Termination of the given JBC could be proven:

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

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 221 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:
Load843(java.lang.Object(IntList(o588, i91))) → Cond_Load843(i91 > 0, java.lang.Object(IntList(o588, i91)))
Cond_Load843(TRUE, java.lang.Object(IntList(o588, i91))) → Load843(java.lang.Object(IntList(o588, i91 - 1)))
The set Q consists of the following terms:
Load843(java.lang.Object(IntList(x0, x1)))
Cond_Load843(TRUE, java.lang.Object(IntList(x0, x1)))

(6) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(7) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
Load843(java.lang.Object(IntList(o588, i91))) → Cond_Load843(i91 > 0, java.lang.Object(IntList(o588, i91)))
Cond_Load843(TRUE, java.lang.Object(IntList(o588, i91))) → Load843(java.lang.Object(IntList(o588, i91 - 1)))

The integer pair graph contains the following rules and edges:
(0): LOAD843(java.lang.Object(IntList(o588[0], i91[0]))) → COND_LOAD843(i91[0] > 0, java.lang.Object(IntList(o588[0], i91[0])))
(1): COND_LOAD843(TRUE, java.lang.Object(IntList(o588[1], i91[1]))) → LOAD843(java.lang.Object(IntList(o588[1], i91[1] - 1)))

(0) -> (1), if ((java.lang.Object(IntList(o588[0], i91[0])) →* java.lang.Object(IntList(o588[1], i91[1])))∧(i91[0] > 0 →* TRUE))


(1) -> (0), if ((java.lang.Object(IntList(o588[1], i91[1] - 1)) →* java.lang.Object(IntList(o588[0], i91[0]))))



The set Q consists of the following terms:
Load843(java.lang.Object(IntList(x0, x1)))
Cond_Load843(TRUE, java.lang.Object(IntList(x0, x1)))

(8) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(9) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD843(java.lang.Object(IntList(o588[0], i91[0]))) → COND_LOAD843(i91[0] > 0, java.lang.Object(IntList(o588[0], i91[0])))
(1): COND_LOAD843(TRUE, java.lang.Object(IntList(o588[1], i91[1]))) → LOAD843(java.lang.Object(IntList(o588[1], i91[1] - 1)))

(0) -> (1), if ((java.lang.Object(IntList(o588[0], i91[0])) →* java.lang.Object(IntList(o588[1], i91[1])))∧(i91[0] > 0 →* TRUE))


(1) -> (0), if ((java.lang.Object(IntList(o588[1], i91[1] - 1)) →* java.lang.Object(IntList(o588[0], i91[0]))))



The set Q consists of the following terms:
Load843(java.lang.Object(IntList(x0, x1)))
Cond_Load843(TRUE, java.lang.Object(IntList(x0, x1)))

(10) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(11) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD843(java.lang.Object(IntList(o588[0], i91[0]))) → COND_LOAD843(i91[0] > 0, java.lang.Object(IntList(o588[0], i91[0])))
(1): COND_LOAD843(TRUE, java.lang.Object(IntList(o588[1], i91[1]))) → LOAD843(java.lang.Object(IntList(o588[1], i91[1] - 1)))

(0) -> (1), if (((o588[0] →* o588[1])∧(i91[0] →* i91[1]))∧(i91[0] > 0 →* TRUE))


(1) -> (0), if (((o588[1] →* o588[0])∧(i91[1] - 1 →* i91[0])))



The set Q consists of the following terms:
Load843(java.lang.Object(IntList(x0, x1)))
Cond_Load843(TRUE, java.lang.Object(IntList(x0, x1)))

(12) IDPNonInfProof (SOUND transformation)

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


For Pair LOAD843(java.lang.Object(IntList(o588, i91))) → COND_LOAD843(>(i91, 0), java.lang.Object(IntList(o588, i91))) the following chains were created:
  • We consider the chain LOAD843(java.lang.Object(IntList(o588[0], i91[0]))) → COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0]))), COND_LOAD843(TRUE, java.lang.Object(IntList(o588[1], i91[1]))) → LOAD843(java.lang.Object(IntList(o588[1], -(i91[1], 1)))) which results in the following constraint:

    (1)    (o588[0]=o588[1]∧i91[0]=i91[1]∧>(i91[0], 0)=TRUE ⇒ LOAD843(java.lang.Object(IntList(o588[0], i91[0])))≥NonInfC∧LOAD843(java.lang.Object(IntList(o588[0], i91[0])))≥COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))∧(UIncreasing(COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))), ≥))



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

    (2)    (>(i91[0], 0)=TRUE ⇒ LOAD843(java.lang.Object(IntList(o588[0], i91[0])))≥NonInfC∧LOAD843(java.lang.Object(IntList(o588[0], i91[0])))≥COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))∧(UIncreasing(COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))), ≥))



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

    (3)    (i91[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]i91[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (4)    (i91[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]i91[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (5)    (i91[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]i91[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (6)    (i91[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))), ≥)∧0 = 0∧[bni_11 + (-1)Bound*bni_11] + [bni_11]i91[0] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)



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

    (7)    (i91[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))), ≥)∧0 = 0∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i91[0] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)







For Pair COND_LOAD843(TRUE, java.lang.Object(IntList(o588, i91))) → LOAD843(java.lang.Object(IntList(o588, -(i91, 1)))) the following chains were created:
  • We consider the chain COND_LOAD843(TRUE, java.lang.Object(IntList(o588[1], i91[1]))) → LOAD843(java.lang.Object(IntList(o588[1], -(i91[1], 1)))) which results in the following constraint:

    (8)    (COND_LOAD843(TRUE, java.lang.Object(IntList(o588[1], i91[1])))≥NonInfC∧COND_LOAD843(TRUE, java.lang.Object(IntList(o588[1], i91[1])))≥LOAD843(java.lang.Object(IntList(o588[1], -(i91[1], 1))))∧(UIncreasing(LOAD843(java.lang.Object(IntList(o588[1], -(i91[1], 1))))), ≥))



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

    (9)    ((UIncreasing(LOAD843(java.lang.Object(IntList(o588[1], -(i91[1], 1))))), ≥)∧[(-1)bso_14] ≥ 0)



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

    (10)    ((UIncreasing(LOAD843(java.lang.Object(IntList(o588[1], -(i91[1], 1))))), ≥)∧[(-1)bso_14] ≥ 0)



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

    (11)    ((UIncreasing(LOAD843(java.lang.Object(IntList(o588[1], -(i91[1], 1))))), ≥)∧[(-1)bso_14] ≥ 0)



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

    (12)    ((UIncreasing(LOAD843(java.lang.Object(IntList(o588[1], -(i91[1], 1))))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_14] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD843(java.lang.Object(IntList(o588, i91))) → COND_LOAD843(>(i91, 0), java.lang.Object(IntList(o588, i91)))
    • (i91[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))), ≥)∧0 = 0∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i91[0] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  • COND_LOAD843(TRUE, java.lang.Object(IntList(o588, i91))) → LOAD843(java.lang.Object(IntList(o588, -(i91, 1))))
    • ((UIncreasing(LOAD843(java.lang.Object(IntList(o588[1], -(i91[1], 1))))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_14] ≥ 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(LOAD843(x1)) = [-1]x1   
POL(java.lang.Object(x1)) = x1   
POL(IntList(x1, x2)) = [-1] + [-1]x2   
POL(COND_LOAD843(x1, x2)) = [-1] + [-1]x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

LOAD843(java.lang.Object(IntList(o588[0], i91[0]))) → COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))

The following pairs are in Pbound:

LOAD843(java.lang.Object(IntList(o588[0], i91[0]))) → COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))

The following pairs are in P≥:

COND_LOAD843(TRUE, java.lang.Object(IntList(o588[1], i91[1]))) → LOAD843(java.lang.Object(IntList(o588[1], -(i91[1], 1))))

There are no usable rules.

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD843(TRUE, java.lang.Object(IntList(o588[1], i91[1]))) → LOAD843(java.lang.Object(IntList(o588[1], i91[1] - 1)))


The set Q consists of the following terms:
Load843(java.lang.Object(IntList(x0, x1)))
Cond_Load843(TRUE, java.lang.Object(IntList(x0, x1)))

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) 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:
Load823(java.lang.Object(ARRAY(i2, a689data)), i75, i79) → Load823ARR1(java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910)))
Load823ARR1(java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) → Cond_Load823ARR1(i75 > 0 && i75 < i2 && i79 > 0 && i75 + 1 > 0, java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910)))
Cond_Load823ARR1(TRUE, java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) → Load823(java.lang.Object(ARRAY(i2, a689data)), i75 + 1, i79 + -1)
The set Q consists of the following terms:
Load823(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load823ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load823ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(17) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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:

Boolean, Integer


The ITRS R consists of the following rules:
Load823(java.lang.Object(ARRAY(i2, a689data)), i75, i79) → Load823ARR1(java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910)))
Load823ARR1(java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) → Cond_Load823ARR1(i75 > 0 && i75 < i2 && i79 > 0 && i75 + 1 > 0, java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910)))
Cond_Load823ARR1(TRUE, java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) → Load823(java.lang.Object(ARRAY(i2, a689data)), i75 + 1, i79 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD823(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0]) → LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))
(1): LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1]))) → COND_LOAD823ARR1(i75[1] > 0 && i75[1] < i2[1] && i79[1] > 0 && i75[1] + 1 > 0, java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))
(2): COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2], i79[2], java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2]))) → LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2] + 1, i79[2] + -1)

(0) -> (1), if ((java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])) →* java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))∧(java.lang.Object(ARRAY(i2[0], a689data[0])) →* java.lang.Object(ARRAY(i2[1], a689data[1])))∧(i79[0] →* i79[1])∧(i75[0] →* i75[1]))


(1) -> (2), if ((java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])) →* java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2])))∧(i75[1] →* i75[2])∧(i75[1] > 0 && i75[1] < i2[1] && i79[1] > 0 && i75[1] + 1 > 0 →* TRUE)∧(java.lang.Object(ARRAY(i2[1], a689data[1])) →* java.lang.Object(ARRAY(i2[2], a689data[2])))∧(i79[1] →* i79[2]))


(2) -> (0), if ((i75[2] + 1 →* i75[0])∧(i79[2] + -1 →* i79[0])∧(java.lang.Object(ARRAY(i2[2], a689data[2])) →* java.lang.Object(ARRAY(i2[0], a689data[0]))))



The set Q consists of the following terms:
Load823(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load823ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load823ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(19) 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.

(20) 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): LOAD823(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0]) → LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))
(1): LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1]))) → COND_LOAD823ARR1(i75[1] > 0 && i75[1] < i2[1] && i79[1] > 0 && i75[1] + 1 > 0, java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))
(2): COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2], i79[2], java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2]))) → LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2] + 1, i79[2] + -1)

(0) -> (1), if ((java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])) →* java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))∧(java.lang.Object(ARRAY(i2[0], a689data[0])) →* java.lang.Object(ARRAY(i2[1], a689data[1])))∧(i79[0] →* i79[1])∧(i75[0] →* i75[1]))


(1) -> (2), if ((java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])) →* java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2])))∧(i75[1] →* i75[2])∧(i75[1] > 0 && i75[1] < i2[1] && i79[1] > 0 && i75[1] + 1 > 0 →* TRUE)∧(java.lang.Object(ARRAY(i2[1], a689data[1])) →* java.lang.Object(ARRAY(i2[2], a689data[2])))∧(i79[1] →* i79[2]))


(2) -> (0), if ((i75[2] + 1 →* i75[0])∧(i79[2] + -1 →* i79[0])∧(java.lang.Object(ARRAY(i2[2], a689data[2])) →* java.lang.Object(ARRAY(i2[0], a689data[0]))))



The set Q consists of the following terms:
Load823(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load823ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load823ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(21) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD823(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0]) → LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))
(1): LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1]))) → COND_LOAD823ARR1(i75[1] > 0 && i75[1] < i2[1] && i79[1] > 0 && i75[1] + 1 > 0, java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))
(2): COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2], i79[2], java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2]))) → LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2] + 1, i79[2] + -1)

(0) -> (1), if (((i138[0] →* i138[1])∧(i137[0] →* i137[1])∧(i139[0] →* i139[1])∧(a910[0] →* a910[1]))∧((i2[0] →* i2[1])∧(a689data[0] →* a689data[1]))∧(i79[0] →* i79[1])∧(i75[0] →* i75[1]))


(1) -> (2), if (((i138[1] →* i138[2])∧(i137[1] →* i137[2])∧(i139[1] →* i139[2])∧(a910[1] →* a910[2]))∧(i75[1] →* i75[2])∧(i75[1] > 0 && i75[1] < i2[1] && i79[1] > 0 && i75[1] + 1 > 0 →* TRUE)∧((i2[1] →* i2[2])∧(a689data[1] →* a689data[2]))∧(i79[1] →* i79[2]))


(2) -> (0), if ((i75[2] + 1 →* i75[0])∧(i79[2] + -1 →* i79[0])∧((i2[2] →* i2[0])∧(a689data[2] →* a689data[0])))



The set Q consists of the following terms:
Load823(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load823ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load823ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(23) 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 LOAD823(java.lang.Object(ARRAY(i2, a689data)), i75, i79) → LOAD823ARR1(java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) the following chains were created:
  • We consider the chain LOAD823(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0]) → LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0]))) which results in the following constraint:

    (1)    (LOAD823(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0])≥NonInfC∧LOAD823(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0])≥LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))∧(UIncreasing(LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))), ≥))



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

    (2)    ((UIncreasing(LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (3)    ((UIncreasing(LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (4)    ((UIncreasing(LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (5)    ((UIncreasing(LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)







For Pair LOAD823ARR1(java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) → COND_LOAD823ARR1(&&(&&(&&(>(i75, 0), <(i75, i2)), >(i79, 0)), >(+(i75, 1), 0)), java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) the following chains were created:
  • We consider the chain LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1]))) → COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1]))), COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2], i79[2], java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2]))) → LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[2], -1)) which results in the following constraint:

    (6)    (i138[1]=i138[2]∧i137[1]=i137[2]∧i139[1]=i139[2]∧a910[1]=a910[2]∧i75[1]=i75[2]∧&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0))=TRUE∧i2[1]=i2[2]∧a689data[1]=a689data[2]∧i79[1]=i79[2] ⇒ LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))≥NonInfC∧LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))≥COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))∧(UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥))



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

    (7)    (>(+(i75[1], 1), 0)=TRUE∧>(i79[1], 0)=TRUE∧>(i75[1], 0)=TRUE∧<(i75[1], i2[1])=TRUE ⇒ LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))≥NonInfC∧LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))≥COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))∧(UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥))



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

    (8)    (i75[1] ≥ 0∧i79[1] + [-1] ≥ 0∧i75[1] + [-1] ≥ 0∧i2[1] + [-1] + [-1]i75[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [(-1)bni_18]i75[1] + [bni_18]i2[1] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (9)    (i75[1] ≥ 0∧i79[1] + [-1] ≥ 0∧i75[1] + [-1] ≥ 0∧i2[1] + [-1] + [-1]i75[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [(-1)bni_18]i75[1] + [bni_18]i2[1] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (10)    (i75[1] ≥ 0∧i79[1] + [-1] ≥ 0∧i75[1] + [-1] ≥ 0∧i2[1] + [-1] + [-1]i75[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [(-1)bni_18]i75[1] + [bni_18]i2[1] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (11)    (i75[1] ≥ 0∧i79[1] + [-1] ≥ 0∧i75[1] + [-1] ≥ 0∧i2[1] + [-1] + [-1]i75[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧0 = 0∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [(-1)bni_18]i75[1] + [bni_18]i2[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] + i75[1] ≥ 0∧i79[1] + [-1] ≥ 0∧i75[1] ≥ 0∧i2[1] + [-2] + [-1]i75[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧0 = 0∧[bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [(-1)bni_18]i75[1] + [bni_18]i2[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] + i75[1] ≥ 0∧i79[1] ≥ 0∧i75[1] ≥ 0∧i2[1] + [-2] + [-1]i75[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧0 = 0∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [(-1)bni_18]i75[1] + [bni_18]i2[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] + i75[1] ≥ 0∧i79[1] ≥ 0∧i75[1] ≥ 0∧i2[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧0 = 0∧[(4)bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [bni_18]i2[1] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)







For Pair COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) → LOAD823(java.lang.Object(ARRAY(i2, a689data)), +(i75, 1), +(i79, -1)) the following chains were created:
  • We consider the chain COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2], i79[2], java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2]))) → LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[2], -1)) which results in the following constraint:

    (15)    (COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2], i79[2], java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2])))≥NonInfC∧COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2], i79[2], java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2])))≥LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[2], -1))∧(UIncreasing(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[2], -1))), ≥))



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

    (16)    ((UIncreasing(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[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(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[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(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[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(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[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.
  • LOAD823(java.lang.Object(ARRAY(i2, a689data)), i75, i79) → LOAD823ARR1(java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910)))
    • ((UIncreasing(LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

  • LOAD823ARR1(java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) → COND_LOAD823ARR1(&&(&&(&&(>(i75, 0), <(i75, i2)), >(i79, 0)), >(+(i75, 1), 0)), java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910)))
    • ([1] + i75[1] ≥ 0∧i79[1] ≥ 0∧i75[1] ≥ 0∧i2[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧0 = 0∧[(4)bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [bni_18]i2[1] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)

  • COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) → LOAD823(java.lang.Object(ARRAY(i2, a689data)), +(i75, 1), +(i79, -1))
    • ((UIncreasing(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[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(LOAD823(x1, x2, x3)) = [1] + x3 + [-1]x2 + [-1]x1   
POL(java.lang.Object(x1)) = x1   
POL(ARRAY(x1, x2)) = [-1] + [-1]x1   
POL(LOAD823ARR1(x1, x2, x3, x4)) = [1] + x3 + [-1]x2 + [-1]x1   
POL(java.lang.String(x1, x2, x3, x4)) = [-1]   
POL(COND_LOAD823ARR1(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_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2], i79[2], java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2]))) → LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[2], -1))

The following pairs are in Pbound:

LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1]))) → COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))

The following pairs are in P≥:

LOAD823(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0]) → LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))
LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1]))) → COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))

There are no usable rules.

(24) Complex Obligation (AND)

(25) 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): LOAD823(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0]) → LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))
(1): LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1]))) → COND_LOAD823ARR1(i75[1] > 0 && i75[1] < i2[1] && i79[1] > 0 && i75[1] + 1 > 0, java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))

(0) -> (1), if (((i138[0] →* i138[1])∧(i137[0] →* i137[1])∧(i139[0] →* i139[1])∧(a910[0] →* a910[1]))∧((i2[0] →* i2[1])∧(a689data[0] →* a689data[1]))∧(i79[0] →* i79[1])∧(i75[0] →* i75[1]))



The set Q consists of the following terms:
Load823(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load823ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load823ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(26) IDependencyGraphProof (EQUIVALENT transformation)

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

(27) TRUE

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD823(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0]) → LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))
(2): COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2], i79[2], java.lang.Object(java.lang.String(i138[2], i137[2], i139[2], a910[2]))) → LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), i75[2] + 1, i79[2] + -1)

(2) -> (0), if ((i75[2] + 1 →* i75[0])∧(i79[2] + -1 →* i79[0])∧((i2[2] →* i2[0])∧(a689data[2] →* a689data[0])))



The set Q consists of the following terms:
Load823(java.lang.Object(ARRAY(x0, x1)), x2, x3)
Load823ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))
Cond_Load823ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, java.lang.Object(java.lang.String(x4, x5, x6, x7)))

(29) IDependencyGraphProof (EQUIVALENT transformation)

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

(30) TRUE