Termination of the given JBC could be proven:

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

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

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

}

class IntList {
  int value;
  IntList next;

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

  public static IntList createIntList() {

    int i = Random.random();
    IntList l = null;

    while (i > 0) {
      l = new IntList(Random.random(), l);
      i--;
    }

    return l;
  }
}


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

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


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 244 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:
Load1599(java.lang.Object(IntList(i108))) → Cond_Load1599(i108 > 0, java.lang.Object(IntList(i108)))
Cond_Load1599(TRUE, java.lang.Object(IntList(i108))) → Load1599(java.lang.Object(IntList(i108 - 1)))
The set Q consists of the following terms:
Load1599(java.lang.Object(IntList(x0)))
Cond_Load1599(TRUE, java.lang.Object(IntList(x0)))

(6) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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:
Load1599(java.lang.Object(IntList(i108))) → Cond_Load1599(i108 > 0, java.lang.Object(IntList(i108)))
Cond_Load1599(TRUE, java.lang.Object(IntList(i108))) → Load1599(java.lang.Object(IntList(i108 - 1)))

The integer pair graph contains the following rules and edges:
(0): LOAD1599(java.lang.Object(IntList(i108[0]))) → COND_LOAD1599(i108[0] > 0, java.lang.Object(IntList(i108[0])))
(1): COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[1]))) → LOAD1599(java.lang.Object(IntList(i108[1] - 1)))

(0) -> (1), if ((java.lang.Object(IntList(i108[0])) →* java.lang.Object(IntList(i108[1])))∧(i108[0] > 0 →* TRUE))


(1) -> (0), if ((java.lang.Object(IntList(i108[1] - 1)) →* java.lang.Object(IntList(i108[0]))))



The set Q consists of the following terms:
Load1599(java.lang.Object(IntList(x0)))
Cond_Load1599(TRUE, java.lang.Object(IntList(x0)))

(8) 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.

(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


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1599(java.lang.Object(IntList(i108[0]))) → COND_LOAD1599(i108[0] > 0, java.lang.Object(IntList(i108[0])))
(1): COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[1]))) → LOAD1599(java.lang.Object(IntList(i108[1] - 1)))

(0) -> (1), if ((java.lang.Object(IntList(i108[0])) →* java.lang.Object(IntList(i108[1])))∧(i108[0] > 0 →* TRUE))


(1) -> (0), if ((java.lang.Object(IntList(i108[1] - 1)) →* java.lang.Object(IntList(i108[0]))))



The set Q consists of the following terms:
Load1599(java.lang.Object(IntList(x0)))
Cond_Load1599(TRUE, java.lang.Object(IntList(x0)))

(10) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(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): LOAD1599(java.lang.Object(IntList(i108[0]))) → COND_LOAD1599(i108[0] > 0, java.lang.Object(IntList(i108[0])))
(1): COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[1]))) → LOAD1599(java.lang.Object(IntList(i108[1] - 1)))

(0) -> (1), if (((i108[0] →* i108[1]))∧(i108[0] > 0 →* TRUE))


(1) -> (0), if (((i108[1] - 1 →* i108[0])))



The set Q consists of the following terms:
Load1599(java.lang.Object(IntList(x0)))
Cond_Load1599(TRUE, java.lang.Object(IntList(x0)))

(12) 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 LOAD1599(java.lang.Object(IntList(i108))) → COND_LOAD1599(>(i108, 0), java.lang.Object(IntList(i108))) the following chains were created:
  • We consider the chain LOAD1599(java.lang.Object(IntList(i108[0]))) → COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0]))), COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[1]))) → LOAD1599(java.lang.Object(IntList(-(i108[1], 1)))) which results in the following constraint:

    (1)    (i108[0]=i108[1]∧>(i108[0], 0)=TRUE ⇒ LOAD1599(java.lang.Object(IntList(i108[0])))≥NonInfC∧LOAD1599(java.lang.Object(IntList(i108[0])))≥COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))∧(UIncreasing(COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))), ≥))



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

    (2)    (>(i108[0], 0)=TRUE ⇒ LOAD1599(java.lang.Object(IntList(i108[0])))≥NonInfC∧LOAD1599(java.lang.Object(IntList(i108[0])))≥COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))∧(UIncreasing(COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))), ≥))



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

    (3)    (i108[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i108[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (4)    (i108[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i108[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (5)    (i108[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i108[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (6)    (i108[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))), ≥)∧[(-1)Bound*bni_13] + [bni_13]i108[0] ≥ 0∧[(-1)bso_14] ≥ 0)







For Pair COND_LOAD1599(TRUE, java.lang.Object(IntList(i108))) → LOAD1599(java.lang.Object(IntList(-(i108, 1)))) the following chains were created:
  • We consider the chain LOAD1599(java.lang.Object(IntList(i108[0]))) → COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0]))), COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[1]))) → LOAD1599(java.lang.Object(IntList(-(i108[1], 1)))), LOAD1599(java.lang.Object(IntList(i108[0]))) → COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0]))) which results in the following constraint:

    (7)    (i108[0]=i108[1]∧>(i108[0], 0)=TRUE∧-(i108[1], 1)=i108[0]1 ⇒ COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[1])))≥NonInfC∧COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[1])))≥LOAD1599(java.lang.Object(IntList(-(i108[1], 1))))∧(UIncreasing(LOAD1599(java.lang.Object(IntList(-(i108[1], 1))))), ≥))



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

    (8)    (>(i108[0], 0)=TRUE ⇒ COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[0])))≥NonInfC∧COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[0])))≥LOAD1599(java.lang.Object(IntList(-(i108[0], 1))))∧(UIncreasing(LOAD1599(java.lang.Object(IntList(-(i108[1], 1))))), ≥))



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

    (9)    (i108[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1599(java.lang.Object(IntList(-(i108[1], 1))))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i108[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (10)    (i108[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1599(java.lang.Object(IntList(-(i108[1], 1))))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i108[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (11)    (i108[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1599(java.lang.Object(IntList(-(i108[1], 1))))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i108[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (12)    (i108[0] ≥ 0 ⇒ (UIncreasing(LOAD1599(java.lang.Object(IntList(-(i108[1], 1))))), ≥)∧[(-1)Bound*bni_15] + [bni_15]i108[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD1599(java.lang.Object(IntList(i108))) → COND_LOAD1599(>(i108, 0), java.lang.Object(IntList(i108)))
    • (i108[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))), ≥)∧[(-1)Bound*bni_13] + [bni_13]i108[0] ≥ 0∧[(-1)bso_14] ≥ 0)

  • COND_LOAD1599(TRUE, java.lang.Object(IntList(i108))) → LOAD1599(java.lang.Object(IntList(-(i108, 1))))
    • (i108[0] ≥ 0 ⇒ (UIncreasing(LOAD1599(java.lang.Object(IntList(-(i108[1], 1))))), ≥)∧[(-1)Bound*bni_15] + [bni_15]i108[0] ≥ 0∧[1 + (-1)bso_16] ≥ 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(LOAD1599(x1)) = [-1] + [-1]x1   
POL(java.lang.Object(x1)) = [1] + [-1]x1   
POL(IntList(x1)) = [1] + x1   
POL(COND_LOAD1599(x1, x2)) = [-1] + [-1]x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[1]))) → LOAD1599(java.lang.Object(IntList(-(i108[1], 1))))

The following pairs are in Pbound:

LOAD1599(java.lang.Object(IntList(i108[0]))) → COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))
COND_LOAD1599(TRUE, java.lang.Object(IntList(i108[1]))) → LOAD1599(java.lang.Object(IntList(-(i108[1], 1))))

The following pairs are in P≥:

LOAD1599(java.lang.Object(IntList(i108[0]))) → COND_LOAD1599(>(i108[0], 0), java.lang.Object(IntList(i108[0])))

There are no usable rules.

(13) Complex Obligation (AND)

(14) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1599(java.lang.Object(IntList(i108[0]))) → COND_LOAD1599(i108[0] > 0, java.lang.Object(IntList(i108[0])))


The set Q consists of the following terms:
Load1599(java.lang.Object(IntList(x0)))
Cond_Load1599(TRUE, java.lang.Object(IntList(x0)))

(15) IDependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE

(17) Obligation:

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


The following domains are used:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load1599(java.lang.Object(IntList(x0)))
Cond_Load1599(TRUE, java.lang.Object(IntList(x0)))

(18) IDependencyGraphProof (EQUIVALENT transformation)

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

(19) TRUE

(20) 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:
Load1562(java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → Load1562ARR1(java.lang.Object(ARRAY(i3, a1780data)), i93, i99, java.lang.Object(java.lang.String(i156)))
Load1562ARR1(java.lang.Object(ARRAY(i3, a1780data)), i93, i99, java.lang.Object(java.lang.String(i156))) → Cond_Load1562ARR1(i93 > 0 && i93 < i3 && i99 > 0 && i93 + 1 > 0, java.lang.Object(ARRAY(i3, a1780data)), i93, i99, java.lang.Object(java.lang.String(i156)))
Cond_Load1562ARR1(TRUE, java.lang.Object(ARRAY(i3, a1780data)), i93, i99, java.lang.Object(java.lang.String(i156))) → Load1562(java.lang.Object(ARRAY(i3, a1780data)), i93 + 1, i99 + -1)
The set Q consists of the following terms:
Load1562(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load1562ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4)))
Cond_Load1562ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4)))

(21) ITRSFilterProcessorProof (SOUND transformation)

We filter according the heuristic IdpCand1ShapeHeuristic
We removed arguments according to the following replacements:

Load1562ARR1(x1, x2, x3, x4) → Load1562ARR1(x1, x2, x3)
java.lang.String(x1) → java.lang.String
Cond_Load1562ARR1(x1, x2, x3, x4, x5) → Cond_Load1562ARR1(x1, x2, x3, x4)

(22) 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:
Load1562(java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → Load1562ARR1(java.lang.Object(ARRAY(i3, a1780data)), i93, i99)
Load1562ARR1(java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → Cond_Load1562ARR1(i93 > 0 && i93 < i3 && i99 > 0 && i93 + 1 > 0, java.lang.Object(ARRAY(i3, a1780data)), i93, i99)
Cond_Load1562ARR1(TRUE, java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → Load1562(java.lang.Object(ARRAY(i3, a1780data)), i93 + 1, i99 + -1)
The set Q consists of the following terms:
Load1562(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load1562ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_Load1562ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)

(23) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(24) Obligation:

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


The following domains are used:

Boolean, Integer


The ITRS R consists of the following rules:
Load1562(java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → Load1562ARR1(java.lang.Object(ARRAY(i3, a1780data)), i93, i99)
Load1562ARR1(java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → Cond_Load1562ARR1(i93 > 0 && i93 < i3 && i99 > 0 && i93 + 1 > 0, java.lang.Object(ARRAY(i3, a1780data)), i93, i99)
Cond_Load1562ARR1(TRUE, java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → Load1562(java.lang.Object(ARRAY(i3, a1780data)), i93 + 1, i99 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0]) → LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])
(1): LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1]) → COND_LOAD1562ARR1(i93[1] > 0 && i93[1] < i3[1] && i99[1] > 0 && i93[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])
(2): COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2], i99[2]) → LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2] + 1, i99[2] + -1)

(0) -> (1), if ((java.lang.Object(ARRAY(i3[0], a1780data[0])) →* java.lang.Object(ARRAY(i3[1], a1780data[1])))∧(i93[0] →* i93[1])∧(i99[0] →* i99[1]))


(1) -> (2), if ((java.lang.Object(ARRAY(i3[1], a1780data[1])) →* java.lang.Object(ARRAY(i3[2], a1780data[2])))∧(i93[1] > 0 && i93[1] < i3[1] && i99[1] > 0 && i93[1] + 1 > 0 →* TRUE)∧(i99[1] →* i99[2])∧(i93[1] →* i93[2]))


(2) -> (0), if ((i93[2] + 1 →* i93[0])∧(i99[2] + -1 →* i99[0])∧(java.lang.Object(ARRAY(i3[2], a1780data[2])) →* java.lang.Object(ARRAY(i3[0], a1780data[0]))))



The set Q consists of the following terms:
Load1562(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load1562ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_Load1562ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)

(25) 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.

(26) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0]) → LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])
(1): LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1]) → COND_LOAD1562ARR1(i93[1] > 0 && i93[1] < i3[1] && i99[1] > 0 && i93[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])
(2): COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2], i99[2]) → LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2] + 1, i99[2] + -1)

(0) -> (1), if ((java.lang.Object(ARRAY(i3[0], a1780data[0])) →* java.lang.Object(ARRAY(i3[1], a1780data[1])))∧(i93[0] →* i93[1])∧(i99[0] →* i99[1]))


(1) -> (2), if ((java.lang.Object(ARRAY(i3[1], a1780data[1])) →* java.lang.Object(ARRAY(i3[2], a1780data[2])))∧(i93[1] > 0 && i93[1] < i3[1] && i99[1] > 0 && i93[1] + 1 > 0 →* TRUE)∧(i99[1] →* i99[2])∧(i93[1] →* i93[2]))


(2) -> (0), if ((i93[2] + 1 →* i93[0])∧(i99[2] + -1 →* i99[0])∧(java.lang.Object(ARRAY(i3[2], a1780data[2])) →* java.lang.Object(ARRAY(i3[0], a1780data[0]))))



The set Q consists of the following terms:
Load1562(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load1562ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_Load1562ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)

(27) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(28) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0]) → LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])
(1): LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1]) → COND_LOAD1562ARR1(i93[1] > 0 && i93[1] < i3[1] && i99[1] > 0 && i93[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])
(2): COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2], i99[2]) → LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2] + 1, i99[2] + -1)

(0) -> (1), if (((i3[0] →* i3[1])∧(a1780data[0] →* a1780data[1]))∧(i93[0] →* i93[1])∧(i99[0] →* i99[1]))


(1) -> (2), if (((i3[1] →* i3[2])∧(a1780data[1] →* a1780data[2]))∧(i93[1] > 0 && i93[1] < i3[1] && i99[1] > 0 && i93[1] + 1 > 0 →* TRUE)∧(i99[1] →* i99[2])∧(i93[1] →* i93[2]))


(2) -> (0), if ((i93[2] + 1 →* i93[0])∧(i99[2] + -1 →* i99[0])∧((i3[2] →* i3[0])∧(a1780data[2] →* a1780data[0])))



The set Q consists of the following terms:
Load1562(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load1562ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_Load1562ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)

(29) 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 LOAD1562(java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → LOAD1562ARR1(java.lang.Object(ARRAY(i3, a1780data)), i93, i99) the following chains were created:
  • We consider the chain LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0]) → LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0]), LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1]) → COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1]) which results in the following constraint:

    (1)    (i3[0]=i3[1]∧a1780data[0]=a1780data[1]∧i93[0]=i93[1]∧i99[0]=i99[1] ⇒ LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])≥NonInfC∧LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])≥LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])∧(UIncreasing(LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])), ≥))



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

    (2)    (LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])≥NonInfC∧LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])≥LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])∧(UIncreasing(LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])), ≥))



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

    (3)    ((UIncreasing(LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])), ≥)∧[(-1)bso_16] ≥ 0)



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

    (4)    ((UIncreasing(LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])), ≥)∧[(-1)bso_16] ≥ 0)



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

    (5)    ((UIncreasing(LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])), ≥)∧[(-1)bso_16] ≥ 0)



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

    (6)    ((UIncreasing(LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)







For Pair LOAD1562ARR1(java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → COND_LOAD1562ARR1(&&(&&(&&(>(i93, 0), <(i93, i3)), >(i99, 0)), >(+(i93, 1), 0)), java.lang.Object(ARRAY(i3, a1780data)), i93, i99) the following chains were created:
  • We consider the chain LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1]) → COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1]), COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2], i99[2]) → LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), +(i93[2], 1), +(i99[2], -1)) which results in the following constraint:

    (7)    (i3[1]=i3[2]∧a1780data[1]=a1780data[2]∧&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0))=TRUE∧i99[1]=i99[2]∧i93[1]=i93[2] ⇒ LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])≥NonInfC∧LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])≥COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])∧(UIncreasing(COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])), ≥))



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

    (8)    (>(+(i93[1], 1), 0)=TRUE∧>(i99[1], 0)=TRUE∧>(i93[1], 0)=TRUE∧<(i93[1], i3[1])=TRUE ⇒ LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])≥NonInfC∧LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])≥COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])∧(UIncreasing(COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])), ≥))



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

    (9)    (i93[1] ≥ 0∧i99[1] + [-1] ≥ 0∧i93[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i93[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i99[1] + [(-1)bni_17]i93[1] + [bni_17]i3[1] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (10)    (i93[1] ≥ 0∧i99[1] + [-1] ≥ 0∧i93[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i93[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i99[1] + [(-1)bni_17]i93[1] + [bni_17]i3[1] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (11)    (i93[1] ≥ 0∧i99[1] + [-1] ≥ 0∧i93[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i93[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i99[1] + [(-1)bni_17]i93[1] + [bni_17]i3[1] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (12)    (i93[1] ≥ 0∧i99[1] + [-1] ≥ 0∧i93[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i93[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])), ≥)∧0 = 0∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i99[1] + [(-1)bni_17]i93[1] + [bni_17]i3[1] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)



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

    (13)    ([1] + i93[1] ≥ 0∧i99[1] + [-1] ≥ 0∧i93[1] ≥ 0∧i3[1] + [-2] + [-1]i93[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])), ≥)∧0 = 0∧[bni_17 + (-1)Bound*bni_17] + [bni_17]i99[1] + [(-1)bni_17]i93[1] + [bni_17]i3[1] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)



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

    (14)    ([1] + i93[1] ≥ 0∧i99[1] ≥ 0∧i93[1] ≥ 0∧i3[1] + [-2] + [-1]i93[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])), ≥)∧0 = 0∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i99[1] + [(-1)bni_17]i93[1] + [bni_17]i3[1] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)



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

    (15)    ([1] + i93[1] ≥ 0∧i99[1] ≥ 0∧i93[1] ≥ 0∧i3[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])), ≥)∧0 = 0∧[(4)bni_17 + (-1)Bound*bni_17] + [bni_17]i99[1] + [bni_17]i3[1] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)







For Pair COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → LOAD1562(java.lang.Object(ARRAY(i3, a1780data)), +(i93, 1), +(i99, -1)) the following chains were created:
  • We consider the chain COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2], i99[2]) → LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), +(i93[2], 1), +(i99[2], -1)) which results in the following constraint:

    (16)    (COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2], i99[2])≥NonInfC∧COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2], i99[2])≥LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), +(i93[2], 1), +(i99[2], -1))∧(UIncreasing(LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), +(i93[2], 1), +(i99[2], -1))), ≥))



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

    (17)    ((UIncreasing(LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), +(i93[2], 1), +(i99[2], -1))), ≥)∧[2 + (-1)bso_20] ≥ 0)



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

    (18)    ((UIncreasing(LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), +(i93[2], 1), +(i99[2], -1))), ≥)∧[2 + (-1)bso_20] ≥ 0)



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

    (19)    ((UIncreasing(LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), +(i93[2], 1), +(i99[2], -1))), ≥)∧[2 + (-1)bso_20] ≥ 0)



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

    (20)    ((UIncreasing(LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), +(i93[2], 1), +(i99[2], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_20] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD1562(java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → LOAD1562ARR1(java.lang.Object(ARRAY(i3, a1780data)), i93, i99)
    • ((UIncreasing(LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

  • LOAD1562ARR1(java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → COND_LOAD1562ARR1(&&(&&(&&(>(i93, 0), <(i93, i3)), >(i99, 0)), >(+(i93, 1), 0)), java.lang.Object(ARRAY(i3, a1780data)), i93, i99)
    • ([1] + i93[1] ≥ 0∧i99[1] ≥ 0∧i93[1] ≥ 0∧i3[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])), ≥)∧0 = 0∧[(4)bni_17 + (-1)Bound*bni_17] + [bni_17]i99[1] + [bni_17]i3[1] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)

  • COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3, a1780data)), i93, i99) → LOAD1562(java.lang.Object(ARRAY(i3, a1780data)), +(i93, 1), +(i99, -1))
    • ((UIncreasing(LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), +(i93[2], 1), +(i99[2], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_20] ≥ 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(LOAD1562(x1, x2, x3)) = [1] + x3 + [-1]x2 + [-1]x1   
POL(java.lang.Object(x1)) = x1   
POL(ARRAY(x1, x2)) = [-1] + [-1]x1   
POL(LOAD1562ARR1(x1, x2, x3)) = [1] + x3 + [-1]x2 + [-1]x1   
POL(COND_LOAD1562ARR1(x1, x2, x3, x4)) = [1] + x4 + [-1]x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(-1) = [-1]   

The following pairs are in P>:

COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2], i99[2]) → LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), +(i93[2], 1), +(i99[2], -1))

The following pairs are in Pbound:

LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1]) → COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])

The following pairs are in P≥:

LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0]) → LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])
LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1]) → COND_LOAD1562ARR1(&&(&&(&&(>(i93[1], 0), <(i93[1], i3[1])), >(i99[1], 0)), >(+(i93[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])

There are no usable rules.

(30) Complex Obligation (AND)

(31) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0]) → LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])
(1): LOAD1562ARR1(java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1]) → COND_LOAD1562ARR1(i93[1] > 0 && i93[1] < i3[1] && i99[1] > 0 && i93[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a1780data[1])), i93[1], i99[1])

(0) -> (1), if (((i3[0] →* i3[1])∧(a1780data[0] →* a1780data[1]))∧(i93[0] →* i93[1])∧(i99[0] →* i99[1]))



The set Q consists of the following terms:
Load1562(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load1562ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_Load1562ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)

(32) IDependencyGraphProof (EQUIVALENT transformation)

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

(33) TRUE

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1562(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0]) → LOAD1562ARR1(java.lang.Object(ARRAY(i3[0], a1780data[0])), i93[0], i99[0])
(2): COND_LOAD1562ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2], i99[2]) → LOAD1562(java.lang.Object(ARRAY(i3[2], a1780data[2])), i93[2] + 1, i99[2] + -1)

(2) -> (0), if ((i93[2] + 1 →* i93[0])∧(i99[2] + -1 →* i99[0])∧((i3[2] →* i3[0])∧(a1780data[2] →* a1780data[0])))



The set Q consists of the following terms:
Load1562(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load1562ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Cond_Load1562ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3)

(35) IDependencyGraphProof (EQUIVALENT transformation)

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

(36) TRUE