Termination proof of /tmp/tmptcsD9r/ListContentArbitrary.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 620 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 AND

    ↳5 JBCTerminationSCC

        ↳6 SCCToIDPv1Proof (⇒, 30 ms)

        ↳7 IDP

        ↳8 IDPNonInfProof (⇒, 70 ms)

        ↳9 AND

            ↳10 IDP

                ↳11 IDependencyGraphProof (⇔, 0 ms)

                ↳12 TRUE

            ↳13 IDP

                ↳14 IDependencyGraphProof (⇔, 0 ms)

                ↳15 TRUE

    ↳16 JBCTerminationSCC

        ↳17 SCCToIDPv1Proof (⇒, 50 ms)

        ↳18 IDP

        ↳19 IDPNonInfProof (⇒, 130 ms)

        ↳20 AND

            ↳21 IDP

                ↳22 IDependencyGraphProof (⇔, 0 ms)

                ↳23 TRUE

            ↳24 IDP

                ↳25 IDependencyGraphProof (⇔, 0 ms)

                ↳26 TRUE

    ↳27 JBCTerminationSCC

        ↳28 SCCToIDPv1Proof (⇒, 150 ms)

        ↳29 IDP

        ↳30 IDPNonInfProof (⇒, 80 ms)

        ↳31 AND

            ↳32 IDP

                ↳33 IDependencyGraphProof (⇔, 0 ms)

                ↳34 TRUE

            ↳35 IDP

                ↳36 IDependencyGraphProof (⇔, 0 ms)

                ↳37 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: ListContentArbitrary

public class ListContentArbitrary{

  public static void main(String[] args) {
    Random.args = args;
    IntList l = IntList.createIntList();
    int n = Random.random();
    int m = l.nth(n);

    while (m > 0) m--;
  }

}

class IntList {
  int value;
  IntList next;

  public IntList(int value, IntList next) {
    this.value = value;
    this.next = next;
  }

  public static IntList createIntList() {

    int i = Random.random();
    IntList l = null;

    while (i > 0) {
      l = new IntList(Random.random(), l);
      i--;
    }

    return l;
  }

  public int nth(int n){

    IntList l = this;

    while (n > 1) {
      n--;
      l = l.next;
    }

    return l.value;
  }
}



public 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:
ListContentArbitrary.main([Ljava/lang/String;)V: Graph of 230 nodes with 2 SCCs.

IntList.createIntList()LIntList;: Graph of 163 nodes with 1 SCC.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 3 SCCss.

(4) Complex Obligation (AND)

(5) Obligation:

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

(6) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 38 rules for P and 0 rules for R.

P rules:
1771_0_createIntList_LE(EOS(STATIC_1771), i482, i482) → 1782_0_createIntList_LE(EOS(STATIC_1782), i482, i482)
1782_0_createIntList_LE(EOS(STATIC_1782), i482, i482) → 1793_0_createIntList_New(EOS(STATIC_1793), i482) | >(i482, 0)
1793_0_createIntList_New(EOS(STATIC_1793), i482) → 1802_0_createIntList_Duplicate(EOS(STATIC_1802), i482)
1802_0_createIntList_Duplicate(EOS(STATIC_1802), i482) → 1810_0_createIntList_InvokeMethod(EOS(STATIC_1810), i482)
1810_0_createIntList_InvokeMethod(EOS(STATIC_1810), i482) → 1821_0_random_FieldAccess(EOS(STATIC_1821), i482)
1821_0_random_FieldAccess(EOS(STATIC_1821), i482) → 1842_0_random_FieldAccess(EOS(STATIC_1842), i482)
1842_0_random_FieldAccess(EOS(STATIC_1842), i482) → 1849_0_random_ArrayAccess(EOS(STATIC_1849), i482)
1849_0_random_ArrayAccess(EOS(STATIC_1849), i482) → 1857_0_random_ArrayAccess(EOS(STATIC_1857), i482)
1857_0_random_ArrayAccess(EOS(STATIC_1857), i482) → 1865_0_random_Store(EOS(STATIC_1865), i482, o742)
1865_0_random_Store(EOS(STATIC_1865), i482, o742) → 1876_0_random_FieldAccess(EOS(STATIC_1876), i482, o742)
1876_0_random_FieldAccess(EOS(STATIC_1876), i482, o742) → 1885_0_random_ConstantStackPush(EOS(STATIC_1885), i482, o742)
1885_0_random_ConstantStackPush(EOS(STATIC_1885), i482, o742) → 1899_0_random_IntArithmetic(EOS(STATIC_1899), i482, o742)
1899_0_random_IntArithmetic(EOS(STATIC_1899), i482, o742) → 1913_0_random_FieldAccess(EOS(STATIC_1913), i482, o742)
1913_0_random_FieldAccess(EOS(STATIC_1913), i482, o742) → 1924_0_random_Load(EOS(STATIC_1924), i482, o742)
1924_0_random_Load(EOS(STATIC_1924), i482, o742) → 1940_0_random_InvokeMethod(EOS(STATIC_1940), i482, o742)
1940_0_random_InvokeMethod(EOS(STATIC_1940), i482, java.lang.Object(o798sub)) → 1952_0_random_InvokeMethod(EOS(STATIC_1952), i482, java.lang.Object(o798sub))
1952_0_random_InvokeMethod(EOS(STATIC_1952), i482, java.lang.Object(o798sub)) → 1963_0_length_Load(EOS(STATIC_1963), i482, java.lang.Object(o798sub), java.lang.Object(o798sub))
1963_0_length_Load(EOS(STATIC_1963), i482, java.lang.Object(o798sub), java.lang.Object(o798sub)) → 1982_0_length_FieldAccess(EOS(STATIC_1982), i482, java.lang.Object(o798sub), java.lang.Object(o798sub))
1982_0_length_FieldAccess(EOS(STATIC_1982), i482, java.lang.Object(java.lang.String(o822sub, i554)), java.lang.Object(java.lang.String(o822sub, i554))) → 1988_0_length_FieldAccess(EOS(STATIC_1988), i482, java.lang.Object(java.lang.String(o822sub, i554)), java.lang.Object(java.lang.String(o822sub, i554))) | &&(>=(i554, 0), >=(i555, 0))
1988_0_length_FieldAccess(EOS(STATIC_1988), i482, java.lang.Object(java.lang.String(o822sub, i554)), java.lang.Object(java.lang.String(o822sub, i554))) → 1997_0_length_Return(EOS(STATIC_1997), i482, java.lang.Object(java.lang.String(o822sub, i554)))
1997_0_length_Return(EOS(STATIC_1997), i482, java.lang.Object(java.lang.String(o822sub, i554))) → 2005_0_random_Return(EOS(STATIC_2005), i482)
2005_0_random_Return(EOS(STATIC_2005), i482) → 2008_0_createIntList_Load(EOS(STATIC_2008), i482)
2008_0_createIntList_Load(EOS(STATIC_2008), i482) → 2017_0_createIntList_InvokeMethod(EOS(STATIC_2017), i482)
2017_0_createIntList_InvokeMethod(EOS(STATIC_2017), i482) → 2028_0_<init>_Load(EOS(STATIC_2028), i482)
2028_0_<init>_Load(EOS(STATIC_2028), i482) → 2048_0_<init>_InvokeMethod(EOS(STATIC_2048), i482)
2048_0_<init>_InvokeMethod(EOS(STATIC_2048), i482) → 2062_0_<init>_Load(EOS(STATIC_2062), i482)
2062_0_<init>_Load(EOS(STATIC_2062), i482) → 2072_0_<init>_Load(EOS(STATIC_2072), i482)
2072_0_<init>_Load(EOS(STATIC_2072), i482) → 2084_0_<init>_FieldAccess(EOS(STATIC_2084), i482)
2084_0_<init>_FieldAccess(EOS(STATIC_2084), i482) → 2096_0_<init>_Load(EOS(STATIC_2096), i482)
2096_0_<init>_Load(EOS(STATIC_2096), i482) → 2107_0_<init>_Load(EOS(STATIC_2107), i482)
2107_0_<init>_Load(EOS(STATIC_2107), i482) → 2117_0_<init>_FieldAccess(EOS(STATIC_2117), i482)
2117_0_<init>_FieldAccess(EOS(STATIC_2117), i482) → 2130_0_<init>_Return(EOS(STATIC_2130), i482)
2130_0_<init>_Return(EOS(STATIC_2130), i482) → 2140_0_createIntList_Store(EOS(STATIC_2140), i482)
2140_0_createIntList_Store(EOS(STATIC_2140), i482) → 2154_0_createIntList_Inc(EOS(STATIC_2154), i482)
2154_0_createIntList_Inc(EOS(STATIC_2154), i482) → 2165_0_createIntList_JMP(EOS(STATIC_2165), +(i482, -1)) | >(i482, 0)
2165_0_createIntList_JMP(EOS(STATIC_2165), i607) → 2171_0_createIntList_Load(EOS(STATIC_2171), i607)
2171_0_createIntList_Load(EOS(STATIC_2171), i607) → 1759_0_createIntList_Load(EOS(STATIC_1759), i607)
1759_0_createIntList_Load(EOS(STATIC_1759), i468) → 1771_0_createIntList_LE(EOS(STATIC_1771), i468, i468)
R rules:

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

P rules:
1771_0_createIntList_LE(EOS(STATIC_1771), x0, x0) → 1771_0_createIntList_LE(EOS(STATIC_1771), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

1771_0_createIntList_LE(x1, x2, x3) → 1771_0_createIntList_LE(x2, x3)
EOS(x1) → EOS
Cond_1771_0_createIntList_LE(x1, x2, x3, x4) → Cond_1771_0_createIntList_LE(x1, x3, x4)

Filtered duplicate args:

1771_0_createIntList_LE(x1, x2) → 1771_0_createIntList_LE(x2)
Cond_1771_0_createIntList_LE(x1, x2, x3) → Cond_1771_0_createIntList_LE(x1, x3)

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

P rules:
1771_0_createIntList_LE(x0) → 1771_0_createIntList_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:
1771_0_CREATEINTLIST_LE(x0) → COND_1771_0_CREATEINTLIST_LE(>(x0, 0), x0)
COND_1771_0_CREATEINTLIST_LE(TRUE, x0) → 1771_0_CREATEINTLIST_LE(+(x0, -1))
R rules:

(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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1771_0_CREATEINTLIST_LE(x0[0]) → COND_1771_0_CREATEINTLIST_LE(x0[0] > 0, x0[0])
(1): COND_1771_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1771_0_CREATEINTLIST_LE(x0[1] + -1)

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

(1) -> (0), if (x0[1] + -1 →^* x0[0])

The set Q is empty.

(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@53bfbe41 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 1771_0_CREATEINTLIST_LE(x0) → COND_1771_0_CREATEINTLIST_LE(>(x0, 0), x0) the following chains were created:

We consider the chain 1771_0_CREATEINTLIST_LE(x0[0]) → COND_1771_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0]), COND_1771_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1771_0_CREATEINTLIST_LE(+(x0[1], -1)) which results in the following constraint:
(1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 1771_0_CREATEINTLIST_LE(x0[0])≥NonInfC∧1771_0_CREATEINTLIST_LE(x0[0])≥COND_1771_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_1771_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE ⇒ 1771_0_CREATEINTLIST_LE(x0[0])≥NonInfC∧1771_0_CREATEINTLIST_LE(x0[0])≥COND_1771_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_1771_0_CREATEINTLIST_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 ⇒ (U^Increasing(COND_1771_0_CREATEINTLIST_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 ⇒ (U^Increasing(COND_1771_0_CREATEINTLIST_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 ⇒ (U^Increasing(COND_1771_0_CREATEINTLIST_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 ⇒ (U^Increasing(COND_1771_0_CREATEINTLIST_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_1771_0_CREATEINTLIST_LE(TRUE, x0) → 1771_0_CREATEINTLIST_LE(+(x0, -1)) the following chains were created:

We consider the chain COND_1771_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1771_0_CREATEINTLIST_LE(+(x0[1], -1)) which results in the following constraint:
(7)    (COND_1771_0_CREATEINTLIST_LE(TRUE, x0[1])≥NonInfC∧COND_1771_0_CREATEINTLIST_LE(TRUE, x0[1])≥1771_0_CREATEINTLIST_LE(+(x0[1], -1))∧(U^Increasing(1771_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(1771_0_CREATEINTLIST_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)    ((U^Increasing(1771_0_CREATEINTLIST_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)    ((U^Increasing(1771_0_CREATEINTLIST_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)    ((U^Increasing(1771_0_CREATEINTLIST_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.

1771_0_CREATEINTLIST_LE(x0) → COND_1771_0_CREATEINTLIST_LE(>(x0, 0), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_1771_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)
COND_1771_0_CREATEINTLIST_LE(TRUE, x0) → 1771_0_CREATEINTLIST_LE(+(x0, -1))
- ((U^Increasing(1771_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 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(1771_0_CREATEINTLIST_LE(x₁)) = [2]x₁
POL(COND_1771_0_CREATEINTLIST_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_1771_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1771_0_CREATEINTLIST_LE(+(x0[1], -1))

The following pairs are in P_bound:

1771_0_CREATEINTLIST_LE(x0[0]) → COND_1771_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

1771_0_CREATEINTLIST_LE(x0[0]) → COND_1771_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])

There are no usable rules.

(9) Complex Obligation (AND)

(10) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1771_0_CREATEINTLIST_LE(x0[0]) → COND_1771_0_CREATEINTLIST_LE(x0[0] > 0, x0[0])

The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE

(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

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1771_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1771_0_CREATEINTLIST_LE(x0[1] + -1)

The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: ListContentArbitrary.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(17) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 6 rules for P and 0 rules for R.

P rules:
1955_0_main_LE(EOS(STATIC_1955), i542, i542) → 1969_0_main_LE(EOS(STATIC_1969), i542, i542)
1969_0_main_LE(EOS(STATIC_1969), i542, i542) → 1976_0_main_Inc(EOS(STATIC_1976), i542) | >(i542, 0)
1976_0_main_Inc(EOS(STATIC_1976), i542) → 1985_0_main_JMP(EOS(STATIC_1985), +(i542, -1)) | >(i542, 0)
1985_0_main_JMP(EOS(STATIC_1985), i550) → 1993_0_main_Load(EOS(STATIC_1993), i550)
1993_0_main_Load(EOS(STATIC_1993), i550) → 1943_0_main_Load(EOS(STATIC_1943), i550)
1943_0_main_Load(EOS(STATIC_1943), i518) → 1955_0_main_LE(EOS(STATIC_1955), i518, i518)
R rules:

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

P rules:
1955_0_main_LE(EOS(STATIC_1955), x0, x0) → 1955_0_main_LE(EOS(STATIC_1955), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

1955_0_main_LE(x1, x2, x3) → 1955_0_main_LE(x2, x3)
EOS(x1) → EOS
Cond_1955_0_main_LE(x1, x2, x3, x4) → Cond_1955_0_main_LE(x1, x3, x4)

Filtered duplicate args:

1955_0_main_LE(x1, x2) → 1955_0_main_LE(x2)
Cond_1955_0_main_LE(x1, x2, x3) → Cond_1955_0_main_LE(x1, x3)

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

P rules:
1955_0_main_LE(x0) → 1955_0_main_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:
1955_0_MAIN_LE(x0) → COND_1955_0_MAIN_LE(>(x0, 0), x0)
COND_1955_0_MAIN_LE(TRUE, x0) → 1955_0_MAIN_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): 1955_0_MAIN_LE(x0[0]) → COND_1955_0_MAIN_LE(x0[0] > 0, x0[0])
(1): COND_1955_0_MAIN_LE(TRUE, x0[1]) → 1955_0_MAIN_LE(x0[1] + -1)

(0) -> (1), if (x0[0] > 0 ∧x0[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@53bfbe41 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 1955_0_MAIN_LE(x0) → COND_1955_0_MAIN_LE(>(x0, 0), x0) the following chains were created:

We consider the chain 1955_0_MAIN_LE(x0[0]) → COND_1955_0_MAIN_LE(>(x0[0], 0), x0[0]), COND_1955_0_MAIN_LE(TRUE, x0[1]) → 1955_0_MAIN_LE(+(x0[1], -1)) which results in the following constraint:
(1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 1955_0_MAIN_LE(x0[0])≥NonInfC∧1955_0_MAIN_LE(x0[0])≥COND_1955_0_MAIN_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_1955_0_MAIN_LE(>(x0[0], 0), x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE ⇒ 1955_0_MAIN_LE(x0[0])≥NonInfC∧1955_0_MAIN_LE(x0[0])≥COND_1955_0_MAIN_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_1955_0_MAIN_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 ⇒ (U^Increasing(COND_1955_0_MAIN_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 ⇒ (U^Increasing(COND_1955_0_MAIN_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 ⇒ (U^Increasing(COND_1955_0_MAIN_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 ⇒ (U^Increasing(COND_1955_0_MAIN_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_1955_0_MAIN_LE(TRUE, x0) → 1955_0_MAIN_LE(+(x0, -1)) the following chains were created:

We consider the chain COND_1955_0_MAIN_LE(TRUE, x0[1]) → 1955_0_MAIN_LE(+(x0[1], -1)) which results in the following constraint:
(7)    (COND_1955_0_MAIN_LE(TRUE, x0[1])≥NonInfC∧COND_1955_0_MAIN_LE(TRUE, x0[1])≥1955_0_MAIN_LE(+(x0[1], -1))∧(U^Increasing(1955_0_MAIN_LE(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(1955_0_MAIN_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)    ((U^Increasing(1955_0_MAIN_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)    ((U^Increasing(1955_0_MAIN_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)    ((U^Increasing(1955_0_MAIN_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.

1955_0_MAIN_LE(x0) → COND_1955_0_MAIN_LE(>(x0, 0), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_1955_0_MAIN_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)
COND_1955_0_MAIN_LE(TRUE, x0) → 1955_0_MAIN_LE(+(x0, -1))
- ((U^Increasing(1955_0_MAIN_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(1955_0_MAIN_LE(x₁)) = [2]x₁
POL(COND_1955_0_MAIN_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_1955_0_MAIN_LE(TRUE, x0[1]) → 1955_0_MAIN_LE(+(x0[1], -1))

The following pairs are in P_bound:

1955_0_MAIN_LE(x0[0]) → COND_1955_0_MAIN_LE(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

1955_0_MAIN_LE(x0[0]) → COND_1955_0_MAIN_LE(>(x0[0], 0), x0[0])

There are no usable rules.

(20) Complex Obligation (AND)

(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): 1955_0_MAIN_LE(x0[0]) → COND_1955_0_MAIN_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.

(23) TRUE

(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_1955_0_MAIN_LE(TRUE, x0[1]) → 1955_0_MAIN_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.

(26) TRUE

(27) Obligation:

(28) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 19 rules for P and 0 rules for R.

P rules:
1837_0_nth_ConstantStackPush(EOS(STATIC_1837), java.lang.Object(o717sub), i495, o716, i495) → 1844_0_nth_LE(EOS(STATIC_1844), java.lang.Object(o717sub), i495, o716, i495, 1)
1844_0_nth_LE(EOS(STATIC_1844), java.lang.Object(o717sub), i508, o716, i508, matching1) → 1852_0_nth_LE(EOS(STATIC_1852), java.lang.Object(o717sub), i508, o716, i508, 1) | =(matching1, 1)
1852_0_nth_LE(EOS(STATIC_1852), java.lang.Object(o717sub), i508, o716, i508, matching1) → 1861_0_nth_Inc(EOS(STATIC_1861), java.lang.Object(o717sub), i508, o716) | &&(>(i508, 1), =(matching1, 1))
1861_0_nth_Inc(EOS(STATIC_1861), java.lang.Object(o717sub), i508, o716) → 1872_0_nth_Load(EOS(STATIC_1872), java.lang.Object(o717sub), +(i508, -1), o716) | >(i508, 0)
1872_0_nth_Load(EOS(STATIC_1872), java.lang.Object(o717sub), i510, o716) → 1881_0_nth_FieldAccess(EOS(STATIC_1881), java.lang.Object(o717sub), i510, o716)
1881_0_nth_FieldAccess(EOS(STATIC_1881), java.lang.Object(o717sub), i510, java.lang.Object(o755sub)) → 1893_0_nth_FieldAccess(EOS(STATIC_1893), java.lang.Object(o717sub), i510, java.lang.Object(o755sub))
1893_0_nth_FieldAccess(EOS(STATIC_1893), java.lang.Object(o717sub), i510, java.lang.Object(o755sub)) → 1907_0_nth_FieldAccess(EOS(STATIC_1907), java.lang.Object(o717sub), i510, java.lang.Object(o755sub))
1893_0_nth_FieldAccess(EOS(STATIC_1893), java.lang.Object(o717sub), i510, java.lang.Object(o717sub)) → 1908_0_nth_FieldAccess(EOS(STATIC_1908), java.lang.Object(o717sub), i510, java.lang.Object(o717sub))
1907_0_nth_FieldAccess(EOS(STATIC_1907), java.lang.Object(o717sub), i510, java.lang.Object(IntList(EOC, o769))) → 1918_0_nth_FieldAccess(EOS(STATIC_1918), java.lang.Object(o717sub), i510, java.lang.Object(IntList(EOC, o769)))
1918_0_nth_FieldAccess(EOS(STATIC_1918), java.lang.Object(o717sub), i510, java.lang.Object(IntList(EOC, o769))) → 1933_0_nth_Store(EOS(STATIC_1933), java.lang.Object(o717sub), i510, o769)
1933_0_nth_Store(EOS(STATIC_1933), java.lang.Object(o717sub), i510, o769) → 1945_0_nth_JMP(EOS(STATIC_1945), java.lang.Object(o717sub), i510, o769)
1945_0_nth_JMP(EOS(STATIC_1945), java.lang.Object(o717sub), i510, o769) → 1958_0_nth_Load(EOS(STATIC_1958), java.lang.Object(o717sub), i510, o769)
1958_0_nth_Load(EOS(STATIC_1958), java.lang.Object(o717sub), i510, o769) → 1828_0_nth_Load(EOS(STATIC_1828), java.lang.Object(o717sub), i510, o769)
1828_0_nth_Load(EOS(STATIC_1828), java.lang.Object(o717sub), i495, o716) → 1837_0_nth_ConstantStackPush(EOS(STATIC_1837), java.lang.Object(o717sub), i495, o716, i495)
1908_0_nth_FieldAccess(EOS(STATIC_1908), java.lang.Object(IntList(EOC, o771)), i510, java.lang.Object(IntList(EOC, o771))) → 1919_0_nth_FieldAccess(EOS(STATIC_1919), java.lang.Object(IntList(EOC, o771)), i510, java.lang.Object(IntList(EOC, o771)))
1919_0_nth_FieldAccess(EOS(STATIC_1919), java.lang.Object(IntList(EOC, o771)), i510, java.lang.Object(IntList(EOC, o771))) → 1935_0_nth_Store(EOS(STATIC_1935), java.lang.Object(IntList(EOC, o771)), i510, o771)
1935_0_nth_Store(EOS(STATIC_1935), java.lang.Object(IntList(EOC, o771)), i510, o771) → 1946_0_nth_JMP(EOS(STATIC_1946), java.lang.Object(IntList(EOC, o771)), i510, o771)
1946_0_nth_JMP(EOS(STATIC_1946), java.lang.Object(IntList(EOC, o771)), i510, o771) → 1960_0_nth_Load(EOS(STATIC_1960), java.lang.Object(IntList(EOC, o771)), i510, o771)
1960_0_nth_Load(EOS(STATIC_1960), java.lang.Object(IntList(EOC, o771)), i510, o771) → 1828_0_nth_Load(EOS(STATIC_1828), java.lang.Object(IntList(EOC, o771)), i510, o771)
R rules:

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

P rules:
1837_0_nth_ConstantStackPush(EOS(STATIC_1837), java.lang.Object(x0), x1, java.lang.Object(IntList(EOC, x2)), x1) → 1837_0_nth_ConstantStackPush(EOS(STATIC_1837), java.lang.Object(x0), +(x1, -1), x2, +(x1, -1)) | >(x1, 1)
1837_0_nth_ConstantStackPush(EOS(STATIC_1837), java.lang.Object(IntList(EOC, x0)), x1, java.lang.Object(IntList(EOC, x0)), x1) → 1837_0_nth_ConstantStackPush(EOS(STATIC_1837), java.lang.Object(IntList(EOC, x0)), +(x1, -1), x0, +(x1, -1)) | >(x1, 1)
R rules:

Filtered ground terms:

1837_0_nth_ConstantStackPush(x1, x2, x3, x4, x5) → 1837_0_nth_ConstantStackPush(x2, x3, x4, x5)
IntList(x1, x2) → IntList(x2)
EOS(x1) → EOS
Cond_1837_0_nth_ConstantStackPush1(x1, x2, x3, x4, x5, x6) → Cond_1837_0_nth_ConstantStackPush1(x1, x3, x4, x5, x6)
Cond_1837_0_nth_ConstantStackPush(x1, x2, x3, x4, x5, x6) → Cond_1837_0_nth_ConstantStackPush(x1, x3, x4, x5, x6)

Filtered duplicate args:

1837_0_nth_ConstantStackPush(x1, x2, x3, x4) → 1837_0_nth_ConstantStackPush(x1, x3, x4)
Cond_1837_0_nth_ConstantStackPush(x1, x2, x3, x4, x5) → Cond_1837_0_nth_ConstantStackPush(x1, x2, x4, x5)
Cond_1837_0_nth_ConstantStackPush1(x1, x2, x3, x4, x5) → Cond_1837_0_nth_ConstantStackPush1(x1, x4, x5)

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

P rules:
1837_0_nth_ConstantStackPush(java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → 1837_0_nth_ConstantStackPush(java.lang.Object(x0), x2, +(x1, -1)) | >(x1, 1)
1837_0_nth_ConstantStackPush(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1837_0_nth_ConstantStackPush(java.lang.Object(IntList(x0)), x0, +(x1, -1)) | >(x1, 1)
R rules:

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

P rules:
1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1, 1), java.lang.Object(x0), java.lang.Object(IntList(x2)), x1)
COND_1837_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), x2, +(x1, -1))
1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1, 1), java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1)
COND_1837_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), x0, +(x1, -1))
R rules:

(29) 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): 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]) → COND_1837_0_NTH_CONSTANTSTACKPUSH(x1[0] > 1, java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])
(1): COND_1837_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1]) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], x1[1] + -1)
(2): 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1837_0_NTH_CONSTANTSTACKPUSH1(x1[2] > 1, java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])
(3): COND_1837_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], x1[3] + -1)

(0) -> (1), if (x1[0] > 1 ∧java.lang.Object(x0[0]) →^* java.lang.Object(x0[1])∧java.lang.Object(IntList(x2[0])) →^* java.lang.Object(IntList(x2[1]))∧x1[0] →^* x1[1])

(1) -> (0), if (java.lang.Object(x0[1]) →^* java.lang.Object(x0[0])∧x2[1] →^* java.lang.Object(IntList(x2[0]))∧x1[1] + -1 →^* x1[0])

(1) -> (2), if (java.lang.Object(x0[1]) →^* java.lang.Object(IntList(x0[2]))∧x2[1] →^* java.lang.Object(IntList(x0[2]))∧x1[1] + -1 →^* x1[2])

(2) -> (3), if (x1[2] > 1 ∧java.lang.Object(IntList(x0[2])) →^* java.lang.Object(IntList(x0[3]))∧x1[2] →^* x1[3])

(3) -> (0), if (java.lang.Object(IntList(x0[3])) →^* java.lang.Object(x0[0])∧x0[3] →^* java.lang.Object(IntList(x2[0]))∧x1[3] + -1 →^* x1[0])

(3) -> (2), if (java.lang.Object(IntList(x0[3])) →^* java.lang.Object(IntList(x0[2]))∧x0[3] →^* java.lang.Object(IntList(x0[2]))∧x1[3] + -1 →^* x1[2])

The set Q is empty.

(30) 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@53bfbe41 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 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1, 1), java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) the following chains were created:

We consider the chain 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]) → COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]), COND_1837_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1]) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1)) which results in the following constraint:
(1)    (>(x1[0], 1)=TRUE∧java.lang.Object(x0[0])=java.lang.Object(x0[1])∧java.lang.Object(IntList(x2[0]))=java.lang.Object(IntList(x2[1]))∧x1[0]=x1[1] ⇒ 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])≥NonInfC∧1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])≥COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])∧(U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
(2)    (>(x1[0], 1)=TRUE ⇒ 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])≥NonInfC∧1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])≥COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])∧(U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])), ≥)∧[(3)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_1837_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), x2, +(x1, -1)) the following chains were created:

We consider the chain COND_1837_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1]) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1)) which results in the following constraint:
(7)    (COND_1837_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1])≥NonInfC∧COND_1837_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1])≥1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))∧(U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧[2 + (-1)bso_15] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧[2 + (-1)bso_15] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧[2 + (-1)bso_15] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧0 = 0∧[2 + (-1)bso_15] ≥ 0)

For Pair 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1, 1), java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) the following chains were created:

We consider the chain 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]), COND_1837_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1)) which results in the following constraint:
(12)    (>(x1[2], 1)=TRUE∧java.lang.Object(IntList(x0[2]))=java.lang.Object(IntList(x0[3]))∧x1[2]=x1[3] ⇒ 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥NonInfC∧1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])∧(U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥))

We simplified constraint (12) using rules (I), (II), (IV) which results in the following new constraint:
(13)    (>(x1[2], 1)=TRUE ⇒ 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥NonInfC∧1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])∧(U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥))

We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14)    (x1[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15)    (x1[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16)    (x1[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(17)    (x1[2] ≥ 0 ⇒ (U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_1837_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), x0, +(x1, -1)) the following chains were created:

We consider the chain COND_1837_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1)) which results in the following constraint:
(18)    (COND_1837_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3])≥NonInfC∧COND_1837_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3])≥1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))∧(U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥))

We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(19)    ((U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧[2 + (-1)bso_19] ≥ 0)

We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(20)    ((U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧[2 + (-1)bso_19] ≥ 0)

We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(21)    ((U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧[2 + (-1)bso_19] ≥ 0)

We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(22)    ((U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1, 1), java.lang.Object(x0), java.lang.Object(IntList(x2)), x1)
- (x1[0] ≥ 0 ⇒ (U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])), ≥)∧[(3)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)
COND_1837_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), x2, +(x1, -1))
- ((U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧0 = 0∧[2 + (-1)bso_15] ≥ 0)
1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1, 1), java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1)
- (x1[2] ≥ 0 ⇒ (U^Increasing(COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)
COND_1837_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), x0, +(x1, -1))
- ((U^Increasing(1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(1837_0_NTH_CONSTANTSTACKPUSH(x₁, x₂, x₃)) = [-1] + [2]x₃
POL(java.lang.Object(x₁)) = [-1]
POL(IntList(x₁)) = [-1]
POL(COND_1837_0_NTH_CONSTANTSTACKPUSH(x₁, x₂, x₃, x₄)) = [-1] + [2]x₄
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(COND_1837_0_NTH_CONSTANTSTACKPUSH1(x₁, x₂, x₃, x₄)) = [-1] + [2]x₄

The following pairs are in P_>:

COND_1837_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1]) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))
COND_1837_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))

The following pairs are in P_bound:

1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]) → COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])
1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])

The following pairs are in P_≥:

1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]) → COND_1837_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])
1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1837_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])

There are no usable rules.

(31) Complex Obligation (AND)

(32) 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

(33) IDependencyGraphProof (EQUIVALENT transformation)

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

(34) TRUE

(35) 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_1837_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1]) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], x1[1] + -1)
(3): COND_1837_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1837_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], x1[3] + -1)

The set Q is empty.

(36) 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) Complex Obligation (AND)

(5) Obligation:

(6) SCCToIDPv1Proof (SOUND transformation)

(7) Obligation:

(8) IDPNonInfProof (SOUND transformation)

(9) Complex Obligation (AND)

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE

(13) Obligation:

(14) IDependencyGraphProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) SCCToIDPv1Proof (SOUND transformation)

(18) Obligation:

(19) IDPNonInfProof (SOUND transformation)

(20) Complex Obligation (AND)

(21) Obligation:

(22) IDependencyGraphProof (EQUIVALENT transformation)

(23) TRUE

(24) Obligation:

(25) IDependencyGraphProof (EQUIVALENT transformation)

(26) TRUE

(27) Obligation:

(28) SCCToIDPv1Proof (SOUND transformation)

(29) Obligation:

(30) IDPNonInfProof (SOUND transformation)

(31) Complex Obligation (AND)

(32) Obligation:

(33) IDependencyGraphProof (EQUIVALENT transformation)

(34) TRUE

(35) Obligation:

(36) IDependencyGraphProof (EQUIVALENT transformation)

(37) TRUE