### (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: PastaB12

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 173 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:
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:
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

The ITRS R consists of the following rules:

The integer pair graph contains the following rules and edges:

(0) -> (1), if ((i50[0]* i50[1])∧(i50[0] > 0* TRUE))

(1) -> (0), if (i50[1] + -1* i50[0])

(1) -> (2), if ((i50[1] + -1* i33[2])∧(0* i43[2]))

(2) -> (3), if ((i43[2] > 0* TRUE)∧(i33[2]* i33[3])∧(i43[2]* i43[3]))

(3) -> (0), if ((i33[3]* i50[0])∧(i43[3] + -1* 0))

(3) -> (2), if ((i33[3]* i33[2])∧(i43[3] + -1* i43[2]))

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

R is empty.

The integer pair graph contains the following rules and edges:

(0) -> (1), if ((i50[0]* i50[1])∧(i50[0] > 0* TRUE))

(1) -> (0), if (i50[1] + -1* i50[0])

(1) -> (2), if ((i50[1] + -1* i33[2])∧(0* i43[2]))

(2) -> (3), if ((i43[2] > 0* TRUE)∧(i33[2]* i33[3])∧(i43[2]* i43[3]))

(3) -> (0), if ((i33[3]* i50[0])∧(i43[3] + -1* 0))

(3) -> (2), if ((i33[3]* i33[2])∧(i43[3] + -1* i43[2]))

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 LOAD344(0, i50) → COND_LOAD344(>(i50, 0), i50) the following chains were created:
• We consider the chain LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0]), COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[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)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(4)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(5)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(6)    (i50[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_LOAD344(TRUE, i50) → LOAD344(0, +(i50, -1)) the following chains were created:
• We consider the chain COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[1], -1)) which results in the following constraint:

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

(8)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧[(-1)bso_13] ≥ 0)

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

(9)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧[(-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧[(-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

For Pair LOAD344(i43, i33) → COND_LOAD3441(>(i43, 0), i43, i33) the following chains were created:
• We consider the chain LOAD344(i43[2], i33[2]) → COND_LOAD3441(>(i43[2], 0), i43[2], i33[2]), COND_LOAD3441(TRUE, i43[3], i33[3]) → LOAD344(+(i43[3], -1), i33[3]) which results in the following constraint:

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

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

(14)    (i43[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(15)    (i43[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(16)    (i43[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(17)    (i43[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧0 = 0∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

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

(18)    (i43[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair COND_LOAD3441(TRUE, i43, i33) → LOAD344(+(i43, -1), i33) the following chains were created:
• We consider the chain COND_LOAD3441(TRUE, i43[3], i33[3]) → LOAD344(+(i43[3], -1), i33[3]) which results in the following constraint:

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

(20)    ((UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(21)    ((UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(22)    ((UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(23)    ((UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i50[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

• ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

• (i43[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

• ((UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 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(LOAD344(x1, x2)) = [-1] + x1
POL(0) = 0
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_LOAD3441(x1, x2, x3)) = [-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

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (0), if (i50[1] + -1* i50[0])

(0) -> (1), if ((i50[0]* i50[1])∧(i50[0] > 0* TRUE))

(1) -> (2), if ((i50[1] + -1* i33[2])∧(0* i43[2]))

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 1 less node.

### (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 (i50[1] + -1* i50[0])

(0) -> (1), if ((i50[0]* i50[1])∧(i50[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_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[1], -1)) the following chains were created:
• We consider the chain COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[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(LOAD344(0, +(i50[1], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(3)    ((UIncreasing(LOAD344(0, +(i50[1], -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(LOAD344(0, +(i50[1], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(5)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

For Pair LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0]) the following chains were created:
• We consider the chain LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0]), COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[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)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i50[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(9)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i50[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(10)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i50[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(11)    (i50[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i50[0] ≥ 0∧[(-1)bso_9] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

• (i50[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i50[0] ≥ 0∧[(-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(COND_LOAD344(x1, x2)) = [1] + x2
POL(LOAD344(x1, x2)) = [1] + x2
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (18) 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:

### (19) IDependencyGraphProof (EQUIVALENT transformation)

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

### (21) 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:

### (22) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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: