Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 AND
    ↳5 ITRS
        ↳6 GroundTermsRemoverProof (⇔)
        ↳7 ITRS
        ↳8 ITRStoIDPProof (⇔)
        ↳9 IDP
        ↳10 UsableRulesProof (⇔)
        ↳11 IDP
        ↳12 ItpfGraphProof (⇔)
        ↳13 IDP
        ↳14 IDPNonInfProof (⇐)
        ↳15 AND
            ↳16 IDP
                ↳17 IDPNonInfProof (⇐)
                ↳18 AND
                    ↳19 IDP
                        ↳20 IDependencyGraphProof (⇔)
                        ↳21 TRUE
                    ↳22 IDP
                        ↳23 IDependencyGraphProof (⇔)
                        ↳24 TRUE
            ↳25 IDP
                ↳26 IDPtoQDPProof (⇐)
                ↳27 QDP
                ↳28 DependencyGraphProof (⇔)
                ↳29 TRUE
    ↳30 ITRS
        ↳31 ITRStoIDPProof (⇔)
        ↳32 IDP
        ↳33 UsableRulesProof (⇔)
        ↳34 IDP
        ↳35 IDPNonInfProof (⇐)
        ↳36 AND
            ↳37 IDP
                ↳38 IDependencyGraphProof (⇔)
                ↳39 TRUE
            ↳40 IDP
                ↳41 IDependencyGraphProof (⇔)
                ↳42 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: ListDuplicate

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 185 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:
Load747(1, java.lang.Object(ObjectList(o203, o202)), 0) → Load747(1, o203, 1)
Load747(1, java.lang.Object(ObjectList(o208, o207)), i37) → Cond_Load747(i37 > 0, 1, java.lang.Object(ObjectList(o208, o207)), i37)
Cond_Load747(TRUE, 1, java.lang.Object(ObjectList(o208, o207)), i37) → Load747(1, java.lang.Object(ObjectList(o208, o207)), 0)
The set Q consists of the following terms:
Load747(1, java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load747(TRUE, 1, java.lang.Object(ObjectList(x0, x1)), x2)

(6) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
  • 1

We removed arguments according to the following replacements:

Load747(x1, x2, x3) → Load747(x2, x3)
Cond_Load747(x1, x2, x3, x4) → Cond_Load747(x1, 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:
Load747(java.lang.Object(ObjectList(o203, o202)), 0) → Load747(o203, 1)
Load747(java.lang.Object(ObjectList(o208, o207)), i37) → Cond_Load747(i37 > 0, java.lang.Object(ObjectList(o208, o207)), i37)
Cond_Load747(TRUE, java.lang.Object(ObjectList(o208, o207)), i37) → Load747(java.lang.Object(ObjectList(o208, o207)), 0)
The set Q consists of the following terms:
Load747(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load747(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(8) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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


The ITRS R consists of the following rules:
Load747(java.lang.Object(ObjectList(o203, o202)), 0) → Load747(o203, 1)
Load747(java.lang.Object(ObjectList(o208, o207)), i37) → Cond_Load747(i37 > 0, java.lang.Object(ObjectList(o208, o207)), i37)
Cond_Load747(TRUE, java.lang.Object(ObjectList(o208, o207)), i37) → Load747(java.lang.Object(ObjectList(o208, o207)), 0)

The integer pair graph contains the following rules and edges:
(0): LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1)
(1): LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(i37[1] > 0, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])
(2): COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)

(0) -> (0), if ((o203[0] →* java.lang.Object(ObjectList(o203[0]', o202[0]')))∧(1 →* 0))


(0) -> (1), if ((o203[0] →* java.lang.Object(ObjectList(o208[1], o207[1])))∧(1 →* i37[1]))


(1) -> (2), if ((i37[1] > 0 →* TRUE)∧(java.lang.Object(ObjectList(o208[1], o207[1])) →* java.lang.Object(ObjectList(o208[2], o207[2])))∧(i37[1] →* i37[2]))


(2) -> (0), if (java.lang.Object(ObjectList(o208[2], o207[2])) →* java.lang.Object(ObjectList(o203[0], o202[0])))


(2) -> (1), if ((0 →* i37[1])∧(java.lang.Object(ObjectList(o208[2], o207[2])) →* java.lang.Object(ObjectList(o208[1], o207[1]))))



The set Q consists of the following terms:
Load747(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load747(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(10) 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.

(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): LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1)
(1): LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(i37[1] > 0, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])
(2): COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)

(0) -> (0), if ((o203[0] →* java.lang.Object(ObjectList(o203[0]', o202[0]')))∧(1 →* 0))


(0) -> (1), if ((o203[0] →* java.lang.Object(ObjectList(o208[1], o207[1])))∧(1 →* i37[1]))


(1) -> (2), if ((i37[1] > 0 →* TRUE)∧(java.lang.Object(ObjectList(o208[1], o207[1])) →* java.lang.Object(ObjectList(o208[2], o207[2])))∧(i37[1] →* i37[2]))


(2) -> (0), if (java.lang.Object(ObjectList(o208[2], o207[2])) →* java.lang.Object(ObjectList(o203[0], o202[0])))


(2) -> (1), if ((0 →* i37[1])∧(java.lang.Object(ObjectList(o208[2], o207[2])) →* java.lang.Object(ObjectList(o208[1], o207[1]))))



The set Q consists of the following terms:
Load747(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load747(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(12) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(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:
(0): LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1)
(1): LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(i37[1] > 0, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])
(2): COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)

(0) -> (1), if ((o203[0] →* java.lang.Object(ObjectList(o208[1], o207[1])))∧(1 →* i37[1]))


(0) -> (0), if ((o203[0] →* java.lang.Object(ObjectList(o203[0]', o202[0]')))∧false)


(1) -> (2), if ((i37[1] > 0 →* TRUE)∧((o208[1] →* o208[2])∧(o207[1] →* o207[2]))∧(i37[1] →* i37[2]))


(2) -> (0), if ((o208[2] →* o203[0])∧(o207[2] →* o202[0]))


(2) -> (1), if ((0 →* i37[1])∧((o208[2] →* o208[1])∧(o207[2] →* o207[1])))



The set Q consists of the following terms:
Load747(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load747(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(14) 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 LOAD747(java.lang.Object(ObjectList(o203, o202)), 0) → LOAD747(o203, 1) the following chains were created:
  • We consider the chain LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1), LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1), LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) which results in the following constraint:

    (1)    (LOAD747(java.lang.Object(ObjectList(o203[0]1, o202[0]1)), 0)≥NonInfC∧LOAD747(java.lang.Object(ObjectList(o203[0]1, o202[0]1)), 0)≥LOAD747(o203[0]1, 1)∧(UIncreasing(LOAD747(o203[0]1, 1)), ≥))



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

    (2)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)



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

    (3)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)



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

    (4)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)



    We simplified constraint (4) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (5)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)



  • We consider the chain COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0), LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1), LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) which results in the following constraint:

    (6)    (o208[2]=o203[0]∧o207[2]=o202[0]∧o203[0]=java.lang.Object(ObjectList(o208[1], o207[1]))∧1=i37[1] ⇒ LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0)≥NonInfC∧LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0)≥LOAD747(o203[0], 1)∧(UIncreasing(LOAD747(o203[0], 1)), ≥))



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

    (7)    (LOAD747(java.lang.Object(ObjectList(java.lang.Object(ObjectList(o208[1], o207[1])), o207[2])), 0)≥NonInfC∧LOAD747(java.lang.Object(ObjectList(java.lang.Object(ObjectList(o208[1], o207[1])), o207[2])), 0)≥LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), 1)∧(UIncreasing(LOAD747(o203[0], 1)), ≥))



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

    (8)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[180 + (-1)bso_14] + [90]o208[1] ≥ 0)



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

    (9)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[180 + (-1)bso_14] + [90]o208[1] ≥ 0)



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

    (10)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[180 + (-1)bso_14] + [90]o208[1] ≥ 0)



    We simplified constraint (10) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (11)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧0 ≥ 0∧0 ≥ 0∧[180 + (-1)bso_14] ≥ 0∧[1] ≥ 0)



  • We consider the chain LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1), LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1), LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1) which results in the following constraint:

    (12)    (LOAD747(java.lang.Object(ObjectList(o203[0]1, o202[0]1)), 0)≥NonInfC∧LOAD747(java.lang.Object(ObjectList(o203[0]1, o202[0]1)), 0)≥LOAD747(o203[0]1, 1)∧(UIncreasing(LOAD747(o203[0]1, 1)), ≥))



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

    (13)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)



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

    (14)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)



    We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (15)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)



    We simplified constraint (15) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (16)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)



  • We consider the chain COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0), LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1), LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1) which results in the following constraint:

    (17)    (LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0)≥NonInfC∧LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0)≥LOAD747(o203[0], 1)∧(UIncreasing(LOAD747(o203[0], 1)), ≥))



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

    (18)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0] ≥ 0)



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

    (19)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0] ≥ 0)



    We simplified constraint (19) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (20)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0] ≥ 0)



    We simplified constraint (20) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (21)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)







For Pair LOAD747(java.lang.Object(ObjectList(o208, o207)), i37) → COND_LOAD747(>(i37, 0), java.lang.Object(ObjectList(o208, o207)), i37) the following chains were created:
  • We consider the chain LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]), COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0) which results in the following constraint:

    (22)    (>(i37[1], 0)=TRUE∧o208[1]=o208[2]∧o207[1]=o207[2]∧i37[1]=i37[2] ⇒ LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥NonInfC∧LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])∧(UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥))



    We simplified constraint (22) using rule (IV) which results in the following new constraint:

    (23)    (>(i37[1], 0)=TRUE ⇒ LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥NonInfC∧LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])∧(UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥))



    We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (24)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[(32)bni_15 + (-1)Bound*bni_15] + [(18)bni_15]o208[1] ≥ 0∧[(-1)bso_16] ≥ 0)



    We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (25)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[(32)bni_15 + (-1)Bound*bni_15] + [(18)bni_15]o208[1] ≥ 0∧[(-1)bso_16] ≥ 0)



    We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (26)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[(32)bni_15 + (-1)Bound*bni_15] + [(18)bni_15]o208[1] ≥ 0∧[(-1)bso_16] ≥ 0)



    We simplified constraint (26) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (27)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_15] ≥ 0∧[(32)bni_15 + (-1)Bound*bni_15] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_16] ≥ 0)







For Pair COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208, o207)), i37) → LOAD747(java.lang.Object(ObjectList(o208, o207)), 0) the following chains were created:
  • We consider the chain LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]), COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0), LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1) which results in the following constraint:

    (28)    (>(i37[1], 0)=TRUE∧o208[1]=o208[2]∧o207[1]=o207[2]∧i37[1]=i37[2]∧o208[2]=o203[0]∧o207[2]=o202[0] ⇒ COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2])≥NonInfC∧COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2])≥LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)∧(UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥))



    We simplified constraint (28) using rules (III), (IV) which results in the following new constraint:

    (29)    (>(i37[1], 0)=TRUE ⇒ COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥NonInfC∧COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), 0)∧(UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥))



    We simplified constraint (29) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (30)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (30) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (31)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (31) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (32)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (32) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (33)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_17] ≥ 0∧[(32)bni_17 + (-1)Bound*bni_17] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_18] ≥ 0)



  • We consider the chain LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]), COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0), LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) which results in the following constraint:

    (34)    (>(i37[1], 0)=TRUE∧o208[1]=o208[2]∧o207[1]=o207[2]∧i37[1]=i37[2]∧0=i37[1]1∧o208[2]=o208[1]1∧o207[2]=o207[1]1 ⇒ COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2])≥NonInfC∧COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2])≥LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)∧(UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥))



    We simplified constraint (34) using rules (III), (IV) which results in the following new constraint:

    (35)    (>(i37[1], 0)=TRUE ⇒ COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥NonInfC∧COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), 0)∧(UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥))



    We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (36)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (37)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (38)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (38) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (39)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_17] ≥ 0∧[(32)bni_17 + (-1)Bound*bni_17] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_18] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD747(java.lang.Object(ObjectList(o203, o202)), 0) → LOAD747(o203, 1)
    • ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)
    • ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧0 ≥ 0∧0 ≥ 0∧[180 + (-1)bso_14] ≥ 0∧[1] ≥ 0)
    • ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)
    • ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)

  • LOAD747(java.lang.Object(ObjectList(o208, o207)), i37) → COND_LOAD747(>(i37, 0), java.lang.Object(ObjectList(o208, o207)), i37)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_15] ≥ 0∧[(32)bni_15 + (-1)Bound*bni_15] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_16] ≥ 0)

  • COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208, o207)), i37) → LOAD747(java.lang.Object(ObjectList(o208, o207)), 0)
    • (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_17] ≥ 0∧[(32)bni_17 + (-1)Bound*bni_17] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_18] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_17] ≥ 0∧[(32)bni_17 + (-1)Bound*bni_17] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_18] ≥ 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 with natural coefficients for all symbols [NONINF][POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(LOAD747(x1, x2)) = [2] + [3]x1   
POL(java.lang.Object(x1)) = [1] + [3]x1   
POL(ObjectList(x1, x2)) = [3] + [2]x1   
POL(0) = 0   
POL(1) = 0   
POL(COND_LOAD747(x1, x2, x3)) = [2] + [3]x2   
POL(>(x1, x2)) = 0   

The following pairs are in P>:

LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1)

The following pairs are in Pbound:

LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])
COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)

The following pairs are in P≥:

LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])
COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)

There are no usable rules.

(15) Complex Obligation (AND)

(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:
(1): LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(i37[1] > 0, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])
(2): COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)

(2) -> (1), if ((0 →* i37[1])∧((o208[2] →* o208[1])∧(o207[2] →* o207[1])))


(1) -> (2), if ((i37[1] > 0 →* TRUE)∧((o208[1] →* o208[2])∧(o207[1] →* o207[2]))∧(i37[1] →* i37[2]))



The set Q consists of the following terms:
Load747(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load747(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(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 LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) the following chains were created:
  • We consider the chain LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]), COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0) which results in the following constraint:

    (1)    (>(i37[1], 0)=TRUE∧o208[1]=o208[2]∧o207[1]=o207[2]∧i37[1]=i37[2] ⇒ LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥NonInfC∧LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])∧(UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥))



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

    (2)    (>(i37[1], 0)=TRUE ⇒ LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥NonInfC∧LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])∧(UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥))



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

    (3)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧[-2 + (-1)bso_17] + [2]i37[1] ≥ 0)



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

    (4)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧[-2 + (-1)bso_17] + [2]i37[1] ≥ 0)



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

    (5)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧[-2 + (-1)bso_17] + [2]i37[1] ≥ 0)



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

    (6)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧0 = 0∧0 = 0∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧0 = 0∧0 = 0∧[-2 + (-1)bso_17] + [2]i37[1] ≥ 0)



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

    (7)    (i37[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧0 = 0∧0 = 0∧[(3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] + [2]i37[1] ≥ 0)







For Pair COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0) the following chains were created:
  • We consider the chain LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]), COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0), LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) which results in the following constraint:

    (8)    (>(i37[1], 0)=TRUE∧o208[1]=o208[2]∧o207[1]=o207[2]∧i37[1]=i37[2]∧0=i37[1]1∧o208[2]=o208[1]1∧o207[2]=o207[1]1 ⇒ COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2])≥NonInfC∧COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2])≥LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)∧(UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥))



    We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:

    (9)    (>(i37[1], 0)=TRUE ⇒ COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥NonInfC∧COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])≥LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), 0)∧(UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥))



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

    (10)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧[2 + (-1)bso_19] ≥ 0)



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

    (11)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧[2 + (-1)bso_19] ≥ 0)



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

    (12)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧[2 + (-1)bso_19] ≥ 0)



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

    (13)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)



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

    (14)    (i37[1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])
    • (i37[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧0 = 0∧0 = 0∧[(3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] + [2]i37[1] ≥ 0)

  • COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)
    • (i37[1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)




The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(LOAD747(x1, x2)) = [-1] + [2]x2 + [2]x1   
POL(java.lang.Object(x1)) = [1]   
POL(ObjectList(x1, x2)) = [-1]x2 + x1   
POL(COND_LOAD747(x1, x2, x3)) = [2] + x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)

The following pairs are in Pbound:

LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])
COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)

The following pairs are in P≥:

LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])

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:
(1): LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(i37[1] > 0, java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])


The set Q consists of the following terms:
Load747(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load747(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load747(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load747(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE

(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:
none


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1)

(0) -> (0), if ((o203[0] →* java.lang.Object(ObjectList(o203[0]', o202[0]')))∧false)



The set Q consists of the following terms:
Load747(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load747(TRUE, java.lang.Object(ObjectList(x0, x1)), x2)

(26) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(27) Obligation:

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

LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), pos(01)) → LOAD747(o203[0], pos(s(01)))

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

Load747(java.lang.Object(ObjectList(x0, x1)), x2)
Cond_Load747(true, java.lang.Object(ObjectList(x0, x1)), x2)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

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

(29) TRUE

(30) 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:
Load418(i29) → Cond_Load418(i29 > 0, i29)
Cond_Load418(TRUE, i29) → Load418(i29 + -1)
The set Q consists of the following terms:
Load418(x0)
Cond_Load418(TRUE, x0)

(31) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(32) 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:
Load418(i29) → Cond_Load418(i29 > 0, i29)
Cond_Load418(TRUE, i29) → Load418(i29 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD418(i29[0]) → COND_LOAD418(i29[0] > 0, i29[0])
(1): COND_LOAD418(TRUE, i29[1]) → LOAD418(i29[1] + -1)

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


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



The set Q consists of the following terms:
Load418(x0)
Cond_Load418(TRUE, x0)

(33) 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.

(34) 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): LOAD418(i29[0]) → COND_LOAD418(i29[0] > 0, i29[0])
(1): COND_LOAD418(TRUE, i29[1]) → LOAD418(i29[1] + -1)

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


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



The set Q consists of the following terms:
Load418(x0)
Cond_Load418(TRUE, x0)

(35) 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 LOAD418(i29) → COND_LOAD418(>(i29, 0), i29) the following chains were created:
  • We consider the chain LOAD418(i29[0]) → COND_LOAD418(>(i29[0], 0), i29[0]), COND_LOAD418(TRUE, i29[1]) → LOAD418(+(i29[1], -1)) which results in the following constraint:

    (1)    (>(i29[0], 0)=TRUE∧i29[0]=i29[1] ⇒ LOAD418(i29[0])≥NonInfC∧LOAD418(i29[0])≥COND_LOAD418(>(i29[0], 0), i29[0])∧(UIncreasing(COND_LOAD418(>(i29[0], 0), i29[0])), ≥))



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

    (2)    (>(i29[0], 0)=TRUE ⇒ LOAD418(i29[0])≥NonInfC∧LOAD418(i29[0])≥COND_LOAD418(>(i29[0], 0), i29[0])∧(UIncreasing(COND_LOAD418(>(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_LOAD418(>(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_LOAD418(>(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_LOAD418(>(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_LOAD418(>(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_LOAD418(TRUE, i29) → LOAD418(+(i29, -1)) the following chains were created:
  • We consider the chain COND_LOAD418(TRUE, i29[1]) → LOAD418(+(i29[1], -1)) which results in the following constraint:

    (7)    (COND_LOAD418(TRUE, i29[1])≥NonInfC∧COND_LOAD418(TRUE, i29[1])≥LOAD418(+(i29[1], -1))∧(UIncreasing(LOAD418(+(i29[1], -1))), ≥))



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

    (8)    ((UIncreasing(LOAD418(+(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(LOAD418(+(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(LOAD418(+(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(LOAD418(+(i29[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_11] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD418(i29) → COND_LOAD418(>(i29, 0), i29)
    • (i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD418(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)

  • COND_LOAD418(TRUE, i29) → LOAD418(+(i29, -1))
    • ((UIncreasing(LOAD418(+(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(LOAD418(x1)) = [2]x1   
POL(COND_LOAD418(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_LOAD418(TRUE, i29[1]) → LOAD418(+(i29[1], -1))

The following pairs are in Pbound:

LOAD418(i29[0]) → COND_LOAD418(>(i29[0], 0), i29[0])

The following pairs are in P≥:

LOAD418(i29[0]) → COND_LOAD418(>(i29[0], 0), i29[0])

There are no usable rules.

(36) Complex Obligation (AND)

(37) 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): LOAD418(i29[0]) → COND_LOAD418(i29[0] > 0, i29[0])


The set Q consists of the following terms:
Load418(x0)
Cond_Load418(TRUE, x0)

(38) IDependencyGraphProof (EQUIVALENT transformation)

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

(39) TRUE

(40) 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_LOAD418(TRUE, i29[1]) → LOAD418(i29[1] + -1)


The set Q consists of the following terms:
Load418(x0)
Cond_Load418(TRUE, x0)

(41) IDependencyGraphProof (EQUIVALENT transformation)

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

(42) TRUE