### (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: AProVEMath

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 215 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:
Load845(i108, i111, i110) → Cond_Load845(i111 > 0 && 1 = i111 % 2, i108, i111, i110)
Load845(i108, i111, i110) → Cond_Load8451(i111 % 2 >= 0 && !(i111 % 2 = 1) && i111 > 0, i108, i111, i110)
Cond_Load8451(TRUE, i108, i111, i110) → Load845(i108 * i108, i111 / 2, i110)
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:
Load845(i108, i111, i110) → Cond_Load845(i111 > 0 && 1 = i111 % 2, i108, i111, i110)
Load845(i108, i111, i110) → Cond_Load8451(i111 % 2 >= 0 && !(i111 % 2 = 1) && i111 > 0, i108, i111, i110)
Cond_Load8451(TRUE, i108, i111, i110) → Load845(i108 * i108, i111 / 2, i110)

The integer pair graph contains the following rules and edges:
(0): LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(i111[0] > 0 && 1 = i111[0] % 2, i108[0], i111[0], i110[0])
(2): LOAD888(i108[2], i111[2], i110[2]) → LOAD845(i108[2] * i108[2], i111[2] / 2, i110[2])
(3): LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(i111[3] % 2 >= 0 && !(i111[3] % 2 = 1) && i111[3] > 0, i108[3], i111[3], i110[3])
(4): COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(i108[4] * i108[4], i111[4] / 2, i110[4])

(0) -> (1), if ((i111[0] > 0 && 1 = i111[0] % 2* TRUE)∧(i111[0]* i111[1])∧(i110[0]* i110[1])∧(i108[0]* i108[1]))

(1) -> (2), if ((i108[1]* i108[2])∧(i110[1] * i108[1]* i110[2])∧(i111[1]* i111[2]))

(2) -> (0), if ((i110[2]* i110[0])∧(i108[2] * i108[2]* i108[0])∧(i111[2] / 2* i111[0]))

(2) -> (3), if ((i110[2]* i110[3])∧(i111[2] / 2* i111[3])∧(i108[2] * i108[2]* i108[3]))

(3) -> (4), if ((i110[3]* i110[4])∧(i108[3]* i108[4])∧(i111[3] % 2 >= 0 && !(i111[3] % 2 = 1) && i111[3] > 0* TRUE)∧(i111[3]* i111[4]))

(4) -> (0), if ((i108[4] * i108[4]* i108[0])∧(i110[4]* i110[0])∧(i111[4] / 2* i111[0]))

(4) -> (3), if ((i111[4] / 2* i111[3])∧(i110[4]* i110[3])∧(i108[4] * i108[4]* i108[3]))

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): LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(i111[0] > 0 && 1 = i111[0] % 2, i108[0], i111[0], i110[0])
(2): LOAD888(i108[2], i111[2], i110[2]) → LOAD845(i108[2] * i108[2], i111[2] / 2, i110[2])
(3): LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(i111[3] % 2 >= 0 && !(i111[3] % 2 = 1) && i111[3] > 0, i108[3], i111[3], i110[3])
(4): COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(i108[4] * i108[4], i111[4] / 2, i110[4])

(0) -> (1), if ((i111[0] > 0 && 1 = i111[0] % 2* TRUE)∧(i111[0]* i111[1])∧(i110[0]* i110[1])∧(i108[0]* i108[1]))

(1) -> (2), if ((i108[1]* i108[2])∧(i110[1] * i108[1]* i110[2])∧(i111[1]* i111[2]))

(2) -> (0), if ((i110[2]* i110[0])∧(i108[2] * i108[2]* i108[0])∧(i111[2] / 2* i111[0]))

(2) -> (3), if ((i110[2]* i110[3])∧(i111[2] / 2* i111[3])∧(i108[2] * i108[2]* i108[3]))

(3) -> (4), if ((i110[3]* i110[4])∧(i108[3]* i108[4])∧(i111[3] % 2 >= 0 && !(i111[3] % 2 = 1) && i111[3] > 0* TRUE)∧(i111[3]* i111[4]))

(4) -> (0), if ((i108[4] * i108[4]* i108[0])∧(i110[4]* i110[0])∧(i111[4] / 2* i111[0]))

(4) -> (3), if ((i111[4] / 2* i111[3])∧(i110[4]* i110[3])∧(i108[4] * i108[4]* i108[3]))

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 LOAD845(i108, i111, i110) → COND_LOAD845(&&(>(i111, 0), =(1, %(i111, 2))), i108, i111, i110) the following chains were created:
• We consider the chain LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0]), COND_LOAD845(TRUE, i108[1], i111[1], i110[1]) → LOAD888(i108[1], i111[1], *(i110[1], i108[1])) which results in the following constraint:

(1)    (&&(>(i111[0], 0), =(1, %(i111[0], 2)))=TRUEi111[0]=i111[1]i110[0]=i110[1]i108[0]=i108[1]LOAD845(i108[0], i111[0], i110[0])≥NonInfC∧LOAD845(i108[0], i111[0], i110[0])≥COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])∧(UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥))

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

(2)    (>(i111[0], 0)=TRUE>=(1, %(i111[0], 2))=TRUE<=(1, %(i111[0], 2))=TRUELOAD845(i108[0], i111[0], i110[0])≥NonInfC∧LOAD845(i108[0], i111[0], i110[0])≥COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])∧(UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥))

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

(3)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧[(-1)Bound*bni_30] + [bni_30]i111[0] ≥ 0∧[(-1)bso_31] ≥ 0)

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

(4)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧[(-1)Bound*bni_30] + [bni_30]i111[0] ≥ 0∧[(-1)bso_31] ≥ 0)

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

(5)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧[(-1)Bound*bni_30] + [bni_30]i111[0] ≥ 0∧[(-1)bso_31] ≥ 0)

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

(6)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30] + [bni_30]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

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

(7)    (i111[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30 + bni_30] + [bni_30]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

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

(8)    (i111[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30 + bni_30] + [bni_30]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

For Pair COND_LOAD845(TRUE, i108, i111, i110) → LOAD888(i108, i111, *(i110, i108)) the following chains were created:
• We consider the chain LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0]), COND_LOAD845(TRUE, i108[1], i111[1], i110[1]) → LOAD888(i108[1], i111[1], *(i110[1], i108[1])) which results in the following constraint:

We simplified constraint (9) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:

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

(11)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧[(-1)Bound*bni_32] + [bni_32]i111[0] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(12)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧[(-1)Bound*bni_32] + [bni_32]i111[0] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(13)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧[(-1)Bound*bni_32] + [bni_32]i111[0] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(14)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32] + [bni_32]i111[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

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

(15)    (i111[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32 + bni_32] + [bni_32]i111[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

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

(16)    (i111[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32 + bni_32] + [bni_32]i111[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

For Pair LOAD888(i108, i111, i110) → LOAD845(*(i108, i108), /(i111, 2), i110) the following chains were created:
• We consider the chain LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0]), COND_LOAD845(TRUE, i108[1], i111[1], i110[1]) → LOAD888(i108[1], i111[1], *(i110[1], i108[1])), LOAD888(i108[2], i111[2], i110[2]) → LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2]) which results in the following constraint:

We simplified constraint (17) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:

(18)    (>(i111[0], 0)=TRUE>=(1, %(i111[0], 2))=TRUE<=(1, %(i111[0], 2))=TRUELOAD888(i108[1], i111[0], *(i110[1], i108[1]))≥NonInfC∧LOAD888(i108[1], i111[0], *(i110[1], i108[1]))≥LOAD845(*(i108[1], i108[1]), /(i111[0], 2), *(i110[1], i108[1]))∧(UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥))

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

(19)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧[(-1)bso_38] + i111[0] + [-1]max{i111[0], [-1]i111[0]} ≥ 0)

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

(20)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧[(-1)bso_38] + i111[0] + [-1]max{i111[0], [-1]i111[0]} ≥ 0)

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

(21)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2]i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧[(-1)bso_38] ≥ 0)

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

(22)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2]i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

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

(23)    (i111[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2] + [2]i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

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

(24)    (i111[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2] + [2]i111[0] ≥ 0∧i108[1] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

(25)    (i111[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2] + [2]i111[0] ≥ 0∧i108[1] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

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

(26)    (i111[0] ≥ 0∧[1] ≥ 0∧i108[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

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

(27)    (i111[0] ≥ 0∧[1] ≥ 0∧i108[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

For Pair LOAD845(i108, i111, i110) → COND_LOAD8451(&&(&&(>=(%(i111, 2), 0), !(=(%(i111, 2), 1))), >(i111, 0)), i108, i111, i110) the following chains were created:
• We consider the chain LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3]), COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4]) which results in the following constraint:

(28)    (i110[3]=i110[4]i108[3]=i108[4]&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0))=TRUEi111[3]=i111[4]LOAD845(i108[3], i111[3], i110[3])≥NonInfC∧LOAD845(i108[3], i111[3], i110[3])≥COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])∧(UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥))

We simplified constraint (28) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:

(29)    (>(i111[3], 0)=TRUE>=(%(i111[3], 2), 0)=TRUE<(%(i111[3], 2), 1)=TRUELOAD845(i108[3], i111[3], i110[3])≥NonInfC∧LOAD845(i108[3], i111[3], i110[3])≥COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])∧(UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥))

(30)    (>(i111[3], 0)=TRUE>=(%(i111[3], 2), 0)=TRUE>(%(i111[3], 2), 1)=TRUELOAD845(i108[3], i111[3], i110[3])≥NonInfC∧LOAD845(i108[3], i111[3], i110[3])≥COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])∧(UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥))

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

(31)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(32)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(33)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(34)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(35)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(36)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)

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

(37)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

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

(38)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

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

(39)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

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

(40)    (i111[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

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

(41)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

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

(42)    (i111[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

For Pair COND_LOAD8451(TRUE, i108, i111, i110) → LOAD845(*(i108, i108), /(i111, 2), i110) the following chains were created:
• We consider the chain LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3]), COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4]) which results in the following constraint:

(43)    (i110[3]=i110[4]i108[3]=i108[4]&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0))=TRUEi111[3]=i111[4]COND_LOAD8451(TRUE, i108[4], i111[4], i110[4])≥NonInfC∧COND_LOAD8451(TRUE, i108[4], i111[4], i110[4])≥LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])∧(UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥))

We simplified constraint (43) using rules (III), (IDP_BOOLEAN) which results in the following new constraints:

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

(46)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] + i111[3] + [-1]max{i111[3], [-1]i111[3]} ≥ 0)

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

(47)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] + i111[3] + [-1]max{i111[3], [-1]i111[3]} ≥ 0)

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

(48)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] + i111[3] + [-1]max{i111[3], [-1]i111[3]} ≥ 0)

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

(49)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] + i111[3] + [-1]max{i111[3], [-1]i111[3]} ≥ 0)

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

(50)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(51)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (50) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(52)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

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

(53)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

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

(54)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

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

(55)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i111[3] ≥ 0∧i108[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

(56)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i111[3] ≥ 0∧i108[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

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

(57)    (i111[3] ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

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

(58)    (i111[3] ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

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

(59)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2] + [2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (59) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(60)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2] + [2]i111[3] ≥ 0∧i108[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

(61)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2] + [2]i111[3] ≥ 0∧i108[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (60) using rule (IDP_POLY_GCD) which results in the following new constraint:

(62)    (i111[3] ≥ 0∧0 ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (61) using rule (IDP_POLY_GCD) which results in the following new constraint:

(63)    (i111[3] ≥ 0∧0 ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD845(i108, i111, i110) → COND_LOAD845(&&(>(i111, 0), =(1, %(i111, 2))), i108, i111, i110)
• (i111[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30 + bni_30] + [bni_30]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

• (i111[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32 + bni_32] + [bni_32]i111[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

• (i111[0] ≥ 0∧[1] ≥ 0∧i108[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
• (i111[0] ≥ 0∧[1] ≥ 0∧i108[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

• LOAD845(i108, i111, i110) → COND_LOAD8451(&&(&&(>=(%(i111, 2), 0), !(=(%(i111, 2), 1))), >(i111, 0)), i108, i111, i110)
• (i111[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)
• (i111[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

• (i111[3] ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)
• (i111[3] ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)
• (i111[3] ≥ 0∧0 ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)
• (i111[3] ≥ 0∧0 ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 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_LOAD845(x1, x2, x3, x4)) = x3
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(=(x1, x2)) = [-1]
POL(1) = [1]
POL(2) = [2]
POL(LOAD888(x1, x2, x3)) = [-1] + x2
POL(*(x1, x2)) = x1·x2
POL(COND_LOAD8451(x1, x2, x3, x4)) = [-1] + x3
POL(>=(x1, x2)) = [-1]
POL(!(x1)) = [-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 @ {LOAD845_3/1}) = max{x1, [-1]x1} + [-1]

The following pairs are in P>:

LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])

The following pairs are in Pbound:

LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])
LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])

The following pairs are in P:

LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])

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

/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): LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(i111[0] > 0 && 1 = i111[0] % 2, i108[0], i111[0], i110[0])
(2): LOAD888(i108[2], i111[2], i110[2]) → LOAD845(i108[2] * i108[2], i111[2] / 2, i110[2])
(4): COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(i108[4] * i108[4], i111[4] / 2, i110[4])

(2) -> (0), if ((i110[2]* i110[0])∧(i108[2] * i108[2]* i108[0])∧(i111[2] / 2* i111[0]))

(4) -> (0), if ((i108[4] * i108[4]* i108[0])∧(i110[4]* i110[0])∧(i111[4] / 2* i111[0]))

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 3 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:

### (15) IDependencyGraphProof (EQUIVALENT transformation)

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