(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))=TRUE658_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])))∧(UIncreasing(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)=TRUE658_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])))∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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))=TRUE658_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))=TRUE658_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))=TRUE658_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)))∧(UIncreasing(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))=TRUE658_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))=TRUE658_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))=TRUE658_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)))∧(UIncreasing(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))=TRUE658_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))=TRUE658_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))=TRUE658_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])))∧(UIncreasing(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))=TRUE658_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))=TRUE658_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))=TRUE658_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])))∧(UIncreasing(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)=TRUECOND_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))))∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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))=TRUE658_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])))∧(UIncreasing(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)=TRUE658_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])))∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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))=TRUE658_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))=TRUE658_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))=TRUE658_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))))∧(UIncreasing(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))=TRUE658_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))=TRUE658_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))=TRUE658_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))))∧(UIncreasing(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)=TRUECOND_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))))∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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))=TRUE658_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))=TRUE658_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))=TRUE658_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))))∧(UIncreasing(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)=TRUECOND_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))))∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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))=TRUE658_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))=TRUE658_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))=TRUE658_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))))∧(UIncreasing(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)=TRUECOND_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))))∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(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 Pbound:

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)1FALSE1
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

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])))

(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))=TRUE658_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])))∧(UIncreasing(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)=TRUECOND_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])))∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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))=TRUE658_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])))∧(UIncreasing(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)=TRUE658_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])))∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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)

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

POL(TRUE) = [1]
POL(FALSE) = [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 Pbound:

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.