### (0) Obligation:

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

Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: StupidArray

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 74 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

### (4) Obligation:

ITRS problem:

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

The TRS R consists of the following rules:
Load179(java.lang.Object(ARRAY(i1, a8data))) → Cond_Load179(i1 + 1 > 0 && i1 + 1 < i1, java.lang.Object(ARRAY(i1, a8data)))
The set Q consists of the following terms:

### (6) Obligation:

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

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
Load179(java.lang.Object(ARRAY(i1, a8data))) → Cond_Load179(i1 + 1 > 0 && i1 + 1 < i1, java.lang.Object(ARRAY(i1, a8data)))

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

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

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

The set Q consists of the following terms:

### (7) UsableRulesProof (EQUIVALENT transformation)

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

### (8) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

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

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

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

The set Q consists of the following terms:

### (9) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

### (10) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

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

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

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

The set Q consists of the following terms:

### (11) IDPNonInfProof (SOUND transformation)

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

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

(1)    (i1[0]=i1[1]a8data[0]=a8data[1]&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0]))=TRUELOAD179(java.lang.Object(ARRAY(i1[0], a8data[0])))≥NonInfC∧LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0])))≥COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))∧(UIncreasing(COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))), ≥))

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

(2)    (>(+(i1[0], 1), 0)=TRUE<(+(i1[0], 1), i1[0])=TRUELOAD179(java.lang.Object(ARRAY(i1[0], a8data[0])))≥NonInfC∧LOAD179(java.lang.Object(ARRAY(i1[0], a8data[0])))≥COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))∧(UIncreasing(COND_LOAD179(&&(>(+(i1[0], 1), 0), <(+(i1[0], 1), i1[0])), java.lang.Object(ARRAY(i1[0], a8data[0])))), ≥))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1, x2)) = [-1]
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(0) = 0
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:
none

There are no usable rules.

### (13) Obligation:

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

The following domains are used:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (16) Obligation:

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

The following domains are used:
none

R is empty.

The integer pair graph contains the following rules and edges: