### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: AProVEMath
`/** * Abstract class to provide some additional mathematical functions * which are not provided by java.lang.Math. * * @author fuhs */public abstract class AProVEMath {  /**   * Returns <code>base<sup>exponent</sup></code>.   * Works considerably faster than java.lang.Math.pow(double, double).   *   * @param base base of the power   * @param exponent non-negative exponent of the power   * @return base<sup>exponent</sup>   */  public static int power (int base, int exponent) {    if (exponent == 0) {      return 1;    }    else if (exponent == 1) {      return base;    }    else if (base == 2) {      return base << (exponent-1);    }    else {      int result = 1;      while (exponent > 0) {        if (exponent % 2 == 1) {          result *= base;        }        base *= base;        exponent /= 2;      }      return result;    }  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    power(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 238 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:
Load1494(i358, i362, i360) → Cond_Load1494(i362 > 0 && 1 = i362 % 2, i358, i362, i360)
Load1494(i358, i362, i360) → Cond_Load14941(i362 % 2 >= 0 && !(i362 % 2 = 1) && i362 > 0, i358, i362, i360)
Cond_Load14941(TRUE, i358, i362, i360) → Load1494(i358 * i358, i362 / 2, i360)
The set Q consists of the following terms:

### (6) 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:
Load1494(i358, i362, i360) → Cond_Load1494(i362 > 0 && 1 = i362 % 2, i358, i362, i360)
Load1494(i358, i362, i360) → Cond_Load14941(i362 % 2 >= 0 && !(i362 % 2 = 1) && i362 > 0, i358, i362, i360)
Cond_Load14941(TRUE, i358, i362, i360) → Load1494(i358 * i358, i362 / 2, i360)

The integer pair graph contains the following rules and edges:
(0): LOAD1494(i358[0], i362[0], i360[0]) → COND_LOAD1494(i362[0] > 0 && 1 = i362[0] % 2, i358[0], i362[0], i360[0])
(2): LOAD1533(i358[2], i362[2], i360[2]) → LOAD1494(i358[2] * i358[2], i362[2] / 2, i360[2])
(3): LOAD1494(i358[3], i362[3], i360[3]) → COND_LOAD14941(i362[3] % 2 >= 0 && !(i362[3] % 2 = 1) && i362[3] > 0, i358[3], i362[3], i360[3])
(4): COND_LOAD14941(TRUE, i358[4], i362[4], i360[4]) → LOAD1494(i358[4] * i358[4], i362[4] / 2, i360[4])

(0) -> (1), if ((i362[0]* i362[1])∧(i362[0] > 0 && 1 = i362[0] % 2* TRUE)∧(i358[0]* i358[1])∧(i360[0]* i360[1]))

(1) -> (2), if ((i360[1] * i358[1]* i360[2])∧(i358[1]* i358[2])∧(i362[1]* i362[2]))

(2) -> (0), if ((i358[2] * i358[2]* i358[0])∧(i362[2] / 2* i362[0])∧(i360[2]* i360[0]))

(2) -> (3), if ((i360[2]* i360[3])∧(i358[2] * i358[2]* i358[3])∧(i362[2] / 2* i362[3]))

(3) -> (4), if ((i358[3]* i358[4])∧(i360[3]* i360[4])∧(i362[3] % 2 >= 0 && !(i362[3] % 2 = 1) && i362[3] > 0* TRUE)∧(i362[3]* i362[4]))

(4) -> (0), if ((i360[4]* i360[0])∧(i362[4] / 2* i362[0])∧(i358[4] * i358[4]* i358[0]))

(4) -> (3), if ((i358[4] * i358[4]* i358[3])∧(i360[4]* i360[3])∧(i362[4] / 2* i362[3]))

The set Q consists of the following terms:

### (7) 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.

### (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

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1494(i358[0], i362[0], i360[0]) → COND_LOAD1494(i362[0] > 0 && 1 = i362[0] % 2, i358[0], i362[0], i360[0])
(2): LOAD1533(i358[2], i362[2], i360[2]) → LOAD1494(i358[2] * i358[2], i362[2] / 2, i360[2])
(3): LOAD1494(i358[3], i362[3], i360[3]) → COND_LOAD14941(i362[3] % 2 >= 0 && !(i362[3] % 2 = 1) && i362[3] > 0, i358[3], i362[3], i360[3])
(4): COND_LOAD14941(TRUE, i358[4], i362[4], i360[4]) → LOAD1494(i358[4] * i358[4], i362[4] / 2, i360[4])

(0) -> (1), if ((i362[0]* i362[1])∧(i362[0] > 0 && 1 = i362[0] % 2* TRUE)∧(i358[0]* i358[1])∧(i360[0]* i360[1]))

(1) -> (2), if ((i360[1] * i358[1]* i360[2])∧(i358[1]* i358[2])∧(i362[1]* i362[2]))

(2) -> (0), if ((i358[2] * i358[2]* i358[0])∧(i362[2] / 2* i362[0])∧(i360[2]* i360[0]))

(2) -> (3), if ((i360[2]* i360[3])∧(i358[2] * i358[2]* i358[3])∧(i362[2] / 2* i362[3]))

(3) -> (4), if ((i358[3]* i358[4])∧(i360[3]* i360[4])∧(i362[3] % 2 >= 0 && !(i362[3] % 2 = 1) && i362[3] > 0* TRUE)∧(i362[3]* i362[4]))

(4) -> (0), if ((i360[4]* i360[0])∧(i362[4] / 2* i362[0])∧(i358[4] * i358[4]* i358[0]))

(4) -> (3), if ((i358[4] * i358[4]* i358[3])∧(i360[4]* i360[3])∧(i362[4] / 2* i362[3]))

The set Q consists of the following terms:

### (9) 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 LOAD1494(i358, i362, i360) → COND_LOAD1494(&&(>(i362, 0), =(1, %(i362, 2))), i358, i362, i360) the following chains were created:
• We consider the chain LOAD1494(i358[0], i362[0], i360[0]) → COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0]), COND_LOAD1494(TRUE, i358[1], i362[1], i360[1]) → LOAD1533(i358[1], i362[1], *(i360[1], i358[1])) which results in the following constraint:

(1)    (i362[0]=i362[1]&&(>(i362[0], 0), =(1, %(i362[0], 2)))=TRUEi358[0]=i358[1]i360[0]=i360[1]LOAD1494(i358[0], i362[0], i360[0])≥NonInfC∧LOAD1494(i358[0], i362[0], i360[0])≥COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])∧(UIncreasing(COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])), ≥))

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

(2)    (>(i362[0], 0)=TRUE>=(1, %(i362[0], 2))=TRUE<=(1, %(i362[0], 2))=TRUELOAD1494(i358[0], i362[0], i360[0])≥NonInfC∧LOAD1494(i358[0], i362[0], i360[0])≥COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])∧(UIncreasing(COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])), ≥))

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

(3)    (i362[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])), ≥)∧[(-1)Bound*bni_30] + [bni_30]i362[0] ≥ 0∧[(-1)bso_31] ≥ 0)

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

(4)    (i362[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])), ≥)∧[(-1)Bound*bni_30] + [bni_30]i362[0] ≥ 0∧[(-1)bso_31] ≥ 0)

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

(5)    (i362[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])), ≥)∧[(-1)Bound*bni_30] + [bni_30]i362[0] ≥ 0∧[(-1)bso_31] ≥ 0)

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

(6)    (i362[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30] + [bni_30]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

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

(7)    (i362[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30 + bni_30] + [bni_30]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

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

(8)    (i362[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30 + bni_30] + [bni_30]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

For Pair COND_LOAD1494(TRUE, i358, i362, i360) → LOAD1533(i358, i362, *(i360, i358)) the following chains were created:
• We consider the chain LOAD1494(i358[0], i362[0], i360[0]) → COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0]), COND_LOAD1494(TRUE, i358[1], i362[1], i360[1]) → LOAD1533(i358[1], i362[1], *(i360[1], i358[1])) which results in the following constraint:

We simplified constraint (9) using rules (III), (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)    (i362[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1533(i358[1], i362[1], *(i360[1], i358[1]))), ≥)∧[(-1)Bound*bni_32] + [bni_32]i362[0] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(12)    (i362[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1533(i358[1], i362[1], *(i360[1], i358[1]))), ≥)∧[(-1)Bound*bni_32] + [bni_32]i362[0] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(13)    (i362[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD1533(i358[1], i362[1], *(i360[1], i358[1]))), ≥)∧[(-1)Bound*bni_32] + [bni_32]i362[0] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(14)    (i362[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD1533(i358[1], i362[1], *(i360[1], i358[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32] + [bni_32]i362[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

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

(15)    (i362[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD1533(i358[1], i362[1], *(i360[1], i358[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32 + bni_32] + [bni_32]i362[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

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

(16)    (i362[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD1533(i358[1], i362[1], *(i360[1], i358[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32 + bni_32] + [bni_32]i362[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

For Pair LOAD1533(i358, i362, i360) → LOAD1494(*(i358, i358), /(i362, 2), i360) the following chains were created:
• We consider the chain LOAD1494(i358[0], i362[0], i360[0]) → COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0]), COND_LOAD1494(TRUE, i358[1], i362[1], i360[1]) → LOAD1533(i358[1], i362[1], *(i360[1], i358[1])), LOAD1533(i358[2], i362[2], i360[2]) → LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2]) which results in the following constraint:

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

(18)    (>(i362[0], 0)=TRUE>=(1, %(i362[0], 2))=TRUE<=(1, %(i362[0], 2))=TRUELOAD1533(i358[1], i362[0], *(i360[1], i358[1]))≥NonInfC∧LOAD1533(i358[1], i362[0], *(i360[1], i358[1]))≥LOAD1494(*(i358[1], i358[1]), /(i362[0], 2), *(i360[1], i358[1]))∧(UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥))

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

(19)    (i362[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧[(-1)bso_38] + i362[0] + [-1]max{i362[0], [-1]i362[0]} ≥ 0)

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

(20)    (i362[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧[(-1)bso_38] + i362[0] + [-1]max{i362[0], [-1]i362[0]} ≥ 0)

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

(21)    (i362[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2]i362[0] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧[(-1)bso_38] ≥ 0)

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

(22)    (i362[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2]i362[0] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

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

(23)    (i362[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2] + [2]i362[0] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

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

(24)    (i362[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2] + [2]i362[0] ≥ 0∧i358[1] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

(25)    (i362[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2] + [2]i362[0] ≥ 0∧i358[1] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

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

(26)    (i362[0] ≥ 0∧[1] ≥ 0∧i358[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[0] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

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

(27)    (i362[0] ≥ 0∧[1] ≥ 0∧i358[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[0] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

For Pair LOAD1494(i358, i362, i360) → COND_LOAD14941(&&(&&(>=(%(i362, 2), 0), !(=(%(i362, 2), 1))), >(i362, 0)), i358, i362, i360) the following chains were created:
• We consider the chain LOAD1494(i358[3], i362[3], i360[3]) → COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3]), COND_LOAD14941(TRUE, i358[4], i362[4], i360[4]) → LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4]) which results in the following constraint:

(28)    (i358[3]=i358[4]i360[3]=i360[4]&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0))=TRUEi362[3]=i362[4]LOAD1494(i358[3], i362[3], i360[3])≥NonInfC∧LOAD1494(i358[3], i362[3], i360[3])≥COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])∧(UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥))

We simplified constraint (28) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:

(29)    (>(i362[3], 0)=TRUE>=(%(i362[3], 2), 0)=TRUE<(%(i362[3], 2), 1)=TRUELOAD1494(i358[3], i362[3], i360[3])≥NonInfC∧LOAD1494(i358[3], i362[3], i360[3])≥COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])∧(UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥))

(30)    (>(i362[3], 0)=TRUE>=(%(i362[3], 2), 0)=TRUE>(%(i362[3], 2), 1)=TRUELOAD1494(i358[3], i362[3], i360[3])≥NonInfC∧LOAD1494(i358[3], i362[3], i360[3])≥COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])∧(UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥))

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

(31)    (i362[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i362[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(32)    (i362[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i362[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(33)    (i362[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i362[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(34)    (i362[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i362[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(35)    (i362[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i362[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(36)    (i362[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i362[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(37)    (i362[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39] + [bni_39]i362[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

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

(38)    (i362[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39] + [bni_39]i362[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

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

(39)    (i362[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i362[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

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

(40)    (i362[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i362[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

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

(41)    (i362[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i362[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

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

(42)    (i362[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i362[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

For Pair COND_LOAD14941(TRUE, i358, i362, i360) → LOAD1494(*(i358, i358), /(i362, 2), i360) the following chains were created:
• We consider the chain LOAD1494(i358[3], i362[3], i360[3]) → COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3]), COND_LOAD14941(TRUE, i358[4], i362[4], i360[4]) → LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4]) which results in the following constraint:

(43)    (i358[3]=i358[4]i360[3]=i360[4]&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0))=TRUEi362[3]=i362[4]COND_LOAD14941(TRUE, i358[4], i362[4], i360[4])≥NonInfC∧COND_LOAD14941(TRUE, i358[4], i362[4], i360[4])≥LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])∧(UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥))

We simplified constraint (43) using rules (III), (IDP_BOOLEAN) which results in the following new constraints:

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

(46)    (i362[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧[(-1)bso_42] + i362[3] + [-1]max{i362[3], [-1]i362[3]} ≥ 0)

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

(47)    (i362[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧[(-1)bso_42] + i362[3] + [-1]max{i362[3], [-1]i362[3]} ≥ 0)

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

(48)    (i362[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧[(-1)bso_42] + i362[3] + [-1]max{i362[3], [-1]i362[3]} ≥ 0)

We simplified constraint (47) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(49)    (i362[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧[(-1)bso_42] + i362[3] + [-1]max{i362[3], [-1]i362[3]} ≥ 0)

We simplified constraint (48) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(50)    (i362[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧[(-1)bso_42] ≥ 0)

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

(51)    (i362[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2]i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧[(-1)bso_42] ≥ 0)

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

(52)    (i362[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

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

(53)    (i362[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2]i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

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

(54)    (i362[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (54) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(55)    (i362[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i362[3] ≥ 0∧i358[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

(56)    (i362[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i362[3] ≥ 0∧i358[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (55) using rule (IDP_POLY_GCD) which results in the following new constraint:

(57)    (i362[3] ≥ 0∧i358[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (56) using rule (IDP_POLY_GCD) which results in the following new constraint:

(58)    (i362[3] ≥ 0∧i358[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

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

(59)    (i362[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2] + [2]i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (59) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(60)    (i362[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2] + [2]i362[3] ≥ 0∧i358[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

(61)    (i362[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2] + [2]i362[3] ≥ 0∧i358[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (60) using rule (IDP_POLY_GCD) which results in the following new constraint:

(62)    (i362[3] ≥ 0∧0 ≥ 0∧i358[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (61) using rule (IDP_POLY_GCD) which results in the following new constraint:

(63)    (i362[3] ≥ 0∧0 ≥ 0∧i358[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD1494(i358, i362, i360) → COND_LOAD1494(&&(>(i362, 0), =(1, %(i362, 2))), i358, i362, i360)
• (i362[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30 + bni_30] + [bni_30]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

• (i362[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD1533(i358[1], i362[1], *(i360[1], i358[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32 + bni_32] + [bni_32]i362[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

• (i362[0] ≥ 0∧[1] ≥ 0∧i358[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[0] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
• (i362[0] ≥ 0∧[1] ≥ 0∧i358[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[0] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[2], i358[2]), /(i362[2], 2), i360[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i362[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

• LOAD1494(i358, i362, i360) → COND_LOAD14941(&&(&&(>=(%(i362, 2), 0), !(=(%(i362, 2), 1))), >(i362, 0)), i358, i362, i360)
• (i362[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i362[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)
• (i362[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i362[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

• (i362[3] ≥ 0∧i358[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)
• (i362[3] ≥ 0∧i358[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)
• (i362[3] ≥ 0∧0 ≥ 0∧i358[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)
• (i362[3] ≥ 0∧0 ≥ 0∧i358[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i362[3] ≥ 0 ⇒ (UIncreasing(LOAD1494(*(i358[4], i358[4]), /(i362[4], 2), i360[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i362[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_LOAD1494(x1, x2, x3, x4)) = x3
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(=(x1, x2)) = [-1]
POL(1) = [1]
POL(2) = [2]
POL(LOAD1533(x1, x2, x3)) = [-1] + x2
POL(*(x1, x2)) = x1·x2
POL(COND_LOAD14941(x1, x2, x3, x4)) = [-1] + x3
POL(>=(x1, x2)) = [-1]
POL(!(x1)) = [-1]

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(%(x1, 2)-1 @ {}) = min{x2, [-1]x2}
POL(%(x1, 2)1 @ {}) = max{x2, [-1]x2}
POL(/(x1, 2)1 @ {LOAD1494_3/1}) = max{x1, [-1]x1} + [-1]

The following pairs are in P>:

LOAD1494(i358[3], i362[3], i360[3]) → COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])

The following pairs are in Pbound:

LOAD1494(i358[0], i362[0], i360[0]) → COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])
LOAD1494(i358[3], i362[3], i360[3]) → COND_LOAD14941(&&(&&(>=(%(i362[3], 2), 0), !(=(%(i362[3], 2), 1))), >(i362[3], 0)), i358[3], i362[3], i360[3])

The following pairs are in P:

LOAD1494(i358[0], i362[0], i360[0]) → COND_LOAD1494(&&(>(i362[0], 0), =(1, %(i362[0], 2))), i358[0], i362[0], i360[0])

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

/1

### (11) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1494(i358[0], i362[0], i360[0]) → COND_LOAD1494(i362[0] > 0 && 1 = i362[0] % 2, i358[0], i362[0], i360[0])
(2): LOAD1533(i358[2], i362[2], i360[2]) → LOAD1494(i358[2] * i358[2], i362[2] / 2, i360[2])
(4): COND_LOAD14941(TRUE, i358[4], i362[4], i360[4]) → LOAD1494(i358[4] * i358[4], i362[4] / 2, i360[4])

(2) -> (0), if ((i358[2] * i358[2]* i358[0])∧(i362[2] / 2* i362[0])∧(i360[2]* i360[0]))

(4) -> (0), if ((i360[4]* i360[0])∧(i362[4] / 2* i362[0])∧(i358[4] * i358[4]* i358[0]))

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

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

### (14) 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: