Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 ITRS
↳5 DuplicateArgsRemoverProof (⇔)
↳6 ITRS
↳7 ITRStoIDPProof (⇔)
↳8 IDP
↳9 UsableRulesProof (⇔)
↳10 IDP
↳11 ItpfGraphProof (⇔)
↳12 IDP
↳13 IDPNonInfProof (⇐)
↳14 AND
    ↳15 IDP
        ↳16 IDependencyGraphProof (⇔)
        ↳17 TRUE
    ↳18 IDP
        ↳19 IDependencyGraphProof (⇔)
        ↳20 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Take 
/**
 * Java can do infinite data objects, too.
 * Here we take the first n elements from an
 * ascending infinite list of integer numbers.
 *
 * @author Carsten Fuhs
 */
public class Take {

    public static int[] take(int n, MyIterator f) {
        int[] result = new int[n];
        for (int i = 0; i < n; ++i) {
            if (f.hasNext()) {
                result[i] = f.next();
            }
            else {
                break;
            }
        }
        return result;
    }

    public static void main(String args[]) {
        int start = args[0].length();
        int howMany = args[1].length();
        From f = new From(start);
        int[] firstHowMany = take(howMany, f);
    }
}

interface MyIterator {
    boolean hasNext();
    int next();
}

class From implements MyIterator {

    private int current;

    public From(int start) {
        this.current = start;
    }

    public boolean hasNext() {
        return true;
    }

    public int next() {
        return current++;
    }
}



(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 246 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) 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:
Load1879(i392, java.lang.Object(From(i393)), i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Cond_Load1879(i394 >= 0 && i394 < i392 && i394 + 1 > 0, i392, java.lang.Object(From(i393)), i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394)
Cond_Load1879(TRUE, i392, java.lang.Object(From(i393)), i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Load1879(i392, java.lang.Object(From(i393 + 1)), i392, java.lang.Object(From(i393 + 1)), java.lang.Object(ARRAY(i392, a1673dataNew)), i394 + 1)
The set Q consists of the following terms:
Load1879(x0, java.lang.Object(From(x1)), x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
We removed arguments according to the following replacements:

Load1879(x1, x2, x3, x4, x5, x6) → Load1879(x3, x4, x5, x6)
Cond_Load1879(x1, x2, x3, x4, x5, x6, x7) → Cond_Load1879(x1, x4, x5, x6, x7)

(6) 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:
Load1879(i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Cond_Load1879(i394 >= 0 && i394 < i392 && i394 + 1 > 0, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394)
Cond_Load1879(TRUE, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Load1879(i392, java.lang.Object(From(i393 + 1)), java.lang.Object(ARRAY(i392, a1673dataNew)), i394 + 1)
The set Q consists of the following terms:
Load1879(x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(8) Obligation:

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


The following domains are used:

Boolean, Integer


The ITRS R consists of the following rules:
Load1879(i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Cond_Load1879(i394 >= 0 && i394 < i392 && i394 + 1 > 0, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394)
Cond_Load1879(TRUE, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Load1879(i392, java.lang.Object(From(i393 + 1)), java.lang.Object(ARRAY(i392, a1673dataNew)), i394 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0, i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])
(1): COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(i393[1] + 1)), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), i394[1] + 1)

(0) -> (1), if ((i394[0] →* i394[1])∧(i392[0] →* i392[1])∧(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0 →* TRUE)∧(java.lang.Object(From(i393[0])) →* java.lang.Object(From(i393[1])))∧(java.lang.Object(ARRAY(i392[0], a1673data[0])) →* java.lang.Object(ARRAY(i392[1], a1673data[1]))))


(1) -> (0), if ((java.lang.Object(ARRAY(i392[1], a1673dataNew[1])) →* java.lang.Object(ARRAY(i392[0], a1673data[0])))∧(i392[1] →* i392[0])∧(i394[1] + 1 →* i394[0])∧(java.lang.Object(From(i393[1] + 1)) →* java.lang.Object(From(i393[0]))))



The set Q consists of the following terms:
Load1879(x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

(9) 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.

(10) 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): LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0, i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])
(1): COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(i393[1] + 1)), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), i394[1] + 1)

(0) -> (1), if ((i394[0] →* i394[1])∧(i392[0] →* i392[1])∧(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0 →* TRUE)∧(java.lang.Object(From(i393[0])) →* java.lang.Object(From(i393[1])))∧(java.lang.Object(ARRAY(i392[0], a1673data[0])) →* java.lang.Object(ARRAY(i392[1], a1673data[1]))))


(1) -> (0), if ((java.lang.Object(ARRAY(i392[1], a1673dataNew[1])) →* java.lang.Object(ARRAY(i392[0], a1673data[0])))∧(i392[1] →* i392[0])∧(i394[1] + 1 →* i394[0])∧(java.lang.Object(From(i393[1] + 1)) →* java.lang.Object(From(i393[0]))))



The set Q consists of the following terms:
Load1879(x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

(11) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(12) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0, i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])
(1): COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(i393[1] + 1)), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), i394[1] + 1)

(0) -> (1), if ((i394[0] →* i394[1])∧(i392[0] →* i392[1])∧(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0 →* TRUE)∧((i393[0] →* i393[1]))∧((i392[0] →* i392[1])∧(a1673data[0] →* a1673data[1])))


(1) -> (0), if (((i392[1] →* i392[0])∧(a1673dataNew[1] →* a1673data[0]))∧(i392[1] →* i392[0])∧(i394[1] + 1 →* i394[0])∧((i393[1] + 1 →* i393[0])))



The set Q consists of the following terms:
Load1879(x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

(13) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair LOAD1879(i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → COND_LOAD1879(&&(&&(>=(i394, 0), <(i394, i392)), >(+(i394, 1), 0)), i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) the following chains were created:
  • We consider the chain LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]), COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1)) which results in the following constraint:

    (1)    (i394[0]=i394[1]∧i392[0]=i392[1]∧&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0))=TRUE∧i393[0]=i393[1]∧a1673data[0]=a1673data[1] ⇒ LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])≥NonInfC∧LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])≥COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])∧(UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥))



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

    (2)    (>(+(i394[0], 1), 0)=TRUE∧>=(i394[0], 0)=TRUE∧<(i394[0], i392[0])=TRUE ⇒ LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])≥NonInfC∧LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])≥COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])∧(UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥))



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

    (3)    (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] + [-1] + [-1]i394[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i394[0] + [bni_16]i392[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (4)    (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] + [-1] + [-1]i394[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i394[0] + [bni_16]i392[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (5)    (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] + [-1] + [-1]i394[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i394[0] + [bni_16]i392[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (6)    (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] + [-1] + [-1]i394[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_16] + [(-1)bni_16]i394[0] + [bni_16]i392[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)



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

    (7)    (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_16 + bni_16] + [bni_16]i392[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)







For Pair COND_LOAD1879(TRUE, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → LOAD1879(i392, java.lang.Object(From(+(i393, 1))), java.lang.Object(ARRAY(i392, a1673dataNew)), +(i394, 1)) the following chains were created:
  • We consider the chain COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1)) which results in the following constraint:

    (8)    (COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1])≥NonInfC∧COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1])≥LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))∧(UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥))



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

    (9)    ((UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (10)    ((UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (11)    ((UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (12)    ((UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD1879(i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → COND_LOAD1879(&&(&&(>=(i394, 0), <(i394, i392)), >(+(i394, 1), 0)), i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394)
    • (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_16 + bni_16] + [bni_16]i392[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

  • COND_LOAD1879(TRUE, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → LOAD1879(i392, java.lang.Object(From(+(i393, 1))), java.lang.Object(ARRAY(i392, a1673dataNew)), +(i394, 1))
    • ((UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-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(LOAD1879(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + x1   
POL(java.lang.Object(x1)) = x1   
POL(From(x1)) = x1   
POL(ARRAY(x1, x2)) = [-1]   
POL(COND_LOAD1879(x1, x2, x3, x4, x5)) = [-1] + [-1]x5 + [-1]x4 + x2   
POL(&&(x1, x2)) = [-1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   

The following pairs are in P>:

COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))

The following pairs are in Pbound:

LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])

The following pairs are in P≥:

LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])

There are no usable rules.

(14) Complex Obligation (AND)

(15) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0, i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])


The set Q consists of the following terms:
Load1879(x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

(16) IDependencyGraphProof (EQUIVALENT transformation)

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

(17) TRUE

(18) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(i393[1] + 1)), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), i394[1] + 1)


The set Q consists of the following terms:
Load1879(x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

(19) IDependencyGraphProof (EQUIVALENT transformation)

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

(20) TRUE