(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: Round3

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 101 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:
Load139(i12) → Cond_Load139(i12 % 3 > 0 && i12 + 1 > 0, i12)
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:
Load139(i12) → Cond_Load139(i12 % 3 > 0 && i12 + 1 > 0, i12)

The integer pair graph contains the following rules and edges:
(0): LOAD139(i12[0]) → COND_LOAD139(i12[0] % 3 > 0 && i12[0] + 1 > 0, i12[0])

(0) -> (1), if ((i12[0]* i12[1])∧(i12[0] % 3 > 0 && i12[0] + 1 > 0* TRUE))

(1) -> (0), if ((i12[1] + 1* i12[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): LOAD139(i12[0]) → COND_LOAD139(i12[0] % 3 > 0 && i12[0] + 1 > 0, i12[0])

(0) -> (1), if ((i12[0]* i12[1])∧(i12[0] % 3 > 0 && i12[0] + 1 > 0* TRUE))

(1) -> (0), if ((i12[1] + 1* i12[0]))

The set Q consists of the following terms:

(9) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(10) Obligation:

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

The TRS R consists of the following rules:

and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(01), pos(01)) → false
greater_int(pos(01), neg(01)) → false
greater_int(neg(01), pos(01)) → false
greater_int(neg(01), neg(01)) → false
greater_int(pos(01), pos(s(y))) → false
greater_int(neg(01), pos(s(y))) → false
greater_int(pos(01), neg(s(y))) → true
greater_int(neg(01), neg(s(y))) → true
greater_int(pos(s(x)), pos(01)) → true
greater_int(neg(s(x)), pos(01)) → false
greater_int(pos(s(x)), neg(01)) → true
greater_int(neg(s(x)), neg(01)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(01, s(x)) → 01
mod_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(true, x, y) → x
if(false, x, y) → y
minus_nat_s(x, 01) → x
minus_nat_s(01, s(y)) → 01
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(01, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(01, 01) → pos(01)
minus_nat(01, s(y)) → neg(s(y))
minus_nat(s(x), 01) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(01, s(x0))
mod_nat(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 01)
minus_nat_s(01, s(x0))
minus_nat_s(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(01, x0)
plus_nat(s(x0), x1)
minus_nat(01, 01)
minus_nat(01, s(x0))
minus_nat(s(x0), 01)
minus_nat(s(x0), s(x1))

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

(11) 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.

(12) Obligation:

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

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(01, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(01, 01) → pos(01)
minus_nat(01, s(y)) → neg(s(y))
minus_nat(s(x), 01) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
greater_int(pos(01), pos(01)) → false
greater_int(neg(01), pos(01)) → false
greater_int(pos(s(x)), pos(01)) → true
greater_int(neg(s(x)), pos(01)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mod_nat(01, s(x)) → 01
mod_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
minus_nat_s(x, 01) → x
minus_nat_s(01, s(y)) → 01
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(01, s(x0))
mod_nat(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 01)
minus_nat_s(01, s(x0))
minus_nat_s(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(01, x0)
plus_nat(s(x0), x1)
minus_nat(01, 01)
minus_nat(01, s(x0))
minus_nat(s(x0), 01)
minus_nat(s(x0), s(x1))

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

(13) 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].

(14) Obligation:

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

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(01, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(01, 01) → pos(01)
minus_nat(01, s(y)) → neg(s(y))
minus_nat(s(x), 01) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
greater_int(pos(01), pos(01)) → false
greater_int(neg(01), pos(01)) → false
greater_int(pos(s(x)), pos(01)) → true
greater_int(neg(s(x)), pos(01)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mod_nat(01, s(x)) → 01
mod_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
minus_nat_s(x, 01) → x
minus_nat_s(01, s(y)) → 01
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(01, s(x0))
mod_nat(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 01)
minus_nat_s(01, s(x0))
minus_nat_s(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(01, x0)
plus_nat(s(x0), x1)
minus_nat(01, 01)
minus_nat(01, s(x0))
minus_nat(s(x0), 01)
minus_nat(s(x0), s(x1))

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(01) = 0
POL(and(x1, x2)) = x1
POL(false) = 0
POL(greater_int(x1, x2)) = x1
POL(greatereq_int(x1, x2)) = 0
POL(if(x1, x2, x3)) = 0
POL(minus_nat(x1, x2)) = 0
POL(minus_nat_s(x1, x2)) = 0
POL(mod_int(x1, x2)) = 0
POL(mod_nat(x1, x2)) = 0
POL(neg(x1)) = 0
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = 0
POL(pos(x1)) = x1
POL(s(x1)) = 1
POL(true) = 1

The following usable rules [FROCOS05] were oriented:

mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
greater_int(pos(01), pos(01)) → false
greater_int(neg(01), pos(01)) → false
greater_int(pos(s(x)), pos(01)) → true
greater_int(neg(s(x)), pos(01)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mod_nat(01, s(x)) → 01
mod_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))

(16) Obligation:

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

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(01, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(01, 01) → pos(01)
minus_nat(01, s(y)) → neg(s(y))
minus_nat(s(x), 01) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
greater_int(pos(01), pos(01)) → false
greater_int(neg(01), pos(01)) → false
greater_int(pos(s(x)), pos(01)) → true
greater_int(neg(s(x)), pos(01)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mod_nat(01, s(x)) → 01
mod_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
minus_nat_s(x, 01) → x
minus_nat_s(01, s(y)) → 01
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(01, s(x0))
mod_nat(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 01)
minus_nat_s(01, s(x0))
minus_nat_s(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(01, x0)
plus_nat(s(x0), x1)
minus_nat(01, 01)
minus_nat(01, s(x0))
minus_nat(s(x0), 01)
minus_nat(s(x0), s(x1))

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

(17) DependencyGraphProof (EQUIVALENT transformation)

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