Termination proof of /tmp/tmpQzcxVz/AProVEMath.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 IDP

↳5 IDPNonInfProof (⇒)

↳6 IDP

↳7 IDependencyGraphProof (⇔)

↳8 TRUE

(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)))=TRUE∧1127_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]))∧(U^Increasing(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))=TRUE ⇒ 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]))∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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)))=TRUE∧1127_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)))∧(U^Increasing(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))=TRUE ⇒ COND_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)))∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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)))=TRUE∧1127_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)))∧(U^Increasing(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))=TRUE ⇒ COND_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)))∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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)))=TRUE∧1127_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)))∧(U^Increasing(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)=TRUE ⇒ COND_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)))∧(U^Increasing(1127_1_MAIN_INVOKEMETHOD(1127_0_power_LE(/(x0[1], 2)))), ≥))

(24)    (>(x0[3], 1)=TRUE∧>(%(x0[3], 2), 1)=TRUE ⇒ COND_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)))∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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)))=TRUE∧1127_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]))∧(U^Increasing(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))=TRUE ⇒ 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]))∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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)))=TRUE∧1127_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]))∧(U^Increasing(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)=TRUE ⇒ 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]))∧(U^Increasing(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)=TRUE ⇒ 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]))∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 P_bound 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 P_bound.
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(Term^CSAR-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 P_bound:

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)¹ ↔ TRUE¹
&&(TRUE, FALSE)¹ ↔ FALSE¹
&&(FALSE, TRUE)¹ ↔ FALSE¹
&&(FALSE, FALSE)¹ ↔ FALSE¹
/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.