### (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: ListDuplicate

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 185 nodes with 2 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

### (5) 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:
Load747(1, java.lang.Object(ObjectList(o208, o207)), i37) → Cond_Load747(i37 > 0, 1, java.lang.Object(ObjectList(o208, o207)), i37)
The set Q consists of the following terms:

### (6) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
• 1

We removed arguments according to the following replacements:

### (7) 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:

### (9) 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) -> (0), if ((o203[0]* java.lang.Object(ObjectList(o203[0]', o202[0]')))∧(1* 0))

(0) -> (1), if ((o203[0]* java.lang.Object(ObjectList(o208[1], o207[1])))∧(1* i37[1]))

(1) -> (2), if ((i37[1] > 0* TRUE)∧(java.lang.Object(ObjectList(o208[1], o207[1])) →* java.lang.Object(ObjectList(o208[2], o207[2])))∧(i37[1]* i37[2]))

(2) -> (0), if (java.lang.Object(ObjectList(o208[2], o207[2])) →* java.lang.Object(ObjectList(o203[0], o202[0])))

(2) -> (1), if ((0* i37[1])∧(java.lang.Object(ObjectList(o208[2], o207[2])) →* java.lang.Object(ObjectList(o208[1], o207[1]))))

The set Q consists of the following terms:

### (10) 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.

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

(0) -> (0), if ((o203[0]* java.lang.Object(ObjectList(o203[0]', o202[0]')))∧(1* 0))

(0) -> (1), if ((o203[0]* java.lang.Object(ObjectList(o208[1], o207[1])))∧(1* i37[1]))

(1) -> (2), if ((i37[1] > 0* TRUE)∧(java.lang.Object(ObjectList(o208[1], o207[1])) →* java.lang.Object(ObjectList(o208[2], o207[2])))∧(i37[1]* i37[2]))

(2) -> (0), if (java.lang.Object(ObjectList(o208[2], o207[2])) →* java.lang.Object(ObjectList(o203[0], o202[0])))

(2) -> (1), if ((0* i37[1])∧(java.lang.Object(ObjectList(o208[2], o207[2])) →* java.lang.Object(ObjectList(o208[1], o207[1]))))

The set Q consists of the following terms:

### (12) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

### (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:

(0) -> (1), if ((o203[0]* java.lang.Object(ObjectList(o208[1], o207[1])))∧(1* i37[1]))

(0) -> (0), if ((o203[0]* java.lang.Object(ObjectList(o203[0]', o202[0]')))∧false)

(1) -> (2), if ((i37[1] > 0* TRUE)∧((o208[1]* o208[2])∧(o207[1]* o207[2]))∧(i37[1]* i37[2]))

(2) -> (0), if ((o208[2]* o203[0])∧(o207[2]* o202[0]))

(2) -> (1), if ((0* i37[1])∧((o208[2]* o208[1])∧(o207[2]* o207[1])))

The set Q consists of the following terms:

### (14) 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 LOAD747(java.lang.Object(ObjectList(o203, o202)), 0) → LOAD747(o203, 1) the following chains were created:

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

(2)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)

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

(3)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)

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

(4)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)

We simplified constraint (4) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(5)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)

• We consider the chain COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0), LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1), LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) which results in the following constraint:

We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:

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

(8)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[180 + (-1)bso_14] + [90]o208[1] ≥ 0)

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

(9)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[180 + (-1)bso_14] + [90]o208[1] ≥ 0)

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

(10)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[180 + (-1)bso_14] + [90]o208[1] ≥ 0)

We simplified constraint (10) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(11)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧0 ≥ 0∧0 ≥ 0∧[180 + (-1)bso_14] ≥ 0∧[1] ≥ 0)

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

(13)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)

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

(14)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)

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

(15)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0]1 ≥ 0)

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

(16)    ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)

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

(18)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0] ≥ 0)

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

(19)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0] ≥ 0)

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

(20)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧[30 + (-1)bso_14] + [15]o203[0] ≥ 0)

We simplified constraint (20) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(21)    ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)

For Pair LOAD747(java.lang.Object(ObjectList(o208, o207)), i37) → COND_LOAD747(>(i37, 0), java.lang.Object(ObjectList(o208, o207)), i37) the following chains were created:
• We consider the chain LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]), COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0) which results in the following constraint:

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

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

(24)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[(32)bni_15 + (-1)Bound*bni_15] + [(18)bni_15]o208[1] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(25)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[(32)bni_15 + (-1)Bound*bni_15] + [(18)bni_15]o208[1] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(26)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[(32)bni_15 + (-1)Bound*bni_15] + [(18)bni_15]o208[1] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(27)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_15] ≥ 0∧[(32)bni_15 + (-1)Bound*bni_15] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_16] ≥ 0)

For Pair COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208, o207)), i37) → LOAD747(java.lang.Object(ObjectList(o208, o207)), 0) the following chains were created:
• We consider the chain LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]), COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0), LOAD747(java.lang.Object(ObjectList(o203[0], o202[0])), 0) → LOAD747(o203[0], 1) which results in the following constraint:

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

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

(30)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(31)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(32)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(33)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_17] ≥ 0∧[(32)bni_17 + (-1)Bound*bni_17] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_18] ≥ 0)

• We consider the chain LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]), COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0), LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) which results in the following constraint:

We simplified constraint (34) using rules (III), (IV) which results in the following new constraint:

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

(36)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(37)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(38)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(32)bni_17 + (-1)Bound*bni_17] + [(18)bni_17]o208[1] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(39)    (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_17] ≥ 0∧[(32)bni_17 + (-1)Bound*bni_17] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)
• ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧0 ≥ 0∧0 ≥ 0∧[180 + (-1)bso_14] ≥ 0∧[1] ≥ 0)
• ((UIncreasing(LOAD747(o203[0]1, 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)
• ((UIncreasing(LOAD747(o203[0], 1)), ≥)∧0 ≥ 0∧[30 + (-1)bso_14] ≥ 0∧[1] ≥ 0)

• (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_15] ≥ 0∧[(32)bni_15 + (-1)Bound*bni_15] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_16] ≥ 0)

• (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_17] ≥ 0∧[(32)bni_17 + (-1)Bound*bni_17] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_18] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 ≥ 0∧0 ≥ 0∧[(18)bni_17] ≥ 0∧[(32)bni_17 + (-1)Bound*bni_17] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_18] ≥ 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 with natural coefficients for all symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(LOAD747(x1, x2)) = [2] + [3]x1
POL(java.lang.Object(x1)) = [1] + [3]x1
POL(ObjectList(x1, x2)) = [3] + [2]x1
POL(0) = 0
POL(1) = 0
POL(COND_LOAD747(x1, x2, x3)) = [2] + [3]x2
POL(>(x1, x2)) = 0

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (16) 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:

(2) -> (1), if ((0* i37[1])∧((o208[2]* o208[1])∧(o207[2]* o207[1])))

(1) -> (2), if ((i37[1] > 0* TRUE)∧((o208[1]* o208[2])∧(o207[1]* o207[2]))∧(i37[1]* i37[2]))

The set Q consists of the following terms:

### (17) 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 LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) the following chains were created:
• We consider the chain LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]), COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0) 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)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧[-2 + (-1)bso_17] + [2]i37[1] ≥ 0)

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

(4)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧[-2 + (-1)bso_17] + [2]i37[1] ≥ 0)

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

(5)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧[-2 + (-1)bso_17] + [2]i37[1] ≥ 0)

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

(6)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧0 = 0∧0 = 0∧[bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧0 = 0∧0 = 0∧[-2 + (-1)bso_17] + [2]i37[1] ≥ 0)

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

(7)    (i37[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧0 = 0∧0 = 0∧[(3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] + [2]i37[1] ≥ 0)

For Pair COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0) the following chains were created:
• We consider the chain LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]), COND_LOAD747(TRUE, java.lang.Object(ObjectList(o208[2], o207[2])), i37[2]) → LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0), LOAD747(java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) → COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1]) which results in the following constraint:

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

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

(10)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧[2 + (-1)bso_19] ≥ 0)

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

(11)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧[2 + (-1)bso_19] ≥ 0)

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

(12)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧[2 + (-1)bso_19] ≥ 0)

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

(13)    (i37[1] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)

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

(14)    (i37[1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i37[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD747(>(i37[1], 0), java.lang.Object(ObjectList(o208[1], o207[1])), i37[1])), ≥)∧0 = 0∧0 = 0∧[(3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]i37[1] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_17] + [2]i37[1] ≥ 0)

• (i37[1] ≥ 0 ⇒ (UIncreasing(LOAD747(java.lang.Object(ObjectList(o208[2], o207[2])), 0)), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] ≥ 0∧0 = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 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(LOAD747(x1, x2)) = [-1] + [2]x2 + [2]x1
POL(java.lang.Object(x1)) = [1]
POL(ObjectList(x1, x2)) = [-1]x2 + x1
POL(COND_LOAD747(x1, x2, x3)) = [2] + x2
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.

### (19) 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:

### (20) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:

### (23) IDependencyGraphProof (EQUIVALENT transformation)

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

### (25) 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 contains the following rules and edges:

(0) -> (0), if ((o203[0]* java.lang.Object(ObjectList(o203[0]', o202[0]')))∧false)

The set Q consists of the following terms:

### (26) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

### (27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

R is empty.
The set Q consists of the following terms:

We have to consider all minimal (P,Q,R)-chains.

### (28) DependencyGraphProof (EQUIVALENT transformation)

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

### (30) 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:

### (32) 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 ((i29[0] > 0* TRUE)∧(i29[0]* i29[1]))

(1) -> (0), if ((i29[1] + -1* i29[0]))

The set Q consists of the following terms:

### (33) 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.

### (34) 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 ((i29[0] > 0* TRUE)∧(i29[0]* i29[1]))

(1) -> (0), if ((i29[1] + -1* i29[0]))

The set Q consists of the following terms:

### (35) 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 LOAD418(i29) → COND_LOAD418(>(i29, 0), i29) the following chains were created:

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)    (i29[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD418(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(4)    (i29[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD418(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(5)    (i29[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD418(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(6)    (i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD418(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_LOAD418(TRUE, i29) → LOAD418(+(i29, -1)) the following chains were created:
• We consider the chain COND_LOAD418(TRUE, i29[1]) → LOAD418(+(i29[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(LOAD418(+(i29[1], -1))), ≥)∧[2 + (-1)bso_11] ≥ 0)

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

(9)    ((UIncreasing(LOAD418(+(i29[1], -1))), ≥)∧[2 + (-1)bso_11] ≥ 0)

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

(10)    ((UIncreasing(LOAD418(+(i29[1], -1))), ≥)∧[2 + (-1)bso_11] ≥ 0)

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

(11)    ((UIncreasing(LOAD418(+(i29[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD418(>(i29[0], 0), i29[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]i29[0] ≥ 0∧[(-1)bso_9] ≥ 0)

• ((UIncreasing(LOAD418(+(i29[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_11] ≥ 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(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (37) 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:

### (38) IDependencyGraphProof (EQUIVALENT transformation)

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

### (40) 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:

### (41) IDependencyGraphProof (EQUIVALENT transformation)

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