### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Int
`public class Int {  // only wrap a primitive int  private int val;  // count up to the value  // in "limit"  public static void count(      Int orig, Int limit) {    if (orig == null        || limit == null) {      return;    }    // introduce sharing    Int copy = orig;    while (orig.val < limit.val) {      copy.val++;    }  }  public static void main(String[] args) {    Random.args = args;    Int x = new Int();    x.val = Random.random();    Int y = new Int();    y.val = Random.random();    count(x, y);  }}public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 216 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:
Load529(java.lang.Object(Int(i9)), java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Cond_Load529(i9 < i15, java.lang.Object(Int(i9)), java.lang.Object(Int(i15)), java.lang.Object(Int(i9)))
Cond_Load529(TRUE, java.lang.Object(Int(i9)), java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Load529(java.lang.Object(Int(i9 + 1)), java.lang.Object(Int(i15)), java.lang.Object(Int(i9 + 1)))
The set Q consists of the following terms:
Cond_Load529(TRUE, java.lang.Object(Int(x0)), java.lang.Object(Int(x1)), java.lang.Object(Int(x0)))

### (5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

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

Cond_Load529(x1, x2, x3, x4) → Cond_Load529(x1, x3, x4)

### (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:
Load529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Cond_Load529(i9 < i15, java.lang.Object(Int(i15)), java.lang.Object(Int(i9)))
Cond_Load529(TRUE, java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Load529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9 + 1)))
The set Q consists of the following terms:

### (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:

Integer

The ITRS R consists of the following rules:
Load529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Cond_Load529(i9 < i15, java.lang.Object(Int(i15)), java.lang.Object(Int(i9)))
Cond_Load529(TRUE, java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Load529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9 + 1)))

The integer pair graph contains the following rules and edges:
(0): LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(i9[0] < i15[0], java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))
(1): COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1] + 1)))

(0) -> (1), if ((java.lang.Object(Int(i15[0])) →* java.lang.Object(Int(i15[1])))∧(i9[0] < i15[0]* TRUE)∧(java.lang.Object(Int(i9[0])) →* java.lang.Object(Int(i9[1]))))

(1) -> (0), if ((java.lang.Object(Int(i15[1])) →* java.lang.Object(Int(i15[0])))∧(java.lang.Object(Int(i9[1] + 1)) →* java.lang.Object(Int(i9[0]))))

The set Q consists of the following terms:

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(i9[0] < i15[0], java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))
(1): COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1] + 1)))

(0) -> (1), if ((java.lang.Object(Int(i15[0])) →* java.lang.Object(Int(i15[1])))∧(i9[0] < i15[0]* TRUE)∧(java.lang.Object(Int(i9[0])) →* java.lang.Object(Int(i9[1]))))

(1) -> (0), if ((java.lang.Object(Int(i15[1])) →* java.lang.Object(Int(i15[0])))∧(java.lang.Object(Int(i9[1] + 1)) →* java.lang.Object(Int(i9[0]))))

The set Q consists of the following terms:

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(i9[0] < i15[0], java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))
(1): COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1] + 1)))

(0) -> (1), if (((i15[0]* i15[1]))∧(i9[0] < i15[0]* TRUE)∧((i9[0]* i9[1])))

(1) -> (0), if (((i15[1]* i15[0]))∧((i9[1] + 1* i9[0])))

The set Q consists of the following terms:

### (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 LOAD529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → COND_LOAD529(<(i9, i15), java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) the following chains were created:
• We consider the chain LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))), COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1)))) which results in the following constraint:

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

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

(3)    (i15[0] + [-1] + [-1]i9[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]i9[0] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(4)    (i15[0] + [-1] + [-1]i9[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]i9[0] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(5)    (i15[0] + [-1] + [-1]i9[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]i9[0] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(6)    (i15[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)Bound*bni_8] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(7)    (i15[0] ≥ 0∧i9[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)Bound*bni_8] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

(8)    (i15[0] ≥ 0∧i9[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)Bound*bni_8] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_LOAD529(TRUE, java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → LOAD529(java.lang.Object(Int(i15)), java.lang.Object(Int(+(i9, 1)))) the following chains were created:
• We consider the chain COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1)))) which results in the following constraint:

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

(10)    ((UIncreasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥)∧[1 + (-1)bso_11] ≥ 0)

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

(11)    ((UIncreasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥)∧[1 + (-1)bso_11] ≥ 0)

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

(12)    ((UIncreasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥)∧[1 + (-1)bso_11] ≥ 0)

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

(13)    ((UIncreasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → COND_LOAD529(<(i9, i15), java.lang.Object(Int(i15)), java.lang.Object(Int(i9)))
• (i15[0] ≥ 0∧i9[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)Bound*bni_8] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)
• (i15[0] ≥ 0∧i9[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)Bound*bni_8] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

• COND_LOAD529(TRUE, java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → LOAD529(java.lang.Object(Int(i15)), java.lang.Object(Int(+(i9, 1))))
• ((UIncreasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥)∧0 = 0∧0 = 0∧[1 + (-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(LOAD529(x1, x2)) = [-1] + [-1]x2 + x1
POL(java.lang.Object(x1)) = x1
POL(Int(x1)) = x1
POL(COND_LOAD529(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]

The following pairs are in P>:

COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))

The following pairs are in Pbound:

LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))

The following pairs are in P:

LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(i9[0] < i15[0], java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))

The set Q consists of the following terms:

### (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_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1] + 1)))

The set Q consists of the following terms: