### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: CountUpRound
`public class CountUpRound{  public static int round (int x) {    if (x % 2 == 0) return x;    else return x+1;  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    while (x > y) {      y = round(y+1);    }  }}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:
CountUpRound.main([Ljava/lang/String;)V: Graph of 170 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 25 rules for P and 2 rules for R.

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

Filtered ground terms:

Filtered duplicate args:

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): 551_0_MAIN_LOAD(x1[0], x0[0]) → COND_551_0_MAIN_LOAD(x1[0] >= 0 && x1[0] < x0[0] && 0 < x1[0] + 1 && !(x1[0] + 1 % 2 = 0), x1[0], x0[0])
(2): 551_0_MAIN_LOAD(x1[2], x0[2]) → COND_551_0_MAIN_LOAD1(x1[2] >= 0 && x1[2] < x0[2] && 0 = x1[2] + 1 % 2, x1[2], x0[2])

(0) -> (1), if ((x1[0] >= 0 && x1[0] < x0[0] && 0 < x1[0] + 1 && !(x1[0] + 1 % 2 = 0) →* TRUE)∧(x1[0]* x1[1])∧(x0[0]* x0[1]))

(1) -> (0), if ((x1[1] + 1 + 1* x1[0])∧(x0[1]* x0[0]))

(1) -> (2), if ((x1[1] + 1 + 1* x1[2])∧(x0[1]* x0[2]))

(2) -> (3), if ((x1[2] >= 0 && x1[2] < x0[2] && 0 = x1[2] + 1 % 2* TRUE)∧(x1[2]* x1[3])∧(x0[2]* x0[3]))

(3) -> (0), if ((x1[3] + 1* x1[0])∧(x0[3]* x0[0]))

(3) -> (2), if ((x1[3] + 1* x1[2])∧(x0[3]* 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 551_0_MAIN_LOAD(x1, x0) → COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1, 0), <(x1, x0)), <(0, +(x1, 1))), !(=(%(+(x1, 1), 2), 0))), x1, x0) the following chains were created:
• We consider the chain 551_0_MAIN_LOAD(x1[0], x0[0]) → COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0]), COND_551_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1]) which results in the following constraint:

(1)    (&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0)))=TRUEx1[0]=x1[1]x0[0]=x0[1]551_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧551_0_MAIN_LOAD(x1[0], x0[0])≥COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])∧(UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥))

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

(2)    (<(0, +(x1[0], 1))=TRUE>=(x1[0], 0)=TRUE<(x1[0], x0[0])=TRUE<(%(+(x1[0], 1), 2), 0)=TRUE551_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧551_0_MAIN_LOAD(x1[0], x0[0])≥COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])∧(UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥))

(3)    (<(0, +(x1[0], 1))=TRUE>=(x1[0], 0)=TRUE<(x1[0], x0[0])=TRUE>(%(+(x1[0], 1), 2), 0)=TRUE551_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧551_0_MAIN_LOAD(x1[0], x0[0])≥COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])∧(UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥))

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

(4)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(5)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(6)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(7)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(8)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(9)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(10)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(11)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(12)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(13)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

For Pair COND_551_0_MAIN_LOAD(TRUE, x1, x0) → 551_0_MAIN_LOAD(+(+(x1, 1), 1), x0) the following chains were created:
• We consider the chain COND_551_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1]) which results in the following constraint:

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(15)    ((UIncreasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥)∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(16)    ((UIncreasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥)∧[1 + (-1)bso_16] ≥ 0)

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

(17)    ((UIncreasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥)∧[1 + (-1)bso_16] ≥ 0)

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

(18)    ((UIncreasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

For Pair 551_0_MAIN_LOAD(x1, x0) → COND_551_0_MAIN_LOAD1(&&(&&(>=(x1, 0), <(x1, x0)), =(0, %(+(x1, 1), 2))), x1, x0) the following chains were created:
• We consider the chain 551_0_MAIN_LOAD(x1[2], x0[2]) → COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2]), COND_551_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 551_0_MAIN_LOAD(+(x1[3], 1), x0[3]) which results in the following constraint:

(19)    (&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2)))=TRUEx1[2]=x1[3]x0[2]=x0[3]551_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧551_0_MAIN_LOAD(x1[2], x0[2])≥COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])∧(UIncreasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥))

We simplified constraint (19) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(20)    (>=(x1[2], 0)=TRUE<(x1[2], x0[2])=TRUE>=(0, %(+(x1[2], 1), 2))=TRUE<=(0, %(+(x1[2], 1), 2))=TRUE551_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧551_0_MAIN_LOAD(x1[2], x0[2])≥COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])∧(UIncreasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), 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] + [-1] + [-1]x1[2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] + [(-1)bni_17]x1[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(22)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] + [(-1)bni_17]x1[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(23)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] + [(-1)bni_17]x1[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(24)    (x1[2] ≥ 0∧x0[2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(25)    (x1[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[(-1)bso_18] ≥ 0)

For Pair COND_551_0_MAIN_LOAD1(TRUE, x1, x0) → 551_0_MAIN_LOAD(+(x1, 1), x0) the following chains were created:
• We consider the chain COND_551_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 551_0_MAIN_LOAD(+(x1[3], 1), x0[3]) which results in the following constraint:

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

(27)    ((UIncreasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)

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

(28)    ((UIncreasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)

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

(29)    ((UIncreasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)

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

(30)    ((UIncreasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 551_0_MAIN_LOAD(x1, x0) → COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1, 0), <(x1, x0)), <(0, +(x1, 1))), !(=(%(+(x1, 1), 2), 0))), x1, x0)
• (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)
• (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

• ((UIncreasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

• 551_0_MAIN_LOAD(x1, x0) → COND_551_0_MAIN_LOAD1(&&(&&(>=(x1, 0), <(x1, x0)), =(0, %(+(x1, 1), 2))), x1, x0)
• (x1[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[(-1)bso_18] ≥ 0)

• ((UIncreasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 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(551_0_MAIN_LOAD(x1, x2)) = [2] + x2 + [-1]x1
POL(COND_551_0_MAIN_LOAD(x1, x2, x3)) = [1] + x3 + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(!(x1)) = [-1]
POL(=(x1, x2)) = [-1]
POL(2) = [2]
POL(COND_551_0_MAIN_LOAD1(x1, x2, x3)) = [2] + x3 + [-1]x2

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(%(x1, 2)-1 @ {}) = min{x2, [-1]x2}
POL(%(x1, 2)1 @ {}) = max{x2, [-1]x2}

The following pairs are in P>:

551_0_MAIN_LOAD(x1[0], x0[0]) → COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])

The following pairs are in Pbound:

551_0_MAIN_LOAD(x1[0], x0[0]) → COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])
551_0_MAIN_LOAD(x1[2], x0[2]) → COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])

The following pairs are in P:

551_0_MAIN_LOAD(x1[2], x0[2]) → COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])

There are no usable rules.

### (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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): 551_0_MAIN_LOAD(x1[2], x0[2]) → COND_551_0_MAIN_LOAD1(x1[2] >= 0 && x1[2] < x0[2] && 0 = x1[2] + 1 % 2, x1[2], x0[2])

The set Q is empty.

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges: