Termination proof of /tmp/tmpRkgDMh/ListContentArbitrary.jar

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

    ↳23 IDP

        ↳24 IDPNonInfProof (⇒)

        ↳25 AND

            ↳26 IDP

                ↳27 IDependencyGraphProof (⇔)

                ↳28 TRUE

            ↳29 IDP

                ↳30 IDependencyGraphProof (⇔)

                ↳31 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) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
ListContentArbitrary.main([Ljava/lang/String;)V: Graph of 206 nodes with 2 SCCs.

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:

2858_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5) → 2858_1_createIntList_InvokeMethod(x1, x2, x3)
IntList(x1) → IntList
2858_0_random_ArrayAccess(x1, x2, x3) → 2858_0_random_ArrayAccess(x2, x3)
Cond_2970_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2970_1_createIntList_InvokeMethod(x1, x2, x3, x4)
2970_0_random_IntArithmetic(x1, x2, x3, x4) → 2970_0_random_IntArithmetic(x2, x3)
2970_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5) → 2970_1_createIntList_InvokeMethod(x1, x2, x3)
Cond_2858_1_createIntList_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2858_1_createIntList_InvokeMethod(x1, x2, x3, x4)

Filtered unneeded arguments:

2858_1_createIntList_InvokeMethod(x1, x2, x3) → 2858_1_createIntList_InvokeMethod(x1, x2)
Cond_2858_1_createIntList_InvokeMethod(x1, x2, x3, x4) → Cond_2858_1_createIntList_InvokeMethod(x1, x2, x3)
2970_1_createIntList_InvokeMethod(x1, x2, x3) → 2970_1_createIntList_InvokeMethod(x1, x2)
Cond_2970_1_createIntList_InvokeMethod(x1, x2, x3, x4) → Cond_2970_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 6 rules for P and 2 rules for R.

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

Filtered ground terms:

3079_0_main_LE(x1, x2, x3) → 3079_0_main_LE(x2, x3)
Cond_3079_0_main_LE(x1, x2, x3, x4) → Cond_3079_0_main_LE(x1, x3, x4)
3161_0_main_Return(x1) → 3161_0_main_Return

Filtered duplicate args:

3079_0_main_LE(x1, x2) → 3079_0_main_LE(x2)
Cond_3079_0_main_LE(x1, x2, x3) → Cond_3079_0_main_LE(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.

Log for SCC 2:

Generated 17 rules for P and 66 rules for R.

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

Filtered ground terms:

IntList(x1, x2) → IntList(x2)
2767_0_nth_ConstantStackPush(x1, x2, x3, x4) → 2767_0_nth_ConstantStackPush(x2, x3, x4)

Filtered duplicate args:

2767_0_nth_ConstantStackPush(x1, x2, x3) → 2767_0_nth_ConstantStackPush(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.

(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): 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2858_1_CREATEINTLIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_2858_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
(2): 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2970_1_CREATEINTLIST_INVOKEMETHOD(x4[2] > 0 && x2[2] > 0 && 0 < x4[2] + -1, 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])
(3): COND_2970_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_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)∧(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) →^* 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]))∧(x3[0] →^* x3[1]))

(1) -> (2), if ((2970_0_random_IntArithmetic(x5[1], x6[1]) →^* 2970_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)∧(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]) →^* 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]))∧(x4[2] →^* x4[3]))

(3) -> (0), if ((2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]) →^* 2858_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 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) the following chains were created:

We consider the chain 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]), COND_2858_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(x5[1], x6[1]), x3[1]) which results in the following constraint:
(1)    (&&(>=(x2[0], 1), <(x2[0], x0[0]))=TRUE∧2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])=2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1])∧x3[0]=x3[1] ⇒ 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(U^Increasing(COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_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 ⇒ 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥NonInfC∧2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])≥COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])∧(U^Increasing(COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_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 ⇒ (U^Increasing(COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_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 ⇒ (U^Increasing(COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_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 ⇒ (U^Increasing(COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_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 ⇒ (U^Increasing(COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_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 ⇒ (U^Increasing(COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_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 ⇒ (U^Increasing(COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_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_2858_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(x5, x6), x3) the following chains were created:

We consider the chain COND_2858_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(x5[1], x6[1]), x3[1]) which results in the following constraint:
(9)    (COND_2858_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥NonInfC∧COND_2858_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1])≥2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(x5[1], x6[1]), x3[1])∧(U^Increasing(2970_1_CREATEINTLIST_INVOKEMETHOD(2970_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)    ((U^Increasing(2970_1_CREATEINTLIST_INVOKEMETHOD(2970_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)    ((U^Increasing(2970_1_CREATEINTLIST_INVOKEMETHOD(2970_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)    ((U^Increasing(2970_1_CREATEINTLIST_INVOKEMETHOD(2970_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)    ((U^Increasing(2970_1_CREATEINTLIST_INVOKEMETHOD(2970_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 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4, 0), >(x2, 0)), <(0, +(x4, -1))), 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) the following chains were created:

We consider the chain 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]), COND_2970_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_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∧2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2])=2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3])∧x4[2]=x4[3] ⇒ 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥NonInfC∧2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])∧(U^Increasing(COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_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 ⇒ 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥NonInfC∧2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])≥COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])∧(U^Increasing(COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_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 ⇒ (U^Increasing(COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_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 ⇒ (U^Increasing(COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_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 ⇒ (U^Increasing(COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_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 ⇒ (U^Increasing(COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_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 ⇒ (U^Increasing(COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_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 ⇒ (U^Increasing(COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_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_2970_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), +(x4, -1)) the following chains were created:

We consider the chain COND_2970_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1)) which results in the following constraint:
(22)    (COND_2970_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3])≥NonInfC∧COND_2970_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3])≥2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))∧(U^Increasing(2858_1_CREATEINTLIST_INVOKEMETHOD(2858_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)    ((U^Increasing(2858_1_CREATEINTLIST_INVOKEMETHOD(2858_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)    ((U^Increasing(2858_1_CREATEINTLIST_INVOKEMETHOD(2858_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)    ((U^Increasing(2858_1_CREATEINTLIST_INVOKEMETHOD(2858_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)    ((U^Increasing(2858_1_CREATEINTLIST_INVOKEMETHOD(2858_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.

2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2, 1), <(x2, x0)), 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
- (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_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_2858_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(x5, x6), x3)
- ((U^Increasing(2970_1_CREATEINTLIST_INVOKEMETHOD(2970_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)
2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4, 0), >(x2, 0)), <(0, +(x4, -1))), 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4)
- (x4[2] ≥ 0∧[1] + x4[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_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_2970_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x6, x7)), x8), +(x4, -1))
- ((U^Increasing(2858_1_CREATEINTLIST_INVOKEMETHOD(2858_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 P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(2858_1_CREATEINTLIST_INVOKEMETHOD(x₁, x₂)) = [-1] + [2]x₂ + [-1]x₁
POL(2858_0_random_ArrayAccess(x₁, x₂)) = [-1] + [-1]x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁, x₂)) = [-1]
POL(COND_2858_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(2970_1_CREATEINTLIST_INVOKEMETHOD(x₁, x₂)) = [-1] + [2]x₂
POL(2970_0_random_IntArithmetic(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(java.lang.String(x₁, x₂)) = [-1]x₂ + [-1]x₁
POL(COND_2970_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_2970_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), +(x4[3], -1))

The following pairs are in P_bound:

2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2])

The following pairs are in P_≥:

2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2858_1_CREATEINTLIST_INVOKEMETHOD(&&(>=(x2[0], 1), <(x2[0], x0[0])), 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
COND_2858_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[2], x1[2])), x2[2]), x4[2]) → COND_2970_1_CREATEINTLIST_INVOKEMETHOD(&&(&&(>(x4[2], 0), >(x2[2], 0)), <(0, +(x4[2], -1))), 2970_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

(1) -> (2), if ((2970_0_random_IntArithmetic(x5[1], x6[1]) →^* 2970_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): 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0]) → COND_2858_1_CREATEINTLIST_INVOKEMETHOD(x2[0] >= 1 && x2[0] < x0[0], 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), x3[0])
(1): COND_2858_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]), x3[1]) → 2970_1_CREATEINTLIST_INVOKEMETHOD(2970_0_random_IntArithmetic(x5[1], x6[1]), x3[1])
(3): COND_2970_1_CREATEINTLIST_INVOKEMETHOD(TRUE, 2970_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[3], x1[3])), x2[3]), x4[3]) → 2858_1_CREATEINTLIST_INVOKEMETHOD(2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]), x4[3] + -1)

(3) -> (0), if ((2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x6[3], x7[3])), x8[3]) →^* 2858_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]))∧(x4[3] + -1 →^* x3[0]))

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:
3079_0_main_LE(x0) → Cond_3079_0_main_LE(x0 <= 0, x0)
Cond_3079_0_main_LE(TRUE, x0) → 3161_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 3079_0_MAIN_LE(x0[0]) → COND_3079_0_MAIN_LE(x0[0] > 0, x0[0])
(1): COND_3079_0_MAIN_LE(TRUE, x0[1]) → 3079_0_MAIN_LE(x0[1] + -1)

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

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

The set Q consists of the following terms:
3079_0_main_LE(x0)
Cond_3079_0_main_LE(TRUE, x0)

(15) IDPNonInfProof (SOUND transformation)

We consider the chain 3079_0_MAIN_LE(x0[0]) → COND_3079_0_MAIN_LE(>(x0[0], 0), x0[0]), COND_3079_0_MAIN_LE(TRUE, x0[1]) → 3079_0_MAIN_LE(+(x0[1], -1)) which results in the following constraint:
(1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 3079_0_MAIN_LE(x0[0])≥NonInfC∧3079_0_MAIN_LE(x0[0])≥COND_3079_0_MAIN_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_3079_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 ⇒ 3079_0_MAIN_LE(x0[0])≥NonInfC∧3079_0_MAIN_LE(x0[0])≥COND_3079_0_MAIN_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_3079_0_MAIN_LE(>(x0[0], 0), x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3079_0_MAIN_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_14] + [(2)bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3079_0_MAIN_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_14] + [(2)bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3079_0_MAIN_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_14] + [(2)bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_3079_0_MAIN_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(2)bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_3079_0_MAIN_LE(TRUE, x0) → 3079_0_MAIN_LE(+(x0, -1)) the following chains were created:

We consider the chain COND_3079_0_MAIN_LE(TRUE, x0[1]) → 3079_0_MAIN_LE(+(x0[1], -1)) which results in the following constraint:
(7)    (COND_3079_0_MAIN_LE(TRUE, x0[1])≥NonInfC∧COND_3079_0_MAIN_LE(TRUE, x0[1])≥3079_0_MAIN_LE(+(x0[1], -1))∧(U^Increasing(3079_0_MAIN_LE(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(3079_0_MAIN_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_17] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(3079_0_MAIN_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_17] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(3079_0_MAIN_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_17] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(3079_0_MAIN_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_17] ≥ 0)

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

3079_0_MAIN_LE(x0) → COND_3079_0_MAIN_LE(>(x0, 0), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_3079_0_MAIN_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(2)bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)
COND_3079_0_MAIN_LE(TRUE, x0) → 3079_0_MAIN_LE(+(x0, -1))
- ((U^Increasing(3079_0_MAIN_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_17] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(3079_0_main_LE(x₁)) = [-1] + [-1]x₁
POL(Cond_3079_0_main_LE(x₁, x₂)) = [-1] + [-1]x₂
POL(<=(x₁, x₂)) = [-1]
POL(0) = 0
POL(3161_0_main_Return) = [-1]
POL(3079_0_MAIN_LE(x₁)) = [2]x₁
POL(COND_3079_0_MAIN_LE(x₁, x₂)) = [2]x₂
POL(>(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_3079_0_MAIN_LE(TRUE, x0[1]) → 3079_0_MAIN_LE(+(x0[1], -1))

The following pairs are in P_bound:

3079_0_MAIN_LE(x0[0]) → COND_3079_0_MAIN_LE(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

3079_0_MAIN_LE(x0[0]) → COND_3079_0_MAIN_LE(>(x0[0], 0), 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:
3079_0_main_LE(x0) → Cond_3079_0_main_LE(x0 <= 0, x0)
Cond_3079_0_main_LE(TRUE, x0) → 3161_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 3079_0_MAIN_LE(x0[0]) → COND_3079_0_MAIN_LE(x0[0] > 0, x0[0])

The set Q consists of the following terms:
3079_0_main_LE(x0)
Cond_3079_0_main_LE(TRUE, x0)

(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:
3079_0_main_LE(x0) → Cond_3079_0_main_LE(x0 <= 0, x0)
Cond_3079_0_main_LE(TRUE, x0) → 3161_0_main_Return

The integer pair graph contains the following rules and edges:
(1): COND_3079_0_MAIN_LE(TRUE, x0[1]) → 3079_0_MAIN_LE(x0[1] + -1)

The set Q consists of the following terms:
3079_0_main_LE(x0)
Cond_3079_0_main_LE(TRUE, x0)

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(22) TRUE

(23) 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): 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0])) → COND_2767_1_MAIN_INVOKEMETHOD(x0[0] > 1, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))
(1): COND_2767_1_MAIN_INVOKEMETHOD(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[1])), x0[1]), java.lang.Object(x2[1])) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], x0[1] + -1), java.lang.Object(x2[1]))
(2): 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2]))) → COND_2767_1_MAIN_INVOKEMETHOD1(x0[2] > 1, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))
(3): COND_2767_1_MAIN_INVOKEMETHOD1(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[3])), x0[3]), java.lang.Object(IntList(x1[3]))) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], x0[3] + -1), java.lang.Object(IntList(x1[3])))

(0) -> (1), if ((x0[0] > 1 →^* TRUE)∧(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]) →^* 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[1])), x0[1]))∧(java.lang.Object(x2[0]) →^* java.lang.Object(x2[1])))

(1) -> (0), if ((2767_0_nth_ConstantStackPush(x1[1], x0[1] + -1) →^* 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]))∧(java.lang.Object(x2[1]) →^* java.lang.Object(x2[0])))

(1) -> (2), if ((2767_0_nth_ConstantStackPush(x1[1], x0[1] + -1) →^* 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]))∧(java.lang.Object(x2[1]) →^* java.lang.Object(IntList(x1[2]))))

(2) -> (3), if ((x0[2] > 1 →^* TRUE)∧(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]) →^* 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[3])), x0[3]))∧(java.lang.Object(IntList(x1[2])) →^* java.lang.Object(IntList(x1[3]))))

(3) -> (0), if ((2767_0_nth_ConstantStackPush(x1[3], x0[3] + -1) →^* 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]))∧(java.lang.Object(IntList(x1[3])) →^* java.lang.Object(x2[0])))

(3) -> (2), if ((2767_0_nth_ConstantStackPush(x1[3], x0[3] + -1) →^* 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]))∧(java.lang.Object(IntList(x1[3])) →^* java.lang.Object(IntList(x1[2]))))

The set Q is empty.

(24) 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 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(x2)) → COND_2767_1_MAIN_INVOKEMETHOD(>(x0, 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(x2)) the following chains were created:

We consider the chain 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0])) → COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0])), COND_2767_1_MAIN_INVOKEMETHOD(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[1])), x0[1]), java.lang.Object(x2[1])) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], +(x0[1], -1)), java.lang.Object(x2[1])) which results in the following constraint:
(1)    (>(x0[0], 1)=TRUE∧2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0])=2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[1])), x0[1])∧java.lang.Object(x2[0])=java.lang.Object(x2[1]) ⇒ 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))≥NonInfC∧2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))≥COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))∧(U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))), ≥))

We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
(2)    (>(x0[0], 1)=TRUE ⇒ 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))≥NonInfC∧2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))≥COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))∧(U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))), ≥)∧0 = 0∧0 = 0∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))), ≥)∧0 = 0∧0 = 0∧[(3)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_13] ≥ 0)

For Pair COND_2767_1_MAIN_INVOKEMETHOD(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(x2)) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1, +(x0, -1)), java.lang.Object(x2)) the following chains were created:

We consider the chain COND_2767_1_MAIN_INVOKEMETHOD(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[1])), x0[1]), java.lang.Object(x2[1])) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], +(x0[1], -1)), java.lang.Object(x2[1])) which results in the following constraint:
(8)    (COND_2767_1_MAIN_INVOKEMETHOD(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[1])), x0[1]), java.lang.Object(x2[1]))≥NonInfC∧COND_2767_1_MAIN_INVOKEMETHOD(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[1])), x0[1]), java.lang.Object(x2[1]))≥2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], +(x0[1], -1)), java.lang.Object(x2[1]))∧(U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], +(x0[1], -1)), java.lang.Object(x2[1]))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], +(x0[1], -1)), java.lang.Object(x2[1]))), ≥)∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], +(x0[1], -1)), java.lang.Object(x2[1]))), ≥)∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], +(x0[1], -1)), java.lang.Object(x2[1]))), ≥)∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], +(x0[1], -1)), java.lang.Object(x2[1]))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

For Pair 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(IntList(x1))) → COND_2767_1_MAIN_INVOKEMETHOD1(>(x0, 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(IntList(x1))) the following chains were created:

We consider the chain 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2]))) → COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2]))), COND_2767_1_MAIN_INVOKEMETHOD1(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[3])), x0[3]), java.lang.Object(IntList(x1[3]))) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], +(x0[3], -1)), java.lang.Object(IntList(x1[3]))) which results in the following constraint:
(13)    (>(x0[2], 1)=TRUE∧2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2])=2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[3])), x0[3])∧java.lang.Object(IntList(x1[2]))=java.lang.Object(IntList(x1[3])) ⇒ 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))≥NonInfC∧2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))≥COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))∧(U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))), ≥))

We simplified constraint (13) using rules (I), (II), (IV) which results in the following new constraint:
(14)    (>(x0[2], 1)=TRUE ⇒ 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))≥NonInfC∧2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))≥COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))∧(U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (x0[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (x0[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (x0[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (17) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(18)    (x0[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))), ≥)∧0 = 0∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    (x0[2] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))), ≥)∧0 = 0∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

For Pair COND_2767_1_MAIN_INVOKEMETHOD1(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(IntList(x1))) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1, +(x0, -1)), java.lang.Object(IntList(x1))) the following chains were created:

We consider the chain COND_2767_1_MAIN_INVOKEMETHOD1(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[3])), x0[3]), java.lang.Object(IntList(x1[3]))) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], +(x0[3], -1)), java.lang.Object(IntList(x1[3]))) which results in the following constraint:
(20)    (COND_2767_1_MAIN_INVOKEMETHOD1(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[3])), x0[3]), java.lang.Object(IntList(x1[3])))≥NonInfC∧COND_2767_1_MAIN_INVOKEMETHOD1(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[3])), x0[3]), java.lang.Object(IntList(x1[3])))≥2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], +(x0[3], -1)), java.lang.Object(IntList(x1[3])))∧(U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], +(x0[3], -1)), java.lang.Object(IntList(x1[3])))), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    ((U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], +(x0[3], -1)), java.lang.Object(IntList(x1[3])))), ≥)∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    ((U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], +(x0[3], -1)), java.lang.Object(IntList(x1[3])))), ≥)∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    ((U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], +(x0[3], -1)), java.lang.Object(IntList(x1[3])))), ≥)∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    ((U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], +(x0[3], -1)), java.lang.Object(IntList(x1[3])))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

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

2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(x2)) → COND_2767_1_MAIN_INVOKEMETHOD(>(x0, 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(x2))
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))), ≥)∧0 = 0∧0 = 0∧[(3)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_13] ≥ 0)
COND_2767_1_MAIN_INVOKEMETHOD(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(x2)) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1, +(x0, -1)), java.lang.Object(x2))
- ((U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], +(x0[1], -1)), java.lang.Object(x2[1]))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)
2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(IntList(x1))) → COND_2767_1_MAIN_INVOKEMETHOD1(>(x0, 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(IntList(x1)))
- (x0[2] ≥ 0 ⇒ (U^Increasing(COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))), ≥)∧0 = 0∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)
COND_2767_1_MAIN_INVOKEMETHOD1(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1)), x0), java.lang.Object(IntList(x1))) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1, +(x0, -1)), java.lang.Object(IntList(x1)))
- ((U^Increasing(2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], +(x0[3], -1)), java.lang.Object(IntList(x1[3])))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(2767_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [2] + x₁
POL(2767_0_nth_ConstantStackPush(x₁, x₂)) = [-1] + x₂
POL(java.lang.Object(x₁)) = x₁
POL(IntList(x₁)) = x₁
POL(COND_2767_1_MAIN_INVOKEMETHOD(x₁, x₂, x₃)) = [2] + x₂
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(COND_2767_1_MAIN_INVOKEMETHOD1(x₁, x₂, x₃)) = [2] + x₂

The following pairs are in P_>:

COND_2767_1_MAIN_INVOKEMETHOD(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[1])), x0[1]), java.lang.Object(x2[1])) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], +(x0[1], -1)), java.lang.Object(x2[1]))
COND_2767_1_MAIN_INVOKEMETHOD1(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[3])), x0[3]), java.lang.Object(IntList(x1[3]))) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], +(x0[3], -1)), java.lang.Object(IntList(x1[3])))

The following pairs are in P_bound:

2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0])) → COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))
2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2]))) → COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))

The following pairs are in P_≥:

2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0])) → COND_2767_1_MAIN_INVOKEMETHOD(>(x0[0], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[0])), x0[0]), java.lang.Object(x2[0]))
2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2]))) → COND_2767_1_MAIN_INVOKEMETHOD1(>(x0[2], 1), 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[2])), x0[2]), java.lang.Object(IntList(x1[2])))

There are no usable rules.

(25) Complex Obligation (AND)

(26) 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

(27) IDependencyGraphProof (EQUIVALENT transformation)

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

(28) TRUE

(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:
(1): COND_2767_1_MAIN_INVOKEMETHOD(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[1])), x0[1]), java.lang.Object(x2[1])) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[1], x0[1] + -1), java.lang.Object(x2[1]))
(3): COND_2767_1_MAIN_INVOKEMETHOD1(TRUE, 2767_0_nth_ConstantStackPush(java.lang.Object(IntList(x1[3])), x0[3]), java.lang.Object(IntList(x1[3]))) → 2767_1_MAIN_INVOKEMETHOD(2767_0_nth_ConstantStackPush(x1[3], x0[3] + -1), java.lang.Object(IntList(x1[3])))

The set Q is empty.

(30) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) IDPNonInfProof (SOUND transformation)

(7) Complex Obligation (AND)

(8) Obligation:

(9) IDependencyGraphProof (EQUIVALENT transformation)

(10) TRUE

(11) Obligation:

(12) IDependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) IDPNonInfProof (SOUND transformation)

(16) Complex Obligation (AND)

(17) Obligation:

(18) IDependencyGraphProof (EQUIVALENT transformation)

(19) TRUE

(20) Obligation:

(21) IDependencyGraphProof (EQUIVALENT transformation)

(22) TRUE

(23) Obligation:

(24) IDPNonInfProof (SOUND transformation)

(25) Complex Obligation (AND)

(26) Obligation:

(27) IDependencyGraphProof (EQUIVALENT transformation)

(28) TRUE

(29) Obligation:

(30) IDependencyGraphProof (EQUIVALENT transformation)

(31) TRUE