### (0) Obligation:

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

Manifest-Version: 1.0 Created-By: 1.6.0_20 (Sun Microsystems Inc.) Main-Class: AG313

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 163 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:
Load665(i44, i113, i44, i109) → Cond_Load665(i44 > 0 && i113 > 0 && i109 + 1 > 0, i44, i113, i44, i109)
Cond_Load665(TRUE, i44, i113, i44, i109) → Load665(i44, i113 - 1 - i44 - 1, i44, i109 + 1)
The set Q consists of the following terms:

### (5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
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:
Load665(i113, i44, i109) → Cond_Load665(i44 > 0 && i113 > 0 && i109 + 1 > 0, i113, i44, i109)
Cond_Load665(TRUE, i113, i44, i109) → Load665(i113 - 1 - i44 - 1, i44, i109 + 1)
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:
Load665(i113, i44, i109) → Cond_Load665(i44 > 0 && i113 > 0 && i109 + 1 > 0, i113, i44, i109)
Cond_Load665(TRUE, i113, i44, i109) → Load665(i113 - 1 - i44 - 1, i44, i109 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(i44[0] > 0 && i113[0] > 0 && i109[0] + 1 > 0, i113[0], i44[0], i109[0])
(1): COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(i113[1] - 1 - i44[1] - 1, i44[1], i109[1] + 1)

(0) -> (1), if ((i109[0]* i109[1])∧(i113[0]* i113[1])∧(i44[0]* i44[1])∧(i44[0] > 0 && i113[0] > 0 && i109[0] + 1 > 0* TRUE))

(1) -> (0), if ((i109[1] + 1* i109[0])∧(i44[1]* i44[0])∧(i113[1] - 1 - i44[1] - 1* i113[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): LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(i44[0] > 0 && i113[0] > 0 && i109[0] + 1 > 0, i113[0], i44[0], i109[0])
(1): COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(i113[1] - 1 - i44[1] - 1, i44[1], i109[1] + 1)

(0) -> (1), if ((i109[0]* i109[1])∧(i113[0]* i113[1])∧(i44[0]* i44[1])∧(i44[0] > 0 && i113[0] > 0 && i109[0] + 1 > 0* TRUE))

(1) -> (0), if ((i109[1] + 1* i109[0])∧(i44[1]* i44[0])∧(i113[1] - 1 - i44[1] - 1* i113[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 LOAD665(i113, i44, i109) → COND_LOAD665(&&(&&(>(i44, 0), >(i113, 0)), >(+(i109, 1), 0)), i113, i44, i109) the following chains were created:
• We consider the chain LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0]), COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1)) which results in the following constraint:

(1)    (i109[0]=i109[1]i113[0]=i113[1]i44[0]=i44[1]&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0))=TRUELOAD665(i113[0], i44[0], i109[0])≥NonInfC∧LOAD665(i113[0], i44[0], i109[0])≥COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])∧(UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥))

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

(2)    (>(+(i109[0], 1), 0)=TRUE>(i44[0], 0)=TRUE>(i113[0], 0)=TRUELOAD665(i113[0], i44[0], i109[0])≥NonInfC∧LOAD665(i113[0], i44[0], i109[0])≥COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])∧(UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥))

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

(3)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i113[0] ≥ 0∧[(-1)bso_15] + i44[0] ≥ 0)

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

(4)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i113[0] ≥ 0∧[(-1)bso_15] + i44[0] ≥ 0)

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

(5)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i113[0] ≥ 0∧[(-1)bso_15] + i44[0] ≥ 0)

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

(6)    (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i113[0] ≥ 0∧[1 + (-1)bso_15] + i44[0] ≥ 0)

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

(7)    (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i113[0] ≥ 0∧[1 + (-1)bso_15] + i44[0] ≥ 0)

For Pair COND_LOAD665(TRUE, i113, i44, i109) → LOAD665(-(-(i113, 1), -(i44, 1)), i44, +(i109, 1)) the following chains were created:
• We consider the chain LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0]), COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1)), LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0]) which results in the following constraint:

(8)    (i109[0]=i109[1]i113[0]=i113[1]i44[0]=i44[1]&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0))=TRUE+(i109[1], 1)=i109[0]1i44[1]=i44[0]1-(-(i113[1], 1), -(i44[1], 1))=i113[0]1COND_LOAD665(TRUE, i113[1], i44[1], i109[1])≥NonInfC∧COND_LOAD665(TRUE, i113[1], i44[1], i109[1])≥LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))∧(UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥))

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

(9)    (>(+(i109[0], 1), 0)=TRUE>(i44[0], 0)=TRUE>(i113[0], 0)=TRUECOND_LOAD665(TRUE, i113[0], i44[0], i109[0])≥NonInfC∧COND_LOAD665(TRUE, i113[0], i44[0], i109[0])≥LOAD665(-(-(i113[0], 1), -(i44[0], 1)), i44[0], +(i109[0], 1))∧(UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥))

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

(10)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(11)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(12)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(13)    (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16 + (-1)bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(14)    (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD665(i113, i44, i109) → COND_LOAD665(&&(&&(>(i44, 0), >(i113, 0)), >(+(i109, 1), 0)), i113, i44, i109)
• (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i113[0] ≥ 0∧[1 + (-1)bso_15] + i44[0] ≥ 0)

• COND_LOAD665(TRUE, i113, i44, i109) → LOAD665(-(-(i113, 1), -(i44, 1)), i44, +(i109, 1))
• (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)

The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = [1]
POL(COND_LOAD665(x1, x2, x3, x4)) = [-1]x3 + x2
POL(&&(x1, x2)) = [1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(-(x1, x2)) = x1 + [-1]x2

The following pairs are in P>:

LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])

The following pairs are in Pbound:

LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])

The following pairs are in P:

COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))

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

### (12) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(i113[1] - 1 - i44[1] - 1, i44[1], i109[1] + 1)

The set Q consists of the following terms: