### (0) Obligation:

JBC Problem based on JBC Program:

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

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 332 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:
Load1436(o772, java.lang.Object(List(o854, java.lang.Object(List(o1487, o1486)))), i127) → Inc2163(o772, java.lang.Object(List(o854, o1487)), i127)
Load1436(o772, java.lang.Object(List(o854, java.lang.Object(o929Sub1234))), i127) → Inc2163(o772, o854, i127)
Load1436(java.lang.Object(List(o858, java.lang.Object(o944Sub1234))), java.lang.Object(List(o858, java.lang.Object(o944Sub1234))), i127) → Inc2163(java.lang.Object(List(o858, java.lang.Object(o944Sub1234))), o858, i127)
Load1436(java.lang.Object(List(o858, NULL)), java.lang.Object(List(o858, NULL)), i127) → Inc2163(java.lang.Object(List(o858, NULL)), o858, i127)
Inc2163(o772, o854, i127) → Load1436(o772, o854, i127 + 1)
Load1436(java.lang.Object(List(o858, java.lang.Object(List(o1496, o1495)))), java.lang.Object(List(o858, java.lang.Object(List(o1496, o1495)))), i127) → Load1436(java.lang.Object(List(o858, o1496)), java.lang.Object(List(o858, o1496)), i127 + 1)
The set Q consists of the following terms:
Inc2163(x0, x1, x2)

### (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:
Load1436(o772, java.lang.Object(List(o854, java.lang.Object(List(o1487, o1486)))), i127) → Inc2163(o772, java.lang.Object(List(o854, o1487)), i127)
Load1436(o772, java.lang.Object(List(o854, java.lang.Object(o929Sub1234))), i127) → Inc2163(o772, o854, i127)
Load1436(java.lang.Object(List(o858, java.lang.Object(o944Sub1234))), java.lang.Object(List(o858, java.lang.Object(o944Sub1234))), i127) → Inc2163(java.lang.Object(List(o858, java.lang.Object(o944Sub1234))), o858, i127)
Load1436(java.lang.Object(List(o858, NULL)), java.lang.Object(List(o858, NULL)), i127) → Inc2163(java.lang.Object(List(o858, NULL)), o858, i127)
Inc2163(o772, o854, i127) → Load1436(o772, o854, i127 + 1)
Load1436(java.lang.Object(List(o858, java.lang.Object(List(o1496, o1495)))), java.lang.Object(List(o858, java.lang.Object(List(o1496, o1495)))), i127) → Load1436(java.lang.Object(List(o858, o1496)), java.lang.Object(List(o858, o1496)), i127 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD1436(o772[0], java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))), i127[0]) → INC2163(o772[0], java.lang.Object(List(o854[0], o1487[0])), i127[0])
(1): LOAD1436(o772[1], java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))), i127[1]) → INC2163(o772[1], o854[1], i127[1])
(2): LOAD1436(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), i127[2]) → INC2163(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), o858[2], i127[2])
(3): LOAD1436(java.lang.Object(List(o858[3], NULL)), java.lang.Object(List(o858[3], NULL)), i127[3]) → INC2163(java.lang.Object(List(o858[3], NULL)), o858[3], i127[3])
(4): INC2163(o772[4], o854[4], i127[4]) → LOAD1436(o772[4], o854[4], i127[4] + 1)
(6): LOAD1436(java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), i127[6]) → LOAD1436(java.lang.Object(List(o858[6], o1496[6])), java.lang.Object(List(o858[6], o1496[6])), i127[6] + 1)

(0) -> (4), if ((o772[0]* o772[4])∧(i127[0]* i127[4])∧(java.lang.Object(List(o854[0], o1487[0])) →* o854[4]))

(1) -> (4), if ((i127[1]* i127[4])∧(o772[1]* o772[4])∧(o854[1]* o854[4]))

(2) -> (4), if ((i127[2]* i127[4])∧(o858[2]* o854[4])∧(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))) →* o772[4]))

(3) -> (4), if ((i127[3]* i127[4])∧(o858[3]* o854[4])∧(java.lang.Object(List(o858[3], NULL)) →* o772[4]))

(4) -> (0), if ((o854[4]* java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))))∧(i127[4] + 1* i127[0])∧(o772[4]* o772[0]))

(4) -> (1), if ((o772[4]* o772[1])∧(i127[4] + 1* i127[1])∧(o854[4]* java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1])))))

(4) -> (2), if ((o772[4]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))))∧(i127[4] + 1* i127[2])∧(o854[4]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2])))))

(4) -> (3), if ((i127[4] + 1* i127[3])∧(o854[4]* java.lang.Object(List(o858[3], NULL)))∧(o772[4]* java.lang.Object(List(o858[3], NULL))))

(4) -> (5), if ((o854[4]* java.lang.Object(List(o854[5], NULL)))∧(i127[4] + 1* i127[5])∧(o772[4]* o772[5]))

(4) -> (6), if ((o772[4]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))))∧(o854[4]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))))∧(i127[4] + 1* i127[6]))

(5) -> (0), if ((o772[5]* o772[0])∧(i127[5] + 1* i127[0])∧(o854[5]* java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0]))))))

(5) -> (1), if ((o854[5]* java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))))∧(o772[5]* o772[1])∧(i127[5] + 1* i127[1]))

(5) -> (2), if ((o854[5]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))))∧(i127[5] + 1* i127[2])∧(o772[5]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2])))))

(5) -> (3), if ((i127[5] + 1* i127[3])∧(o772[5]* java.lang.Object(List(o858[3], NULL)))∧(o854[5]* java.lang.Object(List(o858[3], NULL))))

(5) -> (5), if ((o772[5]* o772[5]')∧(o854[5]* java.lang.Object(List(o854[5]', NULL)))∧(i127[5] + 1* i127[5]'))

(5) -> (6), if ((i127[5] + 1* i127[6])∧(o772[5]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))))∧(o854[5]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6]))))))

(6) -> (0), if ((java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))))∧(java.lang.Object(List(o858[6], o1496[6])) →* o772[0])∧(i127[6] + 1* i127[0]))

(6) -> (1), if ((i127[6] + 1* i127[1])∧(java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))))∧(java.lang.Object(List(o858[6], o1496[6])) →* o772[1]))

(6) -> (2), if ((java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))))∧(i127[6] + 1* i127[2]))

(6) -> (3), if ((i127[6] + 1* i127[3])∧(java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o858[3], NULL))))

(6) -> (5), if ((java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o854[5], NULL)))∧(i127[6] + 1* i127[5])∧(java.lang.Object(List(o858[6], o1496[6])) →* o772[5]))

(6) -> (6), if ((i127[6] + 1* i127[6]')∧(java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o858[6]', java.lang.Object(List(o1496[6]', o1495[6]'))))))

The set Q consists of the following terms:
Inc2163(x0, x1, x2)

### (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): LOAD1436(o772[0], java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))), i127[0]) → INC2163(o772[0], java.lang.Object(List(o854[0], o1487[0])), i127[0])
(1): LOAD1436(o772[1], java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))), i127[1]) → INC2163(o772[1], o854[1], i127[1])
(2): LOAD1436(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), i127[2]) → INC2163(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), o858[2], i127[2])
(3): LOAD1436(java.lang.Object(List(o858[3], NULL)), java.lang.Object(List(o858[3], NULL)), i127[3]) → INC2163(java.lang.Object(List(o858[3], NULL)), o858[3], i127[3])
(4): INC2163(o772[4], o854[4], i127[4]) → LOAD1436(o772[4], o854[4], i127[4] + 1)
(6): LOAD1436(java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), i127[6]) → LOAD1436(java.lang.Object(List(o858[6], o1496[6])), java.lang.Object(List(o858[6], o1496[6])), i127[6] + 1)

(0) -> (4), if ((o772[0]* o772[4])∧(i127[0]* i127[4])∧(java.lang.Object(List(o854[0], o1487[0])) →* o854[4]))

(1) -> (4), if ((i127[1]* i127[4])∧(o772[1]* o772[4])∧(o854[1]* o854[4]))

(2) -> (4), if ((i127[2]* i127[4])∧(o858[2]* o854[4])∧(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))) →* o772[4]))

(3) -> (4), if ((i127[3]* i127[4])∧(o858[3]* o854[4])∧(java.lang.Object(List(o858[3], NULL)) →* o772[4]))

(4) -> (0), if ((o854[4]* java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))))∧(i127[4] + 1* i127[0])∧(o772[4]* o772[0]))

(4) -> (1), if ((o772[4]* o772[1])∧(i127[4] + 1* i127[1])∧(o854[4]* java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1])))))

(4) -> (2), if ((o772[4]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))))∧(i127[4] + 1* i127[2])∧(o854[4]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2])))))

(4) -> (3), if ((i127[4] + 1* i127[3])∧(o854[4]* java.lang.Object(List(o858[3], NULL)))∧(o772[4]* java.lang.Object(List(o858[3], NULL))))

(4) -> (5), if ((o854[4]* java.lang.Object(List(o854[5], NULL)))∧(i127[4] + 1* i127[5])∧(o772[4]* o772[5]))

(4) -> (6), if ((o772[4]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))))∧(o854[4]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))))∧(i127[4] + 1* i127[6]))

(5) -> (0), if ((o772[5]* o772[0])∧(i127[5] + 1* i127[0])∧(o854[5]* java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0]))))))

(5) -> (1), if ((o854[5]* java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))))∧(o772[5]* o772[1])∧(i127[5] + 1* i127[1]))

(5) -> (2), if ((o854[5]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))))∧(i127[5] + 1* i127[2])∧(o772[5]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2])))))

(5) -> (3), if ((i127[5] + 1* i127[3])∧(o772[5]* java.lang.Object(List(o858[3], NULL)))∧(o854[5]* java.lang.Object(List(o858[3], NULL))))

(5) -> (5), if ((o772[5]* o772[5]')∧(o854[5]* java.lang.Object(List(o854[5]', NULL)))∧(i127[5] + 1* i127[5]'))

(5) -> (6), if ((i127[5] + 1* i127[6])∧(o772[5]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))))∧(o854[5]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6]))))))

(6) -> (0), if ((java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))))∧(java.lang.Object(List(o858[6], o1496[6])) →* o772[0])∧(i127[6] + 1* i127[0]))

(6) -> (1), if ((i127[6] + 1* i127[1])∧(java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))))∧(java.lang.Object(List(o858[6], o1496[6])) →* o772[1]))

(6) -> (2), if ((java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))))∧(i127[6] + 1* i127[2]))

(6) -> (3), if ((i127[6] + 1* i127[3])∧(java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o858[3], NULL))))

(6) -> (5), if ((java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o854[5], NULL)))∧(i127[6] + 1* i127[5])∧(java.lang.Object(List(o858[6], o1496[6])) →* o772[5]))

(6) -> (6), if ((i127[6] + 1* i127[6]')∧(java.lang.Object(List(o858[6], o1496[6])) →* java.lang.Object(List(o858[6]', java.lang.Object(List(o1496[6]', o1495[6]'))))))

The set Q consists of the following terms:
Inc2163(x0, x1, x2)

### (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): LOAD1436(o772[0], java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))), i127[0]) → INC2163(o772[0], java.lang.Object(List(o854[0], o1487[0])), i127[0])
(1): LOAD1436(o772[1], java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))), i127[1]) → INC2163(o772[1], o854[1], i127[1])
(2): LOAD1436(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), i127[2]) → INC2163(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), o858[2], i127[2])
(3): LOAD1436(java.lang.Object(List(o858[3], NULL)), java.lang.Object(List(o858[3], NULL)), i127[3]) → INC2163(java.lang.Object(List(o858[3], NULL)), o858[3], i127[3])
(4): INC2163(o772[4], o854[4], i127[4]) → LOAD1436(o772[4], o854[4], i127[4] + 1)
(6): LOAD1436(java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), i127[6]) → LOAD1436(java.lang.Object(List(o858[6], o1496[6])), java.lang.Object(List(o858[6], o1496[6])), i127[6] + 1)

(0) -> (4), if ((o772[0]* o772[4])∧(i127[0]* i127[4])∧(java.lang.Object(List(o854[0], o1487[0])) →* o854[4]))

(1) -> (4), if ((i127[1]* i127[4])∧(o772[1]* o772[4])∧(o854[1]* o854[4]))

(2) -> (4), if ((i127[2]* i127[4])∧(o858[2]* o854[4])∧(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))) →* o772[4]))

(3) -> (4), if ((i127[3]* i127[4])∧(o858[3]* o854[4])∧(java.lang.Object(List(o858[3], NULL)) →* o772[4]))

(4) -> (0), if ((o854[4]* java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))))∧(i127[4] + 1* i127[0])∧(o772[4]* o772[0]))

(4) -> (1), if ((o772[4]* o772[1])∧(i127[4] + 1* i127[1])∧(o854[4]* java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1])))))

(4) -> (2), if ((o772[4]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))))∧(i127[4] + 1* i127[2])∧(o854[4]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2])))))

(4) -> (3), if ((i127[4] + 1* i127[3])∧(o854[4]* java.lang.Object(List(o858[3], NULL)))∧(o772[4]* java.lang.Object(List(o858[3], NULL))))

(4) -> (5), if ((o854[4]* java.lang.Object(List(o854[5], NULL)))∧(i127[4] + 1* i127[5])∧(o772[4]* o772[5]))

(4) -> (6), if ((o772[4]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))))∧(o854[4]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))))∧(i127[4] + 1* i127[6]))

(5) -> (0), if ((o772[5]* o772[0])∧(i127[5] + 1* i127[0])∧(o854[5]* java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0]))))))

(5) -> (1), if ((o854[5]* java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))))∧(o772[5]* o772[1])∧(i127[5] + 1* i127[1]))

(5) -> (2), if ((o854[5]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))))∧(i127[5] + 1* i127[2])∧(o772[5]* java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2])))))

(5) -> (3), if ((i127[5] + 1* i127[3])∧(o772[5]* java.lang.Object(List(o858[3], NULL)))∧(o854[5]* java.lang.Object(List(o858[3], NULL))))

(5) -> (5), if ((o772[5]* o772[5]')∧(o854[5]* java.lang.Object(List(o854[5]', NULL)))∧(i127[5] + 1* i127[5]'))

(5) -> (6), if ((i127[5] + 1* i127[6])∧(o772[5]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))))∧(o854[5]* java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6]))))))

(6) -> (0), if (((o858[6]* o854[0])∧(o1496[6]* java.lang.Object(List(o1487[0], o1486[0]))))∧(java.lang.Object(List(o858[6], o1496[6])) →* o772[0])∧(i127[6] + 1* i127[0]))

(6) -> (1), if ((i127[6] + 1* i127[1])∧((o858[6]* o854[1])∧(o1496[6]* java.lang.Object(o929Sub1234[1])))∧(java.lang.Object(List(o858[6], o1496[6])) →* o772[1]))

(6) -> (2), if (((o858[6]* o858[2])∧(o1496[6]* java.lang.Object(o944Sub1234[2])))∧(i127[6] + 1* i127[2]))

(6) -> (3), if ((i127[6] + 1* i127[3])∧((o858[6]* o858[3])∧(o1496[6]* NULL)))

(6) -> (5), if (((o858[6]* o854[5])∧(o1496[6]* NULL))∧(i127[6] + 1* i127[5])∧(java.lang.Object(List(o858[6], o1496[6])) →* o772[5]))

(6) -> (6), if ((i127[6] + 1* i127[6]')∧((o858[6]* o858[6]')∧(o1496[6]* java.lang.Object(List(o1496[6]', o1495[6]')))))

The set Q consists of the following terms:
Inc2163(x0, x1, x2)

### (12) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

### (13) Obligation:

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

LOAD1436(o772[0], java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))), i127[0]) → INC2163(o772[0], java.lang.Object(List(o854[0], o1487[0])), i127[0])
LOAD1436(o772[1], java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))), i127[1]) → INC2163(o772[1], o854[1], i127[1])
LOAD1436(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), i127[2]) → INC2163(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), o858[2], i127[2])
LOAD1436(java.lang.Object(List(o858[3], NULL)), java.lang.Object(List(o858[3], NULL)), i127[3]) → INC2163(java.lang.Object(List(o858[3], NULL)), o858[3], i127[3])
INC2163(o772[4], o854[4], i127[4]) → LOAD1436(o772[4], o854[4], plus_int(pos(s(0)), i127[4]))
LOAD1436(java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), i127[6]) → LOAD1436(java.lang.Object(List(o858[6], o1496[6])), java.lang.Object(List(o858[6], o1496[6])), plus_int(pos(s(0)), i127[6]))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Inc2163(x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

### (14) 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.

### (15) Obligation:

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

LOAD1436(o772[0], java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))), i127[0]) → INC2163(o772[0], java.lang.Object(List(o854[0], o1487[0])), i127[0])
LOAD1436(o772[1], java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))), i127[1]) → INC2163(o772[1], o854[1], i127[1])
LOAD1436(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), i127[2]) → INC2163(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), o858[2], i127[2])
LOAD1436(java.lang.Object(List(o858[3], NULL)), java.lang.Object(List(o858[3], NULL)), i127[3]) → INC2163(java.lang.Object(List(o858[3], NULL)), o858[3], i127[3])
INC2163(o772[4], o854[4], i127[4]) → LOAD1436(o772[4], o854[4], plus_int(pos(s(0)), i127[4]))
LOAD1436(java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), i127[6]) → LOAD1436(java.lang.Object(List(o858[6], o1496[6])), java.lang.Object(List(o858[6], o1496[6])), plus_int(pos(s(0)), i127[6]))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Inc2163(x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

### (16) 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].

Inc2163(x0, x1, x2)

### (17) Obligation:

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

LOAD1436(o772[0], java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))), i127[0]) → INC2163(o772[0], java.lang.Object(List(o854[0], o1487[0])), i127[0])
LOAD1436(o772[1], java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))), i127[1]) → INC2163(o772[1], o854[1], i127[1])
LOAD1436(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), i127[2]) → INC2163(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), o858[2], i127[2])
LOAD1436(java.lang.Object(List(o858[3], NULL)), java.lang.Object(List(o858[3], NULL)), i127[3]) → INC2163(java.lang.Object(List(o858[3], NULL)), o858[3], i127[3])
INC2163(o772[4], o854[4], i127[4]) → LOAD1436(o772[4], o854[4], plus_int(pos(s(0)), i127[4]))
LOAD1436(java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), i127[6]) → LOAD1436(java.lang.Object(List(o858[6], o1496[6])), java.lang.Object(List(o858[6], o1496[6])), plus_int(pos(s(0)), i127[6]))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

### (18) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

LOAD1436(java.lang.Object(List(o858[3], NULL)), java.lang.Object(List(o858[3], NULL)), i127[3]) → INC2163(java.lang.Object(List(o858[3], NULL)), o858[3], i127[3])
INC2163(o772[4], o854[4], i127[4]) → LOAD1436(o772[4], o854[4], plus_int(pos(s(0)), i127[4]))
LOAD1436(java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), java.lang.Object(List(o858[6], java.lang.Object(List(o1496[6], o1495[6])))), i127[6]) → LOAD1436(java.lang.Object(List(o858[6], o1496[6])), java.lang.Object(List(o858[6], o1496[6])), plus_int(pos(s(0)), i127[6]))

Strictly oriented rules of the TRS R:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(INC2163(x1, x2, x3)) = 4 + x1 + x2 + x3
POL(LOAD1436(x1, x2, x3)) = x1 + x2 + x3
POL(List(x1, x2)) = 4 + x1 + x2
POL(NULL) = 1
POL(java.lang.Object(x1)) = x1
POL(minus_nat(x1, x2)) = 1 + x1 + x2
POL(neg(x1)) = x1
POL(plus_int(x1, x2)) = 1 + x1 + x2
POL(plus_nat(x1, x2)) = 1 + x1 + x2
POL(pos(x1)) = 1 + x1
POL(s(x1)) = 1 + x1

### (19) Obligation:

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

LOAD1436(o772[0], java.lang.Object(List(o854[0], java.lang.Object(List(o1487[0], o1486[0])))), i127[0]) → INC2163(o772[0], java.lang.Object(List(o854[0], o1487[0])), i127[0])
LOAD1436(o772[1], java.lang.Object(List(o854[1], java.lang.Object(o929Sub1234[1]))), i127[1]) → INC2163(o772[1], o854[1], i127[1])
LOAD1436(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), i127[2]) → INC2163(java.lang.Object(List(o858[2], java.lang.Object(o944Sub1234[2]))), o858[2], i127[2])

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

### (20) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

### (22) 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:
Load885(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60) → Load885ARR1(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651)))
Load885ARR1(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651))) → Cond_Load885ARR1(i58 > 0 && i58 < i3 && i60 >= 0 && i60 < i59 && i58 + 1 > 0, java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651))) → Load2533(java.lang.Object(ARRAY(i3, a523data)), i58 + 1, i59, i60, i87)
Load2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213) → Cond_Load2533(i213 > 0, java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213) → Load2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213 + -1)
Load2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, 0) → Load885(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60 + 1)
The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (24) 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:
Load885(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60) → Load885ARR1(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651)))
Load885ARR1(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651))) → Cond_Load885ARR1(i58 > 0 && i58 < i3 && i60 >= 0 && i60 < i59 && i58 + 1 > 0, java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651))) → Load2533(java.lang.Object(ARRAY(i3, a523data)), i58 + 1, i59, i60, i87)
Load2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213) → Cond_Load2533(i213 > 0, java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213) → Load2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213 + -1)
Load2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, 0) → Load885(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD885(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0]) → LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))
(1): LOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1]))) → COND_LOAD885ARR1(i58[1] > 0 && i58[1] < i3[1] && i60[1] >= 0 && i60[1] < i59[1] && i58[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))
(2): COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2], i59[2], i60[2], java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2]))) → LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2] + 1, i59[2], i60[2], i87[2])
(3): LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(i213[3] > 0, java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])
(4): COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4] + -1)
(5): LOAD2533(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5], 0) → LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5] + 1)

(0) -> (1), if ((i60[0]* i60[1])∧(java.lang.Object(ARRAY(i3[0], a523data[0])) →* java.lang.Object(ARRAY(i3[1], a523data[1])))∧(i59[0]* i59[1])∧(i58[0]* i58[1])∧(java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])) →* java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1]))))

(1) -> (2), if ((i58[1]* i58[2])∧(java.lang.Object(ARRAY(i3[1], a523data[1])) →* java.lang.Object(ARRAY(i3[2], a523data[2])))∧(i58[1] > 0 && i58[1] < i3[1] && i60[1] >= 0 && i60[1] < i59[1] && i58[1] + 1 > 0* TRUE)∧(i60[1]* i60[2])∧(i59[1]* i59[2])∧(java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])) →* java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2]))))

(2) -> (3), if ((i87[2]* i213[3])∧(i60[2]* i60[3])∧(i58[2] + 1* i68[3])∧(i59[2]* i59[3])∧(java.lang.Object(ARRAY(i3[2], a523data[2])) →* java.lang.Object(ARRAY(i3[3], a1231data[3]))))

(2) -> (5), if ((java.lang.Object(ARRAY(i3[2], a523data[2])) →* java.lang.Object(ARRAY(i3[5], a1231data[5])))∧(i87[2]* 0)∧(i58[2] + 1* i68[5])∧(i59[2]* i59[5])∧(i60[2]* i60[5]))

(3) -> (4), if ((i59[3]* i59[4])∧(i68[3]* i68[4])∧(i213[3]* i213[4])∧(i60[3]* i60[4])∧(i213[3] > 0* TRUE)∧(java.lang.Object(ARRAY(i3[3], a1231data[3])) →* java.lang.Object(ARRAY(i3[4], a1231data[4]))))

(4) -> (3), if ((i59[4]* i59[3])∧(i60[4]* i60[3])∧(i68[4]* i68[3])∧(i213[4] + -1* i213[3])∧(java.lang.Object(ARRAY(i3[4], a1231data[4])) →* java.lang.Object(ARRAY(i3[3], a1231data[3]))))

(4) -> (5), if ((i213[4] + -1* 0)∧(i59[4]* i59[5])∧(java.lang.Object(ARRAY(i3[4], a1231data[4])) →* java.lang.Object(ARRAY(i3[5], a1231data[5])))∧(i68[4]* i68[5])∧(i60[4]* i60[5]))

(5) -> (0), if ((i60[5] + 1* i60[0])∧(java.lang.Object(ARRAY(i3[5], a1231data[5])) →* java.lang.Object(ARRAY(i3[0], a523data[0])))∧(i59[5]* i59[0])∧(i68[5]* i58[0]))

The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (25) 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.

### (26) 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): LOAD885(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0]) → LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))
(1): LOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1]))) → COND_LOAD885ARR1(i58[1] > 0 && i58[1] < i3[1] && i60[1] >= 0 && i60[1] < i59[1] && i58[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))
(2): COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2], i59[2], i60[2], java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2]))) → LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2] + 1, i59[2], i60[2], i87[2])
(3): LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(i213[3] > 0, java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])
(4): COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4] + -1)
(5): LOAD2533(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5], 0) → LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5] + 1)

(0) -> (1), if ((i60[0]* i60[1])∧(java.lang.Object(ARRAY(i3[0], a523data[0])) →* java.lang.Object(ARRAY(i3[1], a523data[1])))∧(i59[0]* i59[1])∧(i58[0]* i58[1])∧(java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])) →* java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1]))))

(1) -> (2), if ((i58[1]* i58[2])∧(java.lang.Object(ARRAY(i3[1], a523data[1])) →* java.lang.Object(ARRAY(i3[2], a523data[2])))∧(i58[1] > 0 && i58[1] < i3[1] && i60[1] >= 0 && i60[1] < i59[1] && i58[1] + 1 > 0* TRUE)∧(i60[1]* i60[2])∧(i59[1]* i59[2])∧(java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])) →* java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2]))))

(2) -> (3), if ((i87[2]* i213[3])∧(i60[2]* i60[3])∧(i58[2] + 1* i68[3])∧(i59[2]* i59[3])∧(java.lang.Object(ARRAY(i3[2], a523data[2])) →* java.lang.Object(ARRAY(i3[3], a1231data[3]))))

(2) -> (5), if ((java.lang.Object(ARRAY(i3[2], a523data[2])) →* java.lang.Object(ARRAY(i3[5], a1231data[5])))∧(i87[2]* 0)∧(i58[2] + 1* i68[5])∧(i59[2]* i59[5])∧(i60[2]* i60[5]))

(3) -> (4), if ((i59[3]* i59[4])∧(i68[3]* i68[4])∧(i213[3]* i213[4])∧(i60[3]* i60[4])∧(i213[3] > 0* TRUE)∧(java.lang.Object(ARRAY(i3[3], a1231data[3])) →* java.lang.Object(ARRAY(i3[4], a1231data[4]))))

(4) -> (3), if ((i59[4]* i59[3])∧(i60[4]* i60[3])∧(i68[4]* i68[3])∧(i213[4] + -1* i213[3])∧(java.lang.Object(ARRAY(i3[4], a1231data[4])) →* java.lang.Object(ARRAY(i3[3], a1231data[3]))))

(4) -> (5), if ((i213[4] + -1* 0)∧(i59[4]* i59[5])∧(java.lang.Object(ARRAY(i3[4], a1231data[4])) →* java.lang.Object(ARRAY(i3[5], a1231data[5])))∧(i68[4]* i68[5])∧(i60[4]* i60[5]))

(5) -> (0), if ((i60[5] + 1* i60[0])∧(java.lang.Object(ARRAY(i3[5], a1231data[5])) →* java.lang.Object(ARRAY(i3[0], a523data[0])))∧(i59[5]* i59[0])∧(i68[5]* i58[0]))

The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (27) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

### (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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD885(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0]) → LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))
(1): LOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1]))) → COND_LOAD885ARR1(i58[1] > 0 && i58[1] < i3[1] && i60[1] >= 0 && i60[1] < i59[1] && i58[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))
(2): COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2], i59[2], i60[2], java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2]))) → LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2] + 1, i59[2], i60[2], i87[2])
(3): LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(i213[3] > 0, java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])
(4): COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4] + -1)
(5): LOAD2533(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5], 0) → LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5] + 1)

(0) -> (1), if ((i60[0]* i60[1])∧((i3[0]* i3[1])∧(a523data[0]* a523data[1]))∧(i59[0]* i59[1])∧(i58[0]* i58[1])∧((i87[0]* i87[1])∧(i86[0]* i86[1])∧(i88[0]* i88[1])∧(a651[0]* a651[1])))

(1) -> (2), if ((i58[1]* i58[2])∧((i3[1]* i3[2])∧(a523data[1]* a523data[2]))∧(i58[1] > 0 && i58[1] < i3[1] && i60[1] >= 0 && i60[1] < i59[1] && i58[1] + 1 > 0* TRUE)∧(i60[1]* i60[2])∧(i59[1]* i59[2])∧((i87[1]* i87[2])∧(i86[1]* i86[2])∧(i88[1]* i88[2])∧(a651[1]* a651[2])))

(2) -> (3), if ((i87[2]* i213[3])∧(i60[2]* i60[3])∧(i58[2] + 1* i68[3])∧(i59[2]* i59[3])∧((i3[2]* i3[3])∧(a523data[2]* a1231data[3])))

(2) -> (5), if (((i3[2]* i3[5])∧(a523data[2]* a1231data[5]))∧(i87[2]* 0)∧(i58[2] + 1* i68[5])∧(i59[2]* i59[5])∧(i60[2]* i60[5]))

(3) -> (4), if ((i59[3]* i59[4])∧(i68[3]* i68[4])∧(i213[3]* i213[4])∧(i60[3]* i60[4])∧(i213[3] > 0* TRUE)∧((i3[3]* i3[4])∧(a1231data[3]* a1231data[4])))

(4) -> (3), if ((i59[4]* i59[3])∧(i60[4]* i60[3])∧(i68[4]* i68[3])∧(i213[4] + -1* i213[3])∧((i3[4]* i3[3])∧(a1231data[4]* a1231data[3])))

(4) -> (5), if ((i213[4] + -1* 0)∧(i59[4]* i59[5])∧((i3[4]* i3[5])∧(a1231data[4]* a1231data[5]))∧(i68[4]* i68[5])∧(i60[4]* i60[5]))

(5) -> (0), if ((i60[5] + 1* i60[0])∧((i3[5]* i3[0])∧(a1231data[5]* a523data[0]))∧(i59[5]* i59[0])∧(i68[5]* i58[0]))

The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (29) 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 LOAD885(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60) → LOAD885ARR1(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651))) the following chains were created:
• We consider the chain LOAD885(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0]) → LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0]))) which results in the following constraint:

(1)    (LOAD885(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0])≥NonInfC∧LOAD885(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0])≥LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))∧(UIncreasing(LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))), ≥))

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

(2)    ((UIncreasing(LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))), ≥)∧[(-1)bso_22] ≥ 0)

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

(3)    ((UIncreasing(LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))), ≥)∧[(-1)bso_22] ≥ 0)

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

(4)    ((UIncreasing(LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))), ≥)∧[(-1)bso_22] ≥ 0)

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

(5)    ((UIncreasing(LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_22] ≥ 0)

For Pair LOAD885ARR1(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651))) → COND_LOAD885ARR1(&&(&&(&&(&&(>(i58, 0), <(i58, i3)), >=(i60, 0)), <(i60, i59)), >(+(i58, 1), 0)), java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651))) the following chains were created:
• We consider the chain LOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1]))) → COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1]))), COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2], i59[2], i60[2], java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2]))) → LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), +(i58[2], 1), i59[2], i60[2], i87[2]) which results in the following constraint:

(6)    (i58[1]=i58[2]i3[1]=i3[2]a523data[1]=a523data[2]&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0))=TRUEi60[1]=i60[2]i59[1]=i59[2]i87[1]=i87[2]i86[1]=i86[2]i88[1]=i88[2]a651[1]=a651[2]LOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))≥NonInfC∧LOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))≥COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))∧(UIncreasing(COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))), ≥))

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

(7)    (>(+(i58[1], 1), 0)=TRUE<(i60[1], i59[1])=TRUE>=(i60[1], 0)=TRUE>(i58[1], 0)=TRUE<(i58[1], i3[1])=TRUELOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))≥NonInfC∧LOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))≥COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))∧(UIncreasing(COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))), ≥))

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

(8)    (i58[1] ≥ 0∧i59[1] + [-1] + [-1]i60[1] ≥ 0∧i60[1] ≥ 0∧i58[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))), ≥)∧[(-3)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i60[1] + [(2)bni_23]i59[1] + [bni_23]i3[1] + [(-1)bni_23]i58[1] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(9)    (i58[1] ≥ 0∧i59[1] + [-1] + [-1]i60[1] ≥ 0∧i60[1] ≥ 0∧i58[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))), ≥)∧[(-3)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i60[1] + [(2)bni_23]i59[1] + [bni_23]i3[1] + [(-1)bni_23]i58[1] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(10)    (i58[1] ≥ 0∧i59[1] + [-1] + [-1]i60[1] ≥ 0∧i60[1] ≥ 0∧i58[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))), ≥)∧[(-3)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i60[1] + [(2)bni_23]i59[1] + [bni_23]i3[1] + [(-1)bni_23]i58[1] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(11)    (i58[1] ≥ 0∧i59[1] + [-1] + [-1]i60[1] ≥ 0∧i60[1] ≥ 0∧i58[1] + [-1] ≥ 0∧i3[1] + [-1] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))), ≥)∧0 = 0∧[(-3)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i60[1] + [(2)bni_23]i59[1] + [bni_23]i3[1] + [(-1)bni_23]i58[1] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

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

(12)    ([1] + i58[1] ≥ 0∧i59[1] + [-1] + [-1]i60[1] ≥ 0∧i60[1] ≥ 0∧i58[1] ≥ 0∧i3[1] + [-2] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))), ≥)∧0 = 0∧[(-4)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i60[1] + [(2)bni_23]i59[1] + [bni_23]i3[1] + [(-1)bni_23]i58[1] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

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

(13)    ([1] + i58[1] ≥ 0∧i59[1] ≥ 0∧i60[1] ≥ 0∧i58[1] ≥ 0∧i3[1] + [-2] + [-1]i58[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))), ≥)∧0 = 0∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]i60[1] + [(2)bni_23]i59[1] + [bni_23]i3[1] + [(-1)bni_23]i58[1] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

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

(14)    ([1] + i58[1] ≥ 0∧i59[1] ≥ 0∧i60[1] ≥ 0∧i58[1] ≥ 0∧i3[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))), ≥)∧0 = 0∧[(-1)Bound*bni_23] + [bni_23]i60[1] + [(2)bni_23]i59[1] + [bni_23]i3[1] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

For Pair COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651))) → LOAD2533(java.lang.Object(ARRAY(i3, a523data)), +(i58, 1), i59, i60, i87) the following chains were created:
• We consider the chain COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2], i59[2], i60[2], java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2]))) → LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), +(i58[2], 1), i59[2], i60[2], i87[2]) which results in the following constraint:

(15)    (COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2], i59[2], i60[2], java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2])))≥NonInfC∧COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2], i59[2], i60[2], java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2])))≥LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), +(i58[2], 1), i59[2], i60[2], i87[2])∧(UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), +(i58[2], 1), i59[2], i60[2], i87[2])), ≥))

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

(16)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), +(i58[2], 1), i59[2], i60[2], i87[2])), ≥)∧[1 + (-1)bso_26] ≥ 0)

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

(17)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), +(i58[2], 1), i59[2], i60[2], i87[2])), ≥)∧[1 + (-1)bso_26] ≥ 0)

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

(18)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), +(i58[2], 1), i59[2], i60[2], i87[2])), ≥)∧[1 + (-1)bso_26] ≥ 0)

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

(19)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), +(i58[2], 1), i59[2], i60[2], i87[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

For Pair LOAD2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213) → COND_LOAD2533(>(i213, 0), java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213) the following chains were created:
• We consider the chain LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]), COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1)) which results in the following constraint:

(20)    (i59[3]=i59[4]i68[3]=i68[4]i213[3]=i213[4]i60[3]=i60[4]>(i213[3], 0)=TRUEi3[3]=i3[4]a1231data[3]=a1231data[4]LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥NonInfC∧LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])∧(UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥))

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

(21)    (>(i213[3], 0)=TRUELOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥NonInfC∧LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])∧(UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥))

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

(22)    (i213[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-3)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]i68[3] + [(-1)bni_27]i60[3] + [(2)bni_27]i59[3] + [bni_27]i3[3] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(23)    (i213[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-3)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]i68[3] + [(-1)bni_27]i60[3] + [(2)bni_27]i59[3] + [bni_27]i3[3] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(24)    (i213[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-3)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]i68[3] + [(-1)bni_27]i60[3] + [(2)bni_27]i59[3] + [bni_27]i3[3] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(25)    (i213[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-1)bni_27] = 0∧[(-1)bni_27] = 0∧[(2)bni_27] = 0∧0 = 0∧[bni_27] = 0∧[(-3)bni_27 + (-1)Bound*bni_27] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

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

(26)    (i213[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-1)bni_27] = 0∧[(-1)bni_27] = 0∧[(2)bni_27] = 0∧0 = 0∧[bni_27] = 0∧[(-3)bni_27 + (-1)Bound*bni_27] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

For Pair COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213) → LOAD2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, +(i213, -1)) the following chains were created:
• We consider the chain COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1)) which results in the following constraint:

(27)    (COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4])≥NonInfC∧COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4])≥LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))∧(UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥))

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

(28)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧[(-1)bso_30] ≥ 0)

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

(29)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧[(-1)bso_30] ≥ 0)

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

(30)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧[(-1)bso_30] ≥ 0)

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

(31)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

For Pair LOAD2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, 0) → LOAD885(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, +(i60, 1)) the following chains were created:
• We consider the chain LOAD2533(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5], 0) → LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], +(i60[5], 1)) which results in the following constraint:

(32)    (LOAD2533(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5], 0)≥NonInfC∧LOAD2533(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5], 0)≥LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], +(i60[5], 1))∧(UIncreasing(LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], +(i60[5], 1))), ≥))

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

(33)    ((UIncreasing(LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], +(i60[5], 1))), ≥)∧[1 + (-1)bso_32] ≥ 0)

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

(34)    ((UIncreasing(LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], +(i60[5], 1))), ≥)∧[1 + (-1)bso_32] ≥ 0)

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

(35)    ((UIncreasing(LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], +(i60[5], 1))), ≥)∧[1 + (-1)bso_32] ≥ 0)

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

(36)    ((UIncreasing(LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], +(i60[5], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_32] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD885(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60) → LOAD885ARR1(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651)))
• ((UIncreasing(LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_22] ≥ 0)

• LOAD885ARR1(java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651))) → COND_LOAD885ARR1(&&(&&(&&(&&(>(i58, 0), <(i58, i3)), >=(i60, 0)), <(i60, i59)), >(+(i58, 1), 0)), java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651)))
• ([1] + i58[1] ≥ 0∧i59[1] ≥ 0∧i60[1] ≥ 0∧i58[1] ≥ 0∧i3[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))), ≥)∧0 = 0∧[(-1)Bound*bni_23] + [bni_23]i60[1] + [(2)bni_23]i59[1] + [bni_23]i3[1] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

• COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3, a523data)), i58, i59, i60, java.lang.Object(java.lang.String(i87, i86, i88, a651))) → LOAD2533(java.lang.Object(ARRAY(i3, a523data)), +(i58, 1), i59, i60, i87)
• ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), +(i58[2], 1), i59[2], i60[2], i87[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

• LOAD2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213) → COND_LOAD2533(>(i213, 0), java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213)
• (i213[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-1)bni_27] = 0∧[(-1)bni_27] = 0∧[(2)bni_27] = 0∧0 = 0∧[bni_27] = 0∧[(-3)bni_27 + (-1)Bound*bni_27] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

• COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, i213) → LOAD2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, +(i213, -1))
• ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

• LOAD2533(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, i60, 0) → LOAD885(java.lang.Object(ARRAY(i3, a1231data)), i68, i59, +(i60, 1))
• ((UIncreasing(LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], +(i60[5], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_32] ≥ 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(LOAD885(x1, x2, x3, x4)) = [-1] + [-1]x4 + [2]x3 + [-1]x1 + [-1]x2
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1, x2)) = [2] + [-1]x1
POL(LOAD885ARR1(x1, x2, x3, x4, x5)) = [-1] + [-1]x4 + [2]x3 + [-1]x1 + [-1]x2
POL(java.lang.String(x1, x2, x3, x4)) = [-1]
POL(COND_LOAD885ARR1(x1, x2, x3, x4, x5, x6)) = [-1] + [-1]x5 + [2]x4 + [-1]x2 + [-1]x3
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(<(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(LOAD2533(x1, x2, x3, x4, x5)) = [-1] + [-1]x2 + [-1]x4 + [2]x3 + [-1]x1
POL(COND_LOAD2533(x1, x2, x3, x4, x5, x6)) = [-1] + [-1]x5 + [2]x4 + [-1]x3 + [-1]x2
POL(-1) = [-1]

The following pairs are in P>:

COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2], i59[2], i60[2], java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2]))) → LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), +(i58[2], 1), i59[2], i60[2], i87[2])
LOAD2533(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5], 0) → LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], +(i60[5], 1))

The following pairs are in Pbound:

LOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1]))) → COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))

The following pairs are in P:

LOAD885(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0]) → LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))
LOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1]))) → COND_LOAD885ARR1(&&(&&(&&(&&(>(i58[1], 0), <(i58[1], i3[1])), >=(i60[1], 0)), <(i60[1], i59[1])), >(+(i58[1], 1), 0)), java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))
LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])
COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))

There are no usable rules.

### (31) 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): LOAD885(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0]) → LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))
(1): LOAD885ARR1(java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1]))) → COND_LOAD885ARR1(i58[1] > 0 && i58[1] < i3[1] && i60[1] >= 0 && i60[1] < i59[1] && i58[1] + 1 > 0, java.lang.Object(ARRAY(i3[1], a523data[1])), i58[1], i59[1], i60[1], java.lang.Object(java.lang.String(i87[1], i86[1], i88[1], a651[1])))
(3): LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(i213[3] > 0, java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])
(4): COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4] + -1)

(0) -> (1), if ((i60[0]* i60[1])∧((i3[0]* i3[1])∧(a523data[0]* a523data[1]))∧(i59[0]* i59[1])∧(i58[0]* i58[1])∧((i87[0]* i87[1])∧(i86[0]* i86[1])∧(i88[0]* i88[1])∧(a651[0]* a651[1])))

(4) -> (3), if ((i59[4]* i59[3])∧(i60[4]* i60[3])∧(i68[4]* i68[3])∧(i213[4] + -1* i213[3])∧((i3[4]* i3[3])∧(a1231data[4]* a1231data[3])))

(3) -> (4), if ((i59[3]* i59[4])∧(i68[3]* i68[4])∧(i213[3]* i213[4])∧(i60[3]* i60[4])∧(i213[3] > 0* TRUE)∧((i3[3]* i3[4])∧(a1231data[3]* a1231data[4])))

The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (32) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

### (33) 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:
(4): COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4] + -1)
(3): LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(i213[3] > 0, java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])

(4) -> (3), if ((i59[4]* i59[3])∧(i60[4]* i60[3])∧(i68[4]* i68[3])∧(i213[4] + -1* i213[3])∧((i3[4]* i3[3])∧(a1231data[4]* a1231data[3])))

(3) -> (4), if ((i59[3]* i59[4])∧(i68[3]* i68[4])∧(i213[3]* i213[4])∧(i60[3]* i60[4])∧(i213[3] > 0* TRUE)∧((i3[3]* i3[4])∧(a1231data[3]* a1231data[4])))

The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (34) 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 COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1)) the following chains were created:
• We consider the chain COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1)) which results in the following constraint:

(1)    (COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4])≥NonInfC∧COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4])≥LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))∧(UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥))

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

(2)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧[2 + (-1)bso_10] ≥ 0)

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

(3)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧[2 + (-1)bso_10] ≥ 0)

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

(4)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧[2 + (-1)bso_10] ≥ 0)

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

(5)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧0 = 0∧[2 + (-1)bso_10] ≥ 0)

For Pair LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) the following chains were created:
• We consider the chain LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]), COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1)) which results in the following constraint:

(6)    (i59[3]=i59[4]i68[3]=i68[4]i213[3]=i213[4]i60[3]=i60[4]>(i213[3], 0)=TRUEi3[3]=i3[4]a1231data[3]=a1231data[4]LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥NonInfC∧LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])∧(UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥))

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

(7)    (>(i213[3], 0)=TRUELOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥NonInfC∧LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])∧(UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥))

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

(8)    (i213[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i213[3] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(9)    (i213[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i213[3] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(10)    (i213[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i213[3] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(11)    (i213[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i213[3] ≥ 0∧[(-1)bso_12] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))
• ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧0 = 0∧[2 + (-1)bso_10] ≥ 0)

• LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])
• (i213[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i213[3] ≥ 0∧[(-1)bso_12] ≥ 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(COND_LOAD2533(x1, x2, x3, x4, x5, x6)) = [-1] + [2]x6
POL(java.lang.Object(x1)) = [-1]
POL(ARRAY(x1, x2)) = [-1]
POL(LOAD2533(x1, x2, x3, x4, x5)) = [-1] + [2]x5
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))

The following pairs are in Pbound:

LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])

The following pairs are in P:

LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])

There are no usable rules.

### (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

R is empty.

The integer pair graph contains the following rules and edges:
(3): LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(i213[3] > 0, java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])

The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (37) IDependencyGraphProof (EQUIVALENT transformation)

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

### (39) 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:
(4): COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4] + -1)

The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (40) IDependencyGraphProof (EQUIVALENT transformation)

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

### (42) 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): LOAD885(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0]) → LOAD885ARR1(java.lang.Object(ARRAY(i3[0], a523data[0])), i58[0], i59[0], i60[0], java.lang.Object(java.lang.String(i87[0], i86[0], i88[0], a651[0])))
(2): COND_LOAD885ARR1(TRUE, java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2], i59[2], i60[2], java.lang.Object(java.lang.String(i87[2], i86[2], i88[2], a651[2]))) → LOAD2533(java.lang.Object(ARRAY(i3[2], a523data[2])), i58[2] + 1, i59[2], i60[2], i87[2])
(3): LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(i213[3] > 0, java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])
(4): COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4] + -1)
(5): LOAD2533(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5], 0) → LOAD885(java.lang.Object(ARRAY(i3[5], a1231data[5])), i68[5], i59[5], i60[5] + 1)

(5) -> (0), if ((i60[5] + 1* i60[0])∧((i3[5]* i3[0])∧(a1231data[5]* a523data[0]))∧(i59[5]* i59[0])∧(i68[5]* i58[0]))

(2) -> (3), if ((i87[2]* i213[3])∧(i60[2]* i60[3])∧(i58[2] + 1* i68[3])∧(i59[2]* i59[3])∧((i3[2]* i3[3])∧(a523data[2]* a1231data[3])))

(4) -> (3), if ((i59[4]* i59[3])∧(i60[4]* i60[3])∧(i68[4]* i68[3])∧(i213[4] + -1* i213[3])∧((i3[4]* i3[3])∧(a1231data[4]* a1231data[3])))

(3) -> (4), if ((i59[3]* i59[4])∧(i68[3]* i68[4])∧(i213[3]* i213[4])∧(i60[3]* i60[4])∧(i213[3] > 0* TRUE)∧((i3[3]* i3[4])∧(a1231data[3]* a1231data[4])))

(2) -> (5), if (((i3[2]* i3[5])∧(a523data[2]* a1231data[5]))∧(i87[2]* 0)∧(i58[2] + 1* i68[5])∧(i59[2]* i59[5])∧(i60[2]* i60[5]))

(4) -> (5), if ((i213[4] + -1* 0)∧(i59[4]* i59[5])∧((i3[4]* i3[5])∧(a1231data[4]* a1231data[5]))∧(i68[4]* i68[5])∧(i60[4]* i60[5]))

The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (43) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

### (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:
(4): COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4] + -1)
(3): LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(i213[3] > 0, java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])

(4) -> (3), if ((i59[4]* i59[3])∧(i60[4]* i60[3])∧(i68[4]* i68[3])∧(i213[4] + -1* i213[3])∧((i3[4]* i3[3])∧(a1231data[4]* a1231data[3])))

(3) -> (4), if ((i59[3]* i59[4])∧(i68[3]* i68[4])∧(i213[3]* i213[4])∧(i60[3]* i60[4])∧(i213[3] > 0* TRUE)∧((i3[3]* i3[4])∧(a1231data[3]* a1231data[4])))

The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (45) 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 COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1)) the following chains were created:
• We consider the chain COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1)) which results in the following constraint:

(1)    (COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4])≥NonInfC∧COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4])≥LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))∧(UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥))

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

(2)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧[2 + (-1)bso_9] ≥ 0)

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

(3)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧[2 + (-1)bso_9] ≥ 0)

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

(4)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧[2 + (-1)bso_9] ≥ 0)

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

(5)    ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧0 = 0∧[2 + (-1)bso_9] ≥ 0)

For Pair LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) the following chains were created:
• We consider the chain LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]), COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1)) which results in the following constraint:

(6)    (i59[3]=i59[4]i68[3]=i68[4]i213[3]=i213[4]i60[3]=i60[4]>(i213[3], 0)=TRUEi3[3]=i3[4]a1231data[3]=a1231data[4]LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥NonInfC∧LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])∧(UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥))

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

(7)    (>(i213[3], 0)=TRUELOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥NonInfC∧LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])≥COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])∧(UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥))

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

(8)    (i213[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]i213[3] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(9)    (i213[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]i213[3] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(10)    (i213[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]i213[3] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(11)    (i213[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [(2)bni_10]i213[3] ≥ 0∧[(-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))
• ((UIncreasing(LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))), ≥)∧0 = 0∧[2 + (-1)bso_9] ≥ 0)

• LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])
• (i213[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [(2)bni_10]i213[3] ≥ 0∧[(-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(COND_LOAD2533(x1, x2, x3, x4, x5, x6)) = [-1] + [2]x6
POL(java.lang.Object(x1)) = [2]
POL(ARRAY(x1, x2)) = [-1]
POL(LOAD2533(x1, x2, x3, x4, x5)) = [-1] + [2]x5
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], +(i213[4], -1))

The following pairs are in Pbound:

LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])

The following pairs are in P:

LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(>(i213[3], 0), java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])

There are no usable rules.

### (47) 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:
(3): LOAD2533(java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3]) → COND_LOAD2533(i213[3] > 0, java.lang.Object(ARRAY(i3[3], a1231data[3])), i68[3], i59[3], i60[3], i213[3])

The set Q consists of the following terms:
Load885ARR1(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Cond_Load885ARR1(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, java.lang.Object(java.lang.String(x5, x6, x7, x8)))
Load2533(java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)
Cond_Load2533(TRUE, java.lang.Object(ARRAY(x0, x1)), x2, x3, x4, x5)

### (48) IDependencyGraphProof (EQUIVALENT transformation)

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

### (50) 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:
(4): COND_LOAD2533(TRUE, java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4]) → LOAD2533(java.lang.Object(ARRAY(i3[4], a1231data[4])), i68[4], i59[4], i60[4], i213[4] + -1)

The set Q consists of the following terms: