### (0) Obligation:

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

Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: AppE

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 232 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:
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:
• java.lang.Object(AppE(NULL))

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) -> (1), if ((i74[0]* i74[1])∧(i74[0] > 0* TRUE))

(1) -> (2), if ((i74[1] + -1* i56[2])∧(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2]))))))

(2) -> (0), if ((java.lang.Object(AppE(o408Field0[2])) →* java.lang.Object(AppE(NULL)))∧(i56[2]* i74[0]))

(2) -> (3), if ((java.lang.Object(AppE(o408Field0[2])) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))))∧(i56[2]* i56[3]))

(3) -> (0), if ((i56[3]* i74[0])∧(java.lang.Object(AppE(o408Field0[3])) →* java.lang.Object(AppE(NULL))))

(3) -> (3), if ((java.lang.Object(AppE(o408Field0[3])) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]')))))∧(i56[3]* i56[3]'))

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) -> (1), if ((i74[0]* i74[1])∧(i74[0] > 0* TRUE))

(1) -> (2), if ((i74[1] + -1* i56[2])∧(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2]))))))

(2) -> (0), if ((java.lang.Object(AppE(o408Field0[2])) →* java.lang.Object(AppE(NULL)))∧(i56[2]* i74[0]))

(2) -> (3), if ((java.lang.Object(AppE(o408Field0[2])) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3])))))∧(i56[2]* i56[3]))

(3) -> (0), if ((i56[3]* i74[0])∧(java.lang.Object(AppE(o408Field0[3])) →* java.lang.Object(AppE(NULL))))

(3) -> (3), if ((java.lang.Object(AppE(o408Field0[3])) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[3]')))))∧(i56[3]* i56[3]'))

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 ((i74[0]* i74[1])∧(i74[0] > 0* TRUE))

(1) -> (2), if ((i74[1] + -1* i56[2])∧(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2]))))))

(2) -> (0), if (((o408Field0[2]* NULL))∧(i56[2]* i74[0]))

(2) -> (3), if (((o408Field0[2]* java.lang.Object(AppE(o408Field0[3]))))∧(i56[2]* i56[3]))

(3) -> (0), if ((i56[3]* i74[0])∧((o408Field0[3]* NULL)))

(3) -> (3), if (((o408Field0[3]* java.lang.Object(AppE(o408Field0[3]'))))∧(i56[3]* i56[3]'))

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 LOAD684(java.lang.Object(AppE(NULL)), i74) → COND_LOAD684(>(i74, 0), i74) the following chains were created:
• We consider the chain LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]), COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[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)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(2)bni_15]i74[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(4)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(2)bni_15]i74[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(5)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(2)bni_15]i74[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(6)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[(2)bni_15] ≥ 0∧[bni_15 + (-1)Bound*bni_15] ≥ 0∧0 ≥ 0∧[(-1)bso_16] ≥ 0)

For Pair COND_LOAD684(TRUE, i74) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74, -1)) the following chains were created:

We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:

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

(9)    (0 ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i74[0] ≥ 0∧[(-1)bso_18] + [2]i74[0] ≥ 0)

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

(10)    (0 ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i74[0] ≥ 0∧[(-1)bso_18] + [2]i74[0] ≥ 0)

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

(11)    (0 ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i74[0] ≥ 0∧[(-1)bso_18] + [2]i74[0] ≥ 0)

We simplified constraint (11) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(12)    (0 ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[(2)bni_17] ≥ 0∧[bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0∧[1] ≥ 0)

For Pair LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → LOAD684(java.lang.Object(AppE(o408Field0)), i56) the following chains were created:

We simplified constraint (13) using rules (I), (II), (III), (IV), (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint:

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

(15)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[(-1)bso_20] ≥ 0)

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

(16)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[(-1)bso_20] ≥ 0)

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

(17)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[(-1)bso_20] ≥ 0)

We solved constraint (18) using rules (I), (II), (III), (IV).

For Pair LOAD684(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0)))), i56) → LOAD684(java.lang.Object(AppE(o408Field0)), i56) the following chains were created:

We simplified constraint (19) using rules (III), (IV) which results in the following new constraint:

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

(21)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[4 + (-1)bso_22] ≥ 0)

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

(22)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[4 + (-1)bso_22] ≥ 0)

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

(23)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[4 + (-1)bso_22] ≥ 0)

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

(24)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧0 ≥ 0∧[4 + (-1)bso_22] ≥ 0)

We simplified constraint (25) using rules (III), (IV) which results in the following new constraint:

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

(27)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[4 + (-1)bso_22] ≥ 0)

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

(28)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[4 + (-1)bso_22] ≥ 0)

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

(29)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[4 + (-1)bso_22] ≥ 0)

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

(30)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧0 ≥ 0∧[4 + (-1)bso_22] ≥ 0)

We simplified constraint (31) using rules (III), (IV) which results in the following new constraint:

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

(33)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]1 ≥ 0)

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

(34)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]1 ≥ 0)

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

(35)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]1 ≥ 0)

We simplified constraint (35) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(36)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧0 ≥ 0∧[16 + (-1)bso_22] ≥ 0∧[1] ≥ 0)

We simplified constraint (37) using rules (III), (IV) which results in the following new constraint:

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

(39)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]2 ≥ 0)

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

(40)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]2 ≥ 0)

We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(41)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧[16 + (-1)bso_22] + [48]o408Field0[3]2 ≥ 0)

We simplified constraint (41) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(42)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧0 ≥ 0∧[16 + (-1)bso_22] ≥ 0∧[1] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[(2)bni_15] ≥ 0∧[bni_15 + (-1)Bound*bni_15] ≥ 0∧0 ≥ 0∧[(-1)bso_16] ≥ 0)

• (0 ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[(2)bni_17] ≥ 0∧[bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0∧[1] ≥ 0)

• ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[(-1)bso_20] ≥ 0)

• ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧0 ≥ 0∧[4 + (-1)bso_22] ≥ 0)
• ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧0 ≥ 0∧[4 + (-1)bso_22] ≥ 0)
• ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3])), i56[3])), ≥)∧0 ≥ 0∧[16 + (-1)bso_22] ≥ 0∧[1] ≥ 0)
• ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[3]1)), i56[3]1)), ≥)∧0 ≥ 0∧[16 + (-1)bso_22] ≥ 0∧[1] ≥ 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(LOAD684(x1, x2)) = [2]x2 + x1
POL(java.lang.Object(x1)) = [1] + [2]x1
POL(AppE(x1)) = [2]x1
POL(NULL) = 0
POL(COND_LOAD684(x1, x2)) = [1] + [2]x2
POL(>(x1, x2)) = 0
POL(0) = 0
POL(+(x1, x2)) = 0
POL(-1) = 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) -> (0), if (((o408Field0[2]* NULL))∧(i56[2]* i74[0]))

(0) -> (1), if ((i74[0]* i74[1])∧(i74[0] > 0* TRUE))

(1) -> (2), if ((i74[1] + -1* i56[2])∧(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))) →* java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[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 LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]) the following chains were created:
• We consider the chain LOAD684(java.lang.Object(AppE(NULL)), i74[0]) → COND_LOAD684(>(i74[0], 0), i74[0]), COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[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)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]i74[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(4)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]i74[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(5)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]i74[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(6)    (i74[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i74[0] ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_LOAD684(TRUE, i74[1]) → LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1)) the following chains were created:

We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:

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

(9)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]i74[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(10)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]i74[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(11)    (i74[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]i74[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(12)    (i74[0] ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]i74[0] ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(o408Field0[2])))), i56[2]) → LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2]) the following chains were created:

We simplified constraint (13) using rules (I), (II), (III), (IV), (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint:

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

(15)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[1 + (-1)bso_23] ≥ 0)

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

(16)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[1 + (-1)bso_23] ≥ 0)

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

(17)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧[1 + (-1)bso_23] ≥ 0)

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

(18)    ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧0 = 0∧[1 + (-1)bso_23] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i74[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD684(>(i74[0], 0), i74[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]i74[0] ≥ 0∧[(-1)bso_19] ≥ 0)

• (i74[0] ≥ 0 ⇒ (UIncreasing(LOAD796(java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), +(i74[1], -1))), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]i74[0] ≥ 0∧[(-1)bso_21] ≥ 0)

• ((UIncreasing(LOAD684(java.lang.Object(AppE(o408Field0[2])), i56[2])), ≥)∧0 = 0∧[1 + (-1)bso_23] ≥ 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(LOAD684(x1, x2)) = x2 + [-1]x1
POL(java.lang.Object(x1)) = [-1]
POL(AppE(x1)) = [1] + [-1]x1
POL(NULL) = [1]
POL(COND_LOAD684(x1, x2)) = [1] + x2
POL(>(x1, x2)) = [1]
POL(0) = 0
POL(LOAD796(x1, x2)) = [1] + x2 + [-1]x1
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.

### (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:

(0) -> (1), if ((i74[0]* i74[1])∧(i74[0] > 0* TRUE))

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 2 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:
none

R is empty.

The integer pair graph contains the following rules and edges:

The set Q consists of the following terms:

### (23) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:

(2) -> (3), if (((o408Field0[2]* java.lang.Object(AppE(o408Field0[3]))))∧(i56[2]* i56[3]))

(3) -> (3), if (((o408Field0[3]* java.lang.Object(AppE(o408Field0[3]'))))∧(i56[3]* i56[3]'))

The set Q consists of the following terms:

### (26) IDependencyGraphProof (EQUIVALENT transformation)

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

### (27) 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:

(3) -> (3), if (((o408Field0[3]* java.lang.Object(AppE(o408Field0[3]'))))∧(i56[3]* i56[3]'))

The set Q consists of the following terms:

### (28) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

### (29) 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.

### (30) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

### (31) Obligation:

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

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

### (32) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

The graph contains the following edges 1 > 1, 2 >= 2

### (34) 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:

### (36) 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 ((i36[0]* i36[1])∧(i36[0] > 0* TRUE))

(1) -> (0), if ((i36[1] + -1* i36[0]))

The set Q consists of the following terms:

### (37) 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.

### (38) 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 ((i36[0]* i36[1])∧(i36[0] > 0* TRUE))

(1) -> (0), if ((i36[1] + -1* i36[0]))

The set Q consists of the following terms:

### (39) 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 LOAD412(i36) → COND_LOAD412(>(i36, 0), i36) 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)    (i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i36[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(4)    (i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i36[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(5)    (i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]i36[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(6)    (i36[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]i36[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_LOAD412(TRUE, i36) → LOAD412(+(i36, -1)) the following chains were created:
• We consider the chain COND_LOAD412(TRUE, i36[1]) → LOAD412(+(i36[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(LOAD412(+(i36[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(LOAD412(+(i36[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(LOAD412(+(i36[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(LOAD412(+(i36[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i36[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD412(>(i36[0], 0), i36[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]i36[0] ≥ 0∧[(-1)bso_9] ≥ 0)

• ((UIncreasing(LOAD412(+(i36[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.

### (41) 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:

### (42) IDependencyGraphProof (EQUIVALENT transformation)

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

### (44) 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: