Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 690 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 AND
    ↳5 JBCTerminationSCC
        ↳6 SCCToIDPv1Proof (⇒, 120 ms)
        ↳7 IDP
        ↳8 IDPNonInfProof (⇒, 40 ms)
        ↳9 AND
            ↳10 IDP
                ↳11 IDependencyGraphProof (⇔, 0 ms)
                ↳12 TRUE
            ↳13 IDP
                ↳14 IDependencyGraphProof (⇔, 0 ms)
                ↳15 TRUE
    ↳16 JBCTerminationSCC
        ↳17 SCCToIDPv1Proof (⇒, 40 ms)
        ↳18 IDP
        ↳19 IDPNonInfProof (⇒, 150 ms)
        ↳20 AND
            ↳21 IDP
                ↳22 IDependencyGraphProof (⇔, 0 ms)
                ↳23 TRUE
            ↳24 IDP
                ↳25 IDependencyGraphProof (⇔, 0 ms)
                ↳26 TRUE
    ↳27 JBCTerminationSCC
        ↳28 SCCToIDPv1Proof (⇒, 120 ms)
        ↳29 IDP
        ↳30 IDPNonInfProof (⇒, 60 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:
1808_0_createIntList_LE(EOS(STATIC_1808), i490, i490) → 1817_0_createIntList_LE(EOS(STATIC_1817), i490, i490)
1817_0_createIntList_LE(EOS(STATIC_1817), i490, i490) → 1828_0_createIntList_New(EOS(STATIC_1828), i490) | >(i490, 0)
1828_0_createIntList_New(EOS(STATIC_1828), i490) → 1838_0_createIntList_Duplicate(EOS(STATIC_1838), i490)
1838_0_createIntList_Duplicate(EOS(STATIC_1838), i490) → 1846_0_createIntList_InvokeMethod(EOS(STATIC_1846), i490)
1846_0_createIntList_InvokeMethod(EOS(STATIC_1846), i490) → 1855_0_random_FieldAccess(EOS(STATIC_1855), i490)
1855_0_random_FieldAccess(EOS(STATIC_1855), i490) → 1876_0_random_FieldAccess(EOS(STATIC_1876), i490)
1876_0_random_FieldAccess(EOS(STATIC_1876), i490) → 1882_0_random_ArrayAccess(EOS(STATIC_1882), i490)
1882_0_random_ArrayAccess(EOS(STATIC_1882), i490) → 1890_0_random_ArrayAccess(EOS(STATIC_1890), i490)
1890_0_random_ArrayAccess(EOS(STATIC_1890), i490) → 1897_0_random_Store(EOS(STATIC_1897), i490, o742)
1897_0_random_Store(EOS(STATIC_1897), i490, o742) → 1906_0_random_FieldAccess(EOS(STATIC_1906), i490, o742)
1906_0_random_FieldAccess(EOS(STATIC_1906), i490, o742) → 1914_0_random_ConstantStackPush(EOS(STATIC_1914), i490, o742)
1914_0_random_ConstantStackPush(EOS(STATIC_1914), i490, o742) → 1927_0_random_IntArithmetic(EOS(STATIC_1927), i490, o742)
1927_0_random_IntArithmetic(EOS(STATIC_1927), i490, o742) → 1939_0_random_FieldAccess(EOS(STATIC_1939), i490, o742)
1939_0_random_FieldAccess(EOS(STATIC_1939), i490, o742) → 1950_0_random_Load(EOS(STATIC_1950), i490, o742)
1950_0_random_Load(EOS(STATIC_1950), i490, o742) → 1964_0_random_InvokeMethod(EOS(STATIC_1964), i490, o742)
1964_0_random_InvokeMethod(EOS(STATIC_1964), i490, java.lang.Object(o798sub)) → 1978_0_random_InvokeMethod(EOS(STATIC_1978), i490, java.lang.Object(o798sub))
1978_0_random_InvokeMethod(EOS(STATIC_1978), i490, java.lang.Object(o798sub)) → 1989_0_length_Load(EOS(STATIC_1989), i490, java.lang.Object(o798sub), java.lang.Object(o798sub))
1989_0_length_Load(EOS(STATIC_1989), i490, java.lang.Object(o798sub), java.lang.Object(o798sub)) → 2010_0_length_FieldAccess(EOS(STATIC_2010), i490, java.lang.Object(o798sub), java.lang.Object(o798sub))
2010_0_length_FieldAccess(EOS(STATIC_2010), i490, java.lang.Object(java.lang.String(o822sub, i557)), java.lang.Object(java.lang.String(o822sub, i557))) → 2017_0_length_FieldAccess(EOS(STATIC_2017), i490, java.lang.Object(java.lang.String(o822sub, i557)), java.lang.Object(java.lang.String(o822sub, i557))) | &&(>=(i557, 0), >=(i558, 0))
2017_0_length_FieldAccess(EOS(STATIC_2017), i490, java.lang.Object(java.lang.String(o822sub, i557)), java.lang.Object(java.lang.String(o822sub, i557))) → 2029_0_length_Return(EOS(STATIC_2029), i490, java.lang.Object(java.lang.String(o822sub, i557)))
2029_0_length_Return(EOS(STATIC_2029), i490, java.lang.Object(java.lang.String(o822sub, i557))) → 2040_0_random_Return(EOS(STATIC_2040), i490)
2040_0_random_Return(EOS(STATIC_2040), i490) → 2043_0_createIntList_Load(EOS(STATIC_2043), i490)
2043_0_createIntList_Load(EOS(STATIC_2043), i490) → 2052_0_createIntList_InvokeMethod(EOS(STATIC_2052), i490)
2052_0_createIntList_InvokeMethod(EOS(STATIC_2052), i490) → 2063_0_<init>_Load(EOS(STATIC_2063), i490)
2063_0_<init>_Load(EOS(STATIC_2063), i490) → 2082_0_<init>_InvokeMethod(EOS(STATIC_2082), i490)
2082_0_<init>_InvokeMethod(EOS(STATIC_2082), i490) → 2094_0_<init>_Load(EOS(STATIC_2094), i490)
2094_0_<init>_Load(EOS(STATIC_2094), i490) → 2106_0_<init>_Load(EOS(STATIC_2106), i490)
2106_0_<init>_Load(EOS(STATIC_2106), i490) → 2120_0_<init>_FieldAccess(EOS(STATIC_2120), i490)
2120_0_<init>_FieldAccess(EOS(STATIC_2120), i490) → 2132_0_<init>_Load(EOS(STATIC_2132), i490)
2132_0_<init>_Load(EOS(STATIC_2132), i490) → 2143_0_<init>_Load(EOS(STATIC_2143), i490)
2143_0_<init>_Load(EOS(STATIC_2143), i490) → 2156_0_<init>_FieldAccess(EOS(STATIC_2156), i490)
2156_0_<init>_FieldAccess(EOS(STATIC_2156), i490) → 2170_0_<init>_Return(EOS(STATIC_2170), i490)
2170_0_<init>_Return(EOS(STATIC_2170), i490) → 2181_0_createIntList_Store(EOS(STATIC_2181), i490)
2181_0_createIntList_Store(EOS(STATIC_2181), i490) → 2193_0_createIntList_Inc(EOS(STATIC_2193), i490)
2193_0_createIntList_Inc(EOS(STATIC_2193), i490) → 2204_0_createIntList_JMP(EOS(STATIC_2204), +(i490, -1)) | >(i490, 0)
2204_0_createIntList_JMP(EOS(STATIC_2204), i611) → 2211_0_createIntList_Load(EOS(STATIC_2211), i611)
2211_0_createIntList_Load(EOS(STATIC_2211), i611) → 1796_0_createIntList_Load(EOS(STATIC_1796), i611)
1796_0_createIntList_Load(EOS(STATIC_1796), i484) → 1808_0_createIntList_LE(EOS(STATIC_1808), i484, i484)
R rules:

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


P rules:
1808_0_createIntList_LE(EOS(STATIC_1808), x0, x0) → 1808_0_createIntList_LE(EOS(STATIC_1808), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:



1808_0_createIntList_LE(x1, x2, x3) → 1808_0_createIntList_LE(x2, x3)
EOS(x1) → EOS
Cond_1808_0_createIntList_LE(x1, x2, x3, x4) → Cond_1808_0_createIntList_LE(x1, x3, x4)

Filtered duplicate args:



1808_0_createIntList_LE(x1, x2) → 1808_0_createIntList_LE(x2)
Cond_1808_0_createIntList_LE(x1, x2, x3) → Cond_1808_0_createIntList_LE(x1, x3)

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


P rules:
1808_0_createIntList_LE(x0) → 1808_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:
1808_0_CREATEINTLIST_LE(x0) → COND_1808_0_CREATEINTLIST_LE(>(x0, 0), x0)
COND_1808_0_CREATEINTLIST_LE(TRUE, x0) → 1808_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): 1808_0_CREATEINTLIST_LE(x0[0]) → COND_1808_0_CREATEINTLIST_LE(x0[0] > 0, x0[0])
(1): COND_1808_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1808_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@2350ad83 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 1808_0_CREATEINTLIST_LE(x0) → COND_1808_0_CREATEINTLIST_LE(>(x0, 0), x0) the following chains were created:
  • We consider the chain 1808_0_CREATEINTLIST_LE(x0[0]) → COND_1808_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0]), COND_1808_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1808_0_CREATEINTLIST_LE(+(x0[1], -1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 1808_0_CREATEINTLIST_LE(x0[0])≥NonInfC∧1808_0_CREATEINTLIST_LE(x0[0])≥COND_1808_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_1808_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 ⇒ 1808_0_CREATEINTLIST_LE(x0[0])≥NonInfC∧1808_0_CREATEINTLIST_LE(x0[0])≥COND_1808_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_1808_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 ⇒ (UIncreasing(COND_1808_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 ⇒ (UIncreasing(COND_1808_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 ⇒ (UIncreasing(COND_1808_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 ⇒ (UIncreasing(COND_1808_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_1808_0_CREATEINTLIST_LE(TRUE, x0) → 1808_0_CREATEINTLIST_LE(+(x0, -1)) the following chains were created:
  • We consider the chain COND_1808_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1808_0_CREATEINTLIST_LE(+(x0[1], -1)) which results in the following constraint:

    (7)    (COND_1808_0_CREATEINTLIST_LE(TRUE, x0[1])≥NonInfC∧COND_1808_0_CREATEINTLIST_LE(TRUE, x0[1])≥1808_0_CREATEINTLIST_LE(+(x0[1], -1))∧(UIncreasing(1808_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥))



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

    (8)    ((UIncreasing(1808_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)    ((UIncreasing(1808_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)    ((UIncreasing(1808_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)    ((UIncreasing(1808_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.
  • 1808_0_CREATEINTLIST_LE(x0) → COND_1808_0_CREATEINTLIST_LE(>(x0, 0), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_1808_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_1808_0_CREATEINTLIST_LE(TRUE, x0) → 1808_0_CREATEINTLIST_LE(+(x0, -1))
    • ((UIncreasing(1808_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)




The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(1808_0_CREATEINTLIST_LE(x1)) = [2]x1   
POL(COND_1808_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_1808_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1808_0_CREATEINTLIST_LE(+(x0[1], -1))

The following pairs are in Pbound:

1808_0_CREATEINTLIST_LE(x0[0]) → COND_1808_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P≥:

1808_0_CREATEINTLIST_LE(x0[0]) → COND_1808_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): 1808_0_CREATEINTLIST_LE(x0[0]) → COND_1808_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_1808_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1808_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:
1981_0_main_LE(EOS(STATIC_1981), i543, i543) → 1994_0_main_LE(EOS(STATIC_1994), i543, i543)
1994_0_main_LE(EOS(STATIC_1994), i543, i543) → 2005_0_main_Inc(EOS(STATIC_2005), i543) | >(i543, 0)
2005_0_main_Inc(EOS(STATIC_2005), i543) → 2013_0_main_JMP(EOS(STATIC_2013), +(i543, -1)) | >(i543, 0)
2013_0_main_JMP(EOS(STATIC_2013), i553) → 2023_0_main_Load(EOS(STATIC_2023), i553)
2023_0_main_Load(EOS(STATIC_2023), i553) → 1967_0_main_Load(EOS(STATIC_1967), i553)
1967_0_main_Load(EOS(STATIC_1967), i521) → 1981_0_main_LE(EOS(STATIC_1981), i521, i521)
R rules:

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


P rules:
1981_0_main_LE(EOS(STATIC_1981), x0, x0) → 1981_0_main_LE(EOS(STATIC_1981), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:



1981_0_main_LE(x1, x2, x3) → 1981_0_main_LE(x2, x3)
EOS(x1) → EOS
Cond_1981_0_main_LE(x1, x2, x3, x4) → Cond_1981_0_main_LE(x1, x3, x4)

Filtered duplicate args:



1981_0_main_LE(x1, x2) → 1981_0_main_LE(x2)
Cond_1981_0_main_LE(x1, x2, x3) → Cond_1981_0_main_LE(x1, x3)

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


P rules:
1981_0_main_LE(x0) → 1981_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:
1981_0_MAIN_LE(x0) → COND_1981_0_MAIN_LE(>(x0, 0), x0)
COND_1981_0_MAIN_LE(TRUE, x0) → 1981_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): 1981_0_MAIN_LE(x0[0]) → COND_1981_0_MAIN_LE(x0[0] > 0, x0[0])
(1): COND_1981_0_MAIN_LE(TRUE, x0[1]) → 1981_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@2350ad83 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 1981_0_MAIN_LE(x0) → COND_1981_0_MAIN_LE(>(x0, 0), x0) the following chains were created:
  • We consider the chain 1981_0_MAIN_LE(x0[0]) → COND_1981_0_MAIN_LE(>(x0[0], 0), x0[0]), COND_1981_0_MAIN_LE(TRUE, x0[1]) → 1981_0_MAIN_LE(+(x0[1], -1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 1981_0_MAIN_LE(x0[0])≥NonInfC∧1981_0_MAIN_LE(x0[0])≥COND_1981_0_MAIN_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_1981_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 ⇒ 1981_0_MAIN_LE(x0[0])≥NonInfC∧1981_0_MAIN_LE(x0[0])≥COND_1981_0_MAIN_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_1981_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 ⇒ (UIncreasing(COND_1981_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 ⇒ (UIncreasing(COND_1981_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 ⇒ (UIncreasing(COND_1981_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 ⇒ (UIncreasing(COND_1981_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_1981_0_MAIN_LE(TRUE, x0) → 1981_0_MAIN_LE(+(x0, -1)) the following chains were created:
  • We consider the chain COND_1981_0_MAIN_LE(TRUE, x0[1]) → 1981_0_MAIN_LE(+(x0[1], -1)) which results in the following constraint:

    (7)    (COND_1981_0_MAIN_LE(TRUE, x0[1])≥NonInfC∧COND_1981_0_MAIN_LE(TRUE, x0[1])≥1981_0_MAIN_LE(+(x0[1], -1))∧(UIncreasing(1981_0_MAIN_LE(+(x0[1], -1))), ≥))



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

    (8)    ((UIncreasing(1981_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)    ((UIncreasing(1981_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)    ((UIncreasing(1981_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)    ((UIncreasing(1981_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.
  • 1981_0_MAIN_LE(x0) → COND_1981_0_MAIN_LE(>(x0, 0), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_1981_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_1981_0_MAIN_LE(TRUE, x0) → 1981_0_MAIN_LE(+(x0, -1))
    • ((UIncreasing(1981_0_MAIN_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)




The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(1981_0_MAIN_LE(x1)) = [2]x1   
POL(COND_1981_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_1981_0_MAIN_LE(TRUE, x0[1]) → 1981_0_MAIN_LE(+(x0[1], -1))

The following pairs are in Pbound:

1981_0_MAIN_LE(x0[0]) → COND_1981_0_MAIN_LE(>(x0[0], 0), x0[0])

The following pairs are in P≥:

1981_0_MAIN_LE(x0[0]) → COND_1981_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): 1981_0_MAIN_LE(x0[0]) → COND_1981_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_1981_0_MAIN_LE(TRUE, x0[1]) → 1981_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:

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

(28) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 19 rules for P and 0 rules for R.


P rules:
1872_0_nth_ConstantStackPush(EOS(STATIC_1872), java.lang.Object(o717sub), i499, o716, i499) → 1878_0_nth_LE(EOS(STATIC_1878), java.lang.Object(o717sub), i499, o716, i499, 1)
1878_0_nth_LE(EOS(STATIC_1878), java.lang.Object(o717sub), i511, o716, i511, matching1) → 1885_0_nth_LE(EOS(STATIC_1885), java.lang.Object(o717sub), i511, o716, i511, 1) | =(matching1, 1)
1885_0_nth_LE(EOS(STATIC_1885), java.lang.Object(o717sub), i511, o716, i511, matching1) → 1894_0_nth_Inc(EOS(STATIC_1894), java.lang.Object(o717sub), i511, o716) | &&(>(i511, 1), =(matching1, 1))
1894_0_nth_Inc(EOS(STATIC_1894), java.lang.Object(o717sub), i511, o716) → 1902_0_nth_Load(EOS(STATIC_1902), java.lang.Object(o717sub), +(i511, -1), o716) | >(i511, 0)
1902_0_nth_Load(EOS(STATIC_1902), java.lang.Object(o717sub), i514, o716) → 1910_0_nth_FieldAccess(EOS(STATIC_1910), java.lang.Object(o717sub), i514, o716)
1910_0_nth_FieldAccess(EOS(STATIC_1910), java.lang.Object(o717sub), i514, java.lang.Object(o755sub)) → 1922_0_nth_FieldAccess(EOS(STATIC_1922), java.lang.Object(o717sub), i514, java.lang.Object(o755sub))
1922_0_nth_FieldAccess(EOS(STATIC_1922), java.lang.Object(o717sub), i514, java.lang.Object(o755sub)) → 1933_0_nth_FieldAccess(EOS(STATIC_1933), java.lang.Object(o717sub), i514, java.lang.Object(o755sub))
1922_0_nth_FieldAccess(EOS(STATIC_1922), java.lang.Object(o717sub), i514, java.lang.Object(o717sub)) → 1934_0_nth_FieldAccess(EOS(STATIC_1934), java.lang.Object(o717sub), i514, java.lang.Object(o717sub))
1933_0_nth_FieldAccess(EOS(STATIC_1933), java.lang.Object(o717sub), i514, java.lang.Object(IntList(EOC, o769))) → 1944_0_nth_FieldAccess(EOS(STATIC_1944), java.lang.Object(o717sub), i514, java.lang.Object(IntList(EOC, o769)))
1944_0_nth_FieldAccess(EOS(STATIC_1944), java.lang.Object(o717sub), i514, java.lang.Object(IntList(EOC, o769))) → 1958_0_nth_Store(EOS(STATIC_1958), java.lang.Object(o717sub), i514, o769)
1958_0_nth_Store(EOS(STATIC_1958), java.lang.Object(o717sub), i514, o769) → 1969_0_nth_JMP(EOS(STATIC_1969), java.lang.Object(o717sub), i514, o769)
1969_0_nth_JMP(EOS(STATIC_1969), java.lang.Object(o717sub), i514, o769) → 1983_0_nth_Load(EOS(STATIC_1983), java.lang.Object(o717sub), i514, o769)
1983_0_nth_Load(EOS(STATIC_1983), java.lang.Object(o717sub), i514, o769) → 1863_0_nth_Load(EOS(STATIC_1863), java.lang.Object(o717sub), i514, o769)
1863_0_nth_Load(EOS(STATIC_1863), java.lang.Object(o717sub), i499, o716) → 1872_0_nth_ConstantStackPush(EOS(STATIC_1872), java.lang.Object(o717sub), i499, o716, i499)
1934_0_nth_FieldAccess(EOS(STATIC_1934), java.lang.Object(IntList(EOC, o771)), i514, java.lang.Object(IntList(EOC, o771))) → 1945_0_nth_FieldAccess(EOS(STATIC_1945), java.lang.Object(IntList(EOC, o771)), i514, java.lang.Object(IntList(EOC, o771)))
1945_0_nth_FieldAccess(EOS(STATIC_1945), java.lang.Object(IntList(EOC, o771)), i514, java.lang.Object(IntList(EOC, o771))) → 1959_0_nth_Store(EOS(STATIC_1959), java.lang.Object(IntList(EOC, o771)), i514, o771)
1959_0_nth_Store(EOS(STATIC_1959), java.lang.Object(IntList(EOC, o771)), i514, o771) → 1971_0_nth_JMP(EOS(STATIC_1971), java.lang.Object(IntList(EOC, o771)), i514, o771)
1971_0_nth_JMP(EOS(STATIC_1971), java.lang.Object(IntList(EOC, o771)), i514, o771) → 1986_0_nth_Load(EOS(STATIC_1986), java.lang.Object(IntList(EOC, o771)), i514, o771)
1986_0_nth_Load(EOS(STATIC_1986), java.lang.Object(IntList(EOC, o771)), i514, o771) → 1863_0_nth_Load(EOS(STATIC_1863), java.lang.Object(IntList(EOC, o771)), i514, o771)
R rules:

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


P rules:
1872_0_nth_ConstantStackPush(EOS(STATIC_1872), java.lang.Object(x0), x1, java.lang.Object(IntList(EOC, x2)), x1) → 1872_0_nth_ConstantStackPush(EOS(STATIC_1872), java.lang.Object(x0), +(x1, -1), x2, +(x1, -1)) | >(x1, 1)
1872_0_nth_ConstantStackPush(EOS(STATIC_1872), java.lang.Object(IntList(EOC, x0)), x1, java.lang.Object(IntList(EOC, x0)), x1) → 1872_0_nth_ConstantStackPush(EOS(STATIC_1872), java.lang.Object(IntList(EOC, x0)), +(x1, -1), x0, +(x1, -1)) | >(x1, 1)
R rules:

Filtered ground terms:



1872_0_nth_ConstantStackPush(x1, x2, x3, x4, x5) → 1872_0_nth_ConstantStackPush(x2, x3, x4, x5)
IntList(x1, x2) → IntList(x2)
EOS(x1) → EOS
Cond_1872_0_nth_ConstantStackPush1(x1, x2, x3, x4, x5, x6) → Cond_1872_0_nth_ConstantStackPush1(x1, x3, x4, x5, x6)
Cond_1872_0_nth_ConstantStackPush(x1, x2, x3, x4, x5, x6) → Cond_1872_0_nth_ConstantStackPush(x1, x3, x4, x5, x6)

Filtered duplicate args:



1872_0_nth_ConstantStackPush(x1, x2, x3, x4) → 1872_0_nth_ConstantStackPush(x1, x3, x4)
Cond_1872_0_nth_ConstantStackPush(x1, x2, x3, x4, x5) → Cond_1872_0_nth_ConstantStackPush(x1, x2, x4, x5)
Cond_1872_0_nth_ConstantStackPush1(x1, x2, x3, x4, x5) → Cond_1872_0_nth_ConstantStackPush1(x1, x4, x5)

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


P rules:
1872_0_nth_ConstantStackPush(java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → 1872_0_nth_ConstantStackPush(java.lang.Object(x0), x2, +(x1, -1)) | >(x1, 1)
1872_0_nth_ConstantStackPush(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1872_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:
1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → COND_1872_0_NTH_CONSTANTSTACKPUSH(>(x1, 1), java.lang.Object(x0), java.lang.Object(IntList(x2)), x1)
COND_1872_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), x2, +(x1, -1))
1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → COND_1872_0_NTH_CONSTANTSTACKPUSH1(>(x1, 1), java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1)
COND_1872_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1872_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): 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]) → COND_1872_0_NTH_CONSTANTSTACKPUSH(x1[0] > 1, java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])
(1): COND_1872_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1]) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], x1[1] + -1)
(2): 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1872_0_NTH_CONSTANTSTACKPUSH1(x1[2] > 1, java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])
(3): COND_1872_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1872_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@2350ad83 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 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → COND_1872_0_NTH_CONSTANTSTACKPUSH(>(x1, 1), java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) the following chains were created:
  • We consider the chain 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]) → COND_1872_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]), COND_1872_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1]) → 1872_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] ⇒ 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])≥NonInfC∧1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])≥COND_1872_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])∧(UIncreasing(COND_1872_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 ⇒ 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])≥NonInfC∧1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])≥COND_1872_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])∧(UIncreasing(COND_1872_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 ⇒ (UIncreasing(COND_1872_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 ⇒ (UIncreasing(COND_1872_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 ⇒ (UIncreasing(COND_1872_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 ⇒ (UIncreasing(COND_1872_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_1872_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), x2, +(x1, -1)) the following chains were created:
  • We consider the chain COND_1872_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1]) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1)) which results in the following constraint:

    (7)    (COND_1872_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1])≥NonInfC∧COND_1872_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1])≥1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))∧(UIncreasing(1872_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)    ((UIncreasing(1872_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)    ((UIncreasing(1872_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)    ((UIncreasing(1872_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)    ((UIncreasing(1872_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 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → COND_1872_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 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1872_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]), COND_1872_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1872_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] ⇒ 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥NonInfC∧1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥COND_1872_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])∧(UIncreasing(COND_1872_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 ⇒ 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥NonInfC∧1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥COND_1872_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])∧(UIncreasing(COND_1872_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 ⇒ (UIncreasing(COND_1872_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 ⇒ (UIncreasing(COND_1872_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 ⇒ (UIncreasing(COND_1872_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 ⇒ (UIncreasing(COND_1872_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_1872_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), x0, +(x1, -1)) the following chains were created:
  • We consider the chain COND_1872_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1)) which results in the following constraint:

    (18)    (COND_1872_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3])≥NonInfC∧COND_1872_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3])≥1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))∧(UIncreasing(1872_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)    ((UIncreasing(1872_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)    ((UIncreasing(1872_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)    ((UIncreasing(1872_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)    ((UIncreasing(1872_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.
  • 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → COND_1872_0_NTH_CONSTANTSTACKPUSH(>(x1, 1), java.lang.Object(x0), java.lang.Object(IntList(x2)), x1)
    • (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1872_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_1872_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0), java.lang.Object(IntList(x2)), x1) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0), x2, +(x1, -1))
    • ((UIncreasing(1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧0 = 0∧[2 + (-1)bso_15] ≥ 0)

  • 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → COND_1872_0_NTH_CONSTANTSTACKPUSH1(>(x1, 1), java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1)
    • (x1[2] ≥ 0 ⇒ (UIncreasing(COND_1872_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_1872_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), x0, +(x1, -1))
    • ((UIncreasing(1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 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(1872_0_NTH_CONSTANTSTACKPUSH(x1, x2, x3)) = [-1] + [2]x3   
POL(java.lang.Object(x1)) = [-1]   
POL(IntList(x1)) = [-1]   
POL(COND_1872_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_1872_0_NTH_CONSTANTSTACKPUSH1(x1, x2, x3, x4)) = [-1] + [2]x4   

The following pairs are in P>:

COND_1872_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1]) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], +(x1[1], -1))
COND_1872_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))

The following pairs are in Pbound:

1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]) → COND_1872_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])
1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1872_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≥:

1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]) → COND_1872_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])
1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1872_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


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0]) → COND_1872_0_NTH_CONSTANTSTACKPUSH(x1[0] > 1, java.lang.Object(x0[0]), java.lang.Object(IntList(x2[0])), x1[0])
(2): 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1872_0_NTH_CONSTANTSTACKPUSH1(x1[2] > 1, java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])


The set Q is empty.

(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_1872_0_NTH_CONSTANTSTACKPUSH(TRUE, java.lang.Object(x0[1]), java.lang.Object(IntList(x2[1])), x1[1]) → 1872_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(x0[1]), x2[1], x1[1] + -1)
(3): COND_1872_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1872_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.

(37) TRUE