### (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: Duplicate

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 171 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

### (4) Obligation:

ITRS problem:

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

The TRS R consists of the following rules:
Load430(2, i12, i37) → Cond_Load430(i37 > 2 && i12 >= 0 && i12 > i37 && i12 + 1 > 0, 2, i12, i37)
Cond_Load430(TRUE, 2, i12, i37) → Load430(2, i12 + 1, 2 * i37)
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:
• 2

We removed arguments according to the following replacements:

### (6) Obligation:

ITRS problem:

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

The TRS R consists of the following rules:
Load430(i12, i37) → Cond_Load430(i37 > 2 && i12 >= 0 && i12 > i37 && i12 + 1 > 0, i12, i37)
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:
Load430(i12, i37) → Cond_Load430(i37 > 2 && i12 >= 0 && i12 > i37 && i12 + 1 > 0, i12, i37)

The integer pair graph contains the following rules and edges:
(0): LOAD430(i12[0], i37[0]) → COND_LOAD430(i37[0] > 2 && i12[0] >= 0 && i12[0] > i37[0] && i12[0] + 1 > 0, i12[0], i37[0])

(0) -> (1), if ((i37[0]* i37[1])∧(i12[0]* i12[1])∧(i37[0] > 2 && i12[0] >= 0 && i12[0] > i37[0] && i12[0] + 1 > 0* TRUE))

(1) -> (0), if ((i12[1] + 1* i12[0])∧(2 * i37[1]* i37[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): LOAD430(i12[0], i37[0]) → COND_LOAD430(i37[0] > 2 && i12[0] >= 0 && i12[0] > i37[0] && i12[0] + 1 > 0, i12[0], i37[0])

(0) -> (1), if ((i37[0]* i37[1])∧(i12[0]* i12[1])∧(i37[0] > 2 && i12[0] >= 0 && i12[0] > i37[0] && i12[0] + 1 > 0* TRUE))

(1) -> (0), if ((i12[1] + 1* i12[0])∧(2 * i37[1]* i37[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 LOAD430(i12, i37) → COND_LOAD430(&&(&&(&&(>(i37, 2), >=(i12, 0)), >(i12, i37)), >(+(i12, 1), 0)), i12, i37) the following chains were created:
• We consider the chain LOAD430(i12[0], i37[0]) → COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0]), COND_LOAD430(TRUE, i12[1], i37[1]) → LOAD430(+(i12[1], 1), *(2, i37[1])) which results in the following constraint:

(1)    (i37[0]=i37[1]i12[0]=i12[1]&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0))=TRUELOAD430(i12[0], i37[0])≥NonInfC∧LOAD430(i12[0], i37[0])≥COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])∧(UIncreasing(COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])), ≥))

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

(2)    (>(+(i12[0], 1), 0)=TRUE>(i12[0], i37[0])=TRUE>(i37[0], 2)=TRUE>=(i12[0], 0)=TRUELOAD430(i12[0], i37[0])≥NonInfC∧LOAD430(i12[0], i37[0])≥COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])∧(UIncreasing(COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])), ≥))

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

(3)    (i12[0] ≥ 0∧i12[0] + [-1] + [-1]i37[0] ≥ 0∧i37[0] + [-3] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i37[0] + [bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(4)    (i12[0] ≥ 0∧i12[0] + [-1] + [-1]i37[0] ≥ 0∧i37[0] + [-3] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i37[0] + [bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(5)    (i12[0] ≥ 0∧i12[0] + [-1] + [-1]i37[0] ≥ 0∧i37[0] + [-3] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i37[0] + [bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(6)    ([1] + i37[0] + i12[0] ≥ 0∧i12[0] ≥ 0∧i37[0] + [-3] ≥ 0∧[1] + i37[0] + i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(7)    ([4] + i37[0] + i12[0] ≥ 0∧i12[0] ≥ 0∧i37[0] ≥ 0∧[4] + i37[0] + i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)

For Pair COND_LOAD430(TRUE, i12, i37) → LOAD430(+(i12, 1), *(2, i37)) the following chains were created:
• We consider the chain LOAD430(i12[0], i37[0]) → COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0]), COND_LOAD430(TRUE, i12[1], i37[1]) → LOAD430(+(i12[1], 1), *(2, i37[1])), LOAD430(i12[0], i37[0]) → COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0]) which results in the following constraint:

(8)    (i37[0]=i37[1]i12[0]=i12[1]&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0))=TRUE+(i12[1], 1)=i12[0]1*(2, i37[1])=i37[0]1COND_LOAD430(TRUE, i12[1], i37[1])≥NonInfC∧COND_LOAD430(TRUE, i12[1], i37[1])≥LOAD430(+(i12[1], 1), *(2, i37[1]))∧(UIncreasing(LOAD430(+(i12[1], 1), *(2, i37[1]))), ≥))

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

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

(10)    (i12[0] ≥ 0∧i12[0] + [-1] + [-1]i37[0] ≥ 0∧i37[0] + [-3] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(LOAD430(+(i12[1], 1), *(2, i37[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i37[0] + [bni_15]i12[0] ≥ 0∧[-1 + (-1)bso_16] + i37[0] ≥ 0)

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

(11)    (i12[0] ≥ 0∧i12[0] + [-1] + [-1]i37[0] ≥ 0∧i37[0] + [-3] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(LOAD430(+(i12[1], 1), *(2, i37[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i37[0] + [bni_15]i12[0] ≥ 0∧[-1 + (-1)bso_16] + i37[0] ≥ 0)

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

(12)    (i12[0] ≥ 0∧i12[0] + [-1] + [-1]i37[0] ≥ 0∧i37[0] + [-3] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(LOAD430(+(i12[1], 1), *(2, i37[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i37[0] + [bni_15]i12[0] ≥ 0∧[-1 + (-1)bso_16] + i37[0] ≥ 0)

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

(13)    ([1] + i37[0] + i12[0] ≥ 0∧i12[0] ≥ 0∧i37[0] + [-3] ≥ 0∧[1] + i37[0] + i12[0] ≥ 0 ⇒ (UIncreasing(LOAD430(+(i12[1], 1), *(2, i37[1]))), ≥)∧[(-1)Bound*bni_15] + [bni_15]i12[0] ≥ 0∧[-1 + (-1)bso_16] + i37[0] ≥ 0)

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

(14)    ([4] + i37[0] + i12[0] ≥ 0∧i12[0] ≥ 0∧i37[0] ≥ 0∧[4] + i37[0] + i12[0] ≥ 0 ⇒ (UIncreasing(LOAD430(+(i12[1], 1), *(2, i37[1]))), ≥)∧[(-1)Bound*bni_15] + [bni_15]i12[0] ≥ 0∧[2 + (-1)bso_16] + i37[0] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD430(i12, i37) → COND_LOAD430(&&(&&(&&(>(i37, 2), >=(i12, 0)), >(i12, i37)), >(+(i12, 1), 0)), i12, i37)
• ([4] + i37[0] + i12[0] ≥ 0∧i12[0] ≥ 0∧i37[0] ≥ 0∧[4] + i37[0] + i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)

• ([4] + i37[0] + i12[0] ≥ 0∧i12[0] ≥ 0∧i37[0] ≥ 0∧[4] + i37[0] + i12[0] ≥ 0 ⇒ (UIncreasing(LOAD430(+(i12[1], 1), *(2, i37[1]))), ≥)∧[(-1)Bound*bni_15] + [bni_15]i12[0] ≥ 0∧[2 + (-1)bso_16] + i37[0] ≥ 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(LOAD430(x1, x2)) = [-1] + [-1]x2 + x1
POL(COND_LOAD430(x1, x2, x3)) = [-1] + [-1]x3 + x2 + [-1]x1
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-1]
POL(2) = [2]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(*(x1, x2)) = x1·x2

The following pairs are in P>:

The following pairs are in Pbound:

LOAD430(i12[0], i37[0]) → COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])

The following pairs are in P:

LOAD430(i12[0], i37[0]) → COND_LOAD430(&&(&&(&&(>(i37[0], 2), >=(i12[0], 0)), >(i12[0], i37[0])), >(+(i12[0], 1), 0)), i12[0], i37[0])

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

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

### (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): LOAD430(i12[0], i37[0]) → COND_LOAD430(i37[0] > 2 && i12[0] >= 0 && i12[0] > i37[0] && i12[0] + 1 > 0, i12[0], i37[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: