### (0) Obligation:

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

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

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 221 nodes with 2 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

### (5) Obligation:

ITRS problem:

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

The TRS R consists of the following rules:
Load843(java.lang.Object(IntList(o588, i91))) → Cond_Load843(i91 > 0, java.lang.Object(IntList(o588, i91)))
Cond_Load843(TRUE, java.lang.Object(IntList(o588, i91))) → Load843(java.lang.Object(IntList(o588, i91 - 1)))
The set Q consists of the following terms:
Load843(java.lang.Object(IntList(x0, x1)))
Cond_Load843(TRUE, java.lang.Object(IntList(x0, x1)))

### (6) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

### (7) Obligation:

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

The following domains are used:

Integer

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

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

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

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

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

### (8) UsableRulesProof (EQUIVALENT transformation)

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

### (9) Obligation:

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

The following domains are used:

Integer

R is empty.

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

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

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

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

### (10) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

### (11) Obligation:

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

The following domains are used:

Integer

R is empty.

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

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

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

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

### (12) IDPNonInfProof (SOUND transformation)

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

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

(1)    (o588[0]=o588[1]i91[0]=i91[1]>(i91[0], 0)=TRUELOAD843(java.lang.Object(IntList(o588[0], i91[0])))≥NonInfC∧LOAD843(java.lang.Object(IntList(o588[0], i91[0])))≥COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))∧(UIncreasing(COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))), ≥))

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

(2)    (>(i91[0], 0)=TRUELOAD843(java.lang.Object(IntList(o588[0], i91[0])))≥NonInfC∧LOAD843(java.lang.Object(IntList(o588[0], i91[0])))≥COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))∧(UIncreasing(COND_LOAD843(>(i91[0], 0), java.lang.Object(IntList(o588[0], i91[0])))), ≥))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(LOAD843(x1)) = [-1]x1
POL(java.lang.Object(x1)) = x1
POL(IntList(x1, x2)) = [-1] + [-1]x2
POL(COND_LOAD843(x1, x2)) = [-1] + [-1]x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]

The following pairs are in P>:

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

The following pairs are in Pbound:

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

The following pairs are in P:

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

There are no usable rules.

### (13) Obligation:

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

The following domains are used:

Integer

R is empty.

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

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

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (16) Obligation:

ITRS problem:

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

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

### (17) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

### (18) Obligation:

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

The following domains are used:

Boolean, Integer

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

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

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

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

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

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

### (19) UsableRulesProof (EQUIVALENT transformation)

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

### (20) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

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

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

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

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

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

### (21) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

### (22) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

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

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

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

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

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

### (23) IDPNonInfProof (SOUND transformation)

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

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

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

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

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

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

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

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

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

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

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

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

(6)    (i138[1]=i138[2]i137[1]=i137[2]i139[1]=i139[2]a910[1]=a910[2]i75[1]=i75[2]&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0))=TRUEi2[1]=i2[2]a689data[1]=a689data[2]i79[1]=i79[2]LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))≥NonInfC∧LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))≥COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))∧(UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥))

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

(7)    (>(+(i75[1], 1), 0)=TRUE>(i79[1], 0)=TRUE>(i75[1], 0)=TRUE<(i75[1], i2[1])=TRUELOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))≥NonInfC∧LOAD823ARR1(java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))≥COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))∧(UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥))

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

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

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

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

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

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

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

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

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

(12)    ([1] + i75[1] ≥ 0∧i79[1] + [-1] ≥ 0∧i75[1] ≥ 0∧i2[1] + [-2] + [-1]i75[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧0 = 0∧[bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [(-1)bni_18]i75[1] + [bni_18]i2[1] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)

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

(13)    ([1] + i75[1] ≥ 0∧i79[1] ≥ 0∧i75[1] ≥ 0∧i2[1] + [-2] + [-1]i75[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧0 = 0∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [(-1)bni_18]i75[1] + [bni_18]i2[1] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)

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

(14)    ([1] + i75[1] ≥ 0∧i79[1] ≥ 0∧i75[1] ≥ 0∧i2[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD823ARR1(&&(&&(&&(>(i75[1], 0), <(i75[1], i2[1])), >(i79[1], 0)), >(+(i75[1], 1), 0)), java.lang.Object(ARRAY(i2[1], a689data[1])), i75[1], i79[1], java.lang.Object(java.lang.String(i138[1], i137[1], i139[1], a910[1])))), ≥)∧0 = 0∧[(4)bni_18 + (-1)Bound*bni_18] + [bni_18]i79[1] + [bni_18]i2[1] ≥ 0∧0 = 0∧[(-1)bso_19] ≥ 0)

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

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

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

(16)    ((UIncreasing(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[2], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)

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

(17)    ((UIncreasing(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[2], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)

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

(18)    ((UIncreasing(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[2], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)

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

(19)    ((UIncreasing(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[2], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_21] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD823(java.lang.Object(ARRAY(i2, a689data)), i75, i79) → LOAD823ARR1(java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910)))
• ((UIncreasing(LOAD823ARR1(java.lang.Object(ARRAY(i2[0], a689data[0])), i75[0], i79[0], java.lang.Object(java.lang.String(i138[0], i137[0], i139[0], a910[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

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

• COND_LOAD823ARR1(TRUE, java.lang.Object(ARRAY(i2, a689data)), i75, i79, java.lang.Object(java.lang.String(i138, i137, i139, a910))) → LOAD823(java.lang.Object(ARRAY(i2, a689data)), +(i75, 1), +(i79, -1))
• ((UIncreasing(LOAD823(java.lang.Object(ARRAY(i2[2], a689data[2])), +(i75[2], 1), +(i79[2], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_21] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(LOAD823(x1, x2, x3)) = [1] + x3 + [-1]x2 + [-1]x1
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1, x2)) = [-1] + [-1]x1
POL(LOAD823ARR1(x1, x2, x3, x4)) = [1] + x3 + [-1]x2 + [-1]x1
POL(java.lang.String(x1, x2, x3, x4)) = [-1]
POL(COND_LOAD823ARR1(x1, x2, x3, x4, x5)) = [1] + x4 + [-1]x3 + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(-1) = [-1]

The following pairs are in P>:

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

The following pairs are in Pbound:

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

The following pairs are in P:

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

There are no usable rules.

### (25) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

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

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

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

### (26) IDependencyGraphProof (EQUIVALENT transformation)

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

### (28) Obligation:

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

The following domains are used:

Integer

R is empty.

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

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

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

### (29) IDependencyGraphProof (EQUIVALENT transformation)

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