### (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: PastaB3

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 169 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:
Load496(i46, i37) → Cond_Load496(i37 >= 0 && i46 > i37, i46, i37)
Cond_Load496(TRUE, i46, i37) → Load496(i46, i46 + i37)
The set Q consists of the following terms:

### (6) 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:
Load496(i46, i37) → Cond_Load496(i37 >= 0 && i46 > i37, i46, i37)
Cond_Load496(TRUE, i46, i37) → Load496(i46, i46 + i37)

The integer pair graph contains the following rules and edges:
(0): LOAD496(i46[0], i37[0]) → COND_LOAD496(i37[0] >= 0 && i46[0] > i37[0], i46[0], i37[0])
(1): COND_LOAD496(TRUE, i46[1], i37[1]) → LOAD496(i46[1], i46[1] + i37[1])

(0) -> (1), if ((i37[0]* i37[1])∧(i37[0] >= 0 && i46[0] > i37[0]* TRUE)∧(i46[0]* i46[1]))

(1) -> (0), if ((i46[1] + i37[1]* i37[0])∧(i46[1]* i46[0]))

The set Q consists of the following terms:

### (7) 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.

### (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

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD496(i46[0], i37[0]) → COND_LOAD496(i37[0] >= 0 && i46[0] > i37[0], i46[0], i37[0])
(1): COND_LOAD496(TRUE, i46[1], i37[1]) → LOAD496(i46[1], i46[1] + i37[1])

(0) -> (1), if ((i37[0]* i37[1])∧(i37[0] >= 0 && i46[0] > i37[0]* TRUE)∧(i46[0]* i46[1]))

(1) -> (0), if ((i46[1] + i37[1]* i37[0])∧(i46[1]* i46[0]))

The set Q consists of the following terms:

### (9) 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 LOAD496(i46, i37) → COND_LOAD496(&&(>=(i37, 0), >(i46, i37)), i46, i37) the following chains were created:
• We consider the chain LOAD496(i46[0], i37[0]) → COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0]), COND_LOAD496(TRUE, i46[1], i37[1]) → LOAD496(i46[1], +(i46[1], i37[1])) which results in the following constraint:

(1)    (i37[0]=i37[1]&&(>=(i37[0], 0), >(i46[0], i37[0]))=TRUEi46[0]=i46[1]LOAD496(i46[0], i37[0])≥NonInfC∧LOAD496(i46[0], i37[0])≥COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])∧(UIncreasing(COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])), ≥))

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

(2)    (>=(i37[0], 0)=TRUE>(i46[0], i37[0])=TRUELOAD496(i46[0], i37[0])≥NonInfC∧LOAD496(i46[0], i37[0])≥COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])∧(UIncreasing(COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])), ≥))

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

(3)    (i37[0] ≥ 0∧i46[0] + [-1] + [-1]i37[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i37[0] + [bni_13]i46[0] ≥ 0∧[(-1)bso_14] + [-1]i37[0] + i46[0] ≥ 0)

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

(4)    (i37[0] ≥ 0∧i46[0] + [-1] + [-1]i37[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i37[0] + [bni_13]i46[0] ≥ 0∧[(-1)bso_14] + [-1]i37[0] + i46[0] ≥ 0)

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

(5)    (i37[0] ≥ 0∧i46[0] + [-1] + [-1]i37[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i37[0] + [bni_13]i46[0] ≥ 0∧[(-1)bso_14] + [-1]i37[0] + i46[0] ≥ 0)

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

(6)    (i37[0] ≥ 0∧i46[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i46[0] ≥ 0∧[1 + (-1)bso_14] + i46[0] ≥ 0)

For Pair COND_LOAD496(TRUE, i46, i37) → LOAD496(i46, +(i46, i37)) the following chains were created:
• We consider the chain LOAD496(i46[0], i37[0]) → COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0]), COND_LOAD496(TRUE, i46[1], i37[1]) → LOAD496(i46[1], +(i46[1], i37[1])), LOAD496(i46[0], i37[0]) → COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0]) which results in the following constraint:

(7)    (i37[0]=i37[1]&&(>=(i37[0], 0), >(i46[0], i37[0]))=TRUEi46[0]=i46[1]+(i46[1], i37[1])=i37[0]1i46[1]=i46[0]1COND_LOAD496(TRUE, i46[1], i37[1])≥NonInfC∧COND_LOAD496(TRUE, i46[1], i37[1])≥LOAD496(i46[1], +(i46[1], i37[1]))∧(UIncreasing(LOAD496(i46[1], +(i46[1], i37[1]))), ≥))

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)    (i37[0] ≥ 0∧i46[0] + [-1] + [-1]i37[0] ≥ 0 ⇒ (UIncreasing(LOAD496(i46[1], +(i46[1], i37[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] ≥ 0∧[(-1)bso_16] + i37[0] ≥ 0)

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

(10)    (i37[0] ≥ 0∧i46[0] + [-1] + [-1]i37[0] ≥ 0 ⇒ (UIncreasing(LOAD496(i46[1], +(i46[1], i37[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] ≥ 0∧[(-1)bso_16] + i37[0] ≥ 0)

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

(11)    (i37[0] ≥ 0∧i46[0] + [-1] + [-1]i37[0] ≥ 0 ⇒ (UIncreasing(LOAD496(i46[1], +(i46[1], i37[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] ≥ 0∧[(-1)bso_16] + i37[0] ≥ 0)

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

(12)    (i37[0] ≥ 0∧i46[0] ≥ 0 ⇒ (UIncreasing(LOAD496(i46[1], +(i46[1], i37[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] ≥ 0∧[(-1)bso_16] + i37[0] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD496(i46, i37) → COND_LOAD496(&&(>=(i37, 0), >(i46, i37)), i46, i37)
• (i37[0] ≥ 0∧i46[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i46[0] ≥ 0∧[1 + (-1)bso_14] + i46[0] ≥ 0)

• COND_LOAD496(TRUE, i46, i37) → LOAD496(i46, +(i46, i37))
• (i37[0] ≥ 0∧i46[0] ≥ 0 ⇒ (UIncreasing(LOAD496(i46[1], +(i46[1], i37[1]))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] ≥ 0∧[(-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(LOAD496(x1, x2)) = [-1] + [-1]x2 + x1
POL(COND_LOAD496(x1, x2, x3)) = [-1]
POL(&&(x1, x2)) = [1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2

The following pairs are in P>:

LOAD496(i46[0], i37[0]) → COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])

The following pairs are in Pbound:

LOAD496(i46[0], i37[0]) → COND_LOAD496(&&(>=(i37[0], 0), >(i46[0], i37[0])), i46[0], i37[0])
COND_LOAD496(TRUE, i46[1], i37[1]) → LOAD496(i46[1], +(i46[1], i37[1]))

The following pairs are in P:

COND_LOAD496(TRUE, i46[1], i37[1]) → LOAD496(i46[1], +(i46[1], i37[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

### (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:
(1): COND_LOAD496(TRUE, i46[1], i37[1]) → LOAD496(i46[1], i46[1] + i37[1])

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

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

### (14) 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: