Termination proof of /tmp/tmpSz65yf/FibSLR.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 420 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 330 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 190 ms)

↳8 AND

    ↳9 IDP

        ↳10 IDependencyGraphProof (⇔, 0 ms)

        ↳11 TRUE

    ↳12 IDP

        ↳13 IDependencyGraphProof (⇔, 0 ms)

        ↳14 TRUE

(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)
1438_0_random_Return(EOS(STATIC_1438(i253)), i238) → 1440_0_mk_Load(EOS(STATIC_1440(i253)), i238)
1440_0_mk_Load(EOS(STATIC_1440(i253)), i238) → 1445_0_mk_InvokeMethod(EOS(STATIC_1445(i253)), i238)
1445_0_mk_InvokeMethod(EOS(STATIC_1445(i253)), i238) → 1451_0_<init>_Load(EOS(STATIC_1451(i253)), i238)
1451_0_<init>_Load(EOS(STATIC_1451(i253)), i238) → 1458_0_<init>_InvokeMethod(EOS(STATIC_1458(i253)), i238)
1458_0_<init>_InvokeMethod(EOS(STATIC_1458(i253)), i238) → 1467_0_<init>_Load(EOS(STATIC_1467(i253)), i238)
1467_0_<init>_Load(EOS(STATIC_1467(i253)), i238) → 1471_0_<init>_Load(EOS(STATIC_1471(i253)), i238)
1471_0_<init>_Load(EOS(STATIC_1471(i253)), i238) → 1476_0_<init>_FieldAccess(EOS(STATIC_1476(i253)), i238)
1476_0_<init>_FieldAccess(EOS(STATIC_1476(i253)), i238) → 1481_0_<init>_Load(EOS(STATIC_1481(i253)), i238)
1481_0_<init>_Load(EOS(STATIC_1481(i253)), i238) → 1487_0_<init>_Load(EOS(STATIC_1487(i253)), i238)
1487_0_<init>_Load(EOS(STATIC_1487(i253)), i238) → 1492_0_<init>_FieldAccess(EOS(STATIC_1492(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)
1514_0_mk_JMP(EOS(STATIC_1514(i253)), i238) → 1520_0_mk_Load(EOS(STATIC_1520(i253)), i238)
1520_0_mk_Load(EOS(STATIC_1520(i253)), i238) → 1323_0_mk_Load(EOS(STATIC_1323(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)
1368_0_random_Return(EOS(STATIC_1368(i234)), i238) → 1371_0_mk_Load(EOS(STATIC_1371(i234)), i238)
1371_0_mk_Load(EOS(STATIC_1371(i234)), i238) → 1376_0_mk_InvokeMethod(EOS(STATIC_1376(i234)), i238)
1376_0_mk_InvokeMethod(EOS(STATIC_1376(i234)), i238) → 1380_0_<init>_Load(EOS(STATIC_1380(i234)), i238)
1380_0_<init>_Load(EOS(STATIC_1380(i234)), i238) → 1386_0_<init>_InvokeMethod(EOS(STATIC_1386(i234)), i238)
1386_0_<init>_InvokeMethod(EOS(STATIC_1386(i234)), i238) → 1391_0_<init>_Load(EOS(STATIC_1391(i234)), i238)
1391_0_<init>_Load(EOS(STATIC_1391(i234)), i238) → 1394_0_<init>_Load(EOS(STATIC_1394(i234)), i238)
1394_0_<init>_Load(EOS(STATIC_1394(i234)), i238) → 1399_0_<init>_FieldAccess(EOS(STATIC_1399(i234)), i238)
1399_0_<init>_FieldAccess(EOS(STATIC_1399(i234)), i238) → 1404_0_<init>_Load(EOS(STATIC_1404(i234)), i238)
1404_0_<init>_Load(EOS(STATIC_1404(i234)), i238) → 1409_0_<init>_Load(EOS(STATIC_1409(i234)), i238)
1409_0_<init>_Load(EOS(STATIC_1409(i234)), i238) → 1415_0_<init>_FieldAccess(EOS(STATIC_1415(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)
1429_0_mk_JMP(EOS(STATIC_1429(i234)), i238) → 1436_0_mk_Load(EOS(STATIC_1436(i234)), i238)
1436_0_mk_Load(EOS(STATIC_1436(i234)), i238) → 1323_0_mk_Load(EOS(STATIC_1323(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] > -1 ∧EOS(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] > 0 ∧EOS(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))=TRUE∧EOS(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])∧(U^Increasing(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)=TRUE ⇒ 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])∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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))∧(U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)=TRUE∧EOS(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])∧(U^Increasing(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)=TRUE ⇒ 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])∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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))∧(U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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 ⇒ (U^Increasing(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))
- ((U^Increasing(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 ⇒ (U^Increasing(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))
- ((U^Increasing(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 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) = 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 P_bound:

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.

(8) Complex Obligation (AND)

(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

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(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.

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Obligation:

(5) SCCToIDPv1Proof (SOUND transformation)

(6) Obligation:

(7) IDPNonInfProof (SOUND transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) IDependencyGraphProof (EQUIVALENT transformation)

(14) TRUE