Termination of the given JBC could be proven:

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

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 74 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:
Load179(java.lang.Object(ARRAY(i1, a8data))) → Cond_Load179(i1 + 1 > 0 && i1 + 1 < i1, java.lang.Object(ARRAY(i1, a8data)))
Cond_Load179(TRUE, java.lang.Object(ARRAY(i1, a8data))) → Load179(java.lang.Object(ARRAY(i1, a8dataNew)))
The set Q consists of the following terms:
Load179(java.lang.Object(ARRAY(x0, x1)))
Cond_Load179(TRUE, java.lang.Object(ARRAY(x0, x1)))

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(6) 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:
Load179(java.lang.Object(ARRAY(i1, a8data))) → Cond_Load179(i1 + 1 > 0 && i1 + 1 < i1, java.lang.Object(ARRAY(i1, a8data)))
Cond_Load179(TRUE, java.lang.Object(ARRAY(i1, a8data))) → Load179(java.lang.Object(ARRAY(i1, a8dataNew)))

The integer pair graph contains the following rules and edges:
(0): LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0]))) → COND_LOAD179(i1[0] + 1 > 0 && i1[0] + 1 < i1[0], java.lang.Object(ARRAY(i1[0], a8data[0])))
(1): COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1[1], a8data[1]))) → LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))

(0) -> (1), if ((java.lang.Object(ARRAY(i1[0], a8data[0])) →* java.lang.Object(ARRAY(i1[1], a8data[1])))∧(i1[0] + 1 > 0 && i1[0] + 1 < i1[0] →* TRUE))


(1) -> (0), if ((java.lang.Object(ARRAY(i1[1], a8dataNew[1])) →* java.lang.Object(ARRAY(i1[0], a8data[0]))))



The set Q consists of the following terms:
Load179(java.lang.Object(ARRAY(x0, x1)))
Cond_Load179(TRUE, java.lang.Object(ARRAY(x0, x1)))

(7) 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.

(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


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0]))) → COND_LOAD179(i1[0] + 1 > 0 && i1[0] + 1 < i1[0], java.lang.Object(ARRAY(i1[0], a8data[0])))
(1): COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1[1], a8data[1]))) → LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))

(0) -> (1), if ((java.lang.Object(ARRAY(i1[0], a8data[0])) →* java.lang.Object(ARRAY(i1[1], a8data[1])))∧(i1[0] + 1 > 0 && i1[0] + 1 < i1[0] →* TRUE))


(1) -> (0), if ((java.lang.Object(ARRAY(i1[1], a8dataNew[1])) →* java.lang.Object(ARRAY(i1[0], a8data[0]))))



The set Q consists of the following terms:
Load179(java.lang.Object(ARRAY(x0, x1)))
Cond_Load179(TRUE, java.lang.Object(ARRAY(x0, x1)))

(9) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(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): LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0]))) → COND_LOAD179(i1[0] + 1 > 0 && i1[0] + 1 < i1[0], java.lang.Object(ARRAY(i1[0], a8data[0])))
(1): COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1[1], a8data[1]))) → LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))

(0) -> (1), if (((i1[0] →* i1[1])∧(a8data[0] →* a8data[1]))∧(i1[0] + 1 > 0 && i1[0] + 1 < i1[0] →* TRUE))


(1) -> (0), if (((i1[1] →* i1[0])∧(a8dataNew[1] →* a8data[0])))



The set Q consists of the following terms:
Load179(java.lang.Object(ARRAY(x0, x1)))
Cond_Load179(TRUE, java.lang.Object(ARRAY(x0, x1)))

(11) 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 LOAD179(java.lang.Object(ARRAY(i1, a8data))) → COND_LOAD179(&&(>(+(i1, 1), 0), <(+(i1, 1), i1)), java.lang.Object(ARRAY(i1, a8data))) the following chains were created:
  • We consider the chain LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0]))) → COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0]))), COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1[1], a8data[1]))) → LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1]))) which results in the following constraint:

    (1)    (i1[0]=i1[1]∧a8data[0]=a8data[1]∧&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0]))=TRUE ⇒ LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0])))≥NonInfC∧LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0])))≥COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))∧(UIncreasing(COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))), ≥))



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

    (2)    (>(+(i1[0], 1), 0)=TRUE∧<(+(i1[0], 1), i1[0])=TRUE ⇒ LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0])))≥NonInfC∧LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0])))≥COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))∧(UIncreasing(COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))), ≥))



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

    (3)    (i1[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] ≥ 0∧[-2 + (-1)bso_14] ≥ 0)



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

    (4)    (i1[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] ≥ 0∧[-2 + (-1)bso_14] ≥ 0)



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

    (5)    (i1[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] ≥ 0∧[-2 + (-1)bso_14] ≥ 0)



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

    (6)    (i1[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))), ≥)∧0 = 0∧[(-2)bni_13 + (-1)Bound*bni_13] ≥ 0∧0 = 0∧[-2 + (-1)bso_14] ≥ 0)



    We solved constraint (6) using rule (IDP_SMT_SPLIT).




For Pair COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1, a8data))) → LOAD179(java.lang.Object(ARRAY(i1, a8dataNew))) the following chains were created:
  • We consider the chain COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1[1], a8data[1]))) → LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1]))) which results in the following constraint:

    (7)    (COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1[1], a8data[1])))≥NonInfC∧COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1[1], a8data[1])))≥LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))∧(UIncreasing(LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))), ≥))



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

    (8)    ((UIncreasing(LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))), ≥)∧[2 + (-1)bso_16] ≥ 0)



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

    (9)    ((UIncreasing(LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))), ≥)∧[2 + (-1)bso_16] ≥ 0)



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

    (10)    ((UIncreasing(LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))), ≥)∧[2 + (-1)bso_16] ≥ 0)



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

    (11)    ((UIncreasing(LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_16] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD179(java.lang.Object(ARRAY(i1, a8data))) → COND_LOAD179(&&(>(+(i1, 1), 0), <(+(i1, 1), i1)), java.lang.Object(ARRAY(i1, a8data)))

  • COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1, a8data))) → LOAD179(java.lang.Object(ARRAY(i1, a8dataNew)))
    • ((UIncreasing(LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[2 + (-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(LOAD179(x1)) = [-1] + x1   
POL(java.lang.Object(x1)) = x1   
POL(ARRAY(x1, x2)) = [-1]   
POL(COND_LOAD179(x1, x2)) = 0   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   

The following pairs are in P>:

LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0]))) → COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))
COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1[1], a8data[1]))) → LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))

The following pairs are in Pbound:

LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0]))) → COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))

The following pairs are in P≥:
none

There are no usable rules.

(12) Complex Obligation (AND)

(13) Obligation:

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


The following domains are used:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load179(java.lang.Object(ARRAY(x0, x1)))
Cond_Load179(TRUE, java.lang.Object(ARRAY(x0, x1)))

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

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


The following domains are used:
none


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD179(TRUE, java.lang.Object(ARRAY(i1[1], a8data[1]))) → LOAD179(java.lang.Object(ARRAY(i1[1], a8dataNew[1])))


The set Q consists of the following terms:
Load179(java.lang.Object(ARRAY(x0, x1)))
Cond_Load179(TRUE, java.lang.Object(ARRAY(x0, x1)))

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE