### (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:
Graph of 140 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

### (4) Obligation:

ITRS 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 TRS R consists of the following rules:
Load644(i72) → Cond_Load644(i72 > 0 && 0 = i72 % 2, i72)
Load644(i72) → Cond_Load6441(i72 % 2 > 0 && i72 > 0, i72)
The set Q consists of the following terms:

### (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

The ITRS R consists of the following rules:
Load644(i72) → Cond_Load644(i72 > 0 && 0 = i72 % 2, i72)
Load644(i72) → Cond_Load6441(i72 % 2 > 0 && i72 > 0, i72)

The integer pair graph contains the following rules and edges:
(0): LOAD644(i72[0]) → COND_LOAD644(i72[0] > 0 && 0 = i72[0] % 2, i72[0])
(1): COND_LOAD644(TRUE, i72[1]) → LOAD644(i72[1] / 2)
(2): LOAD644(i72[2]) → COND_LOAD6441(i72[2] % 2 > 0 && i72[2] > 0, i72[2])
(3): COND_LOAD6441(TRUE, i72[3]) → LOAD644(i72[3] + -1)

(0) -> (1), if ((i72[0] > 0 && 0 = i72[0] % 2* TRUE)∧(i72[0]* i72[1]))

(1) -> (0), if ((i72[1] / 2* i72[0]))

(1) -> (2), if ((i72[1] / 2* i72[2]))

(2) -> (3), if ((i72[2] % 2 > 0 && i72[2] > 0* TRUE)∧(i72[2]* i72[3]))

(3) -> (0), if ((i72[3] + -1* i72[0]))

(3) -> (2), if ((i72[3] + -1* i72[2]))

The set Q consists of the following terms:

### (7) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

### (8) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD644(i72[0]) → COND_LOAD644(i72[0] > 0 && 0 = i72[0] % 2, i72[0])
(1): COND_LOAD644(TRUE, i72[1]) → LOAD644(i72[1] / 2)
(2): LOAD644(i72[2]) → COND_LOAD6441(i72[2] % 2 > 0 && i72[2] > 0, i72[2])
(3): COND_LOAD6441(TRUE, i72[3]) → LOAD644(i72[3] + -1)

(0) -> (1), if ((i72[0] > 0 && 0 = i72[0] % 2* TRUE)∧(i72[0]* i72[1]))

(1) -> (0), if ((i72[1] / 2* i72[0]))

(1) -> (2), if ((i72[1] / 2* i72[2]))

(2) -> (3), if ((i72[2] % 2 > 0 && i72[2] > 0* TRUE)∧(i72[2]* i72[3]))

(3) -> (0), if ((i72[3] + -1* i72[0]))

(3) -> (2), if ((i72[3] + -1* i72[2]))

The set Q consists of the following terms:

### (9) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair LOAD644(i72) → COND_LOAD644(&&(>(i72, 0), =(0, %(i72, 2))), i72) the following chains were created:
• We consider the chain LOAD644(i72[0]) → COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0]), COND_LOAD644(TRUE, i72[1]) → LOAD644(/(i72[1], 2)) which results in the following constraint:

(1)    (&&(>(i72[0], 0), =(0, %(i72[0], 2)))=TRUEi72[0]=i72[1]LOAD644(i72[0])≥NonInfC∧LOAD644(i72[0])≥COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])∧(UIncreasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥))

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

(2)    (>(i72[0], 0)=TRUE>=(0, %(i72[0], 2))=TRUE<=(0, %(i72[0], 2))=TRUELOAD644(i72[0])≥NonInfC∧LOAD644(i72[0])≥COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])∧(UIncreasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥))

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

(3)    (i72[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(4)    (i72[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(5)    (i72[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(6)    (i72[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(7)    (i72[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_LOAD644(TRUE, i72) → LOAD644(/(i72, 2)) the following chains were created:
• We consider the chain LOAD644(i72[0]) → COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0]), COND_LOAD644(TRUE, i72[1]) → LOAD644(/(i72[1], 2)) which results in the following constraint:

We simplified constraint (8) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:

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

(10)    (i72[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] + i72[0] + [-1]max{i72[0], [-1]i72[0]} ≥ 0)

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

(11)    (i72[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] + i72[0] + [-1]max{i72[0], [-1]i72[0]} ≥ 0)

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

(12)    (i72[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]i72[0] ≥ 0 ⇒ (UIncreasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(13)    (i72[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i72[0] ≥ 0 ⇒ (UIncreasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (13) using rule (IDP_POLY_GCD) which results in the following new constraint:

(14)    (i72[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i72[0] ≥ 0 ⇒ (UIncreasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

For Pair LOAD644(i72) → COND_LOAD6441(&&(>(%(i72, 2), 0), >(i72, 0)), i72) the following chains were created:
• We consider the chain LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2]), COND_LOAD6441(TRUE, i72[3]) → LOAD644(+(i72[3], -1)) which results in the following constraint:

(15)    (&&(>(%(i72[2], 2), 0), >(i72[2], 0))=TRUEi72[2]=i72[3]LOAD644(i72[2])≥NonInfC∧LOAD644(i72[2])≥COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])∧(UIncreasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥))

We simplified constraint (15) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(16)    (>(%(i72[2], 2), 0)=TRUE>(i72[2], 0)=TRUELOAD644(i72[2])≥NonInfC∧LOAD644(i72[2])≥COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])∧(UIncreasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥))

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

(17)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(18)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(19)    (i72[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(20)    (i72[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(21)    (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

For Pair COND_LOAD6441(TRUE, i72) → LOAD644(+(i72, -1)) the following chains were created:
• We consider the chain LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2]), COND_LOAD6441(TRUE, i72[3]) → LOAD644(+(i72[3], -1)), LOAD644(i72[0]) → COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0]) which results in the following constraint:

We simplified constraint (22) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(24)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(25)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(26)    (i72[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(27)    (i72[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(28)    (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

• We consider the chain LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2]), COND_LOAD6441(TRUE, i72[3]) → LOAD644(+(i72[3], -1)), LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2]) which results in the following constraint:

We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(31)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(32)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(33)    (i72[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(34)    (i72[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(35)    (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD644(i72) → COND_LOAD644(&&(>(i72, 0), =(0, %(i72, 2))), i72)
• (i72[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

• (i72[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i72[0] ≥ 0 ⇒ (UIncreasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

• LOAD644(i72) → COND_LOAD6441(&&(>(%(i72, 2), 0), >(i72, 0)), i72)
• (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

• (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
• (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 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) = [3]
POL(LOAD644(x1)) = [-1] + x1
POL(COND_LOAD644(x1, x2)) = [-1] + x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(=(x1, x2)) = [-1]
POL(2) = [2]
POL(COND_LOAD6441(x1, x2)) = [-1] + x2
POL(+(x1, x2)) = x1 + x2
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 @ {LOAD644_1/0}) = max{x1, [-1]x1} + [-1]

The following pairs are in P>:

The following pairs are in Pbound:

LOAD644(i72[0]) → COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])
LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])

The following pairs are in P:

LOAD644(i72[0]) → COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])
LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])

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

FALSE1&&(FALSE, FALSE)1
/1

### (11) 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): LOAD644(i72[0]) → COND_LOAD644(i72[0] > 0 && 0 = i72[0] % 2, i72[0])
(2): LOAD644(i72[2]) → COND_LOAD6441(i72[2] % 2 > 0 && i72[2] > 0, i72[2])

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

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

### (14) Obligation:

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

The following domains are used:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: