### (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: PastaB16

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 167 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:
Load430(i69, i80) → Cond_Load430(i80 > 0 && i69 > 0, i69, i80)
The set Q consists of the following terms:

### (5) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
• 0

We removed arguments according to the following replacements:

### (6) 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:
Load430(i69, i80) → Cond_Load430(i80 > 0 && i69 > 0, i69, i80)
The set Q consists of the following terms:

### (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:

Integer, Boolean

The ITRS R consists of the following rules:
Load430(i69, i80) → Cond_Load430(i80 > 0 && i69 > 0, i69, i80)

The integer pair graph contains the following rules and edges:
(2): LOAD430(i69[2], i80[2]) → COND_LOAD430(i80[2] > 0 && i69[2] > 0, i69[2], i80[2])

(0) -> (1), if ((i80[0]* i80[1])∧(i69[0]* i69[1])∧(i80[0] > 0* TRUE))

(1) -> (0), if ((i69[1]* i69[0])∧(i80[1] + -1* i80[0]))

(1) -> (4), if ((i80[1] + -1* 0)∧(i69[1]* i69[4]))

(2) -> (3), if ((i69[2]* i69[3])∧(i80[2] > 0 && i69[2] > 0* TRUE)∧(i80[2]* i80[3]))

(3) -> (0), if ((i69[3]* i69[0])∧(i80[3] + -1* i80[0]))

(3) -> (4), if ((i80[3] + -1* 0)∧(i69[3]* i69[4]))

(4) -> (2), if ((0* i80[2])∧(i69[4] + -1* i69[2]))

(4) -> (5), if (i69[4] + -1* i69[5])

(5) -> (6), if ((i69[5]* i69[6])∧(i69[5] > 0* TRUE))

(6) -> (2), if ((i69[6] + -1* i69[2])∧(0* i80[2]))

(6) -> (5), if (i69[6] + -1* i69[5])

The set Q consists of the following terms:

### (9) 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.

### (10) Obligation:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD430(i69[2], i80[2]) → COND_LOAD430(i80[2] > 0 && i69[2] > 0, i69[2], i80[2])

(0) -> (1), if ((i80[0]* i80[1])∧(i69[0]* i69[1])∧(i80[0] > 0* TRUE))

(1) -> (0), if ((i69[1]* i69[0])∧(i80[1] + -1* i80[0]))

(1) -> (4), if ((i80[1] + -1* 0)∧(i69[1]* i69[4]))

(2) -> (3), if ((i69[2]* i69[3])∧(i80[2] > 0 && i69[2] > 0* TRUE)∧(i80[2]* i80[3]))

(3) -> (0), if ((i69[3]* i69[0])∧(i80[3] + -1* i80[0]))

(3) -> (4), if ((i80[3] + -1* 0)∧(i69[3]* i69[4]))

(4) -> (2), if ((0* i80[2])∧(i69[4] + -1* i69[2]))

(4) -> (5), if (i69[4] + -1* i69[5])

(5) -> (6), if ((i69[5]* i69[6])∧(i69[5] > 0* TRUE))

(6) -> (2), if ((i69[6] + -1* i69[2])∧(0* i80[2]))

(6) -> (5), if (i69[6] + -1* i69[5])

The set Q consists of the following terms:

### (11) 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 LOAD492(i69, i80) → COND_LOAD492(>(i80, 0), i69, i80) the following chains were created:
• We consider the chain LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0]), COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

We simplified constraint (1) using rule (IV) which results in the following new constraint:

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

(3)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i69[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(4)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i69[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(5)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i69[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(6)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_14] = 0∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

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

(7)    (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_14] = 0∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair COND_LOAD492(TRUE, i69, i80) → LOAD492(i69, +(i80, -1)) the following chains were created:
• We consider the chain COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

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

(9)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(10)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(11)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(12)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

For Pair LOAD430(i69, i80) → COND_LOAD430(&&(>(i80, 0), >(i69, 0)), i69, i80) the following chains were created:
• We consider the chain LOAD430(i69[2], i80[2]) → COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2]), COND_LOAD430(TRUE, i69[3], i80[3]) → LOAD492(i69[3], +(i80[3], -1)) which results in the following constraint:

We simplified constraint (13) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(15)    (i80[2] + [-1] ≥ 0∧i69[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(16)    (i80[2] + [-1] ≥ 0∧i69[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(17)    (i80[2] + [-1] ≥ 0∧i69[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(18)    (i80[2] ≥ 0∧i69[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(19)    (i80[2] ≥ 0∧i69[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_LOAD430(TRUE, i69, i80) → LOAD492(i69, +(i80, -1)) the following chains were created:
• We consider the chain COND_LOAD430(TRUE, i69[3], i80[3]) → LOAD492(i69[3], +(i80[3], -1)) which results in the following constraint:

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

(21)    ((UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥)∧[(-1)bso_21] ≥ 0)

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

(22)    ((UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥)∧[(-1)bso_21] ≥ 0)

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

(23)    ((UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥)∧[(-1)bso_21] ≥ 0)

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

(24)    ((UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

For Pair LOAD492(i69, 0) → LOAD430(+(i69, -1), 0) the following chains were created:
• We consider the chain LOAD492(i69[4], 0) → LOAD430(+(i69[4], -1), 0) which results in the following constraint:

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

(26)    ((UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥)∧[1 + (-1)bso_23] ≥ 0)

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

(27)    ((UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥)∧[1 + (-1)bso_23] ≥ 0)

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

(28)    ((UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥)∧[1 + (-1)bso_23] ≥ 0)

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

(29)    ((UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_23] ≥ 0)

For Pair LOAD430(i69, 0) → COND_LOAD4301(>(i69, 0), i69) the following chains were created:
• We consider the chain LOAD430(i69[5], 0) → COND_LOAD4301(>(i69[5], 0), i69[5]), COND_LOAD4301(TRUE, i69[6]) → LOAD430(+(i69[6], -1), 0) which results in the following constraint:

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

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

(32)    (i69[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i69[5] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(33)    (i69[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i69[5] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(34)    (i69[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i69[5] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(35)    (i69[5] ≥ 0 ⇒ (UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥)∧[(-1)Bound*bni_24] + [bni_24]i69[5] ≥ 0∧[(-1)bso_25] ≥ 0)

For Pair COND_LOAD4301(TRUE, i69) → LOAD430(+(i69, -1), 0) the following chains were created:
• We consider the chain COND_LOAD4301(TRUE, i69[6]) → LOAD430(+(i69[6], -1), 0) which results in the following constraint:

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

(37)    ((UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(38)    ((UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(39)    ((UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(40)    ((UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_14] = 0∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

• ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

• (i80[2] ≥ 0∧i69[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)

• ((UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

• ((UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_23] ≥ 0)

• (i69[5] ≥ 0 ⇒ (UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥)∧[(-1)Bound*bni_24] + [bni_24]i69[5] ≥ 0∧[(-1)bso_25] ≥ 0)

• ((UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_27] ≥ 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(LOAD492(x1, x2)) = [-1] + x1
POL(COND_LOAD492(x1, x2, x3)) = [-1] + x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(LOAD430(x1, x2)) = [-1] + x1
POL(COND_LOAD430(x1, x2, x3)) = [-1] + x2
POL(&&(x1, x2)) = [-1]
POL(COND_LOAD4301(x1, x2)) = [-1] + x2

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (13) Obligation:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD430(i69[2], i80[2]) → COND_LOAD430(i80[2] > 0 && i69[2] > 0, i69[2], i80[2])

(1) -> (0), if ((i69[1]* i69[0])∧(i80[1] + -1* i80[0]))

(3) -> (0), if ((i69[3]* i69[0])∧(i80[3] + -1* i80[0]))

(0) -> (1), if ((i80[0]* i80[1])∧(i69[0]* i69[1])∧(i80[0] > 0* TRUE))

(2) -> (3), if ((i69[2]* i69[3])∧(i80[2] > 0 && i69[2] > 0* TRUE)∧(i80[2]* i80[3]))

The set Q consists of the following terms:

### (14) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

### (15) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (0), if ((i69[1]* i69[0])∧(i80[1] + -1* i80[0]))

(0) -> (1), if ((i80[0]* i80[1])∧(i69[0]* i69[1])∧(i80[0] > 0* TRUE))

The set Q consists of the following terms:

### (16) 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 COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) the following chains were created:
• We consider the chain COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(2)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_7] ≥ 0)

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

(3)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_7] ≥ 0)

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

(4)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_7] ≥ 0)

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

(5)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧[(-1)bso_7] ≥ 0)

For Pair LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0]) the following chains were created:
• We consider the chain LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0]), COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

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

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

(8)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]i80[0] ≥ 0∧[2 + (-1)bso_9] ≥ 0)

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

(9)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]i80[0] ≥ 0∧[2 + (-1)bso_9] ≥ 0)

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

(10)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]i80[0] ≥ 0∧[2 + (-1)bso_9] ≥ 0)

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

(11)    (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(4)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]i80[0] ≥ 0∧[2 + (-1)bso_9] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧[(-1)bso_7] ≥ 0)

• (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(4)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]i80[0] ≥ 0∧[2 + (-1)bso_9] ≥ 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(LOAD492(x1, x2)) = [2] + [2]x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (17) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

The set Q consists of the following terms:

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (20) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (0), if ((i69[1]* i69[0])∧(i80[1] + -1* i80[0]))

(3) -> (0), if ((i69[3]* i69[0])∧(i80[3] + -1* i80[0]))

(0) -> (1), if ((i80[0]* i80[1])∧(i69[0]* i69[1])∧(i80[0] > 0* TRUE))

(1) -> (4), if ((i80[1] + -1* 0)∧(i69[1]* i69[4]))

(3) -> (4), if ((i80[3] + -1* 0)∧(i69[3]* i69[4]))

The set Q consists of the following terms:

### (21) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

### (22) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (0), if ((i69[1]* i69[0])∧(i80[1] + -1* i80[0]))

(0) -> (1), if ((i80[0]* i80[1])∧(i69[0]* i69[1])∧(i80[0] > 0* TRUE))

The set Q consists of the following terms:

### (23) 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 COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) the following chains were created:
• We consider the chain COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(2)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_8] ≥ 0)

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

(3)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_8] ≥ 0)

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

(4)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_8] ≥ 0)

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

(5)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧[(-1)bso_8] ≥ 0)

For Pair LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0]) the following chains were created:
• We consider the chain LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0]), COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

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

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

(8)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i80[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

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

(9)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i80[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

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

(10)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i80[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

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

(11)    (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i80[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧[(-1)bso_8] ≥ 0)

• (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i80[0] ≥ 0∧[2 + (-1)bso_10] ≥ 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_LOAD492(x1, x2, x3)) = [-1] + [2]x3
POL(LOAD492(x1, x2)) = [1] + [2]x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (24) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

The set Q consists of the following terms: