### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Take
`/** * Java can do infinite data objects, too. * Here we take the first n elements from an * ascending infinite list of integer numbers. * * @author Carsten Fuhs */public class Take {    public static int[] take(int n, MyIterator f) {        int[] result = new int[n];        for (int i = 0; i < n; ++i) {            if (f.hasNext()) {                result[i] = f.next();            }            else {                break;            }        }        return result;    }    public static void main(String args[]) {        int start = args[0].length();        int howMany = args[1].length();        From f = new From(start);        int[] firstHowMany = take(howMany, f);    }}interface MyIterator {    boolean hasNext();    int next();}class From implements MyIterator {    private int current;    public From(int start) {        this.current = start;    }    public boolean hasNext() {        return true;    }    public int next() {        return current++;    }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 246 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

### (4) Obligation:

ITRS problem:

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

The TRS R consists of the following rules:
Load1879(i392, java.lang.Object(From(i393)), i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Cond_Load1879(i394 >= 0 && i394 < i392 && i394 + 1 > 0, i392, java.lang.Object(From(i393)), i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394)
Cond_Load1879(TRUE, i392, java.lang.Object(From(i393)), i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Load1879(i392, java.lang.Object(From(i393 + 1)), i392, java.lang.Object(From(i393 + 1)), java.lang.Object(ARRAY(i392, a1673dataNew)), i394 + 1)
The set Q consists of the following terms:
Load1879(x0, java.lang.Object(From(x1)), x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

### (5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
We removed arguments according to the following replacements:

Cond_Load1879(x1, x2, x3, x4, x5, x6, x7) → Cond_Load1879(x1, x4, x5, x6, x7)

### (6) 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:
Load1879(i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Cond_Load1879(i394 >= 0 && i394 < i392 && i394 + 1 > 0, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394)
Cond_Load1879(TRUE, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Load1879(i392, java.lang.Object(From(i393 + 1)), java.lang.Object(ARRAY(i392, a1673dataNew)), i394 + 1)
The set Q consists of the following terms:
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

### (8) Obligation:

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

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
Load1879(i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Cond_Load1879(i394 >= 0 && i394 < i392 && i394 + 1 > 0, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394)
Cond_Load1879(TRUE, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → Load1879(i392, java.lang.Object(From(i393 + 1)), java.lang.Object(ARRAY(i392, a1673dataNew)), i394 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0, i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])
(1): COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(i393[1] + 1)), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), i394[1] + 1)

(0) -> (1), if ((i394[0]* i394[1])∧(i392[0]* i392[1])∧(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0* TRUE)∧(java.lang.Object(From(i393[0])) →* java.lang.Object(From(i393[1])))∧(java.lang.Object(ARRAY(i392[0], a1673data[0])) →* java.lang.Object(ARRAY(i392[1], a1673data[1]))))

(1) -> (0), if ((java.lang.Object(ARRAY(i392[1], a1673dataNew[1])) →* java.lang.Object(ARRAY(i392[0], a1673data[0])))∧(i392[1]* i392[0])∧(i394[1] + 1* i394[0])∧(java.lang.Object(From(i393[1] + 1)) →* java.lang.Object(From(i393[0]))))

The set Q consists of the following terms:
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

### (9) 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.

### (10) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0, i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])
(1): COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(i393[1] + 1)), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), i394[1] + 1)

(0) -> (1), if ((i394[0]* i394[1])∧(i392[0]* i392[1])∧(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0* TRUE)∧(java.lang.Object(From(i393[0])) →* java.lang.Object(From(i393[1])))∧(java.lang.Object(ARRAY(i392[0], a1673data[0])) →* java.lang.Object(ARRAY(i392[1], a1673data[1]))))

(1) -> (0), if ((java.lang.Object(ARRAY(i392[1], a1673dataNew[1])) →* java.lang.Object(ARRAY(i392[0], a1673data[0])))∧(i392[1]* i392[0])∧(i394[1] + 1* i394[0])∧(java.lang.Object(From(i393[1] + 1)) →* java.lang.Object(From(i393[0]))))

The set Q consists of the following terms:
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

### (11) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

### (12) 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): LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0, i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])
(1): COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(i393[1] + 1)), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), i394[1] + 1)

(0) -> (1), if ((i394[0]* i394[1])∧(i392[0]* i392[1])∧(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0* TRUE)∧((i393[0]* i393[1]))∧((i392[0]* i392[1])∧(a1673data[0]* a1673data[1])))

(1) -> (0), if (((i392[1]* i392[0])∧(a1673dataNew[1]* a1673data[0]))∧(i392[1]* i392[0])∧(i394[1] + 1* i394[0])∧((i393[1] + 1* i393[0])))

The set Q consists of the following terms:
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

### (13) 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 LOAD1879(i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → COND_LOAD1879(&&(&&(>=(i394, 0), <(i394, i392)), >(+(i394, 1), 0)), i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) the following chains were created:
• We consider the chain LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]), COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1)) which results in the following constraint:

(1)    (i394[0]=i394[1]i392[0]=i392[1]&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0))=TRUEi393[0]=i393[1]a1673data[0]=a1673data[1]LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])≥NonInfC∧LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])≥COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])∧(UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥))

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

(2)    (>(+(i394[0], 1), 0)=TRUE>=(i394[0], 0)=TRUE<(i394[0], i392[0])=TRUELOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])≥NonInfC∧LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])≥COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])∧(UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥))

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

(3)    (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] + [-1] + [-1]i394[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i394[0] + [bni_16]i392[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(4)    (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] + [-1] + [-1]i394[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i394[0] + [bni_16]i392[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(5)    (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] + [-1] + [-1]i394[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i394[0] + [bni_16]i392[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(6)    (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] + [-1] + [-1]i394[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_16] + [(-1)bni_16]i394[0] + [bni_16]i392[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

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

(7)    (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_16 + bni_16] + [bni_16]i392[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

For Pair COND_LOAD1879(TRUE, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → LOAD1879(i392, java.lang.Object(From(+(i393, 1))), java.lang.Object(ARRAY(i392, a1673dataNew)), +(i394, 1)) the following chains were created:
• We consider the chain COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1)) which results in the following constraint:

(8)    (COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1])≥NonInfC∧COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1])≥LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))∧(UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥))

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

(9)    ((UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(10)    ((UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(11)    ((UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(12)    ((UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD1879(i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → COND_LOAD1879(&&(&&(>=(i394, 0), <(i394, i392)), >(+(i394, 1), 0)), i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394)
• (i394[0] ≥ 0∧i394[0] ≥ 0∧i392[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_16 + bni_16] + [bni_16]i392[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

• COND_LOAD1879(TRUE, i392, java.lang.Object(From(i393)), java.lang.Object(ARRAY(i392, a1673data)), i394) → LOAD1879(i392, java.lang.Object(From(+(i393, 1))), java.lang.Object(ARRAY(i392, a1673dataNew)), +(i394, 1))
• ((UIncreasing(LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(LOAD1879(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + x1
POL(java.lang.Object(x1)) = x1
POL(From(x1)) = x1
POL(ARRAY(x1, x2)) = [-1]
POL(COND_LOAD1879(x1, x2, x3, x4, x5)) = [-1] + [-1]x5 + [-1]x4 + x2
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]

The following pairs are in P>:

COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(+(i393[1], 1))), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), +(i394[1], 1))

The following pairs are in Pbound:

LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])

The following pairs are in P:

LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(&&(&&(>=(i394[0], 0), <(i394[0], i392[0])), >(+(i394[0], 1), 0)), i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])

There are no usable rules.

### (15) 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): LOAD1879(i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0]) → COND_LOAD1879(i394[0] >= 0 && i394[0] < i392[0] && i394[0] + 1 > 0, i392[0], java.lang.Object(From(i393[0])), java.lang.Object(ARRAY(i392[0], a1673data[0])), i394[0])

The set Q consists of the following terms:
Cond_Load1879(TRUE, x0, java.lang.Object(From(x1)), java.lang.Object(ARRAY(x0, x2)), x3)

### (16) IDependencyGraphProof (EQUIVALENT transformation)

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

### (18) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD1879(TRUE, i392[1], java.lang.Object(From(i393[1])), java.lang.Object(ARRAY(i392[1], a1673data[1])), i394[1]) → LOAD1879(i392[1], java.lang.Object(From(i393[1] + 1)), java.lang.Object(ARRAY(i392[1], a1673dataNew[1])), i394[1] + 1)

The set Q consists of the following terms: