### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: DivMinus
`public class DivMinus {  public static int div(int x, int y) {    int res = 0;    while (x >= y && y > 0) {      x = x-y;      res = res + 1;    }    return res;  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    div(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 205 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:
Load1155(i135, i120, i135, i122) → Cond_Load1155(i135 > 0 && i120 >= i135, i135, i120, i135, i122)
Cond_Load1155(TRUE, i135, i120, i135, i122) → Load1155(i135, i120 - i135, i135, i122 + 1)
The set Q consists of the following terms:

### (5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

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

### (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:
Load1155(i120, i135, i122) → Cond_Load1155(i135 > 0 && i120 >= i135, i120, i135, i122)
Cond_Load1155(TRUE, i120, i135, i122) → Load1155(i120 - i135, i135, i122 + 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:

Boolean, Integer

The ITRS R consists of the following rules:
Load1155(i120, i135, i122) → Cond_Load1155(i135 > 0 && i120 >= i135, i120, i135, i122)
Cond_Load1155(TRUE, i120, i135, i122) → Load1155(i120 - i135, i135, i122 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD1155(i120[0], i135[0], i122[0]) → COND_LOAD1155(i135[0] > 0 && i120[0] >= i135[0], i120[0], i135[0], i122[0])
(1): COND_LOAD1155(TRUE, i120[1], i135[1], i122[1]) → LOAD1155(i120[1] - i135[1], i135[1], i122[1] + 1)

(0) -> (1), if ((i122[0]* i122[1])∧(i120[0]* i120[1])∧(i135[0]* i135[1])∧(i135[0] > 0 && i120[0] >= i135[0]* TRUE))

(1) -> (0), if ((i135[1]* i135[0])∧(i122[1] + 1* i122[0])∧(i120[1] - i135[1]* i120[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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1155(i120[0], i135[0], i122[0]) → COND_LOAD1155(i135[0] > 0 && i120[0] >= i135[0], i120[0], i135[0], i122[0])
(1): COND_LOAD1155(TRUE, i120[1], i135[1], i122[1]) → LOAD1155(i120[1] - i135[1], i135[1], i122[1] + 1)

(0) -> (1), if ((i122[0]* i122[1])∧(i120[0]* i120[1])∧(i135[0]* i135[1])∧(i135[0] > 0 && i120[0] >= i135[0]* TRUE))

(1) -> (0), if ((i135[1]* i135[0])∧(i122[1] + 1* i122[0])∧(i120[1] - i135[1]* i120[0]))

The set Q consists of the following terms:

### (11) IDPNonInfProof (SOUND transformation)

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

For Pair LOAD1155(i120, i135, i122) → COND_LOAD1155(&&(>(i135, 0), >=(i120, i135)), i120, i135, i122) the following chains were created:
• We consider the chain LOAD1155(i120[0], i135[0], i122[0]) → COND_LOAD1155(&&(>(i135[0], 0), >=(i120[0], i135[0])), i120[0], i135[0], i122[0]), COND_LOAD1155(TRUE, i120[1], i135[1], i122[1]) → LOAD1155(-(i120[1], i135[1]), i135[1], +(i122[1], 1)) which results in the following constraint:

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

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

(3)    (i135[0] + [-1] ≥ 0∧i120[0] + [-1]i135[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1155(&&(>(i135[0], 0), >=(i120[0], i135[0])), i120[0], i135[0], i122[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i135[0] + [bni_15]i120[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(4)    (i135[0] + [-1] ≥ 0∧i120[0] + [-1]i135[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1155(&&(>(i135[0], 0), >=(i120[0], i135[0])), i120[0], i135[0], i122[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i135[0] + [bni_15]i120[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(5)    (i135[0] + [-1] ≥ 0∧i120[0] + [-1]i135[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1155(&&(>(i135[0], 0), >=(i120[0], i135[0])), i120[0], i135[0], i122[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i135[0] + [bni_15]i120[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(6)    (i135[0] + [-1] ≥ 0∧i120[0] + [-1]i135[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1155(&&(>(i135[0], 0), >=(i120[0], i135[0])), i120[0], i135[0], i122[0])), ≥)∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i135[0] + [bni_15]i120[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

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

(7)    (i135[0] ≥ 0∧i120[0] + [-1] + [-1]i135[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1155(&&(>(i135[0], 0), >=(i120[0], i135[0])), i120[0], i135[0], i122[0])), ≥)∧0 = 0∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i135[0] + [bni_15]i120[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

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

(8)    (i135[0] ≥ 0∧i120[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1155(&&(>(i135[0], 0), >=(i120[0], i135[0])), i120[0], i135[0], i122[0])), ≥)∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i120[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

For Pair COND_LOAD1155(TRUE, i120, i135, i122) → LOAD1155(-(i120, i135), i135, +(i122, 1)) the following chains were created:
• We consider the chain LOAD1155(i120[0], i135[0], i122[0]) → COND_LOAD1155(&&(>(i135[0], 0), >=(i120[0], i135[0])), i120[0], i135[0], i122[0]), COND_LOAD1155(TRUE, i120[1], i135[1], i122[1]) → LOAD1155(-(i120[1], i135[1]), i135[1], +(i122[1], 1)), LOAD1155(i120[0], i135[0], i122[0]) → COND_LOAD1155(&&(>(i135[0], 0), >=(i120[0], i135[0])), i120[0], i135[0], i122[0]) which results in the following constraint:

We simplified constraint (9) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(11)    (i135[0] + [-1] ≥ 0∧i120[0] + [-1]i135[0] ≥ 0 ⇒ (UIncreasing(LOAD1155(-(i120[1], i135[1]), i135[1], +(i122[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i135[0] + [bni_17]i120[0] ≥ 0∧[(-1)bso_18] + i135[0] ≥ 0)

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

(12)    (i135[0] + [-1] ≥ 0∧i120[0] + [-1]i135[0] ≥ 0 ⇒ (UIncreasing(LOAD1155(-(i120[1], i135[1]), i135[1], +(i122[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i135[0] + [bni_17]i120[0] ≥ 0∧[(-1)bso_18] + i135[0] ≥ 0)

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

(13)    (i135[0] + [-1] ≥ 0∧i120[0] + [-1]i135[0] ≥ 0 ⇒ (UIncreasing(LOAD1155(-(i120[1], i135[1]), i135[1], +(i122[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i135[0] + [bni_17]i120[0] ≥ 0∧[(-1)bso_18] + i135[0] ≥ 0)

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

(14)    (i135[0] + [-1] ≥ 0∧i120[0] + [-1]i135[0] ≥ 0 ⇒ (UIncreasing(LOAD1155(-(i120[1], i135[1]), i135[1], +(i122[1], 1))), ≥)∧0 = 0∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i135[0] + [bni_17]i120[0] ≥ 0∧0 = 0∧[(-1)bso_18] + i135[0] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(15)    (i135[0] ≥ 0∧i120[0] + [-1] + [-1]i135[0] ≥ 0 ⇒ (UIncreasing(LOAD1155(-(i120[1], i135[1]), i135[1], +(i122[1], 1))), ≥)∧0 = 0∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i135[0] + [bni_17]i120[0] ≥ 0∧0 = 0∧[1 + (-1)bso_18] + i135[0] ≥ 0)

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

(16)    (i135[0] ≥ 0∧i120[0] ≥ 0 ⇒ (UIncreasing(LOAD1155(-(i120[1], i135[1]), i135[1], +(i122[1], 1))), ≥)∧0 = 0∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i120[0] ≥ 0∧0 = 0∧[1 + (-1)bso_18] + i135[0] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD1155(i120, i135, i122) → COND_LOAD1155(&&(>(i135, 0), >=(i120, i135)), i120, i135, i122)
• (i135[0] ≥ 0∧i120[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1155(&&(>(i135[0], 0), >=(i120[0], i135[0])), i120[0], i135[0], i122[0])), ≥)∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i120[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

• (i135[0] ≥ 0∧i120[0] ≥ 0 ⇒ (UIncreasing(LOAD1155(-(i120[1], i135[1]), i135[1], +(i122[1], 1))), ≥)∧0 = 0∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i120[0] ≥ 0∧0 = 0∧[1 + (-1)bso_18] + i135[0] ≥ 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(LOAD1155(x1, x2, x3)) = [-1] + [-1]x2 + x1
POL(COND_LOAD1155(x1, x2, x3, x4)) = [-1] + [-1]x3 + x2 + [-1]x1
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(>=(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(TRUE, TRUE)1TRUE1
&&(TRUE, FALSE)1FALSE1
&&(FALSE, TRUE)1FALSE1
&&(FALSE, FALSE)1FALSE1

### (13) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1155(i120[0], i135[0], i122[0]) → COND_LOAD1155(i135[0] > 0 && i120[0] >= i135[0], i120[0], i135[0], i122[0])

The set Q consists of the following terms:

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (16) Obligation:

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

The following domains are used:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: