### (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:
AProVEMath.main([Ljava/lang/String;)V: Graph of 215 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 29 rules for P and 6 rules for R.

Combined rules. Obtained 3 rules for P and 0 rules for R.

Filtered ground terms:

1127_0_power_LE(x1, x2, x3) → 1127_0_power_LE(x2, x3)

Filtered duplicate args:

1127_0_power_LE(x1, x2) → 1127_0_power_LE(x2)

Combined rules. Obtained 3 rules for P and 0 rules for R.

Finished conversion. Obtained 3 rules for P and 0 rules for R. System has predefined symbols.

### (4) 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): 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[0])) → COND_1127_1_MAIN_INVOKEMETHOD(x0[0] > 1 && 0 = x0[0] % 2, 1127_0_power_LE(x0[0]))
(1): COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1])) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[1] / 2))
(2): 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[2])) → COND_1127_1_MAIN_INVOKEMETHOD(x0[2] > 1 && 1 = x0[2] % 2, 1127_0_power_LE(x0[2]))
(3): 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3])) → COND_1127_1_MAIN_INVOKEMETHOD(x0[3] > 1 && !(x0[3] % 2 = 1), 1127_0_power_LE(x0[3]))

(0) -> (1), if ((x0[0] > 1 && 0 = x0[0] % 2* TRUE)∧(1127_0_power_LE(x0[0]) →* 1127_0_power_LE(x0[1])))

(1) -> (0), if ((1127_0_power_LE(x0[1] / 2) →* 1127_0_power_LE(x0[0])))

(1) -> (2), if ((1127_0_power_LE(x0[1] / 2) →* 1127_0_power_LE(x0[2])))

(1) -> (3), if ((1127_0_power_LE(x0[1] / 2) →* 1127_0_power_LE(x0[3])))

(2) -> (1), if ((x0[2] > 1 && 1 = x0[2] % 2* TRUE)∧(1127_0_power_LE(x0[2]) →* 1127_0_power_LE(x0[1])))

(3) -> (1), if ((x0[3] > 1 && !(x0[3] % 2 = 1) →* TRUE)∧(1127_0_power_LE(x0[3]) →* 1127_0_power_LE(x0[1])))

The set Q is empty.

### (5) 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 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0)) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0, 1), =(0, %(x0, 2))), 1127_0_power_LE(x0)) the following chains were created:
• We consider the chain 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[0])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0])), COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1])) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2))) which results in the following constraint:

(1)    (&&(>(x0[0], 1), =(0, %(x0[0], 2)))=TRUE1127_0_power_LE(x0[0])=1127_0_power_LE(x0[1]) ⇒ 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[0]))≥NonInfC∧1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[0]))≥COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))∧(UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))), ≥))

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

(2)    (>(x0[0], 1)=TRUE>=(0, %(x0[0], 2))=TRUE<=(0, %(x0[0], 2))=TRUE1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[0]))≥NonInfC∧1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[0]))≥COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))∧(UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))), ≥))

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

(3)    (x0[0] + [-2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(4)    (x0[0] + [-2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(5)    (x0[0] + [-2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(6)    (x0[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(7)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

For Pair COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0)) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0, 2))) the following chains were created:
• We consider the chain 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[0])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0])), COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1])) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2))) which results in the following constraint:

(8)    (&&(>(x0[0], 1), =(0, %(x0[0], 2)))=TRUE1127_0_power_LE(x0[0])=1127_0_power_LE(x0[1]) ⇒ COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1]))≥NonInfC∧COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1]))≥1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))∧(UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥))

We simplified constraint (8) using rules (I), (II), (III), (IDP_BOOLEAN) which results in the following new constraint:

(9)    (>(x0[0], 1)=TRUE>=(0, %(x0[0], 2))=TRUE<=(0, %(x0[0], 2))=TRUECOND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[0]))≥NonInfC∧COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[0]))≥1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[0], 2)))∧(UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥))

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

(10)    (x0[0] + [-2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_20] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)

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

(11)    (x0[0] + [-2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_20] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)

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

(12)    (x0[0] + [-2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]x0[0] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(13)    (x0[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[4] + [2]x0[0] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(14)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

• We consider the chain 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[2])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2])), COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1])) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2))) which results in the following constraint:

(15)    (&&(>(x0[2], 1), =(1, %(x0[2], 2)))=TRUE1127_0_power_LE(x0[2])=1127_0_power_LE(x0[1]) ⇒ COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1]))≥NonInfC∧COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1]))≥1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))∧(UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥))

We simplified constraint (15) using rules (I), (II), (III), (IDP_BOOLEAN) which results in the following new constraint:

(16)    (>(x0[2], 1)=TRUE>=(1, %(x0[2], 2))=TRUE<=(1, %(x0[2], 2))=TRUECOND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[2]))≥NonInfC∧COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[2]))≥1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[2], 2)))∧(UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥))

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

(17)    (x0[2] + [-2] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_20] + x0[2] + [-1]max{x0[2], [-1]x0[2]} ≥ 0)

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

(18)    (x0[2] + [-2] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_20] + x0[2] + [-1]max{x0[2], [-1]x0[2]} ≥ 0)

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

(19)    (x0[2] + [-2] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2]x0[2] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(20)    (x0[2] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[4] + [2]x0[2] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(21)    (x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2] + x0[2] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_20] ≥ 0)

• We consider the chain 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3])), COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1])) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2))) which results in the following constraint:

(22)    (&&(>(x0[3], 1), !(=(%(x0[3], 2), 1)))=TRUE1127_0_power_LE(x0[3])=1127_0_power_LE(x0[1]) ⇒ COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1]))≥NonInfC∧COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1]))≥1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))∧(UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥))

We simplified constraint (22) using rules (I), (II), (III), (IDP_BOOLEAN) which results in the following new constraints:

(23)    (>(x0[3], 1)=TRUE<(%(x0[3], 2), 1)=TRUECOND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[3]))≥NonInfC∧COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[3]))≥1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[3], 2)))∧(UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥))

(24)    (>(x0[3], 1)=TRUE>(%(x0[3], 2), 1)=TRUECOND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[3]))≥NonInfC∧COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[3]))≥1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[3], 2)))∧(UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥))

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

(25)    (x0[3] + [-2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] + x0[3] + [-1]max{x0[3], [-1]x0[3]} ≥ 0)

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

(26)    (x0[3] + [-2] ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] + x0[3] + [-1]max{x0[3], [-1]x0[3]} ≥ 0)

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

(27)    (x0[3] + [-2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] + x0[3] + [-1]max{x0[3], [-1]x0[3]} ≥ 0)

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

(28)    (x0[3] + [-2] ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] + x0[3] + [-1]max{x0[3], [-1]x0[3]} ≥ 0)

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

(29)    (x0[3] + [-2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2]x0[3] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(30)    (x0[3] + [-2] ≥ 0∧[4] ≥ 0∧0 ≥ 0∧[2]x0[3] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(31)    (x0[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[4] + [2]x0[3] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(32)    (x0[3] ≥ 0∧[4] ≥ 0∧0 ≥ 0∧[4] + [2]x0[3] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(33)    (x0[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2] + x0[3] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(34)    (x0[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[2] + x0[3] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] ≥ 0)

For Pair 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0)) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0, 1), =(1, %(x0, 2))), 1127_0_power_LE(x0)) the following chains were created:
• We consider the chain 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[2])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2])), COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1])) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2))) which results in the following constraint:

(35)    (&&(>(x0[2], 1), =(1, %(x0[2], 2)))=TRUE1127_0_power_LE(x0[2])=1127_0_power_LE(x0[1]) ⇒ 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[2]))≥NonInfC∧1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[2]))≥COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))∧(UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))), ≥))

We simplified constraint (35) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(36)    (>(x0[2], 1)=TRUE>=(1, %(x0[2], 2))=TRUE<=(1, %(x0[2], 2))=TRUE1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[2]))≥NonInfC∧1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[2]))≥COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))∧(UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))), ≥))

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

(37)    (x0[2] + [-2] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(38)    (x0[2] + [-2] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(39)    (x0[2] + [-2] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(40)    (x0[2] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))), ≥)∧[bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

We simplified constraint (40) using rule (IDP_POLY_GCD) which results in the following new constraint:

(41)    (x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))), ≥)∧[bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

For Pair 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0)) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0, 1), !(=(%(x0, 2), 1))), 1127_0_power_LE(x0)) the following chains were created:
• We consider the chain 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3])), COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1])) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2))) which results in the following constraint:

(42)    (&&(>(x0[3], 1), !(=(%(x0[3], 2), 1)))=TRUE1127_0_power_LE(x0[3])=1127_0_power_LE(x0[1]) ⇒ 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3]))≥NonInfC∧1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3]))≥COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))∧(UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥))

We simplified constraint (42) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraints:

(43)    (>(x0[3], 1)=TRUE<(%(x0[3], 2), 1)=TRUE1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3]))≥NonInfC∧1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3]))≥COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))∧(UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥))

(44)    (>(x0[3], 1)=TRUE>(%(x0[3], 2), 1)=TRUE1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3]))≥NonInfC∧1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3]))≥COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))∧(UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥))

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

(45)    (x0[3] + [-2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(46)    (x0[3] + [-2] ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(47)    (x0[3] + [-2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(48)    (x0[3] + [-2] ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(49)    (x0[3] + [-2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(50)    (x0[3] + [-2] ≥ 0∧[4] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(51)    (x0[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(52)    (x0[3] ≥ 0∧[4] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(53)    (x0[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(54)    (x0[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0)) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0, 1), =(0, %(x0, 2))), 1127_0_power_LE(x0))
• (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

• COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0)) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0, 2)))
• (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)
• (x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2] + x0[2] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_20] ≥ 0)
• (x0[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2] + x0[3] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] ≥ 0)
• (x0[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[2] + x0[3] ≥ 0 ⇒ (UIncreasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[3] ≥ 0∧[(-1)bso_20] ≥ 0)

• 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0)) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0, 1), =(1, %(x0, 2))), 1127_0_power_LE(x0))
• (x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))), ≥)∧[bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

• 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0)) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0, 1), !(=(%(x0, 2), 1))), 1127_0_power_LE(x0))
• (x0[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
• (x0[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 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) = [1]
POL(FALSE) = [1]
POL(1127_1_MAIN_INVOKEMETHOD(x1)) = x1
POL(1127_0_power_LE(x1)) = [-1] + x1
POL(COND_1127_1_MAIN_INVOKEMETHOD(x1, x2)) = x2 + [-1]x1
POL(&&(x1, x2)) = [1]
POL(>(x1, x2)) = [-1]
POL(1) = [1]
POL(=(x1, x2)) = [-1]
POL(0) = 0
POL(2) = [2]
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 @ {1127_1_MAIN_INVOKEMETHOD_1/0, 1127_0_power_LE_1/0}) = max{x1, [-1]x1} + [-1]

The following pairs are in P>:

1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[0])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))
1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[2])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))
1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))

The following pairs are in Pbound:

1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[0])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), =(0, %(x0[0], 2))), 1127_0_power_LE(x0[0]))
COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1])) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))
1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[2])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[2], 1), =(1, %(x0[2], 2))), 1127_0_power_LE(x0[2]))
1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[3])) → COND_1127_1_MAIN_INVOKEMETHOD(&&(>(x0[3], 1), !(=(%(x0[3], 2), 1))), 1127_0_power_LE(x0[3]))

The following pairs are in P:

COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1])) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))

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
/1

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1127_1_MAIN_INVOKEMETHOD(TRUE, 1127_0_power_LE(x0[1])) → 1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(x0[1] / 2))

The set Q is empty.

### (7) IDependencyGraphProof (EQUIVALENT transformation)

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