### (0) Obligation:

JBC Problem based on JBC Program:
`No human-readable program information known.`

Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB8

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 117 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:
Load292(i41) → Cond_Load292(i41 > 0 && 0 = i41 % 2, i41)
Load292(i41) → Cond_Load2921(i41 % 2 > 0 && i41 > 0, i41)
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:
Load292(i41) → Cond_Load292(i41 > 0 && 0 = i41 % 2, i41)
Load292(i41) → Cond_Load2921(i41 % 2 > 0 && i41 > 0, i41)

The integer pair graph contains the following rules and edges:
(0): LOAD292(i41[0]) → COND_LOAD292(i41[0] > 0 && 0 = i41[0] % 2, i41[0])
(2): LOAD292(i41[2]) → COND_LOAD2921(i41[2] % 2 > 0 && i41[2] > 0, i41[2])

(0) -> (1), if ((i41[0]* i41[1])∧(i41[0] > 0 && 0 = i41[0] % 2* TRUE))

(1) -> (0), if ((i41[1] / 2* i41[0]))

(1) -> (2), if ((i41[1] / 2* i41[2]))

(2) -> (3), if ((i41[2] % 2 > 0 && i41[2] > 0* TRUE)∧(i41[2]* i41[3]))

(3) -> (0), if ((i41[3] + -1* i41[0]))

(3) -> (2), if ((i41[3] + -1* i41[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): LOAD292(i41[0]) → COND_LOAD292(i41[0] > 0 && 0 = i41[0] % 2, i41[0])
(2): LOAD292(i41[2]) → COND_LOAD2921(i41[2] % 2 > 0 && i41[2] > 0, i41[2])

(0) -> (1), if ((i41[0]* i41[1])∧(i41[0] > 0 && 0 = i41[0] % 2* TRUE))

(1) -> (0), if ((i41[1] / 2* i41[0]))

(1) -> (2), if ((i41[1] / 2* i41[2]))

(2) -> (3), if ((i41[2] % 2 > 0 && i41[2] > 0* TRUE)∧(i41[2]* i41[3]))

(3) -> (0), if ((i41[3] + -1* i41[0]))

(3) -> (2), if ((i41[3] + -1* i41[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 LOAD292(i41) → COND_LOAD292(&&(>(i41, 0), =(0, %(i41, 2))), i41) the following chains were created:
• We consider the chain LOAD292(i41[0]) → COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0]), COND_LOAD292(TRUE, i41[1]) → LOAD292(/(i41[1], 2)) which results in the following constraint:

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

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

(3)    (i41[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(4)    (i41[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(5)    (i41[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(6)    (i41[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(7)    (i41[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_LOAD292(TRUE, i41) → LOAD292(/(i41, 2)) the following chains were created:
• We consider the chain LOAD292(i41[0]) → COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0]), COND_LOAD292(TRUE, i41[1]) → LOAD292(/(i41[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)    (i41[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] + i41[0] + [-1]max{i41[0], [-1]i41[0]} ≥ 0)

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

(11)    (i41[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] + i41[0] + [-1]max{i41[0], [-1]i41[0]} ≥ 0)

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

(12)    (i41[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]i41[0] ≥ 0 ⇒ (UIncreasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

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

(13)    (i41[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i41[0] ≥ 0 ⇒ (UIncreasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

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

(14)    (i41[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i41[0] ≥ 0 ⇒ (UIncreasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

For Pair LOAD292(i41) → COND_LOAD2921(&&(>(%(i41, 2), 0), >(i41, 0)), i41) the following chains were created:
• We consider the chain LOAD292(i41[2]) → COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2]), COND_LOAD2921(TRUE, i41[3]) → LOAD292(+(i41[3], -1)) which results in the following constraint:

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

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

(17)    (max{[2], [-2]} + [-1] ≥ 0∧i41[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i41[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∧i41[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(19)    (i41[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(20)    (i41[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(21)    (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)

For Pair COND_LOAD2921(TRUE, i41) → LOAD292(+(i41, -1)) the following chains were created:

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∧i41[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[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∧i41[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[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)    (i41[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(27)    (i41[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(28)    (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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∧i41[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[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∧i41[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[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)    (i41[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(34)    (i41[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(35)    (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i41[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

• (i41[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i41[0] ≥ 0 ⇒ (UIncreasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

• (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)

• (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
• (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[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) = 0
POL(COND_LOAD292(x1, x2)) = [-1] + x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(=(x1, x2)) = [-1]
POL(2) = [2]
POL(COND_LOAD2921(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 @ {LOAD292_1/0}) = max{x1, [-1]x1} + [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

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

FALSE1&&(TRUE, FALSE)1
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): LOAD292(i41[0]) → COND_LOAD292(i41[0] > 0 && 0 = i41[0] % 2, i41[0])
(2): LOAD292(i41[2]) → COND_LOAD2921(i41[2] % 2 > 0 && i41[2] > 0, i41[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: