Termination proof of /tmp/tmpRzpbTq/Log.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 IDP

↳9 IDPNonInfProof (⇒)

↳10 IDP

↳11 IDependencyGraphProof (⇔)

↳12 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Log

public class Log{
  public static int half(int x) {

    int res = 0;

    while (x > 1) {

      x = x-2;
      res++;

    }

    return res;

  }


  public static int log(int x) {

    int res = 0;

    while (x > 1) {

      x = half(x);
      res++;

    }

    return res;

  }


  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    log(x);
  }
}


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:
Log.main([Ljava/lang/String;)V: Graph of 118 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 30 rules for P and 5 rules for R.

Combined rules. Obtained 2 rules for P and 0 rules for R.

Filtered ground terms:

658_0_half_LE(x1, x2, x3, x4, x5) → 658_0_half_LE(x2, x3, x4)

Filtered duplicate args:

658_0_half_LE(x1, x2, x3) → 658_0_half_LE(x2, x3)

Combined rules. Obtained 2 rules for P and 0 rules for R.

Finished conversion. Obtained 2 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): 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(x1[0] > 1 && x0[0] <= 1, 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))
(1): COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))
(2): 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(x1[2] >= 0 && x0[2] > 1, 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))
(3): COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[3] + 1, x0[3] - 2)))

(0) -> (1), if ((x1[0] > 1 && x0[0] <= 1 →^* TRUE)∧(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))))

(1) -> (0), if ((658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))))

(1) -> (2), if ((658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))))

(2) -> (3), if ((x1[2] >= 0 && x0[2] > 1 →^* TRUE)∧(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))))

(3) -> (0), if ((658_1_log_InvokeMethod(658_0_half_LE(x1[3] + 1, x0[3] - 2)) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))))

(3) -> (2), if ((658_1_log_InvokeMethod(658_0_half_LE(x1[3] + 1, x0[3] - 2)) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))))

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 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1, x0))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1, 1), <=(x0, 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1, x0))) the following chains were created:

We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))) which results in the following constraint:
(1)    (&&(>(x1[0], 1), <=(x0[0], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))=658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1])) ⇒ 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))≥NonInfC∧658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))≥COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))∧(U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], 1)=TRUE∧<=(x0[0], 1)=TRUE ⇒ 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))≥NonInfC∧658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))≥COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))∧(U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

(8)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1, x0))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1))) the following chains were created:

We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))) which results in the following constraint:
(9) (&&(>(x1[0], 1), <=(x0[0], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))=658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))∧658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))=658_1_log_InvokeMethod(658_0_half_LE(x1[0]1, x0[0]1))∧&&(>(x1[0]1, 1), <=(x0[0]1, 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0]1, x0[0]1))=658_1_log_InvokeMethod(658_0_half_LE(x1[1]1, x0[1]1))∧658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]1))=658_1_log_InvokeMethod(658_0_half_LE(x1[0]2, x0[0]2))∧&&(>(x1[0]2, 1), <=(x0[0]2, 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0]2, x0[0]2))=658_1_log_InvokeMethod(658_0_half_LE(x1[1]2, x0[1]2)) ⇒ COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1]1, x0[1]1)))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1]1, x0[1]1)))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]1)))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]1)))), ≥))

We solved constraint (9) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))) which results in the following constraint:
(10) (&&(>(x1[0], 1), <=(x0[0], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))=658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))∧658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))=658_1_log_InvokeMethod(658_0_half_LE(x1[0]1, x0[0]1))∧&&(>(x1[0]1, 1), <=(x0[0]1, 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0]1, x0[0]1))=658_1_log_InvokeMethod(658_0_half_LE(x1[1]1, x0[1]1))∧658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]1))=658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))∧&&(>=(x1[2], 0), >(x0[2], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))=658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3])) ⇒ COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1]1, x0[1]1)))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1]1, x0[1]1)))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]1)))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]1)))), ≥))

We solved constraint (10) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))) which results in the following constraint:
(11) (&&(>=(x1[2], 0), >(x0[2], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))=658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))∧658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))∧&&(>(x1[0], 1), <=(x0[0], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))=658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))∧658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))=658_1_log_InvokeMethod(658_0_half_LE(x1[0]1, x0[0]1))∧&&(>(x1[0]1, 1), <=(x0[0]1, 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0]1, x0[0]1))=658_1_log_InvokeMethod(658_0_half_LE(x1[1]1, x0[1]1)) ⇒ COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1])))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1])))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥))

We solved constraint (11) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))) which results in the following constraint:
(12)    (&&(>=(x1[2], 0), >(x0[2], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))=658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))∧658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))∧&&(>(x1[0], 1), <=(x0[0], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))=658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))∧658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))=658_1_log_InvokeMethod(658_0_half_LE(x1[2]1, x0[2]1))∧&&(>=(x1[2]1, 0), >(x0[2]1, 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2]1, x0[2]1))=658_1_log_InvokeMethod(658_0_half_LE(x1[3]1, x0[3]1)) ⇒ COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1])))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1])))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥))

We simplified constraint (12) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:
(13)    (>=(x1[2], 0)=TRUE∧>(x0[2], 1)=TRUE∧>(+(x1[2], 1), 1)=TRUE∧<=(-(x0[2], 2), 1)=TRUE ⇒ COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, +(x1[2], 1))))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥))

We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [-1] ≥ 0∧[3] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x1[2] ≥ 0∧[-2 + (-1)bso_23] + x0[2] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [-1] ≥ 0∧[3] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x1[2] ≥ 0∧[-2 + (-1)bso_23] + x0[2] ≥ 0)

We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [-1] ≥ 0∧[3] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x1[2] ≥ 0∧[-2 + (-1)bso_23] + x0[2] ≥ 0)

We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(17)    ([1] + x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] ≥ 0∧[3] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_22 + bni_22] + [bni_22]x0[2] + [bni_22]x1[2] ≥ 0∧[-2 + (-1)bso_23] + x0[2] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    ([1] + x1[2] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_22 + (3)bni_22] + [bni_22]x0[2] + [bni_22]x1[2] ≥ 0∧[(-1)bso_23] + x0[2] ≥ 0)

For Pair 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1, x0))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1, 0), >(x0, 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1, x0))) the following chains were created:

We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))) which results in the following constraint:
(19)    (&&(>=(x1[2], 0), >(x0[2], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))=658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3])) ⇒ 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))≥NonInfC∧658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))≥COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))∧(U^Increasing(COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))), ≥))

We simplified constraint (19) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(20)    (>=(x1[2], 0)=TRUE∧>(x0[2], 1)=TRUE ⇒ 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))≥NonInfC∧658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))≥COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))∧(U^Increasing(COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (23) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(24)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))), ≥)∧[(3)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)

For Pair COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1, x0))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1, 1), -(x0, 2)))) the following chains were created:

We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))) which results in the following constraint:
(25) (&&(>(x1[0], 1), <=(x0[0], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))=658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))∧658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))=658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))∧&&(>=(x1[2], 0), >(x0[2], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))=658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))∧658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=658_1_log_InvokeMethod(658_0_half_LE(x1[0]1, x0[0]1))∧&&(>(x1[0]1, 1), <=(x0[0]1, 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0]1, x0[0]1))=658_1_log_InvokeMethod(658_0_half_LE(x1[1]1, x0[1]1)) ⇒ COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3])))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3])))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥))

We solved constraint (25) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN).
We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))) which results in the following constraint:
(26)    (&&(>(x1[0], 1), <=(x0[0], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))=658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))∧658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))=658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))∧&&(>=(x1[2], 0), >(x0[2], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))=658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))∧658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=658_1_log_InvokeMethod(658_0_half_LE(x1[2]1, x0[2]1))∧&&(>=(x1[2]1, 0), >(x0[2]1, 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2]1, x0[2]1))=658_1_log_InvokeMethod(658_0_half_LE(x1[3]1, x0[3]1)) ⇒ COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3])))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3])))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥))

We simplified constraint (26) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:
(27)    (>(x1[0], 1)=TRUE∧<=(x0[0], 1)=TRUE∧>(-(x1[0], 2), 1)=TRUE ⇒ COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(0, x1[0])))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(0, x1[0])))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(0, 1), -(x1[0], 2))))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥))

We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(28)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x1[0] + [-4] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(29)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x1[0] + [-4] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(30)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x1[0] + [-4] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧[-2] + x1[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(3)bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(32)    ([2] + x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33)    ([2] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)
We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))) which results in the following constraint:
(34)    (&&(>=(x1[2], 0), >(x0[2], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))=658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))∧658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=658_1_log_InvokeMethod(658_0_half_LE(x1[2]1, x0[2]1))∧&&(>=(x1[2]1, 0), >(x0[2]1, 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2]1, x0[2]1))=658_1_log_InvokeMethod(658_0_half_LE(x1[3]1, x0[3]1))∧658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2)))=658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))∧&&(>(x1[0], 1), <=(x0[0], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))=658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1])) ⇒ COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3]1, x0[3]1)))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3]1, x0[3]1)))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥))

We simplified constraint (34) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(35)    (>=(x1[2], 0)=TRUE∧>(x0[2], 1)=TRUE∧>=(+(x1[2], 1), 0)=TRUE∧>(-(x0[2], 2), 1)=TRUE∧>(+(+(x1[2], 1), 1), 1)=TRUE∧<=(-(-(x0[2], 2), 2), 1)=TRUE ⇒ COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(+(x1[2], 1), 1), -(-(x0[2], 2), 2))))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥))

We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(36)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] + [-4] ≥ 0∧x1[2] ≥ 0∧[5] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(37)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] + [-4] ≥ 0∧x1[2] ≥ 0∧[5] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(38)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] + [-4] ≥ 0∧x1[2] ≥ 0∧[5] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (38) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(39)    (x1[2] ≥ 0∧x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[-2] + x0[2] ≥ 0∧x1[2] ≥ 0∧[3] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26 + (2)bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(40)    (x1[2] ≥ 0∧[2] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26 + (4)bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)
We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))), COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))) which results in the following constraint:
(41)    (&&(>=(x1[2], 0), >(x0[2], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))=658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))∧658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=658_1_log_InvokeMethod(658_0_half_LE(x1[2]1, x0[2]1))∧&&(>=(x1[2]1, 0), >(x0[2]1, 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2]1, x0[2]1))=658_1_log_InvokeMethod(658_0_half_LE(x1[3]1, x0[3]1))∧658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2)))=658_1_log_InvokeMethod(658_0_half_LE(x1[2]2, x0[2]2))∧&&(>=(x1[2]2, 0), >(x0[2]2, 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[2]2, x0[2]2))=658_1_log_InvokeMethod(658_0_half_LE(x1[3]2, x0[3]2)) ⇒ COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3]1, x0[3]1)))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3]1, x0[3]1)))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥))

We simplified constraint (41) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(42)    (>=(x1[2], 0)=TRUE∧>(x0[2], 1)=TRUE∧>=(+(x1[2], 1), 0)=TRUE∧>(-(x0[2], 2), 1)=TRUE∧>=(+(+(x1[2], 1), 1), 0)=TRUE∧>(-(-(x0[2], 2), 2), 1)=TRUE ⇒ COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(+(x1[2], 1), 1), -(-(x0[2], 2), 2))))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥))

We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(43)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] + [-4] ≥ 0∧x1[2] + [2] ≥ 0∧x0[2] + [-6] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (43) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(44)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] + [-4] ≥ 0∧x1[2] + [2] ≥ 0∧x0[2] + [-6] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (44) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(45)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] + [-4] ≥ 0∧x1[2] + [2] ≥ 0∧x0[2] + [-6] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (45) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(46)    (x1[2] ≥ 0∧x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[-2] + x0[2] ≥ 0∧x1[2] + [2] ≥ 0∧[-4] + x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26 + (2)bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(47)    (x1[2] ≥ 0∧[2] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] ≥ 0∧x1[2] + [2] ≥ 0∧[-2] + x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26 + (4)bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(48)    (x1[2] ≥ 0∧[4] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[2] + x0[2] ≥ 0∧x1[2] + [2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26 + (6)bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (48) using rule (IDP_POLY_GCD) which results in the following new constraint:
(49)    (x1[2] ≥ 0∧[4] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[2] + x0[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26 + (6)bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1, x0))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1, 1), <=(x0, 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1, x0)))
- (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)
- (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)
COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1, x0))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1)))
- ([1] + x1[2] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_22 + (3)bni_22] + [bni_22]x0[2] + [bni_22]x1[2] ≥ 0∧[(-1)bso_23] + x0[2] ≥ 0)
658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1, x0))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1, 0), >(x0, 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1, x0)))
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))), ≥)∧[(3)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)
COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1, x0))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1, 1), -(x0, 2))))
- ([2] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)
- (x1[2] ≥ 0∧[2] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26 + (4)bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)
- (x1[2] ≥ 0∧[4] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[2] + x0[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(-1)Bound*bni_26 + (6)bni_26] + [bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 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) = 0
POL(FALSE) = 0
POL(658_2_MAIN_INVOKEMETHOD(x₁)) = [-1] + [-1]x₁
POL(658_1_log_InvokeMethod(x₁)) = [-1] + x₁
POL(658_0_half_LE(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(COND_658_2_MAIN_INVOKEMETHOD(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<=(x₁, x₂)) = [-1]
POL(0) = 0
POL(COND_658_2_MAIN_INVOKEMETHOD1(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(>=(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(2) = [2]

The following pairs are in P_>:

COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))

The following pairs are in P_bound:

COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))
658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))
COND_658_2_MAIN_INVOKEMETHOD1(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[3], x0[3]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(+(x1[3], 1), -(x0[3], 2))))

The following pairs are in P_≥:

658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))
COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))
658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))) → COND_658_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2])))

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

TRUE¹ → &&(TRUE, TRUE)¹
FALSE¹ → &&(TRUE, FALSE)¹
&&(FALSE, TRUE)¹ ↔ FALSE¹
FALSE¹ → &&(FALSE, FALSE)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:

Boolean, Integer

(1) -> (0), if ((658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))))

(0) -> (1), if ((x1[0] > 1 && x0[0] <= 1 →^* TRUE)∧(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))))

(1) -> (2), if ((658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[2], x0[2]))))

The set Q is empty.

(7) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
(1): COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))
(0): 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(x1[0] > 1 && x0[0] <= 1, 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))

(1) -> (0), if ((658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))))

(0) -> (1), if ((x1[0] > 1 && x0[0] <= 1 →^* TRUE)∧(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])) →^* 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))))

The set Q is empty.

(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 COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))) the following chains were created:

We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))), 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) which results in the following constraint:
(1)    (&&(>(x1[0], 1), <=(x0[0], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))=658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))∧658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))=658_1_log_InvokeMethod(658_0_half_LE(x1[0]1, x0[0]1)) ⇒ COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1])))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1])))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥))

We simplified constraint (1) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], 1)=TRUE∧<=(x0[0], 1)=TRUE ⇒ COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))≥NonInfC∧COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))≥658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[0])))∧(U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x0[0] + [bni_16]x1[0] ≥ 0∧[-3 + (-1)bso_17] + [-1]x0[0] + [2]x1[0] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x0[0] + [bni_16]x1[0] ≥ 0∧[-3 + (-1)bso_17] + [-1]x0[0] + [2]x1[0] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x0[0] + [bni_16]x1[0] ≥ 0∧[-3 + (-1)bso_17] + [-1]x0[0] + [2]x1[0] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]x0[0] + [bni_16]x1[0] ≥ 0∧[1 + (-1)bso_17] + [-1]x0[0] + [2]x1[0] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]x0[0] + [bni_16]x1[0] ≥ 0∧[1 + (-1)bso_17] + [-1]x0[0] + [2]x1[0] ≥ 0)

(8)    (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] + [bni_16]x1[0] ≥ 0∧[1 + (-1)bso_17] + x0[0] + [2]x1[0] ≥ 0)

For Pair 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) the following chains were created:

We consider the chain 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))), COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1]))) which results in the following constraint:
(9)    (&&(>(x1[0], 1), <=(x0[0], 1))=TRUE∧658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))=658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1])) ⇒ 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))≥NonInfC∧658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))≥COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))∧(U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥))

We simplified constraint (9) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(10)    (>(x1[0], 1)=TRUE∧<=(x0[0], 1)=TRUE ⇒ 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))≥NonInfC∧658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))≥COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))∧(U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥))

We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x0[0] + [bni_18]x1[0] ≥ 0∧[3 + (-1)bso_19] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x0[0] + [bni_18]x1[0] ≥ 0∧[3 + (-1)bso_19] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x0[0] + [bni_18]x1[0] ≥ 0∧[3 + (-1)bso_19] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x0[0] + [bni_18]x1[0] ≥ 0∧[3 + (-1)bso_19] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(15)    (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x1[0] ≥ 0∧[3 + (-1)bso_19] ≥ 0)

(16)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x0[0] + [bni_18]x1[0] ≥ 0∧[3 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))
- (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]x0[0] + [bni_16]x1[0] ≥ 0∧[1 + (-1)bso_17] + [-1]x0[0] + [2]x1[0] ≥ 0)
- (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] + [bni_16]x1[0] ≥ 0∧[1 + (-1)bso_17] + x0[0] + [2]x1[0] ≥ 0)
658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))
- (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x1[0] ≥ 0∧[3 + (-1)bso_19] ≥ 0)
- (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x0[0] + [bni_18]x1[0] ≥ 0∧[3 + (-1)bso_19] ≥ 0)

POL(TRUE) = [1]
POL(FALSE) = [2]
POL(COND_658_2_MAIN_INVOKEMETHOD(x₁, x₂)) = [-1] + [-1]x₂
POL(658_1_log_InvokeMethod(x₁)) = [-1]x₁
POL(658_0_half_LE(x₁, x₂)) = [-1] + [-1]x₂ + x₁
POL(658_2_MAIN_INVOKEMETHOD(x₁)) = [2] + [-1]x₁
POL(0) = 0
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<=(x₁, x₂)) = [-1]

The following pairs are in P_>:

658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))

The following pairs are in P_bound:

COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))
658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0]))) → COND_658_2_MAIN_INVOKEMETHOD(&&(>(x1[0], 1), <=(x0[0], 1)), 658_1_log_InvokeMethod(658_0_half_LE(x1[0], x0[0])))

The following pairs are in P_≥:

COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))

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

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

(10) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:
none

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_658_2_MAIN_INVOKEMETHOD(TRUE, 658_1_log_InvokeMethod(658_0_half_LE(x1[1], x0[1]))) → 658_2_MAIN_INVOKEMETHOD(658_1_log_InvokeMethod(658_0_half_LE(0, x1[1])))

The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Obligation:

(5) IDPNonInfProof (SOUND transformation)

(6) Obligation:

(7) IDependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) IDPNonInfProof (SOUND transformation)

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE