### (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: LogBuiltIn

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 115 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:
Load352(1, i32, i29) → Cond_Load352(i32 > 1 && i29 + 1 > 0, 1, i32, i29)
Cond_Load352(TRUE, 1, i32, i29) → Load352(1, i32 / 2, i29 + 1)
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:
• 1

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:
Load352(i32, i29) → Cond_Load352(i32 > 1 && i29 + 1 > 0, i32, i29)
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:

Boolean, Integer

The ITRS R consists of the following rules:
Load352(i32, i29) → Cond_Load352(i32 > 1 && i29 + 1 > 0, i32, i29)

The integer pair graph contains the following rules and edges:
(0): LOAD352(i32[0], i29[0]) → COND_LOAD352(i32[0] > 1 && i29[0] + 1 > 0, i32[0], i29[0])

(0) -> (1), if ((i29[0]* i29[1])∧(i32[0]* i32[1])∧(i32[0] > 1 && i29[0] + 1 > 0* TRUE))

(1) -> (0), if ((i32[1] / 2* i32[0])∧(i29[1] + 1* i29[0]))

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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD352(i32[0], i29[0]) → COND_LOAD352(i32[0] > 1 && i29[0] + 1 > 0, i32[0], i29[0])

(0) -> (1), if ((i29[0]* i29[1])∧(i32[0]* i32[1])∧(i32[0] > 1 && i29[0] + 1 > 0* TRUE))

(1) -> (0), if ((i32[1] / 2* i32[0])∧(i29[1] + 1* i29[0]))

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 LOAD352(i32, i29) → COND_LOAD352(&&(>(i32, 1), >(+(i29, 1), 0)), i32, i29) the following chains were created:
• We consider the chain LOAD352(i32[0], i29[0]) → COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0]), COND_LOAD352(TRUE, i32[1], i29[1]) → LOAD352(/(i32[1], 2), +(i29[1], 1)) which results in the following constraint:

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

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

(3)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i32[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(4)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i32[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(5)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i32[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(6)    (i32[0] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]i32[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_LOAD352(TRUE, i32, i29) → LOAD352(/(i32, 2), +(i29, 1)) the following chains were created:
• We consider the chain LOAD352(i32[0], i29[0]) → COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0]), COND_LOAD352(TRUE, i32[1], i29[1]) → LOAD352(/(i32[1], 2), +(i29[1], 1)), LOAD352(i32[0], i29[0]) → COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0]) which results in the following constraint:

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

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

(9)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] + i32[0] + [-1]max{i32[0], [-1]i32[0]} ≥ 0)

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

(10)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] + i32[0] + [-1]max{i32[0], [-1]i32[0]} ≥ 0)

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

(11)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0∧[2]i32[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(12)    (i32[0] ≥ 0∧i29[0] ≥ 0∧[4] + [2]i32[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(13)    (i32[0] ≥ 0∧i29[0] ≥ 0∧[2] + i32[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i32[0] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]i32[0] ≥ 0∧[(-1)bso_13] ≥ 0)

• (i32[0] ≥ 0∧i29[0] ≥ 0∧[2] + i32[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-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[POLO]:

POL(TRUE) = 0
POL(FALSE) = [3]
POL(LOAD352(x1, x2)) = [-1] + x1
POL(COND_LOAD352(x1, x2, x3)) = [-1] + x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(1) = [1]
POL(+(x1, x2)) = x1 + x2
POL(0) = 0
POL(2) = [2]

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, 2)1 @ {LOAD352_2/0}) = max{x1, [-1]x1} + [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

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

TRUE1&&(TRUE, TRUE)1
FALSE1&&(FALSE, FALSE)1
/1

### (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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD352(i32[0], i29[0]) → COND_LOAD352(i32[0] > 1 && i29[0] + 1 > 0, i32[0], i29[0])

The set Q consists of the following terms:

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: