### (0) Obligation:

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

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-2)+1;
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:
LogAG.main([Ljava/lang/String;)V: Graph of 122 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 34 rules for P and 5 rules for R.

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

Filtered ground terms:

983_0_half_LE(x1, x2, x3, x4, x5) → 983_0_half_LE(x2, x3, x4)

Filtered duplicate args:

983_0_half_LE(x1, x2, x3) → 983_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): 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(x1[0] >= 0 && x0[0] <= 1 && 1 < x1[0] + 1, 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))
(1): COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, x1[1] + 1 - 2)))
(2): 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(x1[2] >= 0 && x0[2] > 1, 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))
(3): COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[3] + 1, x0[3] - 2)))

(0) -> (1), if ((x1[0] >= 0 && x0[0] <= 1 && 1 < x1[0] + 1* TRUE)∧(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])) →* 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))))

(1) -> (0), if ((983_1_log_InvokeMethod(983_0_half_LE(0, x1[1] + 1 - 2)) →* 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))))

(1) -> (2), if ((983_1_log_InvokeMethod(983_0_half_LE(0, x1[1] + 1 - 2)) →* 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))))

(2) -> (3), if ((x1[2] >= 0 && x0[2] > 1* TRUE)∧(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])) →* 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))))

(3) -> (0), if ((983_1_log_InvokeMethod(983_0_half_LE(x1[3] + 1, x0[3] - 2)) →* 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))))

(3) -> (2), if ((983_1_log_InvokeMethod(983_0_half_LE(x1[3] + 1, x0[3] - 2)) →* 983_1_log_InvokeMethod(983_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 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1, x0))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1, 0), <=(x0, 1)), <(1, +(x1, 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1, x0))) the following chains were created:
• We consider the chain 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))) which results in the following constraint:

(1)    (&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))=983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1])) ⇒ 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))≥NonInfC∧983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))≥COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))∧(UIncreasing(COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_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)    (<(1, +(x1[0], 1))=TRUE>=(x1[0], 0)=TRUE<=(x0[0], 1)=TRUE983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))≥NonInfC∧983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))≥COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))∧(UIncreasing(COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_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] + [-1] ≥ 0∧x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] + [bni_21]x1[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(4)    (x1[0] + [-1] ≥ 0∧x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] + [bni_21]x1[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(5)    (x1[0] + [-1] ≥ 0∧x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] + [bni_21]x1[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(6)    (x1[0] ≥ 0∧[1] + x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))), ≥)∧[(4)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] + [bni_21]x1[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(7)    (x1[0] ≥ 0∧[1] + x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))), ≥)∧[(4)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] + [bni_21]x1[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

(8)    (x1[0] ≥ 0∧[1] + x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))), ≥)∧[(4)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x0[0] + [bni_21]x1[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

For Pair COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1, x0))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1, 1), 2)))) the following chains were created:
• We consider the chain 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))) which results in the following constraint:

(9)    (&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))=983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))∧983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[0]1, x0[0]1))∧&&(&&(>=(x1[0]1, 0), <=(x0[0]1, 1)), <(1, +(x1[0]1, 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0]1, x0[0]1))=983_1_log_InvokeMethod(983_0_half_LE(x1[1]1, x0[1]1))∧983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1]1, 1), 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[0]2, x0[0]2))∧&&(&&(>=(x1[0]2, 0), <=(x0[0]2, 1)), <(1, +(x1[0]2, 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0]2, x0[0]2))=983_1_log_InvokeMethod(983_0_half_LE(x1[1]2, x0[1]2)) ⇒ COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1]1, x0[1]1)))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1]1, x0[1]1)))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1]1, 1), 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1]1, 1), 2))))), ≥))

We solved constraint (9) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN).
• We consider the chain 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))) which results in the following constraint:

(10)    (&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))=983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))∧983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[0]1, x0[0]1))∧&&(&&(>=(x1[0]1, 0), <=(x0[0]1, 1)), <(1, +(x1[0]1, 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0]1, x0[0]1))=983_1_log_InvokeMethod(983_0_half_LE(x1[1]1, x0[1]1))∧983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1]1, 1), 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))∧&&(>=(x1[2], 0), >(x0[2], 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))=983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3])) ⇒ COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1]1, x0[1]1)))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1]1, x0[1]1)))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1]1, 1), 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1]1, 1), 2))))), ≥))

We solved constraint (10) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN).
• We consider the chain 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))) which results in the following constraint:

(11)    (&&(>=(x1[2], 0), >(x0[2], 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))=983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))∧983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))∧&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))=983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))∧983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[0]1, x0[0]1))∧&&(&&(>=(x1[0]1, 0), <=(x0[0]1, 1)), <(1, +(x1[0]1, 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0]1, x0[0]1))=983_1_log_InvokeMethod(983_0_half_LE(x1[1]1, x0[1]1)) ⇒ COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1])))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1])))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))), ≥))

We solved constraint (11) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN).
• We consider the chain 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))) which results in the following constraint:

(12)    (&&(>=(x1[2], 0), >(x0[2], 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))=983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))∧983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))∧&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))=983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))∧983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[2]1, x0[2]1))∧&&(>=(x1[2]1, 0), >(x0[2]1, 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2]1, x0[2]1))=983_1_log_InvokeMethod(983_0_half_LE(x1[3]1, x0[3]1)) ⇒ COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1])))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1])))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))), ≥))

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<(1, +(+(x1[2], 1), 1))=TRUE>(-(+(+(x1[2], 1), 1), 2), 1)=TRUE>=(+(x1[2], 1), 0)=TRUE<=(-(x0[2], 2), 1)=TRUECOND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(+(x1[2], 1), 1), 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))), ≥))

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] ≥ 0∧x1[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧[3] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] + [bni_23]x1[2] ≥ 0∧[-2 + (-1)bso_24] + 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] ≥ 0∧x1[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧[3] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] + [bni_23]x1[2] ≥ 0∧[-2 + (-1)bso_24] + 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] ≥ 0∧x1[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧[3] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] + [bni_23]x1[2] ≥ 0∧[-2 + (-1)bso_24] + x0[2] ≥ 0)

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

(17)    ([2] + x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧[2] + x1[2] ≥ 0∧x1[2] ≥ 0∧[3] + x1[2] ≥ 0∧[3] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] + [bni_23]x1[2] ≥ 0∧[-2 + (-1)bso_24] + x0[2] ≥ 0)

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

(18)    ([2] + x1[2] ≥ 0∧x0[2] ≥ 0∧[2] + x1[2] ≥ 0∧x1[2] ≥ 0∧[3] + x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] + [bni_23]x1[2] ≥ 0∧[(-1)bso_24] + x0[2] ≥ 0)

For Pair 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1, x0))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1, 0), >(x0, 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1, x0))) the following chains were created:
• We consider the chain 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))) which results in the following constraint:

(19)    (&&(>=(x1[2], 0), >(x0[2], 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))=983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3])) ⇒ 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))≥NonInfC∧983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))≥COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))∧(UIncreasing(COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_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)=TRUE983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))≥NonInfC∧983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))≥COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))∧(UIncreasing(COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_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_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))), ≥)∧[(3)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[2] + [bni_25]x1[2] ≥ 0∧[1 + (-1)bso_26] ≥ 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_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))), ≥)∧[(3)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[2] + [bni_25]x1[2] ≥ 0∧[1 + (-1)bso_26] ≥ 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_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))), ≥)∧[(3)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[2] + [bni_25]x1[2] ≥ 0∧[1 + (-1)bso_26] ≥ 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_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))), ≥)∧[(5)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[2] + [bni_25]x1[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

For Pair COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1, x0))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1, 1), -(x0, 2)))) the following chains were created:
• We consider the chain 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))) which results in the following constraint:

(25)    (&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))=983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))∧983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))∧&&(>=(x1[2], 0), >(x0[2], 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))=983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))∧983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[0]1, x0[0]1))∧&&(&&(>=(x1[0]1, 0), <=(x0[0]1, 1)), <(1, +(x1[0]1, 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0]1, x0[0]1))=983_1_log_InvokeMethod(983_0_half_LE(x1[1]1, x0[1]1)) ⇒ COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3])))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3])))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥))

We simplified constraint (25) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

(26)    (<(1, +(x1[0], 1))=TRUE>(-(+(x1[0], 1), 2), 1)=TRUE>=(x1[0], 0)=TRUE<=(x0[0], 1)=TRUE<=(-(-(+(x1[0], 1), 2), 2), 1)=TRUECOND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[0], 1), 2))))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[0], 1), 2))))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(0, 1), -(-(+(x1[0], 1), 2), 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥))

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

(27)    (x1[0] + [-1] ≥ 0∧x1[0] + [-3] ≥ 0∧x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧[4] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(28)    (x1[0] + [-1] ≥ 0∧x1[0] + [-3] ≥ 0∧x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧[4] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(29)    (x1[0] + [-1] ≥ 0∧x1[0] + [-3] ≥ 0∧x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧[4] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(30)    (x1[0] ≥ 0∧[-2] + x1[0] ≥ 0∧[1] + x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧[3] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(31)    ([2] + x1[0] ≥ 0∧x1[0] ≥ 0∧[3] + x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(4)bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(32)    ([2] + x1[0] ≥ 0∧x1[0] ≥ 0∧[3] + x1[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(4)bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

• We consider the chain 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))) which results in the following constraint:

(33)    (&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))=983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))∧983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))∧&&(>=(x1[2], 0), >(x0[2], 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))=983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))∧983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[2]1, x0[2]1))∧&&(>=(x1[2]1, 0), >(x0[2]1, 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2]1, x0[2]1))=983_1_log_InvokeMethod(983_0_half_LE(x1[3]1, x0[3]1)) ⇒ COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3])))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3])))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥))

We simplified constraint (33) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

(34)    (<(1, +(x1[0], 1))=TRUE>(-(+(x1[0], 1), 2), 1)=TRUE>(-(-(+(x1[0], 1), 2), 2), 1)=TRUE>=(x1[0], 0)=TRUE<=(x0[0], 1)=TRUECOND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[0], 1), 2))))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[0], 1), 2))))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(0, 1), -(-(+(x1[0], 1), 2), 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥))

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

(35)    (x1[0] + [-1] ≥ 0∧x1[0] + [-3] ≥ 0∧x1[0] + [-5] ≥ 0∧x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(36)    (x1[0] + [-1] ≥ 0∧x1[0] + [-3] ≥ 0∧x1[0] + [-5] ≥ 0∧x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(37)    (x1[0] + [-1] ≥ 0∧x1[0] + [-3] ≥ 0∧x1[0] + [-5] ≥ 0∧x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(38)    (x1[0] ≥ 0∧[-2] + x1[0] ≥ 0∧[-4] + x1[0] ≥ 0∧[1] + x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(39)    ([2] + x1[0] ≥ 0∧x1[0] ≥ 0∧[-2] + x1[0] ≥ 0∧[3] + x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(4)bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(40)    ([4] + x1[0] ≥ 0∧[2] + x1[0] ≥ 0∧x1[0] ≥ 0∧[5] + x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(6)bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(41)    ([4] + x1[0] ≥ 0∧[2] + x1[0] ≥ 0∧x1[0] ≥ 0∧[5] + x1[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(6)bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)

• We consider the chain 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))), COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2)))) which results in the following constraint:

(42)    (&&(>=(x1[2], 0), >(x0[2], 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))=983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))∧983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[2]1, x0[2]1))∧&&(>=(x1[2]1, 0), >(x0[2]1, 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2]1, x0[2]1))=983_1_log_InvokeMethod(983_0_half_LE(x1[3]1, x0[3]1))∧983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))∧&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1)))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))=983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1])) ⇒ COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3]1, x0[3]1)))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3]1, x0[3]1)))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥))

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

(43)    (>=(x1[2], 0)=TRUE>(x0[2], 1)=TRUE>=(+(x1[2], 1), 0)=TRUE>(-(x0[2], 2), 1)=TRUE<(1, +(+(+(x1[2], 1), 1), 1))=TRUE>=(+(+(x1[2], 1), 1), 0)=TRUE<=(-(-(x0[2], 2), 2), 1)=TRUECOND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(+(x1[2], 1), 1), -(-(x0[2], 2), 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥))

We simplified constraint (43) using rule (POLY_CONSTRAINTS) 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] + [1] ≥ 0∧x1[2] + [2] ≥ 0∧[5] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (44) using rule (IDP_POLY_SIMPLIFY) 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] + [1] ≥ 0∧x1[2] + [2] ≥ 0∧[5] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(46)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] + [-4] ≥ 0∧x1[2] + [1] ≥ 0∧x1[2] + [2] ≥ 0∧[5] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(47)    (x1[2] ≥ 0∧x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[-2] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧x1[2] + [2] ≥ 0∧[3] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(3)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(48)    (x1[2] ≥ 0∧[2] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧x1[2] + [2] ≥ 0∧[1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(5)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(49)    (x1[2] ≥ 0∧[2] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(5)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

• We consider the chain 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))), 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))), COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))) which results in the following constraint:

(50)    (&&(>=(x1[2], 0), >(x0[2], 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))=983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))∧983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[2]1, x0[2]1))∧&&(>=(x1[2]1, 0), >(x0[2]1, 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2]1, x0[2]1))=983_1_log_InvokeMethod(983_0_half_LE(x1[3]1, x0[3]1))∧983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2)))=983_1_log_InvokeMethod(983_0_half_LE(x1[2]2, x0[2]2))∧&&(>=(x1[2]2, 0), >(x0[2]2, 1))=TRUE983_1_log_InvokeMethod(983_0_half_LE(x1[2]2, x0[2]2))=983_1_log_InvokeMethod(983_0_half_LE(x1[3]2, x0[3]2)) ⇒ COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3]1, x0[3]1)))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3]1, x0[3]1)))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥))

We simplified constraint (50) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(51)    (>=(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_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥NonInfC∧COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(+(x1[2], 1), -(x0[2], 2))))≥983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(+(x1[2], 1), 1), -(-(x0[2], 2), 2))))∧(UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥))

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

(52)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] + [-4] ≥ 0∧x1[2] + [2] ≥ 0∧x0[2] + [-6] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(53)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] + [-4] ≥ 0∧x1[2] + [2] ≥ 0∧x0[2] + [-6] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(54)    (x1[2] ≥ 0∧x0[2] + [-2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] + [-4] ≥ 0∧x1[2] + [2] ≥ 0∧x0[2] + [-6] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(55)    (x1[2] ≥ 0∧x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[-2] + x0[2] ≥ 0∧x1[2] + [2] ≥ 0∧[-4] + x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(3)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(56)    (x1[2] ≥ 0∧[2] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] ≥ 0∧x1[2] + [2] ≥ 0∧[-2] + x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(5)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(57)    (x1[2] ≥ 0∧[4] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[2] + x0[2] ≥ 0∧x1[2] + [2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(7)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (57) using rule (IDP_POLY_GCD) which results in the following new constraint:

(58)    (x1[2] ≥ 0∧[4] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[2] + x0[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(7)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1, x0))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1, 0), <=(x0, 1)), <(1, +(x1, 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1, x0)))
• (x1[0] ≥ 0∧[1] + x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))), ≥)∧[(4)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[0] + [bni_21]x1[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)
• (x1[0] ≥ 0∧[1] + x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))), ≥)∧[(4)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x0[0] + [bni_21]x1[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

• COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1, x0))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1, 1), 2))))
• ([2] + x1[2] ≥ 0∧x0[2] ≥ 0∧[2] + x1[2] ≥ 0∧x1[2] ≥ 0∧[3] + x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))), ≥)∧[(5)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] + [bni_23]x1[2] ≥ 0∧[(-1)bso_24] + x0[2] ≥ 0)

• 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1, x0))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1, 0), >(x0, 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1, x0)))
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))), ≥)∧[(5)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[2] + [bni_25]x1[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

• COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1, x0))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1, 1), -(x0, 2))))
• ([2] + x1[0] ≥ 0∧x1[0] ≥ 0∧[3] + x1[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(4)bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)
• ([4] + x1[0] ≥ 0∧[2] + x1[0] ≥ 0∧x1[0] ≥ 0∧[5] + x1[0] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))), ≥)∧[(6)bni_27 + (-1)Bound*bni_27] + [bni_27]x1[0] ≥ 0∧[(-1)bso_28] ≥ 0)
• (x1[2] ≥ 0∧[2] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(5)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)
• (x1[2] ≥ 0∧[4] + x0[2] ≥ 0∧x1[2] + [1] ≥ 0∧[2] + x0[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3]1, 1), -(x0[3]1, 2))))), ≥)∧[(7)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

POL(TRUE) = [1]
POL(FALSE) = [1]
POL(983_2_MAIN_INVOKEMETHOD(x1)) = [1] + [-1]x1
POL(983_1_log_InvokeMethod(x1)) = [-1] + x1
POL(983_0_half_LE(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(COND_983_2_MAIN_INVOKEMETHOD(x1, x2)) = [1] + [-1]x2 + [-1]x1
POL(&&(x1, x2)) = [1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(<=(x1, x2)) = [-1]
POL(1) = [1]
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-(x1, x2)) = x1 + [-1]x2
POL(2) = [2]
POL(COND_983_2_MAIN_INVOKEMETHOD1(x1, x2)) = [-1]x2
POL(>(x1, x2)) = [-1]

The following pairs are in P>:

983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(&&(&&(>=(x1[0], 0), <=(x0[0], 1)), <(1, +(x1[0], 1))), 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))
983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))

The following pairs are in Pbound:

COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))
983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2]))) → COND_983_2_MAIN_INVOKEMETHOD1(&&(>=(x1[2], 0), >(x0[2], 1)), 983_1_log_InvokeMethod(983_0_half_LE(x1[2], x0[2])))
COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))

The following pairs are in P:

COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, -(+(x1[1], 1), 2))))
COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(+(x1[3], 1), -(x0[3], 2))))

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

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

### (7) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_983_2_MAIN_INVOKEMETHOD(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[1], x0[1]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(0, x1[1] + 1 - 2)))
(3): COND_983_2_MAIN_INVOKEMETHOD1(TRUE, 983_1_log_InvokeMethod(983_0_half_LE(x1[3], x0[3]))) → 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[3] + 1, x0[3] - 2)))

The set Q is empty.

### (8) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

### (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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 983_2_MAIN_INVOKEMETHOD(983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0]))) → COND_983_2_MAIN_INVOKEMETHOD(x1[0] >= 0 && x0[0] <= 1 && 1 < x1[0] + 1, 983_1_log_InvokeMethod(983_0_half_LE(x1[0], x0[0])))

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.