Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 AND
    ↳5 ITRS
        ↳6 ITRStoQTRSProof (⇔)
        ↳7 QTRS
        ↳8 QTRSRRRProof (⇔)
        ↳9 QTRS
        ↳10 RisEmptyProof (⇔)
        ↳11 TRUE
    ↳12 ITRS
        ↳13 ITRStoIDPProof (⇔)
        ↳14 IDP
        ↳15 UsableRulesProof (⇔)
        ↳16 IDP
        ↳17 IDPNonInfProof (⇐)
        ↳18 AND
            ↳19 IDP
                ↳20 IDependencyGraphProof (⇔)
                ↳21 TRUE
            ↳22 IDP
                ↳23 IDependencyGraphProof (⇔)
                ↳24 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: RunningPointers

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 245 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:
Load2188(java.lang.Object(ObjectList(o451, o452)), java.lang.Object(ObjectList(o451, o452)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o563, o562)), o491))) → JMP2850(java.lang.Object(ObjectList(o451, o452)), o451, o563)
JMP2850(java.lang.Object(ObjectList(o451, o452)), o611, o563) → Load2188(java.lang.Object(ObjectList(o451, o452)), o611, o563)
Load2188(java.lang.Object(ObjectList(o451, o452)), java.lang.Object(ObjectList(o611, o610)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o563, o562)), o491))) → Load2188(java.lang.Object(ObjectList(o451, o452)), o611, o563)
The set Q consists of the following terms:
JMP2850(java.lang.Object(ObjectList(x0, x1)), x2, x3)
Load2188(java.lang.Object(ObjectList(x0, x1)), java.lang.Object(ObjectList(x2, x3)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x4, x5)), x6)))

(6) ITRStoQTRSProof (EQUIVALENT transformation)

Represented integers and predefined function symbols by Terms

(7) Obligation:

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

Load2188(java.lang.Object(ObjectList(o451, o452)), java.lang.Object(ObjectList(o451, o452)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o563, o562)), o491))) → JMP2850(java.lang.Object(ObjectList(o451, o452)), o451, o563)
JMP2850(java.lang.Object(ObjectList(o451, o452)), o611, o563) → Load2188(java.lang.Object(ObjectList(o451, o452)), o611, o563)
Load2188(java.lang.Object(ObjectList(o451, o452)), java.lang.Object(ObjectList(o611, o610)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o563, o562)), o491))) → Load2188(java.lang.Object(ObjectList(o451, o452)), o611, o563)

The set Q consists of the following terms:

JMP2850(java.lang.Object(ObjectList(x0, x1)), x2, x3)
Load2188(java.lang.Object(ObjectList(x0, x1)), java.lang.Object(ObjectList(x2, x3)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x4, x5)), x6)))

(8) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(JMP2850(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(Load2188(x1, x2, x3)) = x1 + x2 + x3   
POL(ObjectList(x1, x2)) = 1 + x1 + x2   
POL(java.lang.Object(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

Load2188(java.lang.Object(ObjectList(o451, o452)), java.lang.Object(ObjectList(o451, o452)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o563, o562)), o491))) → JMP2850(java.lang.Object(ObjectList(o451, o452)), o451, o563)
JMP2850(java.lang.Object(ObjectList(o451, o452)), o611, o563) → Load2188(java.lang.Object(ObjectList(o451, o452)), o611, o563)
Load2188(java.lang.Object(ObjectList(o451, o452)), java.lang.Object(ObjectList(o611, o610)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(o563, o562)), o491))) → Load2188(java.lang.Object(ObjectList(o451, o452)), o611, o563)


(9) Obligation:

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

JMP2850(java.lang.Object(ObjectList(x0, x1)), x2, x3)
Load2188(java.lang.Object(ObjectList(x0, x1)), java.lang.Object(ObjectList(x2, x3)), java.lang.Object(ObjectList(java.lang.Object(ObjectList(x4, x5)), x6)))

(10) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(11) TRUE

(12) 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:
Load408(i29) → Cond_Load408(i29 > 0, i29)
Cond_Load408(TRUE, i29) → Load408(i29 + -1)
The set Q consists of the following terms:
Load408(x0)
Cond_Load408(TRUE, x0)

(13) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(14) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
Load408(i29) → Cond_Load408(i29 > 0, i29)
Cond_Load408(TRUE, i29) → Load408(i29 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD408(i29[0]) → COND_LOAD408(i29[0] > 0, i29[0])
(1): COND_LOAD408(TRUE, i29[1]) → LOAD408(i29[1] + -1)

(0) -> (1), if ((i29[0] →* i29[1])∧(i29[0] > 0 →* TRUE))


(1) -> (0), if ((i29[1] + -1 →* i29[0]))



The set Q consists of the following terms:
Load408(x0)
Cond_Load408(TRUE, x0)

(15) 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.

(16) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD408(i29[0]) → COND_LOAD408(i29[0] > 0, i29[0])
(1): COND_LOAD408(TRUE, i29[1]) → LOAD408(i29[1] + -1)

(0) -> (1), if ((i29[0] →* i29[1])∧(i29[0] > 0 →* TRUE))


(1) -> (0), if ((i29[1] + -1 →* i29[0]))



The set Q consists of the following terms:
Load408(x0)
Cond_Load408(TRUE, x0)

(17) IDPNonInfProof (SOUND transformation)

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


For Pair LOAD408(i29) → COND_LOAD408(>(i29, 0), i29) the following chains were created:
  • We consider the chain LOAD408(i29[0]) → COND_LOAD408(>(i29[0], 0), i29[0]), COND_LOAD408(TRUE, i29[1]) → LOAD408(+(i29[1], -1)) which results in the following constraint:

    (1)    (i29[0]=i29[1]∧>(i29[0], 0)=TRUE ⇒ LOAD408(i29[0])≥NonInfC∧LOAD408(i29[0])≥COND_LOAD408(>(i29[0], 0), i29[0])∧(UIncreasing(COND_LOAD408(>(i29[0], 0), i29[0])), ≥))



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

    (2)    (>(i29[0], 0)=TRUE ⇒ LOAD408(i29[0])≥NonInfC∧LOAD408(i29[0])≥COND_LOAD408(>(i29[0], 0), i29[0])∧(UIncreasing(COND_LOAD408(>(i29[0], 0), i29[0])), ≥))



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

    (3)    (i29[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD408(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (4)    (i29[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD408(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (5)    (i29[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD408(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (6)    (i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD408(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)







For Pair COND_LOAD408(TRUE, i29) → LOAD408(+(i29, -1)) the following chains were created:
  • We consider the chain COND_LOAD408(TRUE, i29[1]) → LOAD408(+(i29[1], -1)) which results in the following constraint:

    (7)    (COND_LOAD408(TRUE, i29[1])≥NonInfC∧COND_LOAD408(TRUE, i29[1])≥LOAD408(+(i29[1], -1))∧(UIncreasing(LOAD408(+(i29[1], -1))), ≥))



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

    (8)    ((UIncreasing(LOAD408(+(i29[1], -1))), ≥)∧[2 + (-1)bso_11] ≥ 0)



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

    (9)    ((UIncreasing(LOAD408(+(i29[1], -1))), ≥)∧[2 + (-1)bso_11] ≥ 0)



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

    (10)    ((UIncreasing(LOAD408(+(i29[1], -1))), ≥)∧[2 + (-1)bso_11] ≥ 0)



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

    (11)    ((UIncreasing(LOAD408(+(i29[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_11] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD408(i29) → COND_LOAD408(>(i29, 0), i29)
    • (i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD408(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)

  • COND_LOAD408(TRUE, i29) → LOAD408(+(i29, -1))
    • ((UIncreasing(LOAD408(+(i29[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_11] ≥ 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(LOAD408(x1)) = [2]x1   
POL(COND_LOAD408(x1, x2)) = [2]x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

COND_LOAD408(TRUE, i29[1]) → LOAD408(+(i29[1], -1))

The following pairs are in Pbound:

LOAD408(i29[0]) → COND_LOAD408(>(i29[0], 0), i29[0])

The following pairs are in P≥:

LOAD408(i29[0]) → COND_LOAD408(>(i29[0], 0), i29[0])

There are no usable rules.

(18) Complex Obligation (AND)

(19) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD408(i29[0]) → COND_LOAD408(i29[0] > 0, i29[0])


The set Q consists of the following terms:
Load408(x0)
Cond_Load408(TRUE, x0)

(20) IDependencyGraphProof (EQUIVALENT transformation)

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

(21) TRUE

(22) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD408(TRUE, i29[1]) → LOAD408(i29[1] + -1)


The set Q consists of the following terms:
Load408(x0)
Cond_Load408(TRUE, x0)

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE