Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 AND
    ↳5 ITRS
        ↳6 GroundTermsRemoverProof (⇔)
        ↳7 ITRS
        ↳8 ITRStoIDPProof (⇔)
        ↳9 IDP
        ↳10 UsableRulesProof (⇔)
        ↳11 IDP
        ↳12 ItpfGraphProof (⇔)
        ↳13 IDP
        ↳14 IDPNonInfProof (⇐)
        ↳15 AND
            ↳16 IDP
                ↳17 IDPNonInfProof (⇐)
                ↳18 AND
                    ↳19 IDP
                        ↳20 IDependencyGraphProof (⇔)
                        ↳21 TRUE
                    ↳22 IDP
                        ↳23 IDependencyGraphProof (⇔)
                        ↳24 TRUE
            ↳25 IDP
                ↳26 IDependencyGraphProof (⇔)
                ↳27 TRUE
    ↳28 ITRS
        ↳29 ITRStoIDPProof (⇔)
        ↳30 IDP
        ↳31 UsableRulesProof (⇔)
        ↳32 IDP
        ↳33 IDPNonInfProof (⇐)
        ↳34 IDP
        ↳35 IDependencyGraphProof (⇔)
        ↳36 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: ListDuplicate 
/**
 * This class represents a list, where the function duplicate() can be used to
 * duplicate all elements in the list.
 * @author cotto
 */
public class ListDuplicate {
  /**
   * Walk through the list and, for each original element, copy it and append
   * this copy after the original. This transforms abc to aabbcc.
   */
  public static void duplicate(ObjectList list) {
    ObjectList current = list;
    boolean even = true;
    while (current != null) {
      // only copy the original elements!
      if (even) {
        final ObjectList copy =
          new ObjectList(current.value, current.next);
        current.next = copy;
      }
      current = current.next;
      even = !even;
    }
  }

  public static void main(String[] args) {
    Random.args = args;
    ObjectList list = ObjectList.createList();
    duplicate(list);
  }
}


public class ObjectList {
  Object value;
  ObjectList next;

  public ObjectList(Object value, ObjectList next) {
    this.value = value;
    this.next = next;
  }

  public static ObjectList createList() {
    ObjectList result = null;
    int length = Random.random();
    while (length > 0) {
      result = new ObjectList(new Object(), result);
      length--;
    }
    return result;
  }
}


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:
Graph of 209 nodes with 2 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) Complex Obligation (AND)

(5) Obligation:

ITRS 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 TRS R consists of the following rules:
Load1658(1, java.lang.Object(ObjectList(o751, o749)), 0) → Load1658(1, o751, 1)
Load1658(1, java.lang.Object(ObjectList(o770, o769)), i43) → Cond_Load1658(i43 > 0, 1, java.lang.Object(ObjectList(o770, o769)), i43)
Cond_Load1658(TRUE, 1, java.lang.Object(ObjectList(o770, o769)), i43) → Load1658(1, java.lang.Object(ObjectList(o770, o769)), 0)
The set Q consists of the following terms:
Load1658(1, java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load1658(TRUE, 1, java.lang.Object(ObjectList(x0, x1)), x2)

(6) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
  • 1

We removed arguments according to the following replacements:

Load1658(x1, x2, x3) → Load1658(x2, x3)
Cond_Load1658(x1, x2, x3, x4) → Cond_Load1658(x1, x3, x4)

(7) Obligation:

ITRS 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 TRS R consists of the following rules:
Load1658(java.lang.Object(ObjectList(o751, o749)), 0) → Load1658(o751, 1)
Load1658(java.lang.Object(ObjectList(o770, o769)), i43) → Cond_Load1658(i43 > 0, java.lang.Object(ObjectList(o770, o769)), i43)
Cond_Load1658(TRUE, java.lang.Object(ObjectList(o770, o769)), i43) → Load1658(java.lang.Object(ObjectList(o770, o769)), 0)
The set Q consists of the following terms:
Load1658(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load1658(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(8) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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:
Load1658(java.lang.Object(ObjectList(o751, o749)), 0) → Load1658(o751, 1)
Load1658(java.lang.Object(ObjectList(o770, o769)), i43) → Cond_Load1658(i43 > 0, java.lang.Object(ObjectList(o770, o769)), i43)
Cond_Load1658(TRUE, java.lang.Object(ObjectList(o770, o769)), i43) → Load1658(java.lang.Object(ObjectList(o770, o769)), 0)

The integer pair graph contains the following rules and edges:
(0): LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1)
(1): LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(i43[1] > 0, java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])
(2): COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)

(0) -> (0), if ((o751[0] →* java.lang.Object(ObjectList(o751[0]', o749[0]')))∧(1 →* 0))


(0) -> (1), if ((1 →* i43[1])∧(o751[0] →* java.lang.Object(ObjectList(o770[1], o769[1]))))


(1) -> (2), if ((java.lang.Object(ObjectList(o770[1], o769[1])) →* java.lang.Object(ObjectList(o770[2], o769[2])))∧(i43[1] > 0 →* TRUE)∧(i43[1] →* i43[2]))


(2) -> (0), if (java.lang.Object(ObjectList(o770[2], o769[2])) →* java.lang.Object(ObjectList(o751[0], o749[0])))


(2) -> (1), if ((java.lang.Object(ObjectList(o770[2], o769[2])) →* java.lang.Object(ObjectList(o770[1], o769[1])))∧(0 →* i43[1]))



The set Q consists of the following terms:
Load1658(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load1658(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(10) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1)
(1): LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(i43[1] > 0, java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])
(2): COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)

(0) -> (0), if ((o751[0] →* java.lang.Object(ObjectList(o751[0]', o749[0]')))∧(1 →* 0))


(0) -> (1), if ((1 →* i43[1])∧(o751[0] →* java.lang.Object(ObjectList(o770[1], o769[1]))))


(1) -> (2), if ((java.lang.Object(ObjectList(o770[1], o769[1])) →* java.lang.Object(ObjectList(o770[2], o769[2])))∧(i43[1] > 0 →* TRUE)∧(i43[1] →* i43[2]))


(2) -> (0), if (java.lang.Object(ObjectList(o770[2], o769[2])) →* java.lang.Object(ObjectList(o751[0], o749[0])))


(2) -> (1), if ((java.lang.Object(ObjectList(o770[2], o769[2])) →* java.lang.Object(ObjectList(o770[1], o769[1])))∧(0 →* i43[1]))



The set Q consists of the following terms:
Load1658(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load1658(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(12) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(13) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1)
(1): LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(i43[1] > 0, java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])
(2): COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)

(0) -> (1), if ((1 →* i43[1])∧(o751[0] →* java.lang.Object(ObjectList(o770[1], o769[1]))))


(0) -> (0), if ((o751[0] →* java.lang.Object(ObjectList(o751[0]', o749[0]')))∧false)


(1) -> (2), if (((o770[1] →* o770[2])∧(o769[1] →* o769[2]))∧(i43[1] > 0 →* TRUE)∧(i43[1] →* i43[2]))


(2) -> (0), if ((o770[2] →* o751[0])∧(o769[2] →* o749[0]))


(2) -> (1), if (((o770[2] →* o770[1])∧(o769[2] →* o769[1]))∧(0 →* i43[1]))



The set Q consists of the following terms:
Load1658(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load1658(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(14) 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 LOAD1658(java.lang.Object(ObjectList(o751, o749)), 0) → LOAD1658(o751, 1) the following chains were created:
  • We consider the chain LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1), LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1), LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) which results in the following constraint:

    (1)    (LOAD1658(java.lang.Object(ObjectList(o751[0]1, o749[0]1)), 0)≥LOAD1658(o751[0]1, 1)∧(UIncreasing(LOAD1658(o751[0]1, 1)), ≥))



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

    (2)    ((UIncreasing(LOAD1658(o751[0]1, 1)), ≥)∧[1 + (-1)bso_13] + [4]o749[0]1 + [3]o751[0]1 ≥ 0)



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

    (3)    ((UIncreasing(LOAD1658(o751[0]1, 1)), ≥)∧[1 + (-1)bso_13] + [4]o749[0]1 + [3]o751[0]1 ≥ 0)



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

    (4)    ((UIncreasing(LOAD1658(o751[0]1, 1)), ≥)∧[1 + (-1)bso_13] + [4]o749[0]1 + [3]o751[0]1 ≥ 0)



    We simplified constraint (4) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (5)    ((UIncreasing(LOAD1658(o751[0]1, 1)), ≥)∧[1 + (-1)bso_13] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)



  • We consider the chain COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0), LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1), LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) which results in the following constraint:

    (6)    (o770[2]=o751[0]∧o769[2]=o749[0]∧1=i43[1]∧o751[0]=java.lang.Object(ObjectList(o770[1], o769[1])) ⇒ LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0)≥LOAD1658(o751[0], 1)∧(UIncreasing(LOAD1658(o751[0], 1)), ≥))



    We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:

    (7)    (LOAD1658(java.lang.Object(ObjectList(java.lang.Object(ObjectList(o770[1], o769[1])), o769[2])), 0)≥LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), 1)∧(UIncreasing(LOAD1658(o751[0], 1)), ≥))



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

    (8)    ((UIncreasing(LOAD1658(o751[0], 1)), ≥)∧[13 + (-1)bso_13] + [4]o769[2] + [12]o769[1] + [12]o770[1] ≥ 0)



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

    (9)    ((UIncreasing(LOAD1658(o751[0], 1)), ≥)∧[13 + (-1)bso_13] + [4]o769[2] + [12]o769[1] + [12]o770[1] ≥ 0)



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

    (10)    ((UIncreasing(LOAD1658(o751[0], 1)), ≥)∧[13 + (-1)bso_13] + [4]o769[2] + [12]o769[1] + [12]o770[1] ≥ 0)



    We simplified constraint (10) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (11)    ((UIncreasing(LOAD1658(o751[0], 1)), ≥)∧[13 + (-1)bso_13] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)



  • We consider the chain LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1), LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1), LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1) which results in the following constraint:

    (12)    (LOAD1658(java.lang.Object(ObjectList(o751[0]1, o749[0]1)), 0)≥LOAD1658(o751[0]1, 1)∧(UIncreasing(LOAD1658(o751[0]1, 1)), ≥))



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

    (13)    ((UIncreasing(LOAD1658(o751[0]1, 1)), ≥)∧[1 + (-1)bso_13] + [4]o749[0]1 + [3]o751[0]1 ≥ 0)



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

    (14)    ((UIncreasing(LOAD1658(o751[0]1, 1)), ≥)∧[1 + (-1)bso_13] + [4]o749[0]1 + [3]o751[0]1 ≥ 0)



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

    (15)    ((UIncreasing(LOAD1658(o751[0]1, 1)), ≥)∧[1 + (-1)bso_13] + [4]o749[0]1 + [3]o751[0]1 ≥ 0)



    We simplified constraint (15) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (16)    ((UIncreasing(LOAD1658(o751[0]1, 1)), ≥)∧[1 + (-1)bso_13] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)



  • We consider the chain COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0), LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1), LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1) which results in the following constraint:

    (17)    (LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0)≥LOAD1658(o751[0], 1)∧(UIncreasing(LOAD1658(o751[0], 1)), ≥))



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

    (18)    ((UIncreasing(LOAD1658(o751[0], 1)), ≥)∧[1 + (-1)bso_13] + [4]o749[0] + [3]o751[0] ≥ 0)



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

    (19)    ((UIncreasing(LOAD1658(o751[0], 1)), ≥)∧[1 + (-1)bso_13] + [4]o749[0] + [3]o751[0] ≥ 0)



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

    (20)    ((UIncreasing(LOAD1658(o751[0], 1)), ≥)∧[1 + (-1)bso_13] + [4]o749[0] + [3]o751[0] ≥ 0)



    We simplified constraint (20) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (21)    ((UIncreasing(LOAD1658(o751[0], 1)), ≥)∧[1 + (-1)bso_13] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)







For Pair LOAD1658(java.lang.Object(ObjectList(o770, o769)), i43) → COND_LOAD1658(>(i43, 0), java.lang.Object(ObjectList(o770, o769)), i43) the following chains were created:
  • We consider the chain LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]), COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0) which results in the following constraint:

    (22)    (o770[1]=o770[2]∧o769[1]=o769[2]∧>(i43[1], 0)=TRUE∧i43[1]=i43[2] ⇒ LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])≥COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])∧(UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥))



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

    (23)    (>(i43[1], 0)=TRUE ⇒ LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])≥COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])∧(UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥))



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

    (24)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧[(-1)bso_14] + [3]i43[1] ≥ 0)



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

    (25)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧[(-1)bso_14] + [3]i43[1] ≥ 0)



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

    (26)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧[(-1)bso_14] + [3]i43[1] ≥ 0)



    We simplified constraint (26) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (27)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_14] ≥ 0∧[1] ≥ 0)







For Pair COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770, o769)), i43) → LOAD1658(java.lang.Object(ObjectList(o770, o769)), 0) the following chains were created:
  • We consider the chain LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]), COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0), LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1) which results in the following constraint:

    (28)    (o770[1]=o770[2]∧o769[1]=o769[2]∧>(i43[1], 0)=TRUE∧i43[1]=i43[2]∧o770[2]=o751[0]∧o769[2]=o749[0] ⇒ COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2])≥LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)∧(UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥))



    We simplified constraint (28) using rules (III), (IV) which results in the following new constraint:

    (29)    (>(i43[1], 0)=TRUE ⇒ COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])≥LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), 0)∧(UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥))



    We simplified constraint (29) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (30)    (0 ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧[(-1)bso_15] ≥ 0)



    We simplified constraint (30) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (31)    (0 ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧[(-1)bso_15] ≥ 0)



    We simplified constraint (31) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (32)    (0 ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧[(-1)bso_15] ≥ 0)



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

    (33)    (0 ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_15] ≥ 0)



  • We consider the chain LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]), COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0), LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) which results in the following constraint:

    (34)    (o770[1]=o770[2]∧o769[1]=o769[2]∧>(i43[1], 0)=TRUE∧i43[1]=i43[2]∧o770[2]=o770[1]1∧o769[2]=o769[1]1∧0=i43[1]1 ⇒ COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2])≥LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)∧(UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥))



    We simplified constraint (34) using rules (III), (IV) which results in the following new constraint:

    (35)    (>(i43[1], 0)=TRUE ⇒ COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])≥LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), 0)∧(UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥))



    We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (36)    (0 ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧[(-1)bso_15] ≥ 0)



    We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (37)    (0 ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧[(-1)bso_15] ≥ 0)



    We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (38)    (0 ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧[(-1)bso_15] ≥ 0)



    We simplified constraint (38) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (39)    (0 ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_15] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD1658(java.lang.Object(ObjectList(o751, o749)), 0) → LOAD1658(o751, 1)
    • ((UIncreasing(LOAD1658(o751[0]1, 1)), ≥)∧[1 + (-1)bso_13] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)
    • ((UIncreasing(LOAD1658(o751[0], 1)), ≥)∧[13 + (-1)bso_13] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)
    • ((UIncreasing(LOAD1658(o751[0]1, 1)), ≥)∧[1 + (-1)bso_13] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)
    • ((UIncreasing(LOAD1658(o751[0], 1)), ≥)∧[1 + (-1)bso_13] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)

  • LOAD1658(java.lang.Object(ObjectList(o770, o769)), i43) → COND_LOAD1658(>(i43, 0), java.lang.Object(ObjectList(o770, o769)), i43)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_14] ≥ 0∧[1] ≥ 0)

  • COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770, o769)), i43) → LOAD1658(java.lang.Object(ObjectList(o770, o769)), 0)
    • (0 ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_15] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_15] ≥ 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 all symbols [NONINF][POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(LOAD1658(x1, x2)) = [3]x2 + x1   
POL(java.lang.Object(x1)) = [2]x1   
POL(ObjectList(x1, x2)) = [2] + [2]x2 + [2]x1   
POL(0) = 0   
POL(1) = 0   
POL(COND_LOAD1658(x1, x2, x3)) = x2   
POL(>(x1, x2)) = 0   

The following pairs are in P>:

LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1)

The following pairs are in Pbound:

LOAD1658(java.lang.Object(ObjectList(o751[0], o749[0])), 0) → LOAD1658(o751[0], 1)
LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])
COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)

The following pairs are in P≥:

LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])
COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)

There are no usable rules.

(15) Complex Obligation (AND)

(16) 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): LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(i43[1] > 0, java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])
(2): COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)

(2) -> (1), if (((o770[2] →* o770[1])∧(o769[2] →* o769[1]))∧(0 →* i43[1]))


(1) -> (2), if (((o770[1] →* o770[2])∧(o769[1] →* o769[2]))∧(i43[1] > 0 →* TRUE)∧(i43[1] →* i43[2]))



The set Q consists of the following terms:
Load1658(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load1658(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(17) 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 LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) the following chains were created:
  • We consider the chain LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]), COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0) which results in the following constraint:

    (1)    (o770[1]=o770[2]∧o769[1]=o769[2]∧>(i43[1], 0)=TRUE∧i43[1]=i43[2] ⇒ LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])≥NonInfC∧LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])≥COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])∧(UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥))



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

    (2)    (>(i43[1], 0)=TRUE ⇒ LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])≥NonInfC∧LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])≥COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])∧(UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥))



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

    (3)    (i43[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i43[1] ≥ 0∧[-2 + (-1)bso_17] + [2]i43[1] ≥ 0)



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

    (4)    (i43[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i43[1] ≥ 0∧[-2 + (-1)bso_17] + [2]i43[1] ≥ 0)



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

    (5)    (i43[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i43[1] ≥ 0∧[-2 + (-1)bso_17] + [2]i43[1] ≥ 0)



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

    (6)    (i43[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧0 = 0∧0 = 0∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i43[1] ≥ 0∧0 = 0∧0 = 0∧[-2 + (-1)bso_17] + [2]i43[1] ≥ 0)



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

    (7)    (i43[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧0 = 0∧0 = 0∧[(3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i43[1] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] + [2]i43[1] ≥ 0)







For Pair COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0) the following chains were created:
  • We consider the chain LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]), COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0), LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) which results in the following constraint:

    (8)    (o770[1]=o770[2]∧o769[1]=o769[2]∧>(i43[1], 0)=TRUE∧i43[1]=i43[2]∧o770[2]=o770[1]1∧o769[2]=o769[1]1∧0=i43[1]1 ⇒ COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2])≥NonInfC∧COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2])≥LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)∧(UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥))



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

    (9)    (>(i43[1], 0)=TRUE ⇒ COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])≥NonInfC∧COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])≥LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), 0)∧(UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥))



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

    (10)    (i43[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧[2 + (-1)bso_19] ≥ 0)



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

    (11)    (i43[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧[2 + (-1)bso_19] ≥ 0)



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

    (12)    (i43[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧[2 + (-1)bso_19] ≥ 0)



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

    (13)    (i43[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)



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

    (14)    (i43[1] ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])
    • (i43[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])), ≥)∧0 = 0∧0 = 0∧[(3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i43[1] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] + [2]i43[1] ≥ 0)

  • COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)
    • (i43[1] ≥ 0 ⇒ (UIncreasing(LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)




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

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(LOAD1658(x1, x2)) = [-1] + [2]x2 + [2]x1   
POL(java.lang.Object(x1)) = [1]   
POL(ObjectList(x1, x2)) = [-1]x2 + x1   
POL(COND_LOAD1658(x1, x2, x3)) = [2] + x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)

The following pairs are in Pbound:

LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])
COND_LOAD1658(TRUE, java.lang.Object(ObjectList(o770[2], o769[2])), i43[2]) → LOAD1658(java.lang.Object(ObjectList(o770[2], o769[2])), 0)

The following pairs are in P≥:

LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(>(i43[1], 0), java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])

There are no usable rules.

(18) Complex Obligation (AND)

(19) 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): LOAD1658(java.lang.Object(ObjectList(o770[1], o769[1])), i43[1]) → COND_LOAD1658(i43[1] > 0, java.lang.Object(ObjectList(o770[1], o769[1])), i43[1])


The set Q consists of the following terms:
Load1658(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load1658(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(20) IDependencyGraphProof (EQUIVALENT transformation)

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

(21) TRUE

(22) 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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load1658(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load1658(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE

(25) 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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load1658(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load1658(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(26) IDependencyGraphProof (EQUIVALENT transformation)

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

(27) TRUE

(28) Obligation:

ITRS 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 TRS R consists of the following rules:
Load1080(i35) → Cond_Load1080(i35 > 0, i35)
Cond_Load1080(TRUE, i35) → Load1080(i35 + -1)
The set Q consists of the following terms:
Load1080(x0)
Cond_Load1080(TRUE, x0)

(29) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(30) 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:
Load1080(i35) → Cond_Load1080(i35 > 0, i35)
Cond_Load1080(TRUE, i35) → Load1080(i35 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD1080(i35[0]) → COND_LOAD1080(i35[0] > 0, i35[0])
(1): COND_LOAD1080(TRUE, i35[1]) → LOAD1080(i35[1] + -1)

(0) -> (1), if ((i35[0] > 0 →* TRUE)∧(i35[0] →* i35[1]))


(1) -> (0), if ((i35[1] + -1 →* i35[0]))



The set Q consists of the following terms:
Load1080(x0)
Cond_Load1080(TRUE, x0)

(31) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(32) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1080(i35[0]) → COND_LOAD1080(i35[0] > 0, i35[0])
(1): COND_LOAD1080(TRUE, i35[1]) → LOAD1080(i35[1] + -1)

(0) -> (1), if ((i35[0] > 0 →* TRUE)∧(i35[0] →* i35[1]))


(1) -> (0), if ((i35[1] + -1 →* i35[0]))



The set Q consists of the following terms:
Load1080(x0)
Cond_Load1080(TRUE, x0)

(33) 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 LOAD1080(i35) → COND_LOAD1080(>(i35, 0), i35) the following chains were created:
  • We consider the chain LOAD1080(i35[0]) → COND_LOAD1080(>(i35[0], 0), i35[0]), COND_LOAD1080(TRUE, i35[1]) → LOAD1080(+(i35[1], -1)) which results in the following constraint:

    (1)    (>(i35[0], 0)=TRUE∧i35[0]=i35[1] ⇒ LOAD1080(i35[0])≥NonInfC∧LOAD1080(i35[0])≥COND_LOAD1080(>(i35[0], 0), i35[0])∧(UIncreasing(COND_LOAD1080(>(i35[0], 0), i35[0])), ≥))



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

    (2)    (>(i35[0], 0)=TRUE ⇒ LOAD1080(i35[0])≥NonInfC∧LOAD1080(i35[0])≥COND_LOAD1080(>(i35[0], 0), i35[0])∧(UIncreasing(COND_LOAD1080(>(i35[0], 0), i35[0])), ≥))



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

    (3)    (i35[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1080(>(i35[0], 0), i35[0])), ≥)∧[(-1)Bound*bni_9] + [(2)bni_9]i35[0] ≥ 0∧[1 + (-1)bso_10] ≥ 0)



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

    (4)    (i35[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1080(>(i35[0], 0), i35[0])), ≥)∧[(-1)Bound*bni_9] + [(2)bni_9]i35[0] ≥ 0∧[1 + (-1)bso_10] ≥ 0)



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

    (5)    (i35[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1080(>(i35[0], 0), i35[0])), ≥)∧[(-1)Bound*bni_9] + [(2)bni_9]i35[0] ≥ 0∧[1 + (-1)bso_10] ≥ 0)



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

    (6)    (i35[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1080(>(i35[0], 0), i35[0])), ≥)∧[(-1)Bound*bni_9 + (2)bni_9] + [(2)bni_9]i35[0] ≥ 0∧[1 + (-1)bso_10] ≥ 0)







For Pair COND_LOAD1080(TRUE, i35) → LOAD1080(+(i35, -1)) the following chains were created:
  • We consider the chain LOAD1080(i35[0]) → COND_LOAD1080(>(i35[0], 0), i35[0]), COND_LOAD1080(TRUE, i35[1]) → LOAD1080(+(i35[1], -1)), LOAD1080(i35[0]) → COND_LOAD1080(>(i35[0], 0), i35[0]) which results in the following constraint:

    (7)    (>(i35[0], 0)=TRUE∧i35[0]=i35[1]∧+(i35[1], -1)=i35[0]1 ⇒ COND_LOAD1080(TRUE, i35[1])≥NonInfC∧COND_LOAD1080(TRUE, i35[1])≥LOAD1080(+(i35[1], -1))∧(UIncreasing(LOAD1080(+(i35[1], -1))), ≥))



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

    (8)    (>(i35[0], 0)=TRUE ⇒ COND_LOAD1080(TRUE, i35[0])≥NonInfC∧COND_LOAD1080(TRUE, i35[0])≥LOAD1080(+(i35[0], -1))∧(UIncreasing(LOAD1080(+(i35[1], -1))), ≥))



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

    (9)    (i35[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1080(+(i35[1], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i35[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (10)    (i35[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1080(+(i35[1], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i35[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (11)    (i35[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1080(+(i35[1], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i35[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (12)    (i35[0] ≥ 0 ⇒ (UIncreasing(LOAD1080(+(i35[1], -1))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i35[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD1080(i35) → COND_LOAD1080(>(i35, 0), i35)
    • (i35[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1080(>(i35[0], 0), i35[0])), ≥)∧[(-1)Bound*bni_9 + (2)bni_9] + [(2)bni_9]i35[0] ≥ 0∧[1 + (-1)bso_10] ≥ 0)

  • COND_LOAD1080(TRUE, i35) → LOAD1080(+(i35, -1))
    • (i35[0] ≥ 0 ⇒ (UIncreasing(LOAD1080(+(i35[1], -1))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i35[0] ≥ 0∧[1 + (-1)bso_12] ≥ 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(LOAD1080(x1)) = [2]x1   
POL(COND_LOAD1080(x1, x2)) = [-1] + [2]x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

LOAD1080(i35[0]) → COND_LOAD1080(>(i35[0], 0), i35[0])
COND_LOAD1080(TRUE, i35[1]) → LOAD1080(+(i35[1], -1))

The following pairs are in Pbound:

LOAD1080(i35[0]) → COND_LOAD1080(>(i35[0], 0), i35[0])
COND_LOAD1080(TRUE, i35[1]) → LOAD1080(+(i35[1], -1))

The following pairs are in P≥:
none

There are no usable rules.

(34) 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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load1080(x0)
Cond_Load1080(TRUE, x0)

(35) IDependencyGraphProof (EQUIVALENT transformation)

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

(36) TRUE