Termination of the given JBC could be proven:

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

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: ConvertRec/ConvertRec 
package ConvertRec;

public class ConvertRec {

  public static int convert(List xs, int b) {
    if (xs == null) return 0;
    if (xs.value <= 0) return  b * convert(xs.next, b);
    return 1 + convert(new List(xs.value - 1, xs.next), b);
  }

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

    List xs = createList(Random.random());

    convert(xs, Random.random());
  }

  public static List createList(int l) {
	if (l <= 0) {
		return null;
	} else {
		return new List(Random.random(), createList(l-1));
	}
  }
}


package ConvertRec;

public class List {

  int value;
  List next;

  public List() {}

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

  public static List copy(List x) {
    if (x == null) return null;
    else return new List(x.value,copy(x.next));
  }
}


package ConvertRec;

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

  public static int random() {
    if (args.length <= index) {
      return 0;
    }
    String string = args[index];
    index++;
    if (string == null) {
      return 0;
    }
    return string.length();
  }
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
ConvertRec.ConvertRec.main([Ljava/lang/String;)V: Graph of 279 nodes with 0 SCCs.

ConvertRec.ConvertRec.createList(I)LConvertRec/List;: Graph of 145 nodes with 0 SCCs.

ConvertRec.ConvertRec.convert(LConvertRec/List;I)I: Graph of 74 nodes with 0 SCCs.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 42 rules for P and 33 rules for R.


Combined rules. Obtained 2 rules for P and 10 rules for R.


Filtered ground terms:


2929_1_convert_InvokeMethod(x1, x2, x3) → 2929_1_convert_InvokeMethod(x1, x3)
ConvertRec.List(x1, x2, x3) → ConvertRec.List(x2, x3)
1756_0_convert_NONNULL(x1, x2, x3) → 1756_0_convert_NONNULL(x2, x3)
Cond_1756_0_convert_NONNULL1(x1, x2, x3, x4) → Cond_1756_0_convert_NONNULL1(x1, x3, x4)
Cond_1756_0_convert_NONNULL(x1, x2, x3, x4) → Cond_1756_0_convert_NONNULL(x1, x3, x4)
3418_0_convert_Return(x1) → 3418_0_convert_Return
3660_0_convert_Return(x1, x2) → 3660_0_convert_Return(x2)
3411_0_convert_Return(x1, x2) → 3411_0_convert_Return(x2)
2344_0_convert_Return(x1, x2) → 2344_0_convert_Return(x2)
1908_0_convert_Return(x1, x2) → 1908_0_convert_Return

Filtered duplicate args:


1756_0_convert_NONNULL(x1, x2) → 1756_0_convert_NONNULL(x2)
Cond_1756_0_convert_NONNULL1(x1, x2, x3) → Cond_1756_0_convert_NONNULL1(x1, x3)
Cond_1756_0_convert_NONNULL(x1, x2, x3) → Cond_1756_0_convert_NONNULL(x1, x3)

Combined rules. Obtained 2 rules for P and 10 rules for R.


Finished conversion. Obtained 2 rules for P and 10 rules for R. System has predefined symbols.



Log for SCC 1:

Generated 51 rules for P and 104 rules for R.


Combined rules. Obtained 7 rules for P and 19 rules for R.


Filtered ground terms:


825_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → 825_1_createList_InvokeMethod(x1, x5)
ConvertRec.List(x1) → ConvertRec.List
666_0_createList_GT(x1, x2, x3) → 666_0_createList_GT(x2, x3)
Cond_783_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_783_1_createList_InvokeMethod(x1, x2, x3)
783_0_random_GT(x1, x2, x3) → 783_0_random_GT(x2, x3)
783_1_createList_InvokeMethod(x1, x2, x3, x4) → 783_1_createList_InvokeMethod(x1, x2)
1080_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → 1080_1_createList_InvokeMethod(x1, x4, x5)
Cond_835_1_createList_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_835_1_createList_InvokeMethod1(x1, x2, x3)
835_0_random_IntArithmetic(x1, x2, x3, x4) → 835_0_random_IntArithmetic(x2, x3)
835_1_createList_InvokeMethod(x1, x2, x3, x4) → 835_1_createList_InvokeMethod(x1, x2)
Cond_835_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_835_1_createList_InvokeMethod(x1, x2, x3)
Cond_800_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_800_1_createList_InvokeMethod(x1, x2, x3)
800_0_random_ArrayAccess(x1, x2, x3) → 800_0_random_ArrayAccess(x2, x3)
800_1_createList_InvokeMethod(x1, x2, x3, x4) → 800_1_createList_InvokeMethod(x1, x2)
Cond_781_1_createList_InvokeMethod(x1, x2, x3, x4, x5) → Cond_781_1_createList_InvokeMethod(x1, x2, x3)
781_0_random_GT(x1, x2, x3) → 781_0_random_GT(x2, x3)
781_1_createList_InvokeMethod(x1, x2, x3, x4) → 781_1_createList_InvokeMethod(x1, x2)
Cond_666_0_createList_GT1(x1, x2, x3, x4) → Cond_666_0_createList_GT1(x1, x3, x4)
Cond_666_0_createList_GT(x1, x2, x3, x4) → Cond_666_0_createList_GT(x1, x3, x4)
2865_0_createList_Return(x1, x2) → 2865_0_createList_Return
1904_0_createList_Return(x1, x2) → 1904_0_createList_Return
1812_0_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 1812_0_createList_InvokeMethod(x5, x6)
java.lang.ArrayIndexOutOfBoundsException(x1) → java.lang.ArrayIndexOutOfBoundsException
java.lang.IndexOutOfBoundsException(x1) → java.lang.IndexOutOfBoundsException
java.lang.RuntimeException(x1) → java.lang.RuntimeException
java.lang.Exception(x1) → java.lang.Exception
java.lang.Throwable(x1) → java.lang.Throwable
1627_0_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 1627_0_createList_InvokeMethod(x6)
1365_0_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 1365_0_createList_InvokeMethod(x5, x6)
1353_0_createList_InvokeMethod(x1, x2, x3, x4, x5, x6) → 1353_0_createList_InvokeMethod(x6)
1221_0_createList_InvokeMethod(x1, x2, x3, x4, x5) → 1221_0_createList_InvokeMethod(x3)
954_0_createList_Return(x1, x2) → 954_0_createList_Return
715_0_createList_Return(x1, x2, x3) → 715_0_createList_Return

Filtered duplicate args:


666_0_createList_GT(x1, x2) → 666_0_createList_GT(x2)
Cond_666_0_createList_GT1(x1, x2, x3) → Cond_666_0_createList_GT1(x1, x3)
Cond_666_0_createList_GT(x1, x2, x3) → Cond_666_0_createList_GT(x1, x3)

Filtered unneeded arguments:


1080_1_createList_InvokeMethod(x1, x2, x3) → 1080_1_createList_InvokeMethod(x1, x3)

Filtered all non-integer terms:


835_0_random_IntArithmetic(x1, x2) → 835_0_random_IntArithmetic(x2)

Filtered all free variables:


781_1_createList_InvokeMethod(x1, x2) → 781_1_createList_InvokeMethod(x2)
783_1_createList_InvokeMethod(x1, x2) → 783_1_createList_InvokeMethod(x2)
Cond_781_1_createList_InvokeMethod(x1, x2, x3) → Cond_781_1_createList_InvokeMethod(x1, x3)
800_1_createList_InvokeMethod(x1, x2) → 800_1_createList_InvokeMethod(x2)
Cond_800_1_createList_InvokeMethod(x1, x2, x3) → Cond_800_1_createList_InvokeMethod(x1, x3)
835_1_createList_InvokeMethod(x1, x2) → 835_1_createList_InvokeMethod(x2)
Cond_835_1_createList_InvokeMethod(x1, x2, x3) → Cond_835_1_createList_InvokeMethod(x1, x3)
Cond_835_1_createList_InvokeMethod1(x1, x2, x3) → Cond_835_1_createList_InvokeMethod1(x1, x3)
Cond_783_1_createList_InvokeMethod(x1, x2, x3) → Cond_783_1_createList_InvokeMethod(x1, x3)
1812_0_createList_InvokeMethod(x1, x2) → 1812_0_createList_InvokeMethod(x2)

Combined rules. Obtained 2 rules for P and 19 rules for R.


Finished conversion. Obtained 2 rules for P and 19 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:

Integer


The ITRS R consists of the following rules:
1756_0_convert_NONNULL(NULL) → 1908_0_convert_Return
2252_1_convert_InvokeMethod(1908_0_convert_Return, java.lang.Object(ConvertRec.List(x0, NULL)), NULL) → 2344_0_convert_Return(java.lang.Object(ConvertRec.List(x0, NULL)))
2252_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(x0, NULL))), java.lang.Object(ConvertRec.List(x1, java.lang.Object(ConvertRec.List(x0, NULL)))), java.lang.Object(ConvertRec.List(x0, NULL))) → 3411_0_convert_Return(java.lang.Object(ConvertRec.List(x1, java.lang.Object(ConvertRec.List(x0, NULL)))))
2252_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3411_0_convert_Return(java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))))
2252_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))), java.lang.Object(ConvertRec.List(x1, x2))) → 3660_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))
2252_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3411_0_convert_Return(java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))))
2929_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(0, NULL))), java.lang.Object(ConvertRec.List(0, NULL))) → 3418_0_convert_Return
2929_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3418_0_convert_Return
2929_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x1, x2))) → 3418_0_convert_Return
2929_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3418_0_convert_Return

The integer pair graph contains the following rules and edges:
(0): 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0]))) → COND_1756_0_CONVERT_NONNULL(x0[0] <= 0, java.lang.Object(ConvertRec.List(x0[0], x1[0])))
(1): COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[1], x1[1]))) → 1756_0_CONVERT_NONNULL(x1[1])
(2): 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(x0[2] > 0, java.lang.Object(ConvertRec.List(x0[2], x1[2])))
(3): COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[3] - 1, x1[3])))

(0) -> (1), if ((x0[0] <= 0 →* TRUE)∧(java.lang.Object(ConvertRec.List(x0[0], x1[0])) →* java.lang.Object(ConvertRec.List(x0[1], x1[1]))))


(1) -> (0), if ((x1[1] →* java.lang.Object(ConvertRec.List(x0[0], x1[0]))))


(1) -> (2), if ((x1[1] →* java.lang.Object(ConvertRec.List(x0[2], x1[2]))))


(2) -> (3), if ((x0[2] > 0 →* TRUE)∧(java.lang.Object(ConvertRec.List(x0[2], x1[2])) →* java.lang.Object(ConvertRec.List(x0[3], x1[3]))))


(3) -> (0), if ((java.lang.Object(ConvertRec.List(x0[3] - 1, x1[3])) →* java.lang.Object(ConvertRec.List(x0[0], x1[0]))))


(3) -> (2), if ((java.lang.Object(ConvertRec.List(x0[3] - 1, x1[3])) →* java.lang.Object(ConvertRec.List(x0[2], x1[2]))))



The set Q consists of the following terms:
1756_0_convert_NONNULL(NULL)
2252_1_convert_InvokeMethod(1908_0_convert_Return, java.lang.Object(ConvertRec.List(x0, NULL)), NULL)
2252_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(x0, NULL))), java.lang.Object(ConvertRec.List(x1, java.lang.Object(ConvertRec.List(x0, NULL)))), java.lang.Object(ConvertRec.List(x0, NULL)))
2252_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))
2252_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))), java.lang.Object(ConvertRec.List(x1, x2)))
2252_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))
2929_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(0, NULL))), java.lang.Object(ConvertRec.List(0, NULL)))
2929_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1)))))
2929_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x0, x1)))
2929_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1)))))

(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 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0, x1))) → COND_1756_0_CONVERT_NONNULL(<=(x0, 0), java.lang.Object(ConvertRec.List(x0, x1))) the following chains were created:
  • We consider the chain 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0]))) → COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0]))), COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[1], x1[1]))) → 1756_0_CONVERT_NONNULL(x1[1]) which results in the following constraint:

    (1)    (<=(x0[0], 0)=TRUE∧java.lang.Object(ConvertRec.List(x0[0], x1[0]))=java.lang.Object(ConvertRec.List(x0[1], x1[1])) ⇒ 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0])))≥NonInfC∧1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0])))≥COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))∧(UIncreasing(COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))), ≥))



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

    (2)    (<=(x0[0], 0)=TRUE ⇒ 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0])))≥NonInfC∧1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0])))≥COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))∧(UIncreasing(COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))), ≥))



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

    (3)    (0 ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))), ≥)∧[(7)bni_32 + (-1)Bound*bni_32] + [(3)bni_32]x1[0] + [(3)bni_32]x0[0] ≥ 0∧[2 + (-1)bso_33] ≥ 0)



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

    (4)    (0 ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))), ≥)∧[(7)bni_32 + (-1)Bound*bni_32] + [(3)bni_32]x1[0] + [(3)bni_32]x0[0] ≥ 0∧[2 + (-1)bso_33] ≥ 0)



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

    (5)    (0 ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))), ≥)∧[(7)bni_32 + (-1)Bound*bni_32] + [(3)bni_32]x1[0] + [(3)bni_32]x0[0] ≥ 0∧[2 + (-1)bso_33] ≥ 0)



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

    (6)    (0 ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))), ≥)∧[(3)bni_32] ≥ 0∧[(3)bni_32] ≥ 0∧[(7)bni_32 + (-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[2 + (-1)bso_33] ≥ 0)







For Pair COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0, x1))) → 1756_0_CONVERT_NONNULL(x1) the following chains were created:
  • We consider the chain 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0]))) → COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0]))), COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[1], x1[1]))) → 1756_0_CONVERT_NONNULL(x1[1]), 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0]))) → COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0]))) which results in the following constraint:

    (7)    (<=(x0[0], 0)=TRUE∧java.lang.Object(ConvertRec.List(x0[0], x1[0]))=java.lang.Object(ConvertRec.List(x0[1], x1[1]))∧x1[1]=java.lang.Object(ConvertRec.List(x0[0]1, x1[0]1)) ⇒ COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[1], x1[1])))≥NonInfC∧COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[1], x1[1])))≥1756_0_CONVERT_NONNULL(x1[1])∧(UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥))



    We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:

    (8)    (<=(x0[0], 0)=TRUE ⇒ COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[0], java.lang.Object(ConvertRec.List(x0[0]1, x1[0]1)))))≥NonInfC∧COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[0], java.lang.Object(ConvertRec.List(x0[0]1, x1[0]1)))))≥1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0]1, x1[0]1)))∧(UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥))



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

    (9)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥)∧[(23)bni_34 + (-1)Bound*bni_34] + [(9)bni_34]x1[0]1 + [(9)bni_34]x0[0]1 + [(3)bni_34]x0[0] ≥ 0∧[16 + (-1)bso_35] + [6]x1[0]1 + [6]x0[0]1 + [3]x0[0] ≥ 0)



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

    (10)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥)∧[(23)bni_34 + (-1)Bound*bni_34] + [(9)bni_34]x1[0]1 + [(9)bni_34]x0[0]1 + [(3)bni_34]x0[0] ≥ 0∧[16 + (-1)bso_35] + [6]x1[0]1 + [6]x0[0]1 + [3]x0[0] ≥ 0)



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

    (11)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥)∧[(23)bni_34 + (-1)Bound*bni_34] + [(9)bni_34]x1[0]1 + [(9)bni_34]x0[0]1 + [(3)bni_34]x0[0] ≥ 0∧[16 + (-1)bso_35] + [6]x1[0]1 + [6]x0[0]1 + [3]x0[0] ≥ 0)



    We simplified constraint (11) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (12)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥)∧[(9)bni_34] ≥ 0∧[(9)bni_34] ≥ 0∧[(3)bni_34] ≥ 0∧[(23)bni_34 + (-1)Bound*bni_34] ≥ 0∧[16 + (-1)bso_35] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)



  • We consider the chain 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0]))) → COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0]))), COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[1], x1[1]))) → 1756_0_CONVERT_NONNULL(x1[1]), 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2]))) which results in the following constraint:

    (13)    (<=(x0[0], 0)=TRUE∧java.lang.Object(ConvertRec.List(x0[0], x1[0]))=java.lang.Object(ConvertRec.List(x0[1], x1[1]))∧x1[1]=java.lang.Object(ConvertRec.List(x0[2], x1[2])) ⇒ COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[1], x1[1])))≥NonInfC∧COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[1], x1[1])))≥1756_0_CONVERT_NONNULL(x1[1])∧(UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥))



    We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint:

    (14)    (<=(x0[0], 0)=TRUE ⇒ COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[0], java.lang.Object(ConvertRec.List(x0[2], x1[2])))))≥NonInfC∧COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[0], java.lang.Object(ConvertRec.List(x0[2], x1[2])))))≥1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2])))∧(UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥))



    We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (15)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥)∧[(23)bni_34 + (-1)Bound*bni_34] + [(9)bni_34]x1[2] + [(9)bni_34]x0[2] + [(3)bni_34]x0[0] ≥ 0∧[16 + (-1)bso_35] + [6]x1[2] + [6]x0[2] + [3]x0[0] ≥ 0)



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

    (16)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥)∧[(23)bni_34 + (-1)Bound*bni_34] + [(9)bni_34]x1[2] + [(9)bni_34]x0[2] + [(3)bni_34]x0[0] ≥ 0∧[16 + (-1)bso_35] + [6]x1[2] + [6]x0[2] + [3]x0[0] ≥ 0)



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

    (17)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥)∧[(23)bni_34 + (-1)Bound*bni_34] + [(9)bni_34]x1[2] + [(9)bni_34]x0[2] + [(3)bni_34]x0[0] ≥ 0∧[16 + (-1)bso_35] + [6]x1[2] + [6]x0[2] + [3]x0[0] ≥ 0)



    We simplified constraint (17) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (18)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥)∧[(9)bni_34] ≥ 0∧[(9)bni_34] ≥ 0∧[(3)bni_34] ≥ 0∧[(23)bni_34 + (-1)Bound*bni_34] ≥ 0∧[16 + (-1)bso_35] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)







For Pair 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0, x1))) → COND_1756_0_CONVERT_NONNULL1(>(x0, 0), java.lang.Object(ConvertRec.List(x0, x1))) the following chains were created:
  • We consider the chain 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2]))), COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3]))) which results in the following constraint:

    (19)    (>(x0[2], 0)=TRUE∧java.lang.Object(ConvertRec.List(x0[2], x1[2]))=java.lang.Object(ConvertRec.List(x0[3], x1[3])) ⇒ 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥NonInfC∧1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))∧(UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥))



    We simplified constraint (19) using rules (I), (II), (IV) which results in the following new constraint:

    (20)    (>(x0[2], 0)=TRUE ⇒ 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥NonInfC∧1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))∧(UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥))



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

    (21)    (0 ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧[(7)bni_36 + (-1)Bound*bni_36] + [(3)bni_36]x1[2] + [(3)bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)



    We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (22)    (0 ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧[(7)bni_36 + (-1)Bound*bni_36] + [(3)bni_36]x1[2] + [(3)bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)



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

    (23)    (0 ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧[(7)bni_36 + (-1)Bound*bni_36] + [(3)bni_36]x1[2] + [(3)bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)



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

    (24)    (0 ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧[(3)bni_36] ≥ 0∧[(3)bni_36] ≥ 0∧[(7)bni_36 + (-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)







For Pair COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0, x1))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0, 1), x1))) the following chains were created:
  • We consider the chain 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2]))), COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3]))), 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0]))) → COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0]))) which results in the following constraint:

    (25)    (>(x0[2], 0)=TRUE∧java.lang.Object(ConvertRec.List(x0[2], x1[2]))=java.lang.Object(ConvertRec.List(x0[3], x1[3]))∧java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3]))=java.lang.Object(ConvertRec.List(x0[0], x1[0])) ⇒ COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3])))≥NonInfC∧COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3])))≥1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))∧(UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥))



    We simplified constraint (25) using rules (I), (II), (III), (IV) which results in the following new constraint:

    (26)    (>(x0[2], 0)=TRUE ⇒ COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥NonInfC∧COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[2], 1), x1[2])))∧(UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥))



    We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (27)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(7)bni_38 + (-1)Bound*bni_38] + [(3)bni_38]x1[2] + [(3)bni_38]x0[2] ≥ 0∧[(-1)bso_39] + [3]x0[2] ≥ 0)



    We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (28)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(7)bni_38 + (-1)Bound*bni_38] + [(3)bni_38]x1[2] + [(3)bni_38]x0[2] ≥ 0∧[(-1)bso_39] + [3]x0[2] ≥ 0)



    We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (29)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(7)bni_38 + (-1)Bound*bni_38] + [(3)bni_38]x1[2] + [(3)bni_38]x0[2] ≥ 0∧[(-1)bso_39] + [3]x0[2] ≥ 0)



    We simplified constraint (29) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (30)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(3)bni_38] ≥ 0∧[(3)bni_38] ≥ 0∧[(7)bni_38 + (-1)Bound*bni_38] ≥ 0∧0 ≥ 0∧[(-1)bso_39] ≥ 0∧[1] ≥ 0)



  • We consider the chain 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2]))), COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3]))), 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2]))) which results in the following constraint:

    (31)    (>(x0[2], 0)=TRUE∧java.lang.Object(ConvertRec.List(x0[2], x1[2]))=java.lang.Object(ConvertRec.List(x0[3], x1[3]))∧java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3]))=java.lang.Object(ConvertRec.List(x0[2]1, x1[2]1)) ⇒ COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3])))≥NonInfC∧COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3])))≥1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))∧(UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥))



    We simplified constraint (31) using rules (I), (II), (III), (IV) which results in the following new constraint:

    (32)    (>(x0[2], 0)=TRUE ⇒ COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥NonInfC∧COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[2], 1), x1[2])))∧(UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥))



    We simplified constraint (32) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (33)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(7)bni_38 + (-1)Bound*bni_38] + [(3)bni_38]x1[2] + [(3)bni_38]x0[2] ≥ 0∧[(-1)bso_39] + [3]x0[2] ≥ 0)



    We simplified constraint (33) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (34)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(7)bni_38 + (-1)Bound*bni_38] + [(3)bni_38]x1[2] + [(3)bni_38]x0[2] ≥ 0∧[(-1)bso_39] + [3]x0[2] ≥ 0)



    We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (35)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(7)bni_38 + (-1)Bound*bni_38] + [(3)bni_38]x1[2] + [(3)bni_38]x0[2] ≥ 0∧[(-1)bso_39] + [3]x0[2] ≥ 0)



    We simplified constraint (35) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (36)    (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(3)bni_38] ≥ 0∧[(3)bni_38] ≥ 0∧[(7)bni_38 + (-1)Bound*bni_38] ≥ 0∧0 ≥ 0∧[(-1)bso_39] ≥ 0∧[1] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0, x1))) → COND_1756_0_CONVERT_NONNULL(<=(x0, 0), java.lang.Object(ConvertRec.List(x0, x1)))
    • (0 ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))), ≥)∧[(3)bni_32] ≥ 0∧[(3)bni_32] ≥ 0∧[(7)bni_32 + (-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[2 + (-1)bso_33] ≥ 0)

  • COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0, x1))) → 1756_0_CONVERT_NONNULL(x1)
    • (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥)∧[(9)bni_34] ≥ 0∧[(9)bni_34] ≥ 0∧[(3)bni_34] ≥ 0∧[(23)bni_34 + (-1)Bound*bni_34] ≥ 0∧[16 + (-1)bso_35] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(x1[1])), ≥)∧[(9)bni_34] ≥ 0∧[(9)bni_34] ≥ 0∧[(3)bni_34] ≥ 0∧[(23)bni_34 + (-1)Bound*bni_34] ≥ 0∧[16 + (-1)bso_35] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)

  • 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0, x1))) → COND_1756_0_CONVERT_NONNULL1(>(x0, 0), java.lang.Object(ConvertRec.List(x0, x1)))
    • (0 ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧[(3)bni_36] ≥ 0∧[(3)bni_36] ≥ 0∧[(7)bni_36 + (-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)

  • COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0, x1))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0, 1), x1)))
    • (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(3)bni_38] ≥ 0∧[(3)bni_38] ≥ 0∧[(7)bni_38 + (-1)Bound*bni_38] ≥ 0∧0 ≥ 0∧[(-1)bso_39] ≥ 0∧[1] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(3)bni_38] ≥ 0∧[(3)bni_38] ≥ 0∧[(7)bni_38 + (-1)Bound*bni_38] ≥ 0∧0 ≥ 0∧[(-1)bso_39] ≥ 0∧[1] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(1756_0_convert_NONNULL(x1)) = 0   
POL(NULL) = 0   
POL(1908_0_convert_Return) = 0   
POL(2252_1_convert_InvokeMethod(x1, x2, x3)) = 0   
POL(java.lang.Object(x1)) = [3]x1   
POL(ConvertRec.List(x1, x2)) = [2] + x2 + x1   
POL(2344_0_convert_Return(x1)) = 0   
POL(3411_0_convert_Return(x1)) = 0   
POL(3418_0_convert_Return) = 0   
POL(3660_0_convert_Return(x1)) = 0   
POL(2929_1_convert_InvokeMethod(x1, x2)) = 0   
POL(0) = 0   
POL(1756_0_CONVERT_NONNULL(x1)) = [1] + x1   
POL(COND_1756_0_CONVERT_NONNULL(x1, x2)) = [-1] + x2   
POL(<=(x1, x2)) = 0   
POL(COND_1756_0_CONVERT_NONNULL1(x1, x2)) = [1] + x2   
POL(>(x1, x2)) = 0   
POL(-(x1, x2)) = 0   
POL(1) = 0   

The following pairs are in P>:

1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0]))) → COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))
COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[1], x1[1]))) → 1756_0_CONVERT_NONNULL(x1[1])

The following pairs are in Pbound:

1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[0], x1[0]))) → COND_1756_0_CONVERT_NONNULL(<=(x0[0], 0), java.lang.Object(ConvertRec.List(x0[0], x1[0])))
COND_1756_0_CONVERT_NONNULL(TRUE, java.lang.Object(ConvertRec.List(x0[1], x1[1]))) → 1756_0_CONVERT_NONNULL(x1[1])
1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))
COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))

The following pairs are in P≥:

1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))
COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))

There are no usable rules.

(7) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
1756_0_convert_NONNULL(NULL) → 1908_0_convert_Return
2252_1_convert_InvokeMethod(1908_0_convert_Return, java.lang.Object(ConvertRec.List(x0, NULL)), NULL) → 2344_0_convert_Return(java.lang.Object(ConvertRec.List(x0, NULL)))
2252_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(x0, NULL))), java.lang.Object(ConvertRec.List(x1, java.lang.Object(ConvertRec.List(x0, NULL)))), java.lang.Object(ConvertRec.List(x0, NULL))) → 3411_0_convert_Return(java.lang.Object(ConvertRec.List(x1, java.lang.Object(ConvertRec.List(x0, NULL)))))
2252_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3411_0_convert_Return(java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))))
2252_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))), java.lang.Object(ConvertRec.List(x1, x2))) → 3660_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))
2252_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3411_0_convert_Return(java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))))
2929_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(0, NULL))), java.lang.Object(ConvertRec.List(0, NULL))) → 3418_0_convert_Return
2929_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3418_0_convert_Return
2929_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x1, x2))) → 3418_0_convert_Return
2929_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3418_0_convert_Return

The integer pair graph contains the following rules and edges:
(2): 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(x0[2] > 0, java.lang.Object(ConvertRec.List(x0[2], x1[2])))
(3): COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[3] - 1, x1[3])))

(3) -> (2), if ((java.lang.Object(ConvertRec.List(x0[3] - 1, x1[3])) →* java.lang.Object(ConvertRec.List(x0[2], x1[2]))))


(2) -> (3), if ((x0[2] > 0 →* TRUE)∧(java.lang.Object(ConvertRec.List(x0[2], x1[2])) →* java.lang.Object(ConvertRec.List(x0[3], x1[3]))))



The set Q consists of the following terms:
1756_0_convert_NONNULL(NULL)
2252_1_convert_InvokeMethod(1908_0_convert_Return, java.lang.Object(ConvertRec.List(x0, NULL)), NULL)
2252_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(x0, NULL))), java.lang.Object(ConvertRec.List(x1, java.lang.Object(ConvertRec.List(x0, NULL)))), java.lang.Object(ConvertRec.List(x0, NULL)))
2252_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))
2252_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))), java.lang.Object(ConvertRec.List(x1, x2)))
2252_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))
2929_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(0, NULL))), java.lang.Object(ConvertRec.List(0, NULL)))
2929_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1)))))
2929_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x0, x1)))
2929_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1)))))

(8) 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 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2]))) the following chains were created:
  • We consider the chain 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2]))), COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3]))) which results in the following constraint:

    (1)    (>(x0[2], 0)=TRUE∧java.lang.Object(ConvertRec.List(x0[2], x1[2]))=java.lang.Object(ConvertRec.List(x0[3], x1[3])) ⇒ 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥NonInfC∧1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))∧(UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥))



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

    (2)    (>(x0[2], 0)=TRUE ⇒ 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥NonInfC∧1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))∧(UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥))



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

    (3)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧[(-2)bni_32 + (-1)Bound*bni_32] + [bni_32]x0[2] ≥ 0∧[(-1)bso_33] ≥ 0)



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

    (4)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧[(-2)bni_32 + (-1)Bound*bni_32] + [bni_32]x0[2] ≥ 0∧[(-1)bso_33] ≥ 0)



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

    (5)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧[(-2)bni_32 + (-1)Bound*bni_32] + [bni_32]x0[2] ≥ 0∧[(-1)bso_33] ≥ 0)



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

    (6)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧0 = 0∧[(-2)bni_32 + (-1)Bound*bni_32] + [bni_32]x0[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)



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

    (7)    (x0[2] ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧0 = 0∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x0[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)







For Pair COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3]))) the following chains were created:
  • We consider the chain 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2]))), COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3]))), 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2]))) which results in the following constraint:

    (8)    (>(x0[2], 0)=TRUE∧java.lang.Object(ConvertRec.List(x0[2], x1[2]))=java.lang.Object(ConvertRec.List(x0[3], x1[3]))∧java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3]))=java.lang.Object(ConvertRec.List(x0[2]1, x1[2]1)) ⇒ COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3])))≥NonInfC∧COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3])))≥1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))∧(UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥))



    We simplified constraint (8) using rules (I), (II), (III), (IV) which results in the following new constraint:

    (9)    (>(x0[2], 0)=TRUE ⇒ COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥NonInfC∧COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[2], x1[2])))≥1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[2], 1), x1[2])))∧(UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥))



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

    (10)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(-2)bni_34 + (-1)Bound*bni_34] + [bni_34]x0[2] ≥ 0∧[1 + (-1)bso_35] ≥ 0)



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

    (11)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(-2)bni_34 + (-1)Bound*bni_34] + [bni_34]x0[2] ≥ 0∧[1 + (-1)bso_35] ≥ 0)



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

    (12)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧[(-2)bni_34 + (-1)Bound*bni_34] + [bni_34]x0[2] ≥ 0∧[1 + (-1)bso_35] ≥ 0)



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

    (13)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧0 = 0∧[(-2)bni_34 + (-1)Bound*bni_34] + [bni_34]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_35] ≥ 0)



    We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (14)    (x0[2] ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧0 = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_35] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))
    • (x0[2] ≥ 0 ⇒ (UIncreasing(COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))), ≥)∧0 = 0∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x0[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

  • COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))
    • (x0[2] ≥ 0 ⇒ (UIncreasing(1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))), ≥)∧0 = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_35] ≥ 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) = [2]   
POL(FALSE) = 0   
POL(1756_0_convert_NONNULL(x1)) = [-1] + [-1]x1   
POL(NULL) = [-1]   
POL(1908_0_convert_Return) = [-1]   
POL(2252_1_convert_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1   
POL(java.lang.Object(x1)) = [-1]x1   
POL(ConvertRec.List(x1, x2)) = [-1] + x1   
POL(2344_0_convert_Return(x1)) = [-1] + [-1]x1   
POL(3411_0_convert_Return(x1)) = [-1] + [-1]x1   
POL(3418_0_convert_Return) = [-1]   
POL(3660_0_convert_Return(x1)) = [-1] + [-1]x1   
POL(2929_1_convert_InvokeMethod(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(0) = 0   
POL(1756_0_CONVERT_NONNULL(x1)) = [-1] + [-1]x1   
POL(COND_1756_0_CONVERT_NONNULL1(x1, x2)) = [-1] + [-1]x2   
POL(>(x1, x2)) = [-1]   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))

The following pairs are in Pbound:

1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))
COND_1756_0_CONVERT_NONNULL1(TRUE, java.lang.Object(ConvertRec.List(x0[3], x1[3]))) → 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(-(x0[3], 1), x1[3])))

The following pairs are in P≥:

1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(>(x0[2], 0), java.lang.Object(ConvertRec.List(x0[2], x1[2])))

There are no usable rules.

(9) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
1756_0_convert_NONNULL(NULL) → 1908_0_convert_Return
2252_1_convert_InvokeMethod(1908_0_convert_Return, java.lang.Object(ConvertRec.List(x0, NULL)), NULL) → 2344_0_convert_Return(java.lang.Object(ConvertRec.List(x0, NULL)))
2252_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(x0, NULL))), java.lang.Object(ConvertRec.List(x1, java.lang.Object(ConvertRec.List(x0, NULL)))), java.lang.Object(ConvertRec.List(x0, NULL))) → 3411_0_convert_Return(java.lang.Object(ConvertRec.List(x1, java.lang.Object(ConvertRec.List(x0, NULL)))))
2252_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3411_0_convert_Return(java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))))
2252_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))), java.lang.Object(ConvertRec.List(x1, x2))) → 3660_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))
2252_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3411_0_convert_Return(java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))))
2929_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(0, NULL))), java.lang.Object(ConvertRec.List(0, NULL))) → 3418_0_convert_Return
2929_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3418_0_convert_Return
2929_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x1, x2))) → 3418_0_convert_Return
2929_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x1, x2))))) → 3418_0_convert_Return

The integer pair graph contains the following rules and edges:
(2): 1756_0_CONVERT_NONNULL(java.lang.Object(ConvertRec.List(x0[2], x1[2]))) → COND_1756_0_CONVERT_NONNULL1(x0[2] > 0, java.lang.Object(ConvertRec.List(x0[2], x1[2])))


The set Q consists of the following terms:
1756_0_convert_NONNULL(NULL)
2252_1_convert_InvokeMethod(1908_0_convert_Return, java.lang.Object(ConvertRec.List(x0, NULL)), NULL)
2252_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(x0, NULL))), java.lang.Object(ConvertRec.List(x1, java.lang.Object(ConvertRec.List(x0, NULL)))), java.lang.Object(ConvertRec.List(x0, NULL)))
2252_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))
2252_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))), java.lang.Object(ConvertRec.List(x1, x2)))
2252_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2))))), java.lang.Object(ConvertRec.List(x3, java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))), java.lang.Object(ConvertRec.List(x0, java.lang.Object(ConvertRec.List(x1, x2)))))
2929_1_convert_InvokeMethod(2344_0_convert_Return(java.lang.Object(ConvertRec.List(0, NULL))), java.lang.Object(ConvertRec.List(0, NULL)))
2929_1_convert_InvokeMethod(3411_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1)))))
2929_1_convert_InvokeMethod(3418_0_convert_Return, java.lang.Object(ConvertRec.List(x0, x1)))
2929_1_convert_InvokeMethod(3660_0_convert_Return(java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1))))), java.lang.Object(ConvertRec.List(0, java.lang.Object(ConvertRec.List(x0, x1)))))

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
666_0_createList_GT(0) → 715_0_createList_Return
1080_1_createList_InvokeMethod(715_0_createList_Return, 0) → 1904_0_createList_Return
1080_1_createList_InvokeMethod(954_0_createList_Return, x1) → 2865_0_createList_Return
1080_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0) → 1812_0_createList_InvokeMethod(x0)
1080_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1904_0_createList_Return, x1) → 2865_0_createList_Return
1080_1_createList_InvokeMethod(2865_0_createList_Return, x1) → 2865_0_createList_Return
825_1_createList_InvokeMethod(715_0_createList_Return, 0) → 954_0_createList_Return
825_1_createList_InvokeMethod(954_0_createList_Return, x1) → 2865_0_createList_Return
825_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0) → 1812_0_createList_InvokeMethod(x0)
825_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1904_0_createList_Return, x1) → 2865_0_createList_Return
825_1_createList_InvokeMethod(2865_0_createList_Return, x1) → 2865_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 666_0_CREATELIST_GT(x0[0]) → COND_666_0_CREATELIST_GT(x0[0] > 0, x0[0])
(1): COND_666_0_CREATELIST_GT(TRUE, x0[1]) → 666_0_CREATELIST_GT(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:
666_0_createList_GT(0)
1080_1_createList_InvokeMethod(715_0_createList_Return, 0)
1080_1_createList_InvokeMethod(954_0_createList_Return, x0)
1080_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0)
1080_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x0), x1)
1080_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x2)
1080_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x0), x1)
1080_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x0), x1)
1080_1_createList_InvokeMethod(1904_0_createList_Return, x0)
1080_1_createList_InvokeMethod(2865_0_createList_Return, x0)
825_1_createList_InvokeMethod(715_0_createList_Return, 0)
825_1_createList_InvokeMethod(954_0_createList_Return, x0)
825_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0)
825_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x0), x1)
825_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x2)
825_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x0), x1)
825_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x0), x1)
825_1_createList_InvokeMethod(1904_0_createList_Return, x0)
825_1_createList_InvokeMethod(2865_0_createList_Return, x0)

(13) 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 666_0_CREATELIST_GT(x0) → COND_666_0_CREATELIST_GT(>(x0, 0), x0) the following chains were created:
  • We consider the chain 666_0_CREATELIST_GT(x0[0]) → COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0]), COND_666_0_CREATELIST_GT(TRUE, x0[1]) → 666_0_CREATELIST_GT(-(x0[1], 1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 666_0_CREATELIST_GT(x0[0])≥NonInfC∧666_0_CREATELIST_GT(x0[0])≥COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])∧(UIncreasing(COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])), ≥))



    We simplified constraint (1) using rule (IV) which results in the following new constraint:

    (2)    (>(x0[0], 0)=TRUE ⇒ 666_0_CREATELIST_GT(x0[0])≥NonInfC∧666_0_CREATELIST_GT(x0[0])≥COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])∧(UIncreasing(COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])), ≥))



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

    (3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_20] + [(2)bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_20] + [(2)bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 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_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_20] + [(2)bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_20 + (2)bni_20] + [(2)bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)







For Pair COND_666_0_CREATELIST_GT(TRUE, x0) → 666_0_CREATELIST_GT(-(x0, 1)) the following chains were created:
  • We consider the chain COND_666_0_CREATELIST_GT(TRUE, x0[1]) → 666_0_CREATELIST_GT(-(x0[1], 1)) which results in the following constraint:

    (7)    (COND_666_0_CREATELIST_GT(TRUE, x0[1])≥NonInfC∧COND_666_0_CREATELIST_GT(TRUE, x0[1])≥666_0_CREATELIST_GT(-(x0[1], 1))∧(UIncreasing(666_0_CREATELIST_GT(-(x0[1], 1))), ≥))



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

    (8)    ((UIncreasing(666_0_CREATELIST_GT(-(x0[1], 1))), ≥)∧[2 + (-1)bso_23] ≥ 0)



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

    (9)    ((UIncreasing(666_0_CREATELIST_GT(-(x0[1], 1))), ≥)∧[2 + (-1)bso_23] ≥ 0)



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

    (10)    ((UIncreasing(666_0_CREATELIST_GT(-(x0[1], 1))), ≥)∧[2 + (-1)bso_23] ≥ 0)



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

    (11)    ((UIncreasing(666_0_CREATELIST_GT(-(x0[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_23] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 666_0_CREATELIST_GT(x0) → COND_666_0_CREATELIST_GT(>(x0, 0), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_20 + (2)bni_20] + [(2)bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

  • COND_666_0_CREATELIST_GT(TRUE, x0) → 666_0_CREATELIST_GT(-(x0, 1))
    • ((UIncreasing(666_0_CREATELIST_GT(-(x0[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_23] ≥ 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(666_0_createList_GT(x1)) = [-1]   
POL(0) = 0   
POL(715_0_createList_Return) = [-1]   
POL(1080_1_createList_InvokeMethod(x1, x2)) = [-1]   
POL(1904_0_createList_Return) = [-1]   
POL(954_0_createList_Return) = [-1]   
POL(2865_0_createList_Return) = [-1]   
POL(1221_0_createList_InvokeMethod(x1)) = [-1]   
POL(1812_0_createList_InvokeMethod(x1)) = [-1]   
POL(1353_0_createList_InvokeMethod(x1)) = [-1]   
POL(1365_0_createList_InvokeMethod(x1, x2)) = [-1]   
POL(1627_0_createList_InvokeMethod(x1)) = [-1]   
POL(825_1_createList_InvokeMethod(x1, x2)) = [-1]   
POL(666_0_CREATELIST_GT(x1)) = [2]x1   
POL(COND_666_0_CREATELIST_GT(x1, x2)) = [2]x2   
POL(>(x1, x2)) = [-1]   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

COND_666_0_CREATELIST_GT(TRUE, x0[1]) → 666_0_CREATELIST_GT(-(x0[1], 1))

The following pairs are in Pbound:

666_0_CREATELIST_GT(x0[0]) → COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])

The following pairs are in P≥:

666_0_CREATELIST_GT(x0[0]) → COND_666_0_CREATELIST_GT(>(x0[0], 0), x0[0])

There are no usable rules.

(14) Complex Obligation (AND)

(15) 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:
666_0_createList_GT(0) → 715_0_createList_Return
1080_1_createList_InvokeMethod(715_0_createList_Return, 0) → 1904_0_createList_Return
1080_1_createList_InvokeMethod(954_0_createList_Return, x1) → 2865_0_createList_Return
1080_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0) → 1812_0_createList_InvokeMethod(x0)
1080_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1904_0_createList_Return, x1) → 2865_0_createList_Return
1080_1_createList_InvokeMethod(2865_0_createList_Return, x1) → 2865_0_createList_Return
825_1_createList_InvokeMethod(715_0_createList_Return, 0) → 954_0_createList_Return
825_1_createList_InvokeMethod(954_0_createList_Return, x1) → 2865_0_createList_Return
825_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0) → 1812_0_createList_InvokeMethod(x0)
825_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1904_0_createList_Return, x1) → 2865_0_createList_Return
825_1_createList_InvokeMethod(2865_0_createList_Return, x1) → 2865_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 666_0_CREATELIST_GT(x0[0]) → COND_666_0_CREATELIST_GT(x0[0] > 0, x0[0])


The set Q consists of the following terms:
666_0_createList_GT(0)
1080_1_createList_InvokeMethod(715_0_createList_Return, 0)
1080_1_createList_InvokeMethod(954_0_createList_Return, x0)
1080_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0)
1080_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x0), x1)
1080_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x2)
1080_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x0), x1)
1080_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x0), x1)
1080_1_createList_InvokeMethod(1904_0_createList_Return, x0)
1080_1_createList_InvokeMethod(2865_0_createList_Return, x0)
825_1_createList_InvokeMethod(715_0_createList_Return, 0)
825_1_createList_InvokeMethod(954_0_createList_Return, x0)
825_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0)
825_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x0), x1)
825_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x2)
825_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x0), x1)
825_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x0), x1)
825_1_createList_InvokeMethod(1904_0_createList_Return, x0)
825_1_createList_InvokeMethod(2865_0_createList_Return, x0)

(16) IDependencyGraphProof (EQUIVALENT transformation)

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

(17) TRUE

(18) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
666_0_createList_GT(0) → 715_0_createList_Return
1080_1_createList_InvokeMethod(715_0_createList_Return, 0) → 1904_0_createList_Return
1080_1_createList_InvokeMethod(954_0_createList_Return, x1) → 2865_0_createList_Return
1080_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0) → 1812_0_createList_InvokeMethod(x0)
1080_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
1080_1_createList_InvokeMethod(1904_0_createList_Return, x1) → 2865_0_createList_Return
1080_1_createList_InvokeMethod(2865_0_createList_Return, x1) → 2865_0_createList_Return
825_1_createList_InvokeMethod(715_0_createList_Return, 0) → 954_0_createList_Return
825_1_createList_InvokeMethod(954_0_createList_Return, x1) → 2865_0_createList_Return
825_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0) → 1812_0_createList_InvokeMethod(x0)
825_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x1), x3) → 1812_0_createList_InvokeMethod(x3)
825_1_createList_InvokeMethod(1904_0_createList_Return, x1) → 2865_0_createList_Return
825_1_createList_InvokeMethod(2865_0_createList_Return, x1) → 2865_0_createList_Return

The integer pair graph contains the following rules and edges:
(1): COND_666_0_CREATELIST_GT(TRUE, x0[1]) → 666_0_CREATELIST_GT(x0[1] - 1)


The set Q consists of the following terms:
666_0_createList_GT(0)
1080_1_createList_InvokeMethod(715_0_createList_Return, 0)
1080_1_createList_InvokeMethod(954_0_createList_Return, x0)
1080_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0)
1080_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x0), x1)
1080_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x2)
1080_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x0), x1)
1080_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x0), x1)
1080_1_createList_InvokeMethod(1904_0_createList_Return, x0)
1080_1_createList_InvokeMethod(2865_0_createList_Return, x0)
825_1_createList_InvokeMethod(715_0_createList_Return, 0)
825_1_createList_InvokeMethod(954_0_createList_Return, x0)
825_1_createList_InvokeMethod(1221_0_createList_InvokeMethod(x0), x0)
825_1_createList_InvokeMethod(1353_0_createList_InvokeMethod(x0), x1)
825_1_createList_InvokeMethod(1365_0_createList_InvokeMethod(x0, x1), x2)
825_1_createList_InvokeMethod(1627_0_createList_InvokeMethod(x0), x1)
825_1_createList_InvokeMethod(1812_0_createList_InvokeMethod(x0), x1)
825_1_createList_InvokeMethod(1904_0_createList_Return, x0)
825_1_createList_InvokeMethod(2865_0_createList_Return, x0)

(19) IDependencyGraphProof (EQUIVALENT transformation)

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

(20) TRUE