Termination of the given JBC could be proven:

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

(0) Obligation:

JBC Problem based on JBC Program:
No human-readable program information known.

Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Shuffle

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 280 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:
Load1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → Load1484(java.lang.Object(IntList(o873Field0, o873Field1)), java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))
JMP1821(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) → Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
Load1484(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP1821(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → Load1202(o971)
Load1202(java.lang.Object(IntList(NULL, i196))) → Load1202(NULL)
The set Q consists of the following terms:
Load1202(java.lang.Object(IntList(java.lang.Object(IntList(x0, x1)), x2)))
JMP1821(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x2, x3)), x4)
Load1484(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x2, java.lang.Object(IntList(x3, x4)))
Load1484(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x2, NULL)
Load1202(java.lang.Object(IntList(NULL, x0)))

(6) DuplicateArgsRemoverProof (EQUIVALENT transformation)

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

Load1484(x1, x2, x3, x4) → Load1484(x2, x3, x4)
JMP1821(x1, x2, x3, x4) → JMP1821(x2, 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:
Load1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → Load1484(java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))
JMP1821(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) → Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
Load1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP1821(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → Load1202(o971)
Load1202(java.lang.Object(IntList(NULL, i196))) → Load1202(NULL)
The set Q consists of the following terms:
Load1202(java.lang.Object(IntList(java.lang.Object(IntList(x0, x1)), x2)))
JMP1821(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x2, x3)), x4)
Load1484(java.lang.Object(IntList(x0, x1)), x2, java.lang.Object(IntList(x3, x4)))
Load1484(java.lang.Object(IntList(x0, x1)), x2, NULL)
Load1202(java.lang.Object(IntList(NULL, x0)))

(8) ITRStoQTRSProof (EQUIVALENT transformation)

Represented integers and predefined function symbols by Terms

(9) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

Load1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → Load1484(java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))
JMP1821(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) → Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
Load1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP1821(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → Load1202(o971)
Load1202(java.lang.Object(IntList(NULL, i196))) → Load1202(NULL)

The set Q consists of the following terms:

Load1202(java.lang.Object(IntList(java.lang.Object(IntList(x0, x1)), x2)))
JMP1821(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x2, x3)), x4)
Load1484(java.lang.Object(IntList(x0, x1)), x2, java.lang.Object(IntList(x3, x4)))
Load1484(java.lang.Object(IntList(x0, x1)), x2, NULL)
Load1202(java.lang.Object(IntList(NULL, x0)))

(10) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LOAD1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → LOAD1484(java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))
JMP18211(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
LOAD1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP18211(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → LOAD1202(o971)
LOAD1202(java.lang.Object(IntList(NULL, i196))) → LOAD1202(NULL)

The TRS R consists of the following rules:

Load1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → Load1484(java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))
JMP1821(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) → Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
Load1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP1821(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → Load1202(o971)
Load1202(java.lang.Object(IntList(NULL, i196))) → Load1202(NULL)

The set Q consists of the following terms:

Load1202(java.lang.Object(IntList(java.lang.Object(IntList(x0, x1)), x2)))
JMP1821(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x2, x3)), x4)
Load1484(java.lang.Object(IntList(x0, x1)), x2, java.lang.Object(IntList(x3, x4)))
Load1484(java.lang.Object(IntList(x0, x1)), x2, NULL)
Load1202(java.lang.Object(IntList(NULL, x0)))

We have to consider all minimal (P,Q,R)-chains.

(12) DependencyGraphProof (EQUIVALENT transformation)

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

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
LOAD1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP18211(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
JMP18211(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → LOAD1202(o971)
LOAD1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → LOAD1484(java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))

The TRS R consists of the following rules:

Load1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → Load1484(java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))
JMP1821(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) → Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
Load1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP1821(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
Load1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → Load1202(o971)
Load1202(java.lang.Object(IntList(NULL, i196))) → Load1202(NULL)

The set Q consists of the following terms:

Load1202(java.lang.Object(IntList(java.lang.Object(IntList(x0, x1)), x2)))
JMP1821(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x2, x3)), x4)
Load1484(java.lang.Object(IntList(x0, x1)), x2, java.lang.Object(IntList(x3, x4)))
Load1484(java.lang.Object(IntList(x0, x1)), x2, NULL)
Load1202(java.lang.Object(IntList(NULL, x0)))

We have to consider all minimal (P,Q,R)-chains.

(14) 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.

(15) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
LOAD1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP18211(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
JMP18211(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → LOAD1202(o971)
LOAD1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → LOAD1484(java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))

R is empty.
The set Q consists of the following terms:

Load1202(java.lang.Object(IntList(java.lang.Object(IntList(x0, x1)), x2)))
JMP1821(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x2, x3)), x4)
Load1484(java.lang.Object(IntList(x0, x1)), x2, java.lang.Object(IntList(x3, x4)))
Load1484(java.lang.Object(IntList(x0, x1)), x2, NULL)
Load1202(java.lang.Object(IntList(NULL, x0)))

We have to consider all minimal (P,Q,R)-chains.

(16) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

Load1202(java.lang.Object(IntList(java.lang.Object(IntList(x0, x1)), x2)))
JMP1821(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x2, x3)), x4)
Load1484(java.lang.Object(IntList(x0, x1)), x2, java.lang.Object(IntList(x3, x4)))
Load1484(java.lang.Object(IntList(x0, x1)), x2, NULL)
Load1202(java.lang.Object(IntList(NULL, x0)))

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
LOAD1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP18211(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
JMP18211(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → LOAD1202(o971)
LOAD1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → LOAD1484(java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(18) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule JMP18211(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990) we obtained the following new rules [LPAR04]:

JMP18211(java.lang.Object(IntList(z0, z1)), java.lang.Object(IntList(z2, z1)), z0) → LOAD1484(java.lang.Object(IntList(z0, z1)), java.lang.Object(IntList(z2, z1)), z0)

(19) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
LOAD1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP18211(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → LOAD1202(o971)
LOAD1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → LOAD1484(java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))
JMP18211(java.lang.Object(IntList(z0, z1)), java.lang.Object(IntList(z2, z1)), z0) → LOAD1484(java.lang.Object(IntList(z0, z1)), java.lang.Object(IntList(z2, z1)), z0)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, NULL) → LOAD1202(o971)
LOAD1202(java.lang.Object(IntList(java.lang.Object(IntList(o873Field0, o873Field1)), i196))) → LOAD1484(java.lang.Object(IntList(o873Field0, o873Field1)), NULL, java.lang.Object(IntList(o873Field0, o873Field1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(IntList(x1, x2)) = 1 + x1   
POL(JMP18211(x1, x2, x3)) = 1 + x2 + x3   
POL(LOAD1202(x1)) = x1   
POL(LOAD1484(x1, x2, x3)) = 1 + x2 + x3   
POL(NULL) = 0   
POL(java.lang.Object(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(21) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
LOAD1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP18211(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
JMP18211(java.lang.Object(IntList(z0, z1)), java.lang.Object(IntList(z2, z1)), z0) → LOAD1484(java.lang.Object(IntList(z0, z1)), java.lang.Object(IntList(z2, z1)), z0)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(22) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), o971, java.lang.Object(IntList(o990, i252))) → LOAD1484(java.lang.Object(IntList(o973Field0, o973Field1)), java.lang.Object(IntList(o971, i252)), o990)
    The graph contains the following edges 1 >= 1, 3 > 3

  • LOAD1484(java.lang.Object(IntList(o993, i253)), o971, java.lang.Object(IntList(o993, i253))) → JMP18211(java.lang.Object(IntList(o993, i253)), java.lang.Object(IntList(o971, i253)), o993)
    The graph contains the following edges 1 >= 1, 3 >= 1, 1 > 3, 3 > 3

  • JMP18211(java.lang.Object(IntList(z0, z1)), java.lang.Object(IntList(z2, z1)), z0) → LOAD1484(java.lang.Object(IntList(z0, z1)), java.lang.Object(IntList(z2, z1)), z0)
    The graph contains the following edges 1 >= 1, 2 >= 2, 1 > 3, 3 >= 3

(23) TRUE

(24) 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:
Load716(java.lang.Object(ARRAY(i3, a481data)), i58, i61) → Load716ARR1(java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610)))
Load716ARR1(java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610))) → Cond_Load716ARR1(i58 > 0 && i58 < i3 && i61 > 0 && i58 + 1 > 0, java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610)))
Cond_Load716ARR1(TRUE, java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610))) → Load716(java.lang.Object(ARRAY(i3, a481data)), i58 + 1, i61 + -1)
The set Q consists of the following terms:
Load716(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load716ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load716ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(25) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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


The ITRS R consists of the following rules:
Load716(java.lang.Object(ARRAY(i3, a481data)), i58, i61) → Load716ARR1(java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610)))
Load716ARR1(java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610))) → Cond_Load716ARR1(i58 > 0 && i58 < i3 && i61 > 0 && i58 + 1 > 0, java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610)))
Cond_Load716ARR1(TRUE, java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610))) → Load716(java.lang.Object(ARRAY(i3, a481data)), i58 + 1, i61 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD716(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0]) → LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))
(1): LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1]))) → COND_LOAD716ARR1(i58[1] > 0 && i58[1] < i3[1] && i61[1] > 0 && i58[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))
(2): COND_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2], i61[2], java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2]))) → LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2] + 1, i61[2] + -1)

(0) -> (1), if ((java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])) →* java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))∧(i58[0] →* i58[1])∧(i61[0] →* i61[1])∧(java.lang.Object(ARRAY(i3[0], a481data[0])) →* java.lang.Object(ARRAY(i3[1], a481data[1]))))


(1) -> (2), if ((java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])) →* java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2])))∧(i61[1] →* i61[2])∧(i58[1] →* i58[2])∧(java.lang.Object(ARRAY(i3[1], a481data[1])) →* java.lang.Object(ARRAY(i3[2], a481data[2])))∧(i58[1] > 0 && i58[1] < i3[1] && i61[1] > 0 && i58[1] + 1 > 0 →* TRUE))


(2) -> (0), if ((java.lang.Object(ARRAY(i3[2], a481data[2])) →* java.lang.Object(ARRAY(i3[0], a481data[0])))∧(i61[2] + -1 →* i61[0])∧(i58[2] + 1 →* i58[0]))



The set Q consists of the following terms:
Load716(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load716ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load716ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(27) 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.

(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): LOAD716(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0]) → LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))
(1): LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1]))) → COND_LOAD716ARR1(i58[1] > 0 && i58[1] < i3[1] && i61[1] > 0 && i58[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))
(2): COND_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2], i61[2], java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2]))) → LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2] + 1, i61[2] + -1)

(0) -> (1), if ((java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])) →* java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))∧(i58[0] →* i58[1])∧(i61[0] →* i61[1])∧(java.lang.Object(ARRAY(i3[0], a481data[0])) →* java.lang.Object(ARRAY(i3[1], a481data[1]))))


(1) -> (2), if ((java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])) →* java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2])))∧(i61[1] →* i61[2])∧(i58[1] →* i58[2])∧(java.lang.Object(ARRAY(i3[1], a481data[1])) →* java.lang.Object(ARRAY(i3[2], a481data[2])))∧(i58[1] > 0 && i58[1] < i3[1] && i61[1] > 0 && i58[1] + 1 > 0 →* TRUE))


(2) -> (0), if ((java.lang.Object(ARRAY(i3[2], a481data[2])) →* java.lang.Object(ARRAY(i3[0], a481data[0])))∧(i61[2] + -1 →* i61[0])∧(i58[2] + 1 →* i58[0]))



The set Q consists of the following terms:
Load716(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load716ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load716ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(29) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD716(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0]) → LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))
(1): LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1]))) → COND_LOAD716ARR1(i58[1] > 0 && i58[1] < i3[1] && i61[1] > 0 && i58[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))
(2): COND_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2], i61[2], java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2]))) → LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2] + 1, i61[2] + -1)

(0) -> (1), if (((i91[0] →* i91[1])∧(i90[0] →* i90[1])∧(i92[0] →* i92[1])∧(a610[0] →* a610[1]))∧(i58[0] →* i58[1])∧(i61[0] →* i61[1])∧((i3[0] →* i3[1])∧(a481data[0] →* a481data[1])))


(1) -> (2), if (((i91[1] →* i91[2])∧(i90[1] →* i90[2])∧(i92[1] →* i92[2])∧(a610[1] →* a610[2]))∧(i61[1] →* i61[2])∧(i58[1] →* i58[2])∧((i3[1] →* i3[2])∧(a481data[1] →* a481data[2]))∧(i58[1] > 0 && i58[1] < i3[1] && i61[1] > 0 && i58[1] + 1 > 0 →* TRUE))


(2) -> (0), if (((i3[2] →* i3[0])∧(a481data[2] →* a481data[0]))∧(i61[2] + -1 →* i61[0])∧(i58[2] + 1 →* i58[0]))



The set Q consists of the following terms:
Load716(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load716ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load716ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(31) 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 LOAD716(java.lang.Object(ARRAY(i3, a481data)), i58, i61) → LOAD716ARR1(java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610))) the following chains were created:
  • We consider the chain LOAD716(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0]) → LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0]))) which results in the following constraint:

    (1)    (LOAD716(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0])≥NonInfC∧LOAD716(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0])≥LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))∧(UIncreasing(LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))), ≥))



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

    (2)    ((UIncreasing(LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (3)    ((UIncreasing(LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (4)    ((UIncreasing(LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (5)    ((UIncreasing(LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)







For Pair LOAD716ARR1(java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610))) → COND_LOAD716ARR1(&&(&&(&&(>(i58, 0), <(i58, i3)), >(i61, 0)), >(+(i58, 1), 0)), java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610))) the following chains were created:
  • We consider the chain LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1]))) → COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1]))), COND_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2], i61[2], java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2]))) → LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), +(i58[2], 1), +(i61[2], -1)) which results in the following constraint:

    (6)    (i91[1]=i91[2]∧i90[1]=i90[2]∧i92[1]=i92[2]∧a610[1]=a610[2]∧i61[1]=i61[2]∧i58[1]=i58[2]∧i3[1]=i3[2]∧a481data[1]=a481data[2]∧&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0))=TRUE ⇒ LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))≥NonInfC∧LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))≥COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))∧(UIncreasing(COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))), ≥))



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

    (7)    (>(+(i58[1], 1), 0)=TRUE∧>(i61[1], 0)=TRUE∧>(i58[1], 0)=TRUE∧<(i58[1], i3[1])=TRUE ⇒ LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))≥NonInfC∧LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))≥COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))∧(UIncreasing(COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))), ≥))



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

    (8)    (i58[1] ≥ 0∧i61[1] + [-1] ≥ 0∧i58[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i61[1] + [(-1)bni_18]i58[1] + [bni_18]i3[1] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (9)    (i58[1] ≥ 0∧i61[1] + [-1] ≥ 0∧i58[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i61[1] + [(-1)bni_18]i58[1] + [bni_18]i3[1] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (10)    (i58[1] ≥ 0∧i61[1] + [-1] ≥ 0∧i58[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i61[1] + [(-1)bni_18]i58[1] + [bni_18]i3[1] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (11)    (i58[1] ≥ 0∧i61[1] + [-1] ≥ 0∧i58[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))), ≥)∧0 = 0∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i61[1] + [(-1)bni_18]i58[1] + [bni_18]i3[1] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)



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

    (12)    ([1] + i58[1] ≥ 0∧i61[1] + [-1] ≥ 0∧i58[1] ≥ 0∧i3[1] + [-2] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))), ≥)∧0 = 0∧[bni_18 + (-1)Bound*bni_18] + [bni_18]i61[1] + [(-1)bni_18]i58[1] + [bni_18]i3[1] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)



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

    (13)    ([1] + i58[1] ≥ 0∧i61[1] ≥ 0∧i58[1] ≥ 0∧i3[1] + [-2] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))), ≥)∧0 = 0∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i61[1] + [(-1)bni_18]i58[1] + [bni_18]i3[1] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)



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

    (14)    ([1] + i58[1] ≥ 0∧i61[1] ≥ 0∧i58[1] ≥ 0∧i3[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))), ≥)∧0 = 0∧[(4)bni_18 + (-1)Bound*bni_18] + [bni_18]i61[1] + [bni_18]i3[1] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)







For Pair COND_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610))) → LOAD716(java.lang.Object(ARRAY(i3, a481data)), +(i58, 1), +(i61, -1)) the following chains were created:
  • We consider the chain COND_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2], i61[2], java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2]))) → LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), +(i58[2], 1), +(i61[2], -1)) which results in the following constraint:

    (15)    (COND_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2], i61[2], java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2])))≥NonInfC∧COND_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2], i61[2], java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2])))≥LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), +(i58[2], 1), +(i61[2], -1))∧(UIncreasing(LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), +(i58[2], 1), +(i61[2], -1))), ≥))



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

    (16)    ((UIncreasing(LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), +(i58[2], 1), +(i61[2], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)



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

    (17)    ((UIncreasing(LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), +(i58[2], 1), +(i61[2], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)



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

    (18)    ((UIncreasing(LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), +(i58[2], 1), +(i61[2], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)



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

    (19)    ((UIncreasing(LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), +(i58[2], 1), +(i61[2], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_21] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD716(java.lang.Object(ARRAY(i3, a481data)), i58, i61) → LOAD716ARR1(java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610)))
    • ((UIncreasing(LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

  • LOAD716ARR1(java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610))) → COND_LOAD716ARR1(&&(&&(&&(>(i58, 0), <(i58, i3)), >(i61, 0)), >(+(i58, 1), 0)), java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610)))
    • ([1] + i58[1] ≥ 0∧i61[1] ≥ 0∧i58[1] ≥ 0∧i3[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))), ≥)∧0 = 0∧[(4)bni_18 + (-1)Bound*bni_18] + [bni_18]i61[1] + [bni_18]i3[1] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)

  • COND_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3, a481data)), i58, i61, java.lang.Object(java.lang.String(i91, i90, i92, a610))) → LOAD716(java.lang.Object(ARRAY(i3, a481data)), +(i58, 1), +(i61, -1))
    • ((UIncreasing(LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), +(i58[2], 1), +(i61[2], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_21] ≥ 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(LOAD716(x1, x2, x3)) = [1] + x3 + [-1]x2 + [-1]x1   
POL(java.lang.Object(x1)) = x1   
POL(ARRAY(x1, x2)) = [-1] + [-1]x1   
POL(LOAD716ARR1(x1, x2, x3, x4)) = [1] + x3 + [-1]x2 + [-1]x1   
POL(java.lang.String(x1, x2, x3, x4)) = [-1]   
POL(COND_LOAD716ARR1(x1, x2, x3, x4, x5)) = [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_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2], i61[2], java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2]))) → LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), +(i58[2], 1), +(i61[2], -1))

The following pairs are in Pbound:

LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1]))) → COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))

The following pairs are in P≥:

LOAD716(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0]) → LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))
LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1]))) → COND_LOAD716ARR1(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >(i61[1], 0)), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))

There are no usable rules.

(32) Complex Obligation (AND)

(33) 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): LOAD716(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0]) → LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))
(1): LOAD716ARR1(java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1]))) → COND_LOAD716ARR1(i58[1] > 0 && i58[1] < i3[1] && i61[1] > 0 && i58[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a481data[1])), i58[1], i61[1], java.lang.Object(java.lang.String(i91[1], i90[1], i92[1], a610[1])))

(0) -> (1), if (((i91[0] →* i91[1])∧(i90[0] →* i90[1])∧(i92[0] →* i92[1])∧(a610[0] →* a610[1]))∧(i58[0] →* i58[1])∧(i61[0] →* i61[1])∧((i3[0] →* i3[1])∧(a481data[0] →* a481data[1])))



The set Q consists of the following terms:
Load716(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load716ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load716ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(34) IDependencyGraphProof (EQUIVALENT transformation)

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

(35) TRUE

(36) 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): LOAD716(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0]) → LOAD716ARR1(java.lang.Object(ARRAY(i3[0], a481data[0])), i58[0], i61[0], java.lang.Object(java.lang.String(i91[0], i90[0], i92[0], a610[0])))
(2): COND_LOAD716ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2], i61[2], java.lang.Object(java.lang.String(i91[2], i90[2], i92[2], a610[2]))) → LOAD716(java.lang.Object(ARRAY(i3[2], a481data[2])), i58[2] + 1, i61[2] + -1)

(2) -> (0), if (((i3[2] →* i3[0])∧(a481data[2] →* a481data[0]))∧(i61[2] + -1 →* i61[0])∧(i58[2] + 1 →* i58[0]))



The set Q consists of the following terms:
Load716(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load716ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load716ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(37) IDependencyGraphProof (EQUIVALENT transformation)

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

(38) TRUE