### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: GCD4
`public class GCD4 {  public static int mod(int a, int b) {    while(a>=b && b > 0) {      a -= b;    }    return a;  }  public static int gcd(int a, int b) {    int tmp;    while(b > 0 && a > 0) {      tmp = b;      b = mod(a, b);      a = tmp;    }    return a;  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    gcd(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 218 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:
Load1303(i427, i421) → Cond_Load1303(i427 > 0 && i421 > 0, i427, i421)
Load1438(i421, i421, i421, i507, i421) → Cond_Load1438(i421 > 0 && i507 >= i421, i421, i421, i421, i507, i421)
Cond_Load1438(TRUE, i421, i421, i421, i507, i421) → Load1438(i421, i421, i421, i507 - i421, i421)
Load1438(i421, i421, i421, i507, i421) → Cond_Load14381(i507 < i421, i421, i421, i421, i507, i421)
The set Q consists of the following terms:
Cond_Load1438(TRUE, x0, x0, x0, x1, x0)
Cond_Load14381(TRUE, x0, x0, x0, x1, x0)

### (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:
Load1303(i427, i421) → Cond_Load1303(i427 > 0 && i421 > 0, i427, i421)
Load1438(i507, i421) → Cond_Load1438(i421 > 0 && i507 >= i421, i507, i421)
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:
Load1303(i427, i421) → Cond_Load1303(i427 > 0 && i421 > 0, i427, i421)
Load1438(i507, i421) → Cond_Load1438(i421 > 0 && i507 >= i421, i507, i421)

The integer pair graph contains the following rules and edges:
(0): LOAD1303(i427[0], i421[0]) → COND_LOAD1303(i427[0] > 0 && i421[0] > 0, i427[0], i421[0])
(2): LOAD1438(i507[2], i421[2]) → COND_LOAD1438(i421[2] > 0 && i507[2] >= i421[2], i507[2], i421[2])

(0) -> (1), if ((i427[0] > 0 && i421[0] > 0* TRUE)∧(i421[0]* i421[1])∧(i427[0]* i427[1]))

(1) -> (2), if ((i427[1]* i507[2])∧(i421[1]* i421[2]))

(1) -> (4), if ((i421[1]* i421[4])∧(i427[1]* i507[4]))

(2) -> (3), if ((i507[2]* i507[3])∧(i421[2] > 0 && i507[2] >= i421[2]* TRUE)∧(i421[2]* i421[3]))

(3) -> (2), if ((i421[3]* i421[2])∧(i507[3] - i421[3]* i507[2]))

(3) -> (4), if ((i421[3]* i421[4])∧(i507[3] - i421[3]* i507[4]))

(4) -> (5), if ((i421[4]* i421[5])∧(i507[4]* i507[5])∧(i507[4] < i421[4]* TRUE))

(5) -> (0), if ((i507[5]* i421[0])∧(i421[5]* i427[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): LOAD1303(i427[0], i421[0]) → COND_LOAD1303(i427[0] > 0 && i421[0] > 0, i427[0], i421[0])
(2): LOAD1438(i507[2], i421[2]) → COND_LOAD1438(i421[2] > 0 && i507[2] >= i421[2], i507[2], i421[2])

(0) -> (1), if ((i427[0] > 0 && i421[0] > 0* TRUE)∧(i421[0]* i421[1])∧(i427[0]* i427[1]))

(1) -> (2), if ((i427[1]* i507[2])∧(i421[1]* i421[2]))

(1) -> (4), if ((i421[1]* i421[4])∧(i427[1]* i507[4]))

(2) -> (3), if ((i507[2]* i507[3])∧(i421[2] > 0 && i507[2] >= i421[2]* TRUE)∧(i421[2]* i421[3]))

(3) -> (2), if ((i421[3]* i421[2])∧(i507[3] - i421[3]* i507[2]))

(3) -> (4), if ((i421[3]* i421[4])∧(i507[3] - i421[3]* i507[4]))

(4) -> (5), if ((i421[4]* i421[5])∧(i507[4]* i507[5])∧(i507[4] < i421[4]* TRUE))

(5) -> (0), if ((i507[5]* i421[0])∧(i421[5]* i427[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 LOAD1303(i427, i421) → COND_LOAD1303(&&(>(i427, 0), >(i421, 0)), i427, i421) the following chains were created:
• We consider the chain LOAD1303(i427[0], i421[0]) → COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0]), COND_LOAD1303(TRUE, i427[1], i421[1]) → LOAD1438(i427[1], i421[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)    (i427[0] + [-1] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i421[0] + [bni_25]i427[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(4)    (i427[0] + [-1] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i421[0] + [bni_25]i427[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(5)    (i427[0] + [-1] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i421[0] + [bni_25]i427[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(6)    (i427[0] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)Bound*bni_25] + [bni_25]i421[0] + [bni_25]i427[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(7)    (i427[0] ≥ 0∧i421[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i421[0] + [bni_25]i427[0] ≥ 0∧[(-1)bso_26] ≥ 0)

For Pair COND_LOAD1303(TRUE, i427, i421) → LOAD1438(i427, i421) the following chains were created:
• We consider the chain COND_LOAD1303(TRUE, i427[1], i421[1]) → LOAD1438(i427[1], i421[1]), LOAD1438(i507[2], i421[2]) → COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2]) which results in the following constraint:

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

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

(10)    ((UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)bso_28] ≥ 0)

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

(11)    ((UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)bso_28] ≥ 0)

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

(12)    ((UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)bso_28] ≥ 0)

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

(13)    ((UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

• We consider the chain COND_LOAD1303(TRUE, i427[1], i421[1]) → LOAD1438(i427[1], i421[1]), LOAD1438(i507[4], i421[4]) → COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4]) which results in the following constraint:

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

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

(16)    ((UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)bso_28] ≥ 0)

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

(17)    ((UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)bso_28] ≥ 0)

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

(18)    ((UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)bso_28] ≥ 0)

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

(19)    ((UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

For Pair LOAD1438(i507, i421) → COND_LOAD1438(&&(>(i421, 0), >=(i507, i421)), i507, i421) the following chains were created:
• We consider the chain LOAD1438(i507[2], i421[2]) → COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2]), COND_LOAD1438(TRUE, i507[3], i421[3]) → LOAD1438(-(i507[3], i421[3]), i421[3]) which results in the following constraint:

We simplified constraint (20) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(22)    (i421[2] + [-1] ≥ 0∧i507[2] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2])), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i421[2] + [bni_29]i507[2] ≥ 0∧[-1 + (-1)bso_30] + i421[2] ≥ 0)

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

(23)    (i421[2] + [-1] ≥ 0∧i507[2] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2])), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i421[2] + [bni_29]i507[2] ≥ 0∧[-1 + (-1)bso_30] + i421[2] ≥ 0)

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

(24)    (i421[2] + [-1] ≥ 0∧i507[2] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2])), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i421[2] + [bni_29]i507[2] ≥ 0∧[-1 + (-1)bso_30] + i421[2] ≥ 0)

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

(25)    (i421[2] ≥ 0∧i507[2] + [-1] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2])), ≥)∧[(-1)Bound*bni_29] + [bni_29]i421[2] + [bni_29]i507[2] ≥ 0∧[(-1)bso_30] + i421[2] ≥ 0)

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

(26)    (i421[2] ≥ 0∧i507[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2])), ≥)∧[(-1)Bound*bni_29 + bni_29] + [(2)bni_29]i421[2] + [bni_29]i507[2] ≥ 0∧[(-1)bso_30] + i421[2] ≥ 0)

For Pair COND_LOAD1438(TRUE, i507, i421) → LOAD1438(-(i507, i421), i421) the following chains were created:
• We consider the chain LOAD1438(i507[2], i421[2]) → COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2]), COND_LOAD1438(TRUE, i507[3], i421[3]) → LOAD1438(-(i507[3], i421[3]), i421[3]), LOAD1438(i507[2], i421[2]) → COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2]) which results in the following constraint:

We simplified constraint (27) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(29)    (i421[2] + [-1] ≥ 0∧i507[2] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

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

(30)    (i421[2] + [-1] ≥ 0∧i507[2] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

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

(31)    (i421[2] + [-1] ≥ 0∧i507[2] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(32)    (i421[2] ≥ 0∧i507[2] + [-1] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

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

(33)    (i421[2] ≥ 0∧i507[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [bni_31]i421[2] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

• We consider the chain LOAD1438(i507[2], i421[2]) → COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2]), COND_LOAD1438(TRUE, i507[3], i421[3]) → LOAD1438(-(i507[3], i421[3]), i421[3]), LOAD1438(i507[4], i421[4]) → COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4]) which results in the following constraint:

We simplified constraint (34) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(36)    (i421[2] + [-1] ≥ 0∧i507[2] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(37)    (i421[2] + [-1] ≥ 0∧i507[2] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(38)    (i421[2] + [-1] ≥ 0∧i507[2] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

We simplified constraint (38) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(39)    (i421[2] ≥ 0∧i507[2] + [-1] + [-1]i421[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(40)    (i421[2] ≥ 0∧i507[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [bni_31]i421[2] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

For Pair LOAD1438(i507, i421) → COND_LOAD14381(<(i507, i421), i507, i421) the following chains were created:
• We consider the chain LOAD1438(i507[4], i421[4]) → COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4]), COND_LOAD14381(TRUE, i507[5], i421[5]) → LOAD1303(i421[5], i507[5]) which results in the following constraint:

We simplified constraint (41) using rule (IV) which results in the following new constraint:

We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(43)    (i421[4] + [-1] + [-1]i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i421[4] + [bni_33]i507[4] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (43) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(44)    (i421[4] + [-1] + [-1]i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i421[4] + [bni_33]i507[4] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (44) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(45)    (i421[4] + [-1] + [-1]i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i421[4] + [bni_33]i507[4] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (45) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(46)    (i421[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)Bound*bni_33] + [(2)bni_33]i507[4] + [bni_33]i421[4] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(47)    (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)Bound*bni_33] + [(-2)bni_33]i507[4] + [bni_33]i421[4] ≥ 0∧[(-1)bso_34] ≥ 0)

(48)    (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)Bound*bni_33] + [(2)bni_33]i507[4] + [bni_33]i421[4] ≥ 0∧[(-1)bso_34] ≥ 0)

For Pair COND_LOAD14381(TRUE, i507, i421) → LOAD1303(i421, i507) the following chains were created:
• We consider the chain COND_LOAD14381(TRUE, i507[5], i421[5]) → LOAD1303(i421[5], i507[5]), LOAD1303(i427[0], i421[0]) → COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0]) which results in the following constraint:

We simplified constraint (49) using rule (IV) which results in the following new constraint:

We simplified constraint (50) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(51)    ((UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)bso_36] ≥ 0)

We simplified constraint (51) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(52)    ((UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)bso_36] ≥ 0)

We simplified constraint (52) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(53)    ((UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)bso_36] ≥ 0)

We simplified constraint (53) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(54)    ((UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i427[0] ≥ 0∧i421[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i421[0] + [bni_25]i427[0] ≥ 0∧[(-1)bso_26] ≥ 0)

• ((UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)
• ((UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

• (i421[2] ≥ 0∧i507[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1438(&&(>(i421[2], 0), >=(i507[2], i421[2])), i507[2], i421[2])), ≥)∧[(-1)Bound*bni_29 + bni_29] + [(2)bni_29]i421[2] + [bni_29]i507[2] ≥ 0∧[(-1)bso_30] + i421[2] ≥ 0)

• (i421[2] ≥ 0∧i507[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [bni_31]i421[2] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)
• (i421[2] ≥ 0∧i507[2] ≥ 0 ⇒ (UIncreasing(LOAD1438(-(i507[3], i421[3]), i421[3])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [bni_31]i421[2] + [bni_31]i507[2] ≥ 0∧[1 + (-1)bso_32] ≥ 0)

• (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)Bound*bni_33] + [(-2)bni_33]i507[4] + [bni_33]i421[4] ≥ 0∧[(-1)bso_34] ≥ 0)
• (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)Bound*bni_33] + [(2)bni_33]i507[4] + [bni_33]i421[4] ≥ 0∧[(-1)bso_34] ≥ 0)

• ((UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 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) = [1]
POL(LOAD1303(x1, x2)) = [-1] + x2 + x1
POL(COND_LOAD1303(x1, x2, x3)) = [-1] + x3 + x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(LOAD1438(x1, x2)) = [-1] + x2 + x1
POL(>=(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(COND_LOAD14381(x1, x2, x3)) = [-1] + x3 + x2
POL(<(x1, x2)) = [-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:

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

### (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): LOAD1303(i427[0], i421[0]) → COND_LOAD1303(i427[0] > 0 && i421[0] > 0, i427[0], i421[0])
(2): LOAD1438(i507[2], i421[2]) → COND_LOAD1438(i421[2] > 0 && i507[2] >= i421[2], i507[2], i421[2])

(5) -> (0), if ((i507[5]* i421[0])∧(i421[5]* i427[0]))

(0) -> (1), if ((i427[0] > 0 && i421[0] > 0* TRUE)∧(i421[0]* i421[1])∧(i427[0]* i427[1]))

(1) -> (2), if ((i427[1]* i507[2])∧(i421[1]* i421[2]))

(1) -> (4), if ((i421[1]* i421[4])∧(i427[1]* i507[4]))

(4) -> (5), if ((i421[4]* i421[5])∧(i507[4]* i507[5])∧(i507[4] < i421[4]* TRUE))

The set Q consists of the following terms:

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1303(i427[0], i421[0]) → COND_LOAD1303(i427[0] > 0 && i421[0] > 0, i427[0], i421[0])

(5) -> (0), if ((i507[5]* i421[0])∧(i421[5]* i427[0]))

(0) -> (1), if ((i427[0] > 0 && i421[0] > 0* TRUE)∧(i421[0]* i421[1])∧(i427[0]* i427[1]))

(1) -> (4), if ((i421[1]* i421[4])∧(i427[1]* i507[4]))

(4) -> (5), if ((i421[4]* i421[5])∧(i507[4]* i507[5])∧(i507[4] < i421[4]* TRUE))

The set Q consists of the following terms:

### (16) 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_LOAD14381(TRUE, i507[5], i421[5]) → LOAD1303(i421[5], i507[5]) the following chains were created:

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

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

(3)    (i421[4] + [-1] + [-1]i507[4] ≥ 0 ⇒ (UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]i421[4] + [(-1)bni_20]i507[4] ≥ 0∧[(-1)bso_21] + [2]i421[4] + [-2]i507[4] ≥ 0)

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

(4)    (i421[4] + [-1] + [-1]i507[4] ≥ 0 ⇒ (UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]i421[4] + [(-1)bni_20]i507[4] ≥ 0∧[(-1)bso_21] + [2]i421[4] + [-2]i507[4] ≥ 0)

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

(5)    (i421[4] + [-1] + [-1]i507[4] ≥ 0 ⇒ (UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]i421[4] + [(-1)bni_20]i507[4] ≥ 0∧[(-1)bso_21] + [2]i421[4] + [-2]i507[4] ≥ 0)

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

(6)    (i421[4] ≥ 0 ⇒ (UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i421[4] ≥ 0∧[2 + (-1)bso_21] + [2]i421[4] ≥ 0)

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

(7)    (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i421[4] ≥ 0∧[2 + (-1)bso_21] + [2]i421[4] ≥ 0)

(8)    (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i421[4] ≥ 0∧[2 + (-1)bso_21] + [2]i421[4] ≥ 0)

For Pair LOAD1438(i507[4], i421[4]) → COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4]) the following chains were created:

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

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

(11)    (i421[4] + [-1] + [-1]i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i421[4] + [(-1)bni_22]i507[4] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(12)    (i421[4] + [-1] + [-1]i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i421[4] + [(-1)bni_22]i507[4] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(13)    (i421[4] + [-1] + [-1]i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i421[4] + [(-1)bni_22]i507[4] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(14)    (i421[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)Bound*bni_22] + [bni_22]i421[4] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(15)    (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)Bound*bni_22] + [bni_22]i421[4] ≥ 0∧[(-1)bso_23] ≥ 0)

(16)    (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)Bound*bni_22] + [bni_22]i421[4] ≥ 0∧[(-1)bso_23] ≥ 0)

For Pair COND_LOAD1303(TRUE, i427[1], i421[1]) → LOAD1438(i427[1], i421[1]) the following chains were created:

We simplified constraint (17) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(19)    (i427[0] + [-1] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i421[0] + [(-1)bni_24]i427[0] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(20)    (i427[0] + [-1] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i421[0] + [(-1)bni_24]i427[0] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(21)    (i427[0] + [-1] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i421[0] + [(-1)bni_24]i427[0] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(22)    (i427[0] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [bni_24]i421[0] + [(-1)bni_24]i427[0] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(23)    (i427[0] ≥ 0∧i421[0] ≥ 0 ⇒ (UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)Bound*bni_24 + (-1)bni_24] + [bni_24]i421[0] + [(-1)bni_24]i427[0] ≥ 0∧[(-1)bso_25] ≥ 0)

For Pair LOAD1303(i427[0], i421[0]) → COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0]) the following chains were created:

We simplified constraint (24) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(26)    (i427[0] + [-1] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i421[0] + [(-1)bni_26]i427[0] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (26) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(27)    (i427[0] + [-1] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i421[0] + [(-1)bni_26]i427[0] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(28)    (i427[0] + [-1] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i421[0] + [(-1)bni_26]i427[0] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(29)    (i427[0] ≥ 0∧i421[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-2)bni_26 + (-1)Bound*bni_26] + [bni_26]i421[0] + [(-1)bni_26]i427[0] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(30)    (i427[0] ≥ 0∧i421[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)Bound*bni_26 + (-1)bni_26] + [bni_26]i421[0] + [(-1)bni_26]i427[0] ≥ 0∧[(-1)bso_27] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i421[4] ≥ 0∧[2 + (-1)bso_21] + [2]i421[4] ≥ 0)
• (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(LOAD1303(i421[5], i507[5])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i421[4] ≥ 0∧[2 + (-1)bso_21] + [2]i421[4] ≥ 0)

• (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)Bound*bni_22] + [bni_22]i421[4] ≥ 0∧[(-1)bso_23] ≥ 0)
• (i421[4] ≥ 0∧i507[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD14381(<(i507[4], i421[4]), i507[4], i421[4])), ≥)∧[(-1)Bound*bni_22] + [bni_22]i421[4] ≥ 0∧[(-1)bso_23] ≥ 0)

• (i427[0] ≥ 0∧i421[0] ≥ 0 ⇒ (UIncreasing(LOAD1438(i427[1], i421[1])), ≥)∧[(-1)Bound*bni_24 + (-1)bni_24] + [bni_24]i421[0] + [(-1)bni_24]i427[0] ≥ 0∧[(-1)bso_25] ≥ 0)

• (i427[0] ≥ 0∧i421[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1303(&&(>(i427[0], 0), >(i421[0], 0)), i427[0], i421[0])), ≥)∧[(-1)Bound*bni_26 + (-1)bni_26] + [bni_26]i421[0] + [(-1)bni_26]i427[0] ≥ 0∧[(-1)bso_27] ≥ 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) = [2]
POL(FALSE) = [3]
POL(COND_LOAD14381(x1, x2, x3)) = [-1] + x3 + [-1]x2
POL(LOAD1303(x1, x2)) = [-1] + x2 + [-1]x1
POL(LOAD1438(x1, x2)) = [-1] + x2 + [-1]x1
POL(<(x1, x2)) = [-1]
POL(COND_LOAD1303(x1, x2, x3)) = [-1] + x3 + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

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:

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

### (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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1303(i427[0], i421[0]) → COND_LOAD1303(i427[0] > 0 && i421[0] > 0, i427[0], i421[0])

(0) -> (1), if ((i427[0] > 0 && i421[0] > 0* TRUE)∧(i421[0]* i421[1])∧(i427[0]* i427[1]))

(1) -> (4), if ((i421[1]* i421[4])∧(i427[1]* i507[4]))

The set Q consists of the following terms:

### (19) IDependencyGraphProof (EQUIVALENT transformation)

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

### (21) 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): LOAD1303(i427[0], i421[0]) → COND_LOAD1303(i427[0] > 0 && i421[0] > 0, i427[0], i421[0])

(0) -> (1), if ((i427[0] > 0 && i421[0] > 0* TRUE)∧(i421[0]* i421[1])∧(i427[0]* i427[1]))

The set Q consists of the following terms:

### (22) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (4), if ((i421[1]* i421[4])∧(i427[1]* i507[4]))

(4) -> (5), if ((i421[4]* i421[5])∧(i507[4]* i507[5])∧(i507[4] < i421[4]* TRUE))

The set Q consists of the following terms: