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 IDependencyGraphProof (⇔)
                ↳27 IDP
                ↳28 IDPtoQDPProof (⇐)
                ↳29 QDP
                ↳30 QReductionProof (⇔)
                ↳31 QDP
                ↳32 QDPSizeChangeProof (⇔)
                ↳33 TRUE
    ↳34 ITRS
        ↳35 ITRStoIDPProof (⇔)
        ↳36 IDP
        ↳37 UsableRulesProof (⇔)
        ↳38 IDP
        ↳39 IDPNonInfProof (⇐)
        ↳40 AND
            ↳41 IDP
                ↳42 IDependencyGraphProof (⇔)
                ↳43 TRUE
            ↳44 IDP
                ↳45 IDependencyGraphProof (⇔)
                ↳46 TRUE

(0) Obligation:

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

Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: AppE

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 232 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:
Load684(java.lang.Object(AppE(NULL)), i74) → Cond_Load684(i74 > 0, java.lang.Object(AppE(NULL)), i74)
Cond_Load684(TRUE, java.lang.Object(AppE(NULL)), i74) → Load796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i74 + -1)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → Load684(java.lang.Object(AppE(o408Field0)), i56)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → Load684(java.lang.Object(AppE(o408Field0)), i56)
The set Q consists of the following terms:
Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(TRUE, java.lang.Object(AppE(NULL)), x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(6) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
  • java.lang.Object(AppE(NULL))

We removed arguments according to the following replacements:

Cond_Load684(x1, x2, x3) → Cond_Load684(x1, x3)

(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:
Load684(java.lang.Object(AppE(NULL)), i74) → Cond_Load684(i74 > 0, i74)
Cond_Load684(TRUE, i74) → Load796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i74 + -1)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → Load684(java.lang.Object(AppE(o408Field0)), i56)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → Load684(java.lang.Object(AppE(o408Field0)), i56)
The set Q consists of the following terms:
Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(TRUE, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(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:
Load684(java.lang.Object(AppE(NULL)), i74) → Cond_Load684(i74 > 0, i74)
Cond_Load684(TRUE, i74) → Load796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i74 + -1)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → Load684(java.lang.Object(AppE(o408Field0)), i56)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → Load684(java.lang.Object(AppE(o408Field0)), i56)

The integer pair graph contains the following rules and edges:
(0): LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(i74[0] > 0, i74[0])
(1): COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i74[1] + -1)
(2): LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])
(3): LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])

(0) -> (1), if ((i74[0] →* i74[1])∧(i74[0] > 0 →* TRUE))


(1) -> (2), if ((i74[1] + -1 →* i56[2])∧(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2]))))))


(2) -> (0), if ((java.lang.Object(AppE(o408Field0[2])) →* java.lang.Object(AppE(NULL)))∧(i56[2] →* i74[0]))


(2) -> (3), if ((java.lang.Object(AppE(o408Field0[2])) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))))∧(i56[2] →* i56[3]))


(3) -> (0), if ((i56[3] →* i74[0])∧(java.lang.Object(AppE(o408Field0[3])) →* java.lang.Object(AppE(NULL))))


(3) -> (3), if ((java.lang.Object(AppE(o408Field0[3])) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]')))))∧(i56[3] →* i56[3]'))



The set Q consists of the following terms:
Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(TRUE, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(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): LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(i74[0] > 0, i74[0])
(1): COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i74[1] + -1)
(2): LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])
(3): LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])

(0) -> (1), if ((i74[0] →* i74[1])∧(i74[0] > 0 →* TRUE))


(1) -> (2), if ((i74[1] + -1 →* i56[2])∧(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2]))))))


(2) -> (0), if ((java.lang.Object(AppE(o408Field0[2])) →* java.lang.Object(AppE(NULL)))∧(i56[2] →* i74[0]))


(2) -> (3), if ((java.lang.Object(AppE(o408Field0[2])) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))))∧(i56[2] →* i56[3]))


(3) -> (0), if ((i56[3] →* i74[0])∧(java.lang.Object(AppE(o408Field0[3])) →* java.lang.Object(AppE(NULL))))


(3) -> (3), if ((java.lang.Object(AppE(o408Field0[3])) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]')))))∧(i56[3] →* i56[3]'))



The set Q consists of the following terms:
Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(TRUE, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(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): LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(i74[0] > 0, i74[0])
(1): COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i74[1] + -1)
(2): LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])
(3): LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])

(0) -> (1), if ((i74[0] →* i74[1])∧(i74[0] > 0 →* TRUE))


(1) -> (2), if ((i74[1] + -1 →* i56[2])∧(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2]))))))


(2) -> (0), if (((o408Field0[2] →* NULL))∧(i56[2] →* i74[0]))


(2) -> (3), if (((o408Field0[2] →* java.lang.Object(AppE(o408Field0[3]))))∧(i56[2] →* i56[3]))


(3) -> (0), if ((i56[3] →* i74[0])∧((o408Field0[3] →* NULL)))


(3) -> (3), if (((o408Field0[3] →* java.lang.Object(AppE(o408Field0[3]'))))∧(i56[3] →* i56[3]'))



The set Q consists of the following terms:
Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(TRUE, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(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 LOAD684(java.lang.Object(AppE(NULL)), i74) → COND_LOAD684(>(i74, 0), i74) the following chains were created:
  • We consider the chain LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]), COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1)) which results in the following constraint:

    (1)    (i74[0]=i74[1]∧>(i74[0], 0)=TRUE ⇒ LOAD684(java.lang.Object(AppE(NULL)), i74[0])≥NonInfC∧LOAD684(java.lang.Object(AppE(NULL)), i74[0])≥COND_LOAD684(>(i74[0], 0), i74[0])∧(UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥))



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

    (2)    (>(i74[0], 0)=TRUE ⇒ LOAD684(java.lang.Object(AppE(NULL)), i74[0])≥NonInfC∧LOAD684(java.lang.Object(AppE(NULL)), i74[0])≥COND_LOAD684(>(i74[0], 0), i74[0])∧(UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥))



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

    (3)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(2)bni_15]i74[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (4)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(2)bni_15]i74[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (5)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(2)bni_15]i74[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (6)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[(2)bni_15] ≥ 0∧[bni_15 + (-1)Bound*bni_15] ≥ 0∧0 ≥ 0∧[(-1)bso_16] ≥ 0)







For Pair COND_LOAD684(TRUE, i74) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74, -1)) the following chains were created:
  • We consider the chain LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]), COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1)), LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2]) which results in the following constraint:

    (7)    (i74[0]=i74[1]∧>(i74[0], 0)=TRUE∧+(i74[1], -1)=i56[2]∧java.lang.Object(AppE(java.lang.Object(AppE(NULL))))=java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))) ⇒ COND_LOAD684(TRUE, i74[1])≥NonInfC∧COND_LOAD684(TRUE, i74[1])≥LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))∧(UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥))



    We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:

    (8)    (>(i74[0], 0)=TRUE ⇒ COND_LOAD684(TRUE, i74[0])≥NonInfC∧COND_LOAD684(TRUE, i74[0])≥LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[0], -1))∧(UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥))



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

    (9)    (0 ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i74[0] ≥ 0∧[(-1)bso_18] + [2]i74[0] ≥ 0)



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

    (10)    (0 ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i74[0] ≥ 0∧[(-1)bso_18] + [2]i74[0] ≥ 0)



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

    (11)    (0 ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i74[0] ≥ 0∧[(-1)bso_18] + [2]i74[0] ≥ 0)



    We simplified constraint (11) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (12)    (0 ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[(2)bni_17] ≥ 0∧[bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0∧[1] ≥ 0)







For Pair LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → LOAD684(java.lang.Object(AppE(o408Field0)), i56) the following chains were created:
  • We consider the chain COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1)), LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2]), LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]) which results in the following constraint:

    (13)    (+(i74[1], -1)=i56[2]∧java.lang.Object(AppE(java.lang.Object(AppE(NULL))))=java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2]))))∧o408Field0[2]=NULL∧i56[2]=i74[0] ⇒ LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2])≥NonInfC∧LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2])≥LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥))



    We simplified constraint (13) using rules (I), (II), (III), (IV), (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint:

    (14)    (LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))≥NonInfC∧LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))≥LOAD684(java.lang.Object(AppE(NULL)), +(i74[1], -1))∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥))



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

    (15)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[(-1)bso_20] ≥ 0)



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

    (16)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[(-1)bso_20] ≥ 0)



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

    (17)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[(-1)bso_20] ≥ 0)



  • We consider the chain COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1)), LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2]), LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3]) which results in the following constraint:

    (18)    (+(i74[1], -1)=i56[2]∧java.lang.Object(AppE(java.lang.Object(AppE(NULL))))=java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2]))))∧o408Field0[2]=java.lang.Object(AppE(o408Field0[3]))∧i56[2]=i56[3] ⇒ LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2])≥NonInfC∧LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2])≥LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥))



    We solved constraint (18) using rules (I), (II), (III), (IV).




For Pair LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → LOAD684(java.lang.Object(AppE(o408Field0)), i56) the following chains were created:
  • We consider the chain LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2]), LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3]), LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]) which results in the following constraint:

    (19)    (o408Field0[2]=java.lang.Object(AppE(o408Field0[3]))∧i56[2]=i56[3]∧i56[3]=i74[0]∧o408Field0[3]=NULL ⇒ LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3])≥NonInfC∧LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3])≥LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥))



    We simplified constraint (19) using rules (III), (IV) which results in the following new constraint:

    (20)    (LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i56[2])≥NonInfC∧LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i56[2])≥LOAD684(java.lang.Object(AppE(NULL)), i56[2])∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥))



    We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (21)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[4 + (-1)bso_22] ≥ 0)



    We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (22)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[4 + (-1)bso_22] ≥ 0)



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

    (23)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[4 + (-1)bso_22] ≥ 0)



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

    (24)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧0 ≥ 0∧[4 + (-1)bso_22] ≥ 0)



  • We consider the chain LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3]), LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3]), LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]) which results in the following constraint:

    (25)    (o408Field0[3]=java.lang.Object(AppE(o408Field0[3]1))∧i56[3]=i56[3]1∧i56[3]1=i74[0]∧o408Field0[3]1=NULL ⇒ LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]1)))), i56[3]1)≥NonInfC∧LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]1)))), i56[3]1)≥LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥))



    We simplified constraint (25) using rules (III), (IV) which results in the following new constraint:

    (26)    (LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i56[3])≥NonInfC∧LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i56[3])≥LOAD684(java.lang.Object(AppE(NULL)), i56[3])∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥))



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

    (27)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[4 + (-1)bso_22] ≥ 0)



    We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (28)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[4 + (-1)bso_22] ≥ 0)



    We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (29)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[4 + (-1)bso_22] ≥ 0)



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

    (30)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧0 ≥ 0∧[4 + (-1)bso_22] ≥ 0)



  • We consider the chain LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2]), LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3]), LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3]) which results in the following constraint:

    (31)    (o408Field0[2]=java.lang.Object(AppE(o408Field0[3]))∧i56[2]=i56[3]∧o408Field0[3]=java.lang.Object(AppE(o408Field0[3]1))∧i56[3]=i56[3]1 ⇒ LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3])≥NonInfC∧LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3])≥LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥))



    We simplified constraint (31) using rules (III), (IV) which results in the following new constraint:

    (32)    (LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]1)))))), i56[2])≥NonInfC∧LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]1)))))), i56[2])≥LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]1)))), i56[2])∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥))



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

    (33)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]1 ≥ 0)



    We simplified constraint (33) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (34)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]1 ≥ 0)



    We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (35)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]1 ≥ 0)



    We simplified constraint (35) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (36)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧0 ≥ 0∧[16 + (-1)bso_22] ≥ 0∧[1] ≥ 0)



  • We consider the chain LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3]), LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3]), LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3]) which results in the following constraint:

    (37)    (o408Field0[3]=java.lang.Object(AppE(o408Field0[3]1))∧i56[3]=i56[3]1∧o408Field0[3]1=java.lang.Object(AppE(o408Field0[3]2))∧i56[3]1=i56[3]2 ⇒ LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]1)))), i56[3]1)≥NonInfC∧LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]1)))), i56[3]1)≥LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥))



    We simplified constraint (37) using rules (III), (IV) which results in the following new constraint:

    (38)    (LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]2)))))), i56[3])≥NonInfC∧LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]2)))))), i56[3])≥LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]2)))), i56[3])∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥))



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

    (39)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]2 ≥ 0)



    We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (40)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]2 ≥ 0)



    We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (41)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]2 ≥ 0)



    We simplified constraint (41) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (42)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧0 ≥ 0∧[16 + (-1)bso_22] ≥ 0∧[1] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD684(java.lang.Object(AppE(NULL)), i74) → COND_LOAD684(>(i74, 0), i74)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[(2)bni_15] ≥ 0∧[bni_15 + (-1)Bound*bni_15] ≥ 0∧0 ≥ 0∧[(-1)bso_16] ≥ 0)

  • COND_LOAD684(TRUE, i74) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74, -1))
    • (0 ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[(2)bni_17] ≥ 0∧[bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0∧[1] ≥ 0)

  • LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → LOAD684(java.lang.Object(AppE(o408Field0)), i56)
    • ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[(-1)bso_20] ≥ 0)

  • LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → LOAD684(java.lang.Object(AppE(o408Field0)), i56)
    • ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧0 ≥ 0∧[4 + (-1)bso_22] ≥ 0)
    • ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧0 ≥ 0∧[4 + (-1)bso_22] ≥ 0)
    • ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧0 ≥ 0∧[16 + (-1)bso_22] ≥ 0∧[1] ≥ 0)
    • ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧0 ≥ 0∧[16 + (-1)bso_22] ≥ 0∧[1] ≥ 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(LOAD684(x1, x2)) = [2]x2 + x1   
POL(java.lang.Object(x1)) = [1] + [2]x1   
POL(AppE(x1)) = [2]x1   
POL(NULL) = 0   
POL(COND_LOAD684(x1, x2)) = [1] + [2]x2   
POL(>(x1, x2)) = 0   
POL(0) = 0   
POL(LOAD796(x1, x2)) = [1]   
POL(+(x1, x2)) = 0   
POL(-1) = 0   

The following pairs are in P>:

LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])

The following pairs are in Pbound:

LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0])
COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))

The following pairs are in P≥:

LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0])
COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))
LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])

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:
(0): LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(i74[0] > 0, i74[0])
(1): COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i74[1] + -1)
(2): LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])

(2) -> (0), if (((o408Field0[2] →* NULL))∧(i56[2] →* i74[0]))


(0) -> (1), if ((i74[0] →* i74[1])∧(i74[0] > 0 →* TRUE))


(1) -> (2), if ((i74[1] + -1 →* i56[2])∧(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2]))))))



The set Q consists of the following terms:
Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(TRUE, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(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 LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]) the following chains were created:
  • We consider the chain LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]), COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1)) which results in the following constraint:

    (1)    (i74[0]=i74[1]∧>(i74[0], 0)=TRUE ⇒ LOAD684(java.lang.Object(AppE(NULL)), i74[0])≥NonInfC∧LOAD684(java.lang.Object(AppE(NULL)), i74[0])≥COND_LOAD684(>(i74[0], 0), i74[0])∧(UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥))



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

    (2)    (>(i74[0], 0)=TRUE ⇒ LOAD684(java.lang.Object(AppE(NULL)), i74[0])≥NonInfC∧LOAD684(java.lang.Object(AppE(NULL)), i74[0])≥COND_LOAD684(>(i74[0], 0), i74[0])∧(UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥))



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

    (3)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]i74[0] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (4)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]i74[0] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (5)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]i74[0] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (6)    (i74[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i74[0] ≥ 0∧[(-1)bso_19] ≥ 0)







For Pair COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1)) the following chains were created:
  • We consider the chain LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]), COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1)), LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2]) which results in the following constraint:

    (7)    (i74[0]=i74[1]∧>(i74[0], 0)=TRUE∧+(i74[1], -1)=i56[2]∧java.lang.Object(AppE(java.lang.Object(AppE(NULL))))=java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))) ⇒ COND_LOAD684(TRUE, i74[1])≥NonInfC∧COND_LOAD684(TRUE, i74[1])≥LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))∧(UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥))



    We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:

    (8)    (>(i74[0], 0)=TRUE ⇒ COND_LOAD684(TRUE, i74[0])≥NonInfC∧COND_LOAD684(TRUE, i74[0])≥LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[0], -1))∧(UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥))



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

    (9)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]i74[0] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (10)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]i74[0] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (11)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]i74[0] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (12)    (i74[0] ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]i74[0] ≥ 0∧[(-1)bso_21] ≥ 0)







For Pair LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2]) the following chains were created:
  • We consider the chain COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1)), LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2]), LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]) which results in the following constraint:

    (13)    (+(i74[1], -1)=i56[2]∧java.lang.Object(AppE(java.lang.Object(AppE(NULL))))=java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2]))))∧o408Field0[2]=NULL∧i56[2]=i74[0] ⇒ LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2])≥NonInfC∧LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2])≥LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥))



    We simplified constraint (13) using rules (I), (II), (III), (IV), (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint:

    (14)    (LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))≥NonInfC∧LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))≥LOAD684(java.lang.Object(AppE(NULL)), +(i74[1], -1))∧(UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥))



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

    (15)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[1 + (-1)bso_23] ≥ 0)



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

    (16)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[1 + (-1)bso_23] ≥ 0)



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

    (17)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[1 + (-1)bso_23] ≥ 0)



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

    (18)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧0 = 0∧[1 + (-1)bso_23] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0])
    • (i74[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i74[0] ≥ 0∧[(-1)bso_19] ≥ 0)

  • COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))
    • (i74[0] ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]i74[0] ≥ 0∧[(-1)bso_21] ≥ 0)

  • LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])
    • ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧0 = 0∧[1 + (-1)bso_23] ≥ 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(LOAD684(x1, x2)) = x2 + [-1]x1   
POL(java.lang.Object(x1)) = [-1]   
POL(AppE(x1)) = [1] + [-1]x1   
POL(NULL) = [1]   
POL(COND_LOAD684(x1, x2)) = [1] + x2   
POL(>(x1, x2)) = [1]   
POL(0) = 0   
POL(LOAD796(x1, x2)) = [1] + x2 + [-1]x1   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])

The following pairs are in Pbound:

LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0])
COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))

The following pairs are in P≥:

LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0])
COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -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:
(0): LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(i74[0] > 0, i74[0])
(1): COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), i74[1] + -1)

(0) -> (1), if ((i74[0] →* i74[1])∧(i74[0] > 0 →* TRUE))



The set Q consists of the following terms:
Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(TRUE, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(20) IDependencyGraphProof (EQUIVALENT transformation)

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

(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 contains the following rules and edges:
(2): LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])


The set Q consists of the following terms:
Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(TRUE, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
(2): LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])
(3): LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])

(2) -> (3), if (((o408Field0[2] →* java.lang.Object(AppE(o408Field0[3]))))∧(i56[2] →* i56[3]))


(3) -> (3), if (((o408Field0[3] →* java.lang.Object(AppE(o408Field0[3]'))))∧(i56[3] →* i56[3]'))



The set Q consists of the following terms:
Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(TRUE, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(26) IDependencyGraphProof (EQUIVALENT transformation)

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

(27) 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:
(3): LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])

(3) -> (3), if (((o408Field0[3] →* java.lang.Object(AppE(o408Field0[3]'))))∧(i56[3] →* i56[3]'))



The set Q consists of the following terms:
Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(TRUE, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(28) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(29) Obligation:

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

LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])

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

Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(true, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

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

(30) QReductionProof (EQUIVALENT transformation)

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

Load684(java.lang.Object(AppE(NULL)), x0)
Cond_Load684(true, x0)
Load796(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)
Load684(java.lang.Object(AppE(java.lang.Object(AppE(x0)))), x1)

(31) Obligation:

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

LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])

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

(32) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))), i56[3]) → LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])
    The graph contains the following edges 1 > 1, 2 >= 2

(33) TRUE

(34) 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:
Load412(i36) → Cond_Load412(i36 > 0, i36)
Cond_Load412(TRUE, i36) → Load412(i36 + -1)
The set Q consists of the following terms:
Load412(x0)
Cond_Load412(TRUE, x0)

(35) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(36) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
Load412(i36) → Cond_Load412(i36 > 0, i36)
Cond_Load412(TRUE, i36) → Load412(i36 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD412(i36[0]) → COND_LOAD412(i36[0] > 0, i36[0])
(1): COND_LOAD412(TRUE, i36[1]) → LOAD412(i36[1] + -1)

(0) -> (1), if ((i36[0] →* i36[1])∧(i36[0] > 0 →* TRUE))


(1) -> (0), if ((i36[1] + -1 →* i36[0]))



The set Q consists of the following terms:
Load412(x0)
Cond_Load412(TRUE, x0)

(37) 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.

(38) 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): LOAD412(i36[0]) → COND_LOAD412(i36[0] > 0, i36[0])
(1): COND_LOAD412(TRUE, i36[1]) → LOAD412(i36[1] + -1)

(0) -> (1), if ((i36[0] →* i36[1])∧(i36[0] > 0 →* TRUE))


(1) -> (0), if ((i36[1] + -1 →* i36[0]))



The set Q consists of the following terms:
Load412(x0)
Cond_Load412(TRUE, x0)

(39) 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 LOAD412(i36) → COND_LOAD412(>(i36, 0), i36) the following chains were created:
  • We consider the chain LOAD412(i36[0]) → COND_LOAD412(>(i36[0], 0), i36[0]), COND_LOAD412(TRUE, i36[1]) → LOAD412(+(i36[1], -1)) which results in the following constraint:

    (1)    (i36[0]=i36[1]∧>(i36[0], 0)=TRUE ⇒ LOAD412(i36[0])≥NonInfC∧LOAD412(i36[0])≥COND_LOAD412(>(i36[0], 0), i36[0])∧(UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥))



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

    (2)    (>(i36[0], 0)=TRUE ⇒ LOAD412(i36[0])≥NonInfC∧LOAD412(i36[0])≥COND_LOAD412(>(i36[0], 0), i36[0])∧(UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥))



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

    (3)    (i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i36[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (4)    (i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i36[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (5)    (i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i36[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (6)    (i36[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]i36[0] ≥ 0∧[(-1)bso_9] ≥ 0)







For Pair COND_LOAD412(TRUE, i36) → LOAD412(+(i36, -1)) the following chains were created:
  • We consider the chain COND_LOAD412(TRUE, i36[1]) → LOAD412(+(i36[1], -1)) which results in the following constraint:

    (7)    (COND_LOAD412(TRUE, i36[1])≥NonInfC∧COND_LOAD412(TRUE, i36[1])≥LOAD412(+(i36[1], -1))∧(UIncreasing(LOAD412(+(i36[1], -1))), ≥))



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

    (8)    ((UIncreasing(LOAD412(+(i36[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(LOAD412(+(i36[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(LOAD412(+(i36[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(LOAD412(+(i36[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_11] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD412(i36) → COND_LOAD412(>(i36, 0), i36)
    • (i36[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]i36[0] ≥ 0∧[(-1)bso_9] ≥ 0)

  • COND_LOAD412(TRUE, i36) → LOAD412(+(i36, -1))
    • ((UIncreasing(LOAD412(+(i36[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(LOAD412(x1)) = [2]x1   
POL(COND_LOAD412(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_LOAD412(TRUE, i36[1]) → LOAD412(+(i36[1], -1))

The following pairs are in Pbound:

LOAD412(i36[0]) → COND_LOAD412(>(i36[0], 0), i36[0])

The following pairs are in P≥:

LOAD412(i36[0]) → COND_LOAD412(>(i36[0], 0), i36[0])

There are no usable rules.

(40) Complex Obligation (AND)

(41) 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): LOAD412(i36[0]) → COND_LOAD412(i36[0] > 0, i36[0])


The set Q consists of the following terms:
Load412(x0)
Cond_Load412(TRUE, x0)

(42) IDependencyGraphProof (EQUIVALENT transformation)

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

(43) TRUE

(44) 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_LOAD412(TRUE, i36[1]) → LOAD412(i36[1] + -1)


The set Q consists of the following terms:
Load412(x0)
Cond_Load412(TRUE, x0)

(45) IDependencyGraphProof (EQUIVALENT transformation)

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

(46) TRUE