### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB8
`/** * Example taken from "A Term Rewriting Approach to the Automated Termination * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf) * and converted to Java. */public class PastaB8 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        if (x > 0) {            while (x != 0) {                if (x % 2 == 0) {                    x = x/2;                } else {                    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:
PastaB8.main([Ljava/lang/String;)V: Graph of 103 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 19 rules for P and 3 rules for R.

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

Filtered ground terms:

332_0_main_EQ(x1, x2, x3) → 332_0_main_EQ(x2, x3)
Cond_332_0_main_EQ1(x1, x2, x3, x4) → Cond_332_0_main_EQ1(x1, x3, x4)
Cond_332_0_main_EQ(x1, x2, x3, x4) → Cond_332_0_main_EQ(x1, x3, x4)

Filtered duplicate args:

332_0_main_EQ(x1, x2) → 332_0_main_EQ(x2)
Cond_332_0_main_EQ1(x1, x2, x3) → Cond_332_0_main_EQ1(x1, x3)
Cond_332_0_main_EQ(x1, x2, x3) → Cond_332_0_main_EQ(x1, 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): 332_0_MAIN_EQ(x0[0]) → COND_332_0_MAIN_EQ(x0[0] > 0 && !(x0[0] % 2 = 0), x0[0])
(1): COND_332_0_MAIN_EQ(TRUE, x0[1]) → 332_0_MAIN_EQ(x0[1] + -1)
(2): 332_0_MAIN_EQ(x0[2]) → COND_332_0_MAIN_EQ1(x0[2] >= 1 && !(x0[2] = 0) && 0 = x0[2] % 2, x0[2])
(3): COND_332_0_MAIN_EQ1(TRUE, x0[3]) → 332_0_MAIN_EQ(x0[3] / 2)

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

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

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

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

(3) -> (0), if ((x0[3] / 2* x0[0]))

(3) -> (2), if ((x0[3] / 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 332_0_MAIN_EQ(x0) → COND_332_0_MAIN_EQ(&&(>(x0, 0), !(=(%(x0, 2), 0))), x0) the following chains were created:
• We consider the chain 332_0_MAIN_EQ(x0[0]) → COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0]), COND_332_0_MAIN_EQ(TRUE, x0[1]) → 332_0_MAIN_EQ(+(x0[1], -1)) which results in the following constraint:

(1)    (&&(>(x0[0], 0), !(=(%(x0[0], 2), 0)))=TRUEx0[0]=x0[1]332_0_MAIN_EQ(x0[0])≥NonInfC∧332_0_MAIN_EQ(x0[0])≥COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])∧(UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE<(%(x0[0], 2), 0)=TRUE332_0_MAIN_EQ(x0[0])≥NonInfC∧332_0_MAIN_EQ(x0[0])≥COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])∧(UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥))

(3)    (>(x0[0], 0)=TRUE>(%(x0[0], 2), 0)=TRUE332_0_MAIN_EQ(x0[0])≥NonInfC∧332_0_MAIN_EQ(x0[0])≥COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])∧(UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥))

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

(4)    (x0[0] + [-1] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(6)    (x0[0] + [-1] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(7)    (x0[0] + [-1] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(8)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(9)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(10)    (x0[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(11)    (x0[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(12)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(13)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_332_0_MAIN_EQ(TRUE, x0) → 332_0_MAIN_EQ(+(x0, -1)) the following chains were created:
• We consider the chain 332_0_MAIN_EQ(x0[0]) → COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0]), COND_332_0_MAIN_EQ(TRUE, x0[1]) → 332_0_MAIN_EQ(+(x0[1], -1)), 332_0_MAIN_EQ(x0[0]) → COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0]) which results in the following constraint:

(14)    (&&(>(x0[0], 0), !(=(%(x0[0], 2), 0)))=TRUEx0[0]=x0[1]+(x0[1], -1)=x0[0]1COND_332_0_MAIN_EQ(TRUE, x0[1])≥NonInfC∧COND_332_0_MAIN_EQ(TRUE, x0[1])≥332_0_MAIN_EQ(+(x0[1], -1))∧(UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥))

We simplified constraint (14) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraints:

(15)    (>(x0[0], 0)=TRUE<(%(x0[0], 2), 0)=TRUECOND_332_0_MAIN_EQ(TRUE, x0[0])≥NonInfC∧COND_332_0_MAIN_EQ(TRUE, x0[0])≥332_0_MAIN_EQ(+(x0[0], -1))∧(UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥))

(16)    (>(x0[0], 0)=TRUE>(%(x0[0], 2), 0)=TRUECOND_332_0_MAIN_EQ(TRUE, x0[0])≥NonInfC∧COND_332_0_MAIN_EQ(TRUE, x0[0])≥332_0_MAIN_EQ(+(x0[0], -1))∧(UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥))

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

(17)    (x0[0] + [-1] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(18)    (x0[0] + [-1] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(19)    (x0[0] + [-1] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (18) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(20)    (x0[0] + [-1] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (19) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(21)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(22)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(23)    (x0[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(24)    (x0[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(25)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(26)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

• We consider the chain 332_0_MAIN_EQ(x0[0]) → COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0]), COND_332_0_MAIN_EQ(TRUE, x0[1]) → 332_0_MAIN_EQ(+(x0[1], -1)), 332_0_MAIN_EQ(x0[2]) → COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2]) which results in the following constraint:

(27)    (&&(>(x0[0], 0), !(=(%(x0[0], 2), 0)))=TRUEx0[0]=x0[1]+(x0[1], -1)=x0[2]COND_332_0_MAIN_EQ(TRUE, x0[1])≥NonInfC∧COND_332_0_MAIN_EQ(TRUE, x0[1])≥332_0_MAIN_EQ(+(x0[1], -1))∧(UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥))

We simplified constraint (27) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraints:

(28)    (>(x0[0], 0)=TRUE<(%(x0[0], 2), 0)=TRUECOND_332_0_MAIN_EQ(TRUE, x0[0])≥NonInfC∧COND_332_0_MAIN_EQ(TRUE, x0[0])≥332_0_MAIN_EQ(+(x0[0], -1))∧(UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥))

(29)    (>(x0[0], 0)=TRUE>(%(x0[0], 2), 0)=TRUECOND_332_0_MAIN_EQ(TRUE, x0[0])≥NonInfC∧COND_332_0_MAIN_EQ(TRUE, x0[0])≥332_0_MAIN_EQ(+(x0[0], -1))∧(UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥))

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

(30)    (x0[0] + [-1] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(31)    (x0[0] + [-1] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(32)    (x0[0] + [-1] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(33)    (x0[0] + [-1] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(34)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (33) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(35)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(36)    (x0[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(37)    (x0[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(38)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(39)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

For Pair 332_0_MAIN_EQ(x0) → COND_332_0_MAIN_EQ1(&&(&&(>=(x0, 1), !(=(x0, 0))), =(0, %(x0, 2))), x0) the following chains were created:
• We consider the chain 332_0_MAIN_EQ(x0[2]) → COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2]), COND_332_0_MAIN_EQ1(TRUE, x0[3]) → 332_0_MAIN_EQ(/(x0[3], 2)) which results in the following constraint:

(40)    (&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2)))=TRUEx0[2]=x0[3]332_0_MAIN_EQ(x0[2])≥NonInfC∧332_0_MAIN_EQ(x0[2])≥COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])∧(UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥))

We simplified constraint (40) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:

(41)    (>=(x0[2], 1)=TRUE>=(0, %(x0[2], 2))=TRUE<=(0, %(x0[2], 2))=TRUE<(x0[2], 0)=TRUE332_0_MAIN_EQ(x0[2])≥NonInfC∧332_0_MAIN_EQ(x0[2])≥COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])∧(UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥))

(42)    (>=(x0[2], 1)=TRUE>=(0, %(x0[2], 2))=TRUE<=(0, %(x0[2], 2))=TRUE>(x0[2], 0)=TRUE332_0_MAIN_EQ(x0[2])≥NonInfC∧332_0_MAIN_EQ(x0[2])≥COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])∧(UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥))

We simplified constraint (41) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(43)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0∧[-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(44)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (43) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(45)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0∧[-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (44) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(46)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_21] ≥ 0)

We solved constraint (45) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(47)    (x0[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(48)    (x0[2] ≥ 0∧x0[2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(49)    (x0[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair COND_332_0_MAIN_EQ1(TRUE, x0) → 332_0_MAIN_EQ(/(x0, 2)) the following chains were created:
• We consider the chain 332_0_MAIN_EQ(x0[2]) → COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2]), COND_332_0_MAIN_EQ1(TRUE, x0[3]) → 332_0_MAIN_EQ(/(x0[3], 2)) which results in the following constraint:

(50)    (&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2)))=TRUEx0[2]=x0[3]COND_332_0_MAIN_EQ1(TRUE, x0[3])≥NonInfC∧COND_332_0_MAIN_EQ1(TRUE, x0[3])≥332_0_MAIN_EQ(/(x0[3], 2))∧(UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥))

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

(51)    (>=(x0[2], 1)=TRUE>=(0, %(x0[2], 2))=TRUE<=(0, %(x0[2], 2))=TRUE<(x0[2], 0)=TRUECOND_332_0_MAIN_EQ1(TRUE, x0[2])≥NonInfC∧COND_332_0_MAIN_EQ1(TRUE, x0[2])≥332_0_MAIN_EQ(/(x0[2], 2))∧(UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥))

(52)    (>=(x0[2], 1)=TRUE>=(0, %(x0[2], 2))=TRUE<=(0, %(x0[2], 2))=TRUE>(x0[2], 0)=TRUECOND_332_0_MAIN_EQ1(TRUE, x0[2])≥NonInfC∧COND_332_0_MAIN_EQ1(TRUE, x0[2])≥332_0_MAIN_EQ(/(x0[2], 2))∧(UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥))

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

(53)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0∧[-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[1 + (-1)bso_26] + x0[2] + [-1]max{x0[2], [-1]x0[2]} ≥ 0)

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

(54)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[1 + (-1)bso_26] + x0[2] + [-1]max{x0[2], [-1]x0[2]} ≥ 0)

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

(55)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0∧[-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[1 + (-1)bso_26] + x0[2] + [-1]max{x0[2], [-1]x0[2]} ≥ 0)

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

(56)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[1 + (-1)bso_26] + x0[2] + [-1]max{x0[2], [-1]x0[2]} ≥ 0)

We solved constraint (55) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (56) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(57)    (x0[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]x0[2] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

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

(58)    (x0[2] ≥ 0∧x0[2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]x0[2] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

We simplified constraint (58) using rule (IDP_POLY_GCD) which results in the following new constraint:

(59)    (x0[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + x0[2] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 332_0_MAIN_EQ(x0) → COND_332_0_MAIN_EQ(&&(>(x0, 0), !(=(%(x0, 2), 0))), x0)
• (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)
• (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

• COND_332_0_MAIN_EQ(TRUE, x0) → 332_0_MAIN_EQ(+(x0, -1))
• (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)
• (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)
• (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)
• (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

• 332_0_MAIN_EQ(x0) → COND_332_0_MAIN_EQ1(&&(&&(>=(x0, 1), !(=(x0, 0))), =(0, %(x0, 2))), x0)
• (x0[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_21] ≥ 0)

• COND_332_0_MAIN_EQ1(TRUE, x0) → 332_0_MAIN_EQ(/(x0, 2))
• (x0[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + x0[2] ≥ 0 ⇒ (UIncreasing(332_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[1 + (-1)bso_26] ≥ 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) = [2]
POL(FALSE) = [1]
POL(332_0_MAIN_EQ(x1)) = [-1] + x1
POL(COND_332_0_MAIN_EQ(x1, x2)) = [-1] + x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(!(x1)) = [-1]
POL(=(x1, x2)) = [-1]
POL(2) = [2]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_332_0_MAIN_EQ1(x1, x2)) = [-1] + x2
POL(>=(x1, x2)) = [-1]
POL(1) = [1]

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}
POL(/(x1, 2)1 @ {332_0_MAIN_EQ_1/0}) = max{x1, [-1]x1} + [-1]

The following pairs are in P>:

COND_332_0_MAIN_EQ(TRUE, x0[1]) → 332_0_MAIN_EQ(+(x0[1], -1))
COND_332_0_MAIN_EQ1(TRUE, x0[3]) → 332_0_MAIN_EQ(/(x0[3], 2))

The following pairs are in Pbound:

332_0_MAIN_EQ(x0[0]) → COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])
COND_332_0_MAIN_EQ(TRUE, x0[1]) → 332_0_MAIN_EQ(+(x0[1], -1))
332_0_MAIN_EQ(x0[2]) → COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])
COND_332_0_MAIN_EQ1(TRUE, x0[3]) → 332_0_MAIN_EQ(/(x0[3], 2))

The following pairs are in P:

332_0_MAIN_EQ(x0[0]) → COND_332_0_MAIN_EQ(&&(>(x0[0], 0), !(=(%(x0[0], 2), 0))), x0[0])
332_0_MAIN_EQ(x0[2]) → COND_332_0_MAIN_EQ1(&&(&&(>=(x0[2], 1), !(=(x0[2], 0))), =(0, %(x0[2], 2))), x0[2])

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

TRUE1&&(TRUE, TRUE)1
FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)1
FALSE1&&(FALSE, FALSE)1
/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): 332_0_MAIN_EQ(x0[0]) → COND_332_0_MAIN_EQ(x0[0] > 0 && !(x0[0] % 2 = 0), x0[0])
(2): 332_0_MAIN_EQ(x0[2]) → COND_332_0_MAIN_EQ1(x0[2] >= 1 && !(x0[2] = 0) && 0 = x0[2] % 2, x0[2])

The set Q is empty.

### (7) IDependencyGraphProof (EQUIVALENT transformation)

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