Termination proof of /tmp/tmp5xSu

Termination of the given ITRSProblem could not be shown:

↳ ITRS

  ↳ ITRStoQTRSProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

cond(TRUE, x, y) → 1@z
cond(FALSE, x, y) → *@z(2@z, log(x, *@z(y, y)))
logNat(TRUE, x, y) → cond(<=@z(x, y), x, y)
log(x, y) → logNat(&&(>=@z(x, 0@z), >=@z(y, 2@z)), x, y)

The set Q consists of the following terms:

cond(TRUE, x0, x1)
cond(FALSE, x0, x1)
logNat(TRUE, x0, x1)
log(x0, x1)

Represented integers and predefined function symbols by Terms

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

cond(true, x, y) → pos(s(0))
cond(false, x, y) → mult_int(pos(s(s(0))), log(x, mult_int(y, y)))
logNat(true, x, y) → cond(lesseq_int(x, y), x, y)
log(x, y) → logNat(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

COND(false, x, y) → MULT_INT(pos(s(s(0))), log(x, mult_int(y, y)))
COND(false, x, y) → LOG(x, mult_int(y, y))
COND(false, x, y) → MULT_INT(y, y)
LOGNAT(true, x, y) → COND(lesseq_int(x, y), x, y)
LOGNAT(true, x, y) → LESSEQ_INT(x, y)
LOG(x, y) → LOGNAT(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
LOG(x, y) → AND(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0)))))
LOG(x, y) → GREATEREQ_INT(x, pos(0))
LOG(x, y) → GREATEREQ_INT(y, pos(s(s(0))))
MULT_INT(pos(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(pos(x), neg(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), neg(y)) → MULT_NAT(x, y)
MULT_NAT(s(x), s(y)) → PLUS_NAT(mult_nat(x, s(y)), s(y))
MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
LESSEQ_INT(pos(s(x)), pos(s(y))) → LESSEQ_INT(pos(x), pos(y))
LESSEQ_INT(neg(s(x)), neg(s(y))) → LESSEQ_INT(neg(x), neg(y))
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

The TRS R consists of the following rules:

cond(true, x, y) → pos(s(0))
cond(false, x, y) → mult_int(pos(s(s(0))), log(x, mult_int(y, y)))
logNat(true, x, y) → cond(lesseq_int(x, y), x, y)
log(x, y) → logNat(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

COND(false, x, y) → MULT_INT(pos(s(s(0))), log(x, mult_int(y, y)))
COND(false, x, y) → LOG(x, mult_int(y, y))
COND(false, x, y) → MULT_INT(y, y)
LOGNAT(true, x, y) → COND(lesseq_int(x, y), x, y)
LOGNAT(true, x, y) → LESSEQ_INT(x, y)
LOG(x, y) → LOGNAT(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
LOG(x, y) → AND(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0)))))
LOG(x, y) → GREATEREQ_INT(x, pos(0))
LOG(x, y) → GREATEREQ_INT(y, pos(s(s(0))))
MULT_INT(pos(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(pos(x), neg(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), neg(y)) → MULT_NAT(x, y)
MULT_NAT(s(x), s(y)) → PLUS_NAT(mult_nat(x, s(y)), s(y))
MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
LESSEQ_INT(pos(s(x)), pos(s(y))) → LESSEQ_INT(pos(x), pos(y))
LESSEQ_INT(neg(s(x)), neg(s(y))) → LESSEQ_INT(neg(x), neg(y))
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

The TRS R consists of the following rules:

cond(true, x, y) → pos(s(0))
cond(false, x, y) → mult_int(pos(s(s(0))), log(x, mult_int(y, y)))
logNat(true, x, y) → cond(lesseq_int(x, y), x, y)
log(x, y) → logNat(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 11 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

The TRS R consists of the following rules:

cond(true, x, y) → pos(s(0))
cond(false, x, y) → mult_int(pos(s(s(0))), log(x, mult_int(y, y)))
logNat(true, x, y) → cond(lesseq_int(x, y), x, y)
log(x, y) → logNat(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

R is empty.
The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATEREQ_INT(x₁, x₂)) = 2·x₁ + x₂
POL(neg(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))

The TRS R consists of the following rules:

cond(true, x, y) → pos(s(0))
cond(false, x, y) → mult_int(pos(s(s(0))), log(x, mult_int(y, y)))
logNat(true, x, y) → cond(lesseq_int(x, y), x, y)
log(x, y) → logNat(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))

R is empty.
The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATEREQ_INT(x₁, x₂)) = 2·x₁ + x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LESSEQ_INT(neg(s(x)), neg(s(y))) → LESSEQ_INT(neg(x), neg(y))

The TRS R consists of the following rules:

cond(true, x, y) → pos(s(0))
cond(false, x, y) → mult_int(pos(s(s(0))), log(x, mult_int(y, y)))
logNat(true, x, y) → cond(lesseq_int(x, y), x, y)
log(x, y) → logNat(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LESSEQ_INT(neg(s(x)), neg(s(y))) → LESSEQ_INT(neg(x), neg(y))

R is empty.
The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LESSEQ_INT(neg(s(x)), neg(s(y))) → LESSEQ_INT(neg(x), neg(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESSEQ_INT(x₁, x₂)) = 2·x₁ + x₂
POL(neg(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LESSEQ_INT(pos(s(x)), pos(s(y))) → LESSEQ_INT(pos(x), pos(y))

The TRS R consists of the following rules:

cond(true, x, y) → pos(s(0))
cond(false, x, y) → mult_int(pos(s(s(0))), log(x, mult_int(y, y)))
logNat(true, x, y) → cond(lesseq_int(x, y), x, y)
log(x, y) → logNat(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LESSEQ_INT(pos(s(x)), pos(s(y))) → LESSEQ_INT(pos(x), pos(y))

R is empty.
The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LESSEQ_INT(pos(s(x)), pos(s(y))) → LESSEQ_INT(pos(x), pos(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESSEQ_INT(x₁, x₂)) = 2·x₁ + x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

cond(true, x, y) → pos(s(0))
cond(false, x, y) → mult_int(pos(s(s(0))), log(x, mult_int(y, y)))
logNat(true, x, y) → cond(lesseq_int(x, y), x, y)
log(x, y) → logNat(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

R is empty.
The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

The TRS R consists of the following rules:

cond(true, x, y) → pos(s(0))
cond(false, x, y) → mult_int(pos(s(s(0))), log(x, mult_int(y, y)))
logNat(true, x, y) → cond(lesseq_int(x, y), x, y)
log(x, y) → logNat(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

R is empty.
The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

From the DPs we obtained the following set of size-change graphs:

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))
The graph contains the following edges 1 > 1, 2 >= 2

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(x, y) → LOGNAT(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
LOGNAT(true, x, y) → COND(lesseq_int(x, y), x, y)

The TRS R consists of the following rules:

cond(true, x, y) → pos(s(0))
cond(false, x, y) → mult_int(pos(s(s(0))), log(x, mult_int(y, y)))
logNat(true, x, y) → cond(lesseq_int(x, y), x, y)
log(x, y) → logNat(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(x, y) → LOGNAT(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
LOGNAT(true, x, y) → COND(lesseq_int(x, y), x, y)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))

The set Q consists of the following terms:

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

cond(true, x0, x1)
cond(false, x0, x1)
logNat(true, x0, x1)
log(x0, x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

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

COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(x, y) → LOGNAT(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
LOGNAT(true, x, y) → COND(lesseq_int(x, y), x, y)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(s(0))) is replaced by the fresh variable x_removed.
Pair: LOG(x, y) → LOGNAT(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y)
Positions in right side of the pair:

[0,1,1]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                        ↳ RemovalProof

                        ↳ Narrowing

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

COND(false, x, y, x_removed) → LOG(x, mult_int(y, y), x_removed)
LOG(x, y, x_removed) → LOGNAT(and(greatereq_int(x, pos(0)), greatereq_int(y, x_removed)), x, y, x_removed)
LOGNAT(true, x, y, x_removed) → COND(lesseq_int(x, y), x, y, x_removed)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

[0,1,1]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                          ↳ QDP

                        ↳ Narrowing

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

COND(false, x, y, x_removed) → LOG(x, mult_int(y, y), x_removed)
LOG(x, y, x_removed) → LOGNAT(and(greatereq_int(x, pos(0)), greatereq_int(y, x_removed)), x, y, x_removed)
LOGNAT(true, x, y, x_removed) → COND(lesseq_int(x, y), x, y, x_removed)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(x, y) → LOGNAT(and(greatereq_int(x, pos(0)), greatereq_int(y, pos(s(s(0))))), x, y) at position [0] we obtained the following new rules [LPAR04]:

LOG(neg(0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), neg(0), y1)
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOG(y0, pos(0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(neg(s(x0)), y1) → LOGNAT(and(false, greatereq_int(y1, pos(s(s(0))))), neg(s(x0)), y1)
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

COND(false, x, y) → LOG(x, mult_int(y, y))
LOGNAT(true, x, y) → COND(lesseq_int(x, y), x, y)
LOG(neg(0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), neg(0), y1)
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOG(y0, pos(0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(neg(s(x0)), y1) → LOGNAT(and(false, greatereq_int(y1, pos(s(s(0))))), neg(s(x0)), y1)
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOGNAT(true, x, y) → COND(lesseq_int(x, y), x, y) at position [0] we obtained the following new rules [LPAR04]:

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOGNAT(true, pos(0), neg(0)) → COND(true, pos(0), neg(0))
LOGNAT(true, neg(x0), pos(x1)) → COND(true, neg(x0), pos(x1))
LOGNAT(true, pos(0), pos(x0)) → COND(true, pos(0), pos(x0))
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOGNAT(true, neg(x0), neg(0)) → COND(true, neg(x0), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

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

COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(neg(0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), neg(0), y1)
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOG(y0, pos(0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(neg(s(x0)), y1) → LOGNAT(and(false, greatereq_int(y1, pos(s(s(0))))), neg(s(x0)), y1)
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOGNAT(true, pos(0), neg(0)) → COND(true, pos(0), neg(0))
LOGNAT(true, neg(x0), pos(x1)) → COND(true, neg(x0), pos(x1))
LOGNAT(true, pos(0), pos(x0)) → COND(true, pos(0), pos(x0))
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOGNAT(true, neg(x0), neg(0)) → COND(true, neg(x0), neg(0))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

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

LOG(neg(0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), neg(0), y1)
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(y0, pos(0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOG(neg(s(x0)), y1) → LOGNAT(and(false, greatereq_int(y1, pos(s(s(0))))), neg(s(x0)), y1)
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

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

LOG(neg(0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), neg(0), y1)
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(y0, pos(0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOG(neg(s(x0)), y1) → LOGNAT(and(false, greatereq_int(y1, pos(s(s(0))))), neg(s(x0)), y1)
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(neg(0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), neg(0), y1) at position [0] we obtained the following new rules [LPAR04]:

LOG(neg(0), pos(0)) → LOGNAT(and(true, false), neg(0), pos(0))
LOG(neg(0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), neg(0), pos(s(x0)))
LOG(neg(0), neg(x0)) → LOGNAT(and(true, false), neg(0), neg(x0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

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

LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(y0, pos(0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOG(neg(s(x0)), y1) → LOGNAT(and(false, greatereq_int(y1, pos(s(s(0))))), neg(s(x0)), y1)
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOG(neg(0), pos(0)) → LOGNAT(and(true, false), neg(0), pos(0))
LOG(neg(0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), neg(0), pos(s(x0)))
LOG(neg(0), neg(x0)) → LOGNAT(and(true, false), neg(0), neg(x0))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

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

COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(y0, pos(0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOG(neg(s(x0)), y1) → LOGNAT(and(false, greatereq_int(y1, pos(s(s(0))))), neg(s(x0)), y1)
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(y0, pos(0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, pos(0)) at position [0] we obtained the following new rules [LPAR04]:

LOG(neg(0), pos(0)) → LOGNAT(and(true, false), neg(0), pos(0))
LOG(pos(x0), pos(0)) → LOGNAT(and(true, false), pos(x0), pos(0))
LOG(neg(s(x0)), pos(0)) → LOGNAT(and(false, false), neg(s(x0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

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

COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOG(neg(s(x0)), y1) → LOGNAT(and(false, greatereq_int(y1, pos(s(s(0))))), neg(s(x0)), y1)
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
LOG(neg(0), pos(0)) → LOGNAT(and(true, false), neg(0), pos(0))
LOG(pos(x0), pos(0)) → LOGNAT(and(true, false), pos(x0), pos(0))
LOG(neg(s(x0)), pos(0)) → LOGNAT(and(false, false), neg(s(x0)), pos(0))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

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

LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(neg(s(x0)), y1) → LOGNAT(and(false, greatereq_int(y1, pos(s(s(0))))), neg(s(x0)), y1)
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(neg(s(x0)), y1) → LOGNAT(and(false, greatereq_int(y1, pos(s(s(0))))), neg(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

LOG(neg(s(y0)), pos(s(x0))) → LOGNAT(and(false, greatereq_int(pos(x0), pos(s(0)))), neg(s(y0)), pos(s(x0)))
LOG(neg(s(y0)), pos(0)) → LOGNAT(and(false, false), neg(s(y0)), pos(0))
LOG(neg(s(y0)), neg(x0)) → LOGNAT(and(false, false), neg(s(y0)), neg(x0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

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

LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND(false, x, y) → LOG(x, mult_int(y, y))
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
LOG(neg(s(y0)), pos(s(x0))) → LOGNAT(and(false, greatereq_int(pos(x0), pos(s(0)))), neg(s(y0)), pos(s(x0)))
LOG(neg(s(y0)), pos(0)) → LOGNAT(and(false, false), neg(s(y0)), pos(0))
LOG(neg(s(y0)), neg(x0)) → LOGNAT(and(false, false), neg(s(y0)), neg(x0))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1)
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(pos(x0), y1) → LOGNAT(and(true, greatereq_int(y1, pos(s(s(0))))), pos(x0), y1) at position [0] we obtained the following new rules [LPAR04]:

LOG(pos(y0), pos(0)) → LOGNAT(and(true, false), pos(y0), pos(0))
LOG(pos(y0), neg(x0)) → LOGNAT(and(true, false), pos(y0), neg(x0))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(0)) → COND(false, pos(s(x0)), pos(0))
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(y0), pos(0)) → LOGNAT(and(true, false), pos(y0), pos(0))
LOG(pos(y0), neg(x0)) → LOGNAT(and(true, false), pos(y0), neg(x0))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

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

COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

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

COND(false, x, y) → LOG(x, mult_int(y, y))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND(false, x, y) → LOG(x, mult_int(y, y)) at position [1] we obtained the following new rules [LPAR04]:

COND(false, y0, pos(x0)) → LOG(y0, pos(mult_nat(x0, x0)))
COND(false, y0, neg(x0)) → LOG(y0, pos(mult_nat(x0, x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

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

LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(y0, neg(x0)) → LOGNAT(and(greatereq_int(y0, pos(0)), false), y0, neg(x0))
LOGNAT(true, pos(x0), neg(s(x1))) → COND(false, pos(x0), neg(s(x1)))
LOGNAT(true, pos(s(x0)), neg(x1)) → COND(false, pos(s(x0)), neg(x1))
LOGNAT(true, neg(0), neg(s(x0))) → COND(false, neg(0), neg(s(x0)))
LOGNAT(true, neg(s(x0)), neg(s(x1))) → COND(lesseq_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))
COND(false, y0, pos(x0)) → LOG(y0, pos(mult_nat(x0, x0)))
COND(false, y0, neg(x0)) → LOG(y0, pos(mult_nat(x0, x0)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND(false, y0, pos(x0)) → LOG(y0, pos(mult_nat(x0, x0)))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND(false, y0, pos(x0)) → LOG(y0, pos(mult_nat(x0, x0)))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))

The set Q consists of the following terms:

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND(false, y0, pos(x0)) → LOG(y0, pos(mult_nat(x0, x0)))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND(false, y0, pos(x0)) → LOG(y0, pos(mult_nat(x0, x0))) we obtained the following new rules [LPAR04]:

COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(mult_nat(s(z1), s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))
COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(mult_nat(s(z1), s(z1))))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(mult_nat(s(z1), s(z1)))) at position [1,0] we obtained the following new rules [LPAR04]:

COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(plus_nat(mult_nat(z1, s(z1)), s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))
COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(plus_nat(mult_nat(z1, s(z1)), s(z1))))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule LOG(y0, pos(s(x0))) → LOGNAT(and(greatereq_int(y0, pos(0)), greatereq_int(pos(x0), pos(s(0)))), y0, pos(s(x0))) we obtained the following new rules [LPAR04]:

LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(greatereq_int(pos(s(z0)), pos(0)), greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))
COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(plus_nat(mult_nat(z1, s(z1)), s(z1))))
LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(greatereq_int(pos(s(z0)), pos(0)), greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))
COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(plus_nat(mult_nat(z1, s(z1)), s(z1))))
LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(greatereq_int(pos(s(z0)), pos(0)), greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(greatereq_int(pos(s(z0)), pos(0)), greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1))) at position [0,0] we obtained the following new rules [LPAR04]:

LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(true, greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))
COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(plus_nat(mult_nat(z1, s(z1)), s(z1))))
LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(true, greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0)))
COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(plus_nat(mult_nat(z1, s(z1)), s(z1))))
LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(true, greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule LOG(pos(y0), pos(s(x0))) → LOGNAT(and(true, greatereq_int(pos(x0), pos(s(0)))), pos(y0), pos(s(x0))) we obtained the following new rules [LPAR04]:

LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(true, greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ RemovalProof

                                                                                                                        ↳ RemovalProof

                                                                                                                        ↳ QReductionProof

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(plus_nat(mult_nat(z1, s(z1)), s(z1))))
LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(true, greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(true, greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1)))
Positions in right side of the pair:

[0,1,1]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ RemovalProof

                                                                                                                          ↳ QDP

                                                                                                                        ↳ RemovalProof

                                                                                                                        ↳ QReductionProof

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

LOGNAT(true, pos(s(x0)), pos(s(x1)), x_removed) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)), x_removed)
COND(false, pos(s(z0)), pos(s(z1)), x_removed) → LOG(pos(s(z0)), pos(plus_nat(mult_nat(z1, s(z1)), s(z1))), x_removed)
LOG(pos(s(z0)), pos(s(x1)), x_removed) → LOGNAT(and(true, greatereq_int(pos(x1), x_removed)), pos(s(z0)), pos(s(x1)), x_removed)

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

[0,1,1]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ RemovalProof

                                                                                                                        ↳ RemovalProof

                                                                                                                          ↳ QDP

                                                                                                                        ↳ QReductionProof

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

LOGNAT(true, pos(s(x0)), pos(s(x1)), x_removed) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)), x_removed)
COND(false, pos(s(z0)), pos(s(z1)), x_removed) → LOG(pos(s(z0)), pos(plus_nat(mult_nat(z1, s(z1)), s(z1))), x_removed)
LOG(pos(s(z0)), pos(s(x1)), x_removed) → LOGNAT(and(true, greatereq_int(pos(x1), x_removed)), pos(s(z0)), pos(s(x1)), x_removed)

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].

lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(neg(s(x0)), neg(s(x1)))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ RemovalProof

                                                                                                                        ↳ RemovalProof

                                                                                                                        ↳ QReductionProof

                                                                                                                          ↳ QDP

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

LOGNAT(true, pos(s(x0)), pos(s(x1))) → COND(lesseq_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND(false, pos(s(z0)), pos(s(z1))) → LOG(pos(s(z0)), pos(plus_nat(mult_nat(z1, s(z1)), s(z1))))
LOG(pos(s(z0)), pos(s(x1))) → LOGNAT(and(true, greatereq_int(pos(x1), pos(s(0)))), pos(s(z0)), pos(s(x1)))

The TRS R consists of the following rules:

lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
mult_nat(0, y) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), pos(s(x1)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(pos(s(x0)), pos(s(x1)))

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