inc(EOS(STATIC_1326(x0)), x1) → 1326_0_mk_Inc(EOS(STATIC_1326(+(x0, 1))), +(x1, -1)) | &&(>(x1, 0), >(x0, -1))
1326_0_mk_Inc(EOS(STATIC_1326(x0)), x1) → 1326_0_mk_Inc(EOS(STATIC_1326(x0)), +(x1, -1)) | >(x1, 0)
R rules:

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.

P rules:
1326_0_MK_INC(EOS(STATIC_1326(x0)), x1) → COND_1326_0_MK_INC(&&(>(x1, 0), >(x0, -1)), EOS(STATIC_1326(x0)), x1)
COND_1326_0_MK_INC(TRUE, EOS(STATIC_1326(x0)), x1) → 1326_0_MK_INC(EOS(STATIC_1326(+(x0, 1))), +(x1, -1))
1326_0_MK_INC(EOS(STATIC_1326(x0)), x1) → COND_1326_0_MK_INC1(>(x1, 0), EOS(STATIC_1326(x0)), x1)
COND_1326_0_MK_INC1(TRUE, EOS(STATIC_1326(x0)), x1) → 1326_0_MK_INC(EOS(STATIC_1326(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1326_0_MK_INC(EOS(STATIC_1326(x0[0])), x1[0]) → COND_1326_0_MK_INC(x1[0] > 0 && x0[0] > -1, EOS(STATIC_1326(x0[0])), x1[0])
(1): COND_1326_0_MK_INC(TRUE, EOS(STATIC_1326(x0[1])), x1[1]) → 1326_0_MK_INC(EOS(STATIC_1326(x0[1] + 1)), x1[1] + -1)
(2): 1326_0_MK_INC(EOS(STATIC_1326(x0[2])), x1[2]) → COND_1326_0_MK_INC1(x1[2] > 0, EOS(STATIC_1326(x0[2])), x1[2])
(3): COND_1326_0_MK_INC1(TRUE, EOS(STATIC_1326(x0[3])), x1[3]) → 1326_0_MK_INC(EOS(STATIC_1326(x0[3])), x1[3] + -1)

(0) -> (1), if (x1[0] > 0 && x0[0] > -1EOS(STATIC_1326(x0[0])) →* EOS(STATIC_1326(x0[1]))∧x1[0]* x1[1])

(1) -> (0), if (EOS(STATIC_1326(x0[1] + 1)) →* EOS(STATIC_1326(x0[0]))∧x1[1] + -1* x1[0])

(1) -> (2), if (EOS(STATIC_1326(x0[1] + 1)) →* EOS(STATIC_1326(x0[2]))∧x1[1] + -1* x1[2])

(2) -> (3), if (x1[2] > 0EOS(STATIC_1326(x0[2])) →* EOS(STATIC_1326(x0[3]))∧x1[2]* x1[3])

(3) -> (0), if (EOS(STATIC_1326(x0[3])) →* EOS(STATIC_1326(x0[0]))∧x1[3] + -1* x1[0])

(3) -> (2), if (EOS(STATIC_1326(x0[3])) →* EOS(STATIC_1326(x0[2]))∧x1[3] + -1* x1[2])

The set Q is empty.

### (7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@224b44f0 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 1326_0_MK_INC(EOS(STATIC_1326(x0)), x1) → COND_1326_0_MK_INC(&&(>(x1, 0), >(x0, -1)), EOS(STATIC_1326(x0)), x1) the following chains were created:
• We consider the chain 1326_0_MK_INC(EOS(STATIC_1326(x0[0])), x1[0]) → COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0]), COND_1326_0_MK_INC(TRUE, EOS(STATIC_1326(x0[1])), x1[1]) → 1326_0_MK_INC(EOS(STATIC_1326(+(x0[1], 1))), +(x1[1], -1)) which results in the following constraint:

(1)    (&&(>(x1[0], 0), >(x0[0], -1))=TRUEEOS(STATIC_1326(x0[0]))=EOS(STATIC_1326(x0[1]))∧x1[0]=x1[1]1326_0_MK_INC(EOS(STATIC_1326(x0[0])), x1[0])≥NonInfC∧1326_0_MK_INC(EOS(STATIC_1326(x0[0])), x1[0])≥COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])∧(UIncreasing(COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])), ≥))

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

(2)    (>(x1[0], 0)=TRUE>(x0[0], -1)=TRUE1326_0_MK_INC(EOS(STATIC_1326(x0[0])), x1[0])≥NonInfC∧1326_0_MK_INC(EOS(STATIC_1326(x0[0])), x1[0])≥COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])∧(UIncreasing(COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])), ≥))

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

(3)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(4)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(5)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(6)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_1326_0_MK_INC(TRUE, EOS(STATIC_1326(x0)), x1) → 1326_0_MK_INC(EOS(STATIC_1326(+(x0, 1))), +(x1, -1)) the following chains were created:
• We consider the chain COND_1326_0_MK_INC(TRUE, EOS(STATIC_1326(x0[1])), x1[1]) → 1326_0_MK_INC(EOS(STATIC_1326(+(x0[1], 1))), +(x1[1], -1)) which results in the following constraint:

(7)    (COND_1326_0_MK_INC(TRUE, EOS(STATIC_1326(x0[1])), x1[1])≥NonInfC∧COND_1326_0_MK_INC(TRUE, EOS(STATIC_1326(x0[1])), x1[1])≥1326_0_MK_INC(EOS(STATIC_1326(+(x0[1], 1))), +(x1[1], -1))∧(UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(+(x0[1], 1))), +(x1[1], -1))), ≥))

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

(8)    ((UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(+(x0[1], 1))), +(x1[1], -1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

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

(9)    ((UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(+(x0[1], 1))), +(x1[1], -1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(+(x0[1], 1))), +(x1[1], -1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(+(x0[1], 1))), +(x1[1], -1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

For Pair 1326_0_MK_INC(EOS(STATIC_1326(x0)), x1) → COND_1326_0_MK_INC1(>(x1, 0), EOS(STATIC_1326(x0)), x1) the following chains were created:
• We consider the chain 1326_0_MK_INC(EOS(STATIC_1326(x0[2])), x1[2]) → COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2]), COND_1326_0_MK_INC1(TRUE, EOS(STATIC_1326(x0[3])), x1[3]) → 1326_0_MK_INC(EOS(STATIC_1326(x0[3])), +(x1[3], -1)) which results in the following constraint:

(12)    (>(x1[2], 0)=TRUEEOS(STATIC_1326(x0[2]))=EOS(STATIC_1326(x0[3]))∧x1[2]=x1[3]1326_0_MK_INC(EOS(STATIC_1326(x0[2])), x1[2])≥NonInfC∧1326_0_MK_INC(EOS(STATIC_1326(x0[2])), x1[2])≥COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])∧(UIncreasing(COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])), ≥))

We simplified constraint (12) using rules (I), (II), (IV) which results in the following new constraint:

(13)    (>(x1[2], 0)=TRUE1326_0_MK_INC(EOS(STATIC_1326(x0[2])), x1[2])≥NonInfC∧1326_0_MK_INC(EOS(STATIC_1326(x0[2])), x1[2])≥COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])∧(UIncreasing(COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])), ≥))

We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(14)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(15)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(16)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(17)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])), ≥)∧0 = 0∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

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

(18)    (x1[2] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])), ≥)∧0 = 0∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair COND_1326_0_MK_INC1(TRUE, EOS(STATIC_1326(x0)), x1) → 1326_0_MK_INC(EOS(STATIC_1326(x0)), +(x1, -1)) the following chains were created:
• We consider the chain COND_1326_0_MK_INC1(TRUE, EOS(STATIC_1326(x0[3])), x1[3]) → 1326_0_MK_INC(EOS(STATIC_1326(x0[3])), +(x1[3], -1)) which results in the following constraint:

(19)    (COND_1326_0_MK_INC1(TRUE, EOS(STATIC_1326(x0[3])), x1[3])≥NonInfC∧COND_1326_0_MK_INC1(TRUE, EOS(STATIC_1326(x0[3])), x1[3])≥1326_0_MK_INC(EOS(STATIC_1326(x0[3])), +(x1[3], -1))∧(UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(x0[3])), +(x1[3], -1))), ≥))

We simplified constraint (19) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(20)    ((UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(x0[3])), +(x1[3], -1))), ≥)∧[bni_16] = 0∧[1 + (-1)bso_17] ≥ 0)

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

(21)    ((UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(x0[3])), +(x1[3], -1))), ≥)∧[bni_16] = 0∧[1 + (-1)bso_17] ≥ 0)

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

(22)    ((UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(x0[3])), +(x1[3], -1))), ≥)∧[bni_16] = 0∧[1 + (-1)bso_17] ≥ 0)

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

(23)    ((UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(x0[3])), +(x1[3], -1))), ≥)∧[bni_16] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1326_0_MK_INC(EOS(STATIC_1326(x0)), x1) → COND_1326_0_MK_INC(&&(>(x1, 0), >(x0, -1)), EOS(STATIC_1326(x0)), x1)
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

• COND_1326_0_MK_INC(TRUE, EOS(STATIC_1326(x0)), x1) → 1326_0_MK_INC(EOS(STATIC_1326(+(x0, 1))), +(x1, -1))
• ((UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(+(x0[1], 1))), +(x1[1], -1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

• 1326_0_MK_INC(EOS(STATIC_1326(x0)), x1) → COND_1326_0_MK_INC1(>(x1, 0), EOS(STATIC_1326(x0)), x1)
• (x1[2] ≥ 0 ⇒ (UIncreasing(COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])), ≥)∧0 = 0∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

• COND_1326_0_MK_INC1(TRUE, EOS(STATIC_1326(x0)), x1) → 1326_0_MK_INC(EOS(STATIC_1326(x0)), +(x1, -1))
• ((UIncreasing(1326_0_MK_INC(EOS(STATIC_1326(x0[3])), +(x1[3], -1))), ≥)∧[bni_16] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 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(1326_0_MK_INC(x1, x2)) = [1] + x2
POL(EOS(x1)) = x1
POL(STATIC_1326(x1)) = x1
POL(COND_1326_0_MK_INC(x1, x2, x3)) = [1] + x3
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(-1) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(COND_1326_0_MK_INC1(x1, x2, x3)) = [1] + x3

The following pairs are in P>:

COND_1326_0_MK_INC(TRUE, EOS(STATIC_1326(x0[1])), x1[1]) → 1326_0_MK_INC(EOS(STATIC_1326(+(x0[1], 1))), +(x1[1], -1))
COND_1326_0_MK_INC1(TRUE, EOS(STATIC_1326(x0[3])), x1[3]) → 1326_0_MK_INC(EOS(STATIC_1326(x0[3])), +(x1[3], -1))

The following pairs are in Pbound:

1326_0_MK_INC(EOS(STATIC_1326(x0[0])), x1[0]) → COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])
1326_0_MK_INC(EOS(STATIC_1326(x0[2])), x1[2]) → COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])

The following pairs are in P:

1326_0_MK_INC(EOS(STATIC_1326(x0[0])), x1[0]) → COND_1326_0_MK_INC(&&(>(x1[0], 0), >(x0[0], -1)), EOS(STATIC_1326(x0[0])), x1[0])
1326_0_MK_INC(EOS(STATIC_1326(x0[2])), x1[2]) → COND_1326_0_MK_INC1(>(x1[2], 0), EOS(STATIC_1326(x0[2])), x1[2])

There are no usable rules.

### (9) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1326_0_MK_INC(EOS(STATIC_1326(x0[0])), x1[0]) → COND_1326_0_MK_INC(x1[0] > 0 && x0[0] > -1, EOS(STATIC_1326(x0[0])), x1[0])
(2): 1326_0_MK_INC(EOS(STATIC_1326(x0[2])), x1[2]) → COND_1326_0_MK_INC1(x1[2] > 0, EOS(STATIC_1326(x0[2])), x1[2])

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

### (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:
(1): COND_1326_0_MK_INC(TRUE, EOS(STATIC_1326(x0[1])), x1[1]) → 1326_0_MK_INC(EOS(STATIC_1326(x0[1] + 1)), x1[1] + -1)
(3): COND_1326_0_MK_INC1(TRUE, EOS(STATIC_1326(x0[3])), x1[3]) → 1326_0_MK_INC(EOS(STATIC_1326(x0[3])), x1[3] + -1)

The set Q is empty.

### (13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.