Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇒)
↳2 JBCTerminationGraph
↳3 FIGtoITRSProof (⇒)
↳4 AND
    ↳5 IDP
        ↳6 IDPNonInfProof (⇒)
        ↳7 AND
            ↳8 IDP
                ↳9 IDependencyGraphProof (⇔)
                ↳10 TRUE
            ↳11 IDP
                ↳12 IDependencyGraphProof (⇔)
                ↳13 TRUE
    ↳14 IDP
        ↳15 IDPNonInfProof (⇒)
        ↳16 AND
            ↳17 IDP
                ↳18 IDependencyGraphProof (⇔)
                ↳19 TRUE
            ↳20 IDP
                ↳21 IDependencyGraphProof (⇔)
                ↳22 TRUE

(0) Obligation:

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

  public static void main(String[] args) {
    Random.args = args;
    IntList l = IntList.createIntList();

    while (l.value > 0) l.value--;
  }

}

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 class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
ListContent.main([Ljava/lang/String;)V: Graph of 68 nodes with 1 SCC.

IntList.createIntList()LIntList;: Graph of 162 nodes with 1 SCC.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 38 rules for P and 49 rules for R.


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


Filtered ground terms:


1988_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5) → 1988_1_createIntList_InvokeMethod(x1, x2, x3)
IntList(x1) → IntList
1988_0_random_ArrayAccess(x1, x2, x3) → 1988_0_random_ArrayAccess(x2, x3)
Cond_2018_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2018_1_createIntList_InvokeMethod(x1, x2, x3, x4)
2018_0_random_IntArithmetic(x1, x2, x3, x4) → 2018_0_random_IntArithmetic(x2, x3)
2018_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5) → 2018_1_createIntList_InvokeMethod(x1, x2, x3)
Cond_1988_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_1988_1_createIntList_InvokeMethod(x1, x2, x3, x4)

Filtered unneeded arguments:


1988_1_createIntList_InvokeMethod(x1, x2, x3) → 1988_1_createIntList_InvokeMethod(x1, x2)
Cond_1988_1_createIntList_InvokeMethod(x1, x2, x3, x4) → Cond_1988_1_createIntList_InvokeMethod(x1, x2, x3)
2018_1_createIntList_InvokeMethod(x1, x2, x3) → 2018_1_createIntList_InvokeMethod(x1, x2)
Cond_2018_1_createIntList_InvokeMethod(x1, x2, x3, x4) → Cond_2018_1_createIntList_InvokeMethod(x1, x2, x3)

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


Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.



Log for SCC 1:

Generated 14 rules for P and 23 rules for R.


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


Filtered ground terms:


1490_0_main_FieldAccess(x1, x2, x3) → 1490_0_main_FieldAccess(x2, x3)
IntList(x1, x2) → IntList(x2)
Cond_1490_0_main_FieldAccess(x1, x2, x3, x4) → Cond_1490_0_main_FieldAccess(x1, x3, x4)
1755_0_main_FieldAccess(x1, x2, x3, x4) → 1755_0_main_FieldAccess
java.lang.NullPointerException(x1) → java.lang.NullPointerException
java.lang.RuntimeException(x1) → java.lang.RuntimeException
java.lang.Exception(x1) → java.lang.Exception
java.lang.Throwable(x1) → java.lang.Throwable

Filtered duplicate args:


1490_0_main_FieldAccess(x1, x2) → 1490_0_main_FieldAccess(x2)
Cond_1490_0_main_FieldAccess(x1, x2, x3) → Cond_1490_0_main_FieldAccess(x1, x3)

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


Finished conversion. Obtained 1 rules for P and 1 rules for R. System has predefined symbols.


(4) Complex Obligation (AND)

(5) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_1988_1_CREATEINTLIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_1988_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
(2): 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2018_1_CREATEINTLIST_INVOKEMETHOD(x4[2] > 0 && x2[2] > 0 && 0 < x4[2] + -1, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])
(3): COND_2018_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), x4[3] + -1)

(0) -> (1), if ((x2[0] >= 1 && x2[0] < x0[0] →* TRUE)∧(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →* 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0] →* x3[1]))


(1) -> (2), if ((2018_0_random_IntArithmetic(x5[1], x6[1]) →* 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1] →* x4[2]))


(2) -> (3), if ((x4[2] > 0 && x2[2] > 0 && 0 < x4[2] + -1 →* TRUE)∧(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]) →* 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]))∧(x4[2] →* x4[3]))


(3) -> (0), if ((1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]) →* 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] + -1 →* x3[0]))



The set Q is empty.

(6) IDPNonInfProof (SOUND transformation)

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 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) the following chains were created:
  • We consider the chain 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]), COND_1988_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1]) which results in the following constraint:

    (1)    (&&(>=(x2[0], 1), <(x2[0], x0[0]))=TRUE∧1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])=1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1])∧x3[0]=x3[1] ⇒ 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(UIncreasing(COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥))



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

    (2)    (>=(x2[0], 1)=TRUE∧<(x2[0], x0[0])=TRUE ⇒ 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(UIncreasing(COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥))



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

    (3)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x3[0] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (4)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x3[0] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (5)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]x3[0] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (6)    (x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_26] = 0∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (7)    (x2[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_26] = 0∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)



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

    (8)    (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_26] = 0∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)







For Pair COND_1988_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5, x6), x3) the following chains were created:
  • We consider the chain COND_1988_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1]) which results in the following constraint:

    (9)    (COND_1988_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥NonInfC∧COND_1988_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])∧(UIncreasing(2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥))



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

    (10)    ((UIncreasing(2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥)∧[(-1)bso_29] ≥ 0)



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

    (11)    ((UIncreasing(2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥)∧[(-1)bso_29] ≥ 0)



    We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (12)    ((UIncreasing(2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥)∧[(-1)bso_29] ≥ 0)



    We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (13)    ((UIncreasing(2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)







For Pair 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4, 0), >(x2, 0)), <(0, +(x4, -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) the following chains were created:
  • We consider the chain 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]), COND_2018_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1)) which results in the following constraint:

    (14)    (&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1)))=TRUE∧2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2])=2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3])∧x4[2]=x4[3] ⇒ 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥NonInfC∧2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])∧(UIncreasing(COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥))



    We simplified constraint (14) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (15)    (<(0, +(x4[2], -1))=TRUE∧>(x4[2], 0)=TRUE∧>(x2[2], 0)=TRUE ⇒ 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥NonInfC∧2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])∧(UIncreasing(COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥))



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

    (16)    (x4[2] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (17)    (x4[2] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (18)    (x4[2] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧[(-1)bso_31] ≥ 0)



    We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (19)    (x4[2] + [-2] ≥ 0∧x4[2] + [-1] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)



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

    (20)    (x4[2] ≥ 0∧[1] + x4[2] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧0 = 0∧0 = 0∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)



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

    (21)    (x4[2] ≥ 0∧[1] + x4[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧0 = 0∧0 = 0∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)







For Pair COND_2018_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), +(x4, -1)) the following chains were created:
  • We consider the chain COND_2018_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1)) which results in the following constraint:

    (22)    (COND_2018_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3])≥NonInfC∧COND_2018_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3])≥1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))∧(UIncreasing(1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥))



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

    (23)    ((UIncreasing(1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥)∧[2 + (-1)bso_33] ≥ 0)



    We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (24)    ((UIncreasing(1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥)∧[2 + (-1)bso_33] ≥ 0)



    We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (25)    ((UIncreasing(1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥)∧[2 + (-1)bso_33] ≥ 0)



    We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (26)    ((UIncreasing(1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_33] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
    • (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])), ≥)∧[(2)bni_26] = 0∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

  • COND_1988_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5, x6), x3)
    • ((UIncreasing(2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

  • 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4, 0), >(x2, 0)), <(0, +(x4, -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4)
    • (x4[2] ≥ 0∧[1] + x4[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])), ≥)∧0 = 0∧0 = 0∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x4[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

  • COND_2018_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), +(x4, -1))
    • ((UIncreasing(1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_33] ≥ 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(1988_1_CREATEINTLIST_INVOKEMETHOD(x1, x2)) = [-1] + [2]x2 + [-1]x1   
POL(1988_0_random_ArrayAccess(x1, x2)) = [-1] + [-1]x1   
POL(java.lang.Object(x1)) = x1   
POL(ARRAY(x1, x2)) = [-1]   
POL(COND_1988_1_CREATEINTLIST_INVOKEMETHOD(x1, x2, x3)) = [-1] + [2]x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>=(x1, x2)) = [-1]   
POL(1) = [1]   
POL(<(x1, x2)) = [-1]   
POL(2018_1_CREATEINTLIST_INVOKEMETHOD(x1, x2)) = [-1] + [2]x2   
POL(2018_0_random_IntArithmetic(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(java.lang.String(x1, x2)) = [-1]x2 + [-1]x1   
POL(COND_2018_1_CREATEINTLIST_INVOKEMETHOD(x1, x2, x3)) = [-1] + [2]x3   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

COND_2018_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))

The following pairs are in Pbound:

2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])

The following pairs are in P≥:

1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_1988_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
COND_1988_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2018_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])

There are no usable rules.

(7) Complex Obligation (AND)

(8) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_1988_1_CREATEINTLIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_1988_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
(2): 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2018_1_CREATEINTLIST_INVOKEMETHOD(x4[2] > 0 && x2[2] > 0 && 0 < x4[2] + -1, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])

(0) -> (1), if ((x2[0] >= 1 && x2[0] < x0[0] →* TRUE)∧(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →* 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0] →* x3[1]))


(1) -> (2), if ((2018_0_random_IntArithmetic(x5[1], x6[1]) →* 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]))∧(x3[1] →* x4[2]))



The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(10) TRUE

(11) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_1988_1_CREATEINTLIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_1988_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2018_1_CREATEINTLIST_INVOKEMETHOD(2018_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
(3): COND_2018_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2018_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 1988_1_CREATEINTLIST_INVOKEMETHOD(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), x4[3] + -1)

(3) -> (0), if ((1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]) →* 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] + -1 →* x3[0]))


(0) -> (1), if ((x2[0] >= 1 && x2[0] < x0[0] →* TRUE)∧(1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →* 1988_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0] →* x3[1]))



The set Q is empty.

(12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(13) TRUE

(14) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
1490_0_main_FieldAccess(NULL) → 1755_0_main_FieldAccess

The integer pair graph contains the following rules and edges:
(0): 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[0]))) → COND_1490_0_MAIN_FIELDACCESS(x0[0] > 0, java.lang.Object(IntList(x0[0])))
(1): COND_1490_0_MAIN_FIELDACCESS(TRUE, java.lang.Object(IntList(x0[1]))) → 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[1] - 1)))

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


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



The set Q consists of the following terms:
1490_0_main_FieldAccess(NULL)

(15) IDPNonInfProof (SOUND transformation)

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 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0))) → COND_1490_0_MAIN_FIELDACCESS(>(x0, 0), java.lang.Object(IntList(x0))) the following chains were created:
  • We consider the chain 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[0]))) → COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0]))), COND_1490_0_MAIN_FIELDACCESS(TRUE, java.lang.Object(IntList(x0[1]))) → 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0[1], 1)))) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧java.lang.Object(IntList(x0[0]))=java.lang.Object(IntList(x0[1])) ⇒ 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[0])))≥NonInfC∧1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[0])))≥COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))∧(UIncreasing(COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))), ≥))



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

    (2)    (>(x0[0], 0)=TRUE ⇒ 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[0])))≥NonInfC∧1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[0])))≥COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))∧(UIncreasing(COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))), ≥))



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

    (3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))), ≥)∧[(-1)Bound*bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))), ≥)∧[(-1)Bound*bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 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_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))), ≥)∧[(-1)Bound*bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))), ≥)∧[(-1)Bound*bni_11 + (2)bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)







For Pair COND_1490_0_MAIN_FIELDACCESS(TRUE, java.lang.Object(IntList(x0))) → 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0, 1)))) the following chains were created:
  • We consider the chain COND_1490_0_MAIN_FIELDACCESS(TRUE, java.lang.Object(IntList(x0[1]))) → 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0[1], 1)))) which results in the following constraint:

    (7)    (COND_1490_0_MAIN_FIELDACCESS(TRUE, java.lang.Object(IntList(x0[1])))≥NonInfC∧COND_1490_0_MAIN_FIELDACCESS(TRUE, java.lang.Object(IntList(x0[1])))≥1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0[1], 1))))∧(UIncreasing(1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0[1], 1))))), ≥))



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

    (8)    ((UIncreasing(1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0[1], 1))))), ≥)∧[2 + (-1)bso_14] ≥ 0)



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

    (9)    ((UIncreasing(1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0[1], 1))))), ≥)∧[2 + (-1)bso_14] ≥ 0)



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

    (10)    ((UIncreasing(1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0[1], 1))))), ≥)∧[2 + (-1)bso_14] ≥ 0)



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

    (11)    ((UIncreasing(1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0[1], 1))))), ≥)∧0 = 0∧[2 + (-1)bso_14] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0))) → COND_1490_0_MAIN_FIELDACCESS(>(x0, 0), java.lang.Object(IntList(x0)))
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))), ≥)∧[(-1)Bound*bni_11 + (2)bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

  • COND_1490_0_MAIN_FIELDACCESS(TRUE, java.lang.Object(IntList(x0))) → 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0, 1))))
    • ((UIncreasing(1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0[1], 1))))), ≥)∧0 = 0∧[2 + (-1)bso_14] ≥ 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(1490_0_main_FieldAccess(x1)) = [-1]   
POL(NULL) = [-1]   
POL(1755_0_main_FieldAccess) = [-1]   
POL(1490_0_MAIN_FIELDACCESS(x1)) = [2]x1   
POL(java.lang.Object(x1)) = x1   
POL(IntList(x1)) = x1   
POL(COND_1490_0_MAIN_FIELDACCESS(x1, x2)) = [2]x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

COND_1490_0_MAIN_FIELDACCESS(TRUE, java.lang.Object(IntList(x0[1]))) → 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(-(x0[1], 1))))

The following pairs are in Pbound:

1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[0]))) → COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))

The following pairs are in P≥:

1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[0]))) → COND_1490_0_MAIN_FIELDACCESS(>(x0[0], 0), java.lang.Object(IntList(x0[0])))

There are no usable rules.

(16) Complex Obligation (AND)

(17) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
1490_0_main_FieldAccess(NULL) → 1755_0_main_FieldAccess

The integer pair graph contains the following rules and edges:
(0): 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[0]))) → COND_1490_0_MAIN_FIELDACCESS(x0[0] > 0, java.lang.Object(IntList(x0[0])))


The set Q consists of the following terms:
1490_0_main_FieldAccess(NULL)

(18) IDependencyGraphProof (EQUIVALENT transformation)

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

(19) TRUE

(20) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
1490_0_main_FieldAccess(NULL) → 1755_0_main_FieldAccess

The integer pair graph contains the following rules and edges:
(1): COND_1490_0_MAIN_FIELDACCESS(TRUE, java.lang.Object(IntList(x0[1]))) → 1490_0_MAIN_FIELDACCESS(java.lang.Object(IntList(x0[1] - 1)))


The set Q consists of the following terms:
1490_0_main_FieldAccess(NULL)

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(22) TRUE