Termination proof of /tmp/tmpGp3DIo/round.itrs

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_f(TRUE, x, y) → f(x, round(x))
round(x) → +@z(x, 1@z)
round(x) → x
f(x, y) → Cond_f(&&(>=@z(x, 1@z), =@z(y, -@z(x, 1@z))), x, y)

The set Q consists of the following terms:

Cond_f(TRUE, x0, x1)
round(x0)
f(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_f(true, x, y) → f(x, round(x))
round(x) → plus_int(pos(s(0)), x)
round(x) → x
f(x, y) → Cond_f(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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_F(true, x, y) → F(x, round(x))
COND_F(true, x, y) → ROUND(x)
ROUND(x) → PLUS_INT(pos(s(0)), x)
F(x, y) → COND_F(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
F(x, y) → AND(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0)))))
F(x, y) → GREATEREQ_INT(x, pos(s(0)))
F(x, y) → EQUAL_INT(y, minus_int(x, pos(s(0))))
F(x, y) → MINUS_INT(x, pos(s(0)))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, 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))
EQUAL_INT(pos(s(x)), pos(s(y))) → EQUAL_INT(pos(x), pos(y))
EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_INT(neg(x), neg(y))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_f(true, x, y) → f(x, round(x))
round(x) → plus_int(pos(s(0)), x)
round(x) → x
f(x, y) → Cond_f(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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_F(true, x, y) → F(x, round(x))
COND_F(true, x, y) → ROUND(x)
ROUND(x) → PLUS_INT(pos(s(0)), x)
F(x, y) → COND_F(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
F(x, y) → AND(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0)))))
F(x, y) → GREATEREQ_INT(x, pos(s(0)))
F(x, y) → EQUAL_INT(y, minus_int(x, pos(s(0))))
F(x, y) → MINUS_INT(x, pos(s(0)))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, 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))
EQUAL_INT(pos(s(x)), pos(s(y))) → EQUAL_INT(pos(x), pos(y))
EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_INT(neg(x), neg(y))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_f(true, x, y) → f(x, round(x))
round(x) → plus_int(pos(s(0)), x)
round(x) → x
f(x, y) → Cond_f(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 14 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:

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

The TRS R consists of the following rules:

Cond_f(true, x, y) → f(x, round(x))
round(x) → plus_int(pos(s(0)), x)
round(x) → x
f(x, y) → Cond_f(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

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

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

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_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:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(EQUAL_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:

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

The TRS R consists of the following rules:

Cond_f(true, x, y) → f(x, round(x))
round(x) → plus_int(pos(s(0)), x)
round(x) → x
f(x, y) → Cond_f(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

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

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

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(EQUAL_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:

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

The TRS R consists of the following rules:

Cond_f(true, x, y) → f(x, round(x))
round(x) → plus_int(pos(s(0)), x)
round(x) → x
f(x, y) → Cond_f(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

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_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

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

              ↳ 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:

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

The TRS R consists of the following rules:

Cond_f(true, x, y) → f(x, round(x))
round(x) → plus_int(pos(s(0)), x)
round(x) → x
f(x, y) → Cond_f(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

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_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

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

              ↳ 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:

MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_f(true, x, y) → f(x, round(x))
round(x) → plus_int(pos(s(0)), x)
round(x) → x
f(x, y) → Cond_f(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

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

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

MINUS_NAT(s(x), s(y)) → MINUS_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:

MINUS_NAT(s(x), s(y)) → MINUS_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:

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_f(true, x, y) → f(x, round(x))
round(x) → plus_int(pos(s(0)), x)
round(x) → x
f(x, y) → Cond_f(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

              ↳ QDP

                ↳ UsableRulesProof

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

F(x, y) → COND_F(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
COND_F(true, x, y) → F(x, round(x))

The TRS R consists of the following rules:

Cond_f(true, x, y) → f(x, round(x))
round(x) → plus_int(pos(s(0)), x)
round(x) → x
f(x, y) → Cond_f(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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:

F(x, y) → COND_F(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
COND_F(true, x, y) → F(x, round(x))

The TRS R consists of the following rules:

round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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

The set Q consists of the following terms:

Cond_f(true, x0, x1)
round(x0)
f(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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_f(true, x0, x1)
f(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:

F(x, y) → COND_F(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
COND_F(true, x, y) → F(x, round(x))

The TRS R consists of the following rules:

round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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: F(x, y) → COND_F(and(greatereq_int(x, pos(s(0))), equal_int(y, minus_int(x, pos(s(0))))), x, y)
Positions in right side of the pair:

[0,0,1]
[0,1,1,1]

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:

F(x, y, x_removed) → COND_F(and(greatereq_int(x, x_removed), equal_int(y, minus_int(x, x_removed))), x, y, x_removed)
COND_F(true, x, y, x_removed) → F(x, round(x), x_removed)

The TRS R consists of the following rules:

round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))

[0,0,1]
[0,1,1,1]

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:

F(x, y, x_removed) → COND_F(and(greatereq_int(x, x_removed), equal_int(y, minus_int(x, x_removed))), x, y, x_removed)
COND_F(true, x, y, x_removed) → F(x, round(x), x_removed)

The TRS R consists of the following rules:

round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(pos(s(x0)), y1) → COND_F(and(greatereq_int(pos(x0), pos(0)), equal_int(y1, minus_int(pos(s(x0)), pos(s(0))))), pos(s(x0)), y1)
F(neg(x0), y1) → COND_F(and(greatereq_int(neg(x0), pos(s(0))), equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, minus_int(neg(x0), pos(s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_int(pos(0), pos(s(0))))), pos(0), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(pos(s(x0)), y1) → COND_F(and(greatereq_int(pos(x0), pos(0)), equal_int(y1, minus_int(pos(s(x0)), pos(s(0))))), pos(s(x0)), y1)
F(neg(x0), y1) → COND_F(and(greatereq_int(neg(x0), pos(s(0))), equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, minus_int(neg(x0), pos(s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_int(pos(0), pos(s(0))))), pos(0), y1)

The TRS R consists of the following rules:

round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))

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

F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_int(pos(s(x0)), pos(s(0))))), pos(s(x0)), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(greatereq_int(neg(x0), pos(s(0))), equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, minus_int(neg(x0), pos(s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_int(pos(0), pos(s(0))))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_int(pos(s(x0)), pos(s(0))))), pos(s(x0)), y1)

The TRS R consists of the following rules:

round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))

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

F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, minus_int(neg(x0), pos(s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_int(pos(0), pos(s(0))))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_int(pos(s(x0)), pos(s(0))))), pos(s(x0)), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)

The TRS R consists of the following rules:

round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, minus_int(neg(x0), pos(s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_int(pos(0), pos(s(0))))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_int(pos(s(x0)), pos(s(0))))), pos(s(x0)), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(neg(x0), y1) → COND_F(and(false, equal_int(y1, minus_int(neg(x0), pos(s(0))))), neg(x0), y1) at position [0,1,1] we obtained the following new rules [LPAR04]:

F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_int(pos(0), pos(s(0))))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_int(pos(s(x0)), pos(s(0))))), pos(s(x0)), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_int(pos(0), pos(s(0))))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_int(pos(s(x0)), pos(s(0))))), pos(s(x0)), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_int(pos(0), pos(s(0))))), pos(0), y1) at position [0,1,1] we obtained the following new rules [LPAR04]:

F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_nat(0, s(0)))), pos(0), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_int(pos(s(x0)), pos(s(0))))), pos(s(x0)), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_nat(0, s(0)))), pos(0), y1)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_int(pos(s(x0)), pos(s(0))))), pos(s(x0)), y1) at position [0,1,1] we obtained the following new rules [LPAR04]:

F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(s(x0), s(0)))), pos(s(x0)), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_nat(0, s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(s(x0), s(0)))), pos(s(x0)), y1)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_nat(0, s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(s(x0), s(0)))), pos(s(x0)), y1)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(pos(x), pos(0)) → true
round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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].

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_nat(0, s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(s(x0), s(0)))), pos(s(x0)), y1)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(pos(x), pos(0)) → true
round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(pos(0), y1) → COND_F(and(false, equal_int(y1, minus_nat(0, s(0)))), pos(0), y1) at position [0,1,1] we obtained the following new rules [LPAR04]:

F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(s(x0), s(0)))), pos(s(x0)), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(pos(x), pos(0)) → true
round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(s(x0), s(0)))), pos(s(x0)), y1) at position [0,1,1] we obtained the following new rules [LPAR04]:

F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

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

COND_F(true, x, y) → F(x, round(x))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(pos(x), pos(0)) → true
round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, x0, y1) → F(x0, plus_int(pos(s(0)), x0))
COND_F(true, x0, y1) → F(x0, x0)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, plus_int(pos(s(0)), x0))
COND_F(true, x0, y1) → F(x0, x0)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(pos(x), pos(0)) → true
round(x) → plus_int(pos(s(0)), x)
round(x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, plus_int(pos(s(0)), x0))
COND_F(true, x0, y1) → F(x0, x0)

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

round(x0)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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].

round(x0)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, plus_int(pos(s(0)), x0))
COND_F(true, x0, y1) → F(x0, x0)

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, x0, y1) → F(x0, plus_int(pos(s(0)), x0)) we obtained the following new rules [LPAR04]:

COND_F(true, pos(z0), z1) → F(pos(z0), plus_int(pos(s(0)), pos(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), plus_int(pos(s(0)), pos(s(z0))))
COND_F(true, neg(z0), z1) → F(neg(z0), plus_int(pos(s(0)), neg(z0)))
COND_F(true, pos(0), z0) → F(pos(0), plus_int(pos(s(0)), pos(0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, pos(z0), z1) → F(pos(z0), plus_int(pos(s(0)), pos(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), plus_int(pos(s(0)), pos(s(z0))))
COND_F(true, neg(z0), z1) → F(neg(z0), plus_int(pos(s(0)), neg(z0)))
COND_F(true, pos(0), z0) → F(pos(0), plus_int(pos(s(0)), pos(0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(plus_nat(s(0), z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), plus_int(pos(s(0)), pos(s(z0))))
COND_F(true, neg(z0), z1) → F(neg(z0), plus_int(pos(s(0)), neg(z0)))
COND_F(true, pos(0), z0) → F(pos(0), plus_int(pos(s(0)), pos(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(plus_nat(s(0), z0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(plus_nat(s(0), s(z0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, neg(z0), z1) → F(neg(z0), plus_int(pos(s(0)), neg(z0)))
COND_F(true, pos(0), z0) → F(pos(0), plus_int(pos(s(0)), pos(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(plus_nat(s(0), z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(plus_nat(s(0), s(z0))))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, pos(0), z0) → F(pos(0), plus_int(pos(s(0)), pos(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(plus_nat(s(0), z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(plus_nat(s(0), s(z0))))
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, pos(0), z0) → F(pos(0), plus_int(pos(s(0)), pos(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(plus_nat(s(0), z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(plus_nat(s(0), s(z0))))
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))

The TRS R consists of the following rules:

plus_int(pos(x), pos(y)) → pos(plus_nat(x, 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(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, pos(0), z0) → F(pos(0), pos(plus_nat(s(0), 0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, pos(z0), z1) → F(pos(z0), pos(plus_nat(s(0), z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(plus_nat(s(0), s(z0))))
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(plus_nat(s(0), 0)))

The TRS R consists of the following rules:

plus_int(pos(x), pos(y)) → pos(plus_nat(x, 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(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, pos(z0), z1) → F(pos(z0), pos(plus_nat(s(0), z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(plus_nat(s(0), s(z0))))
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(plus_nat(s(0), 0)))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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].

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, pos(z0), z1) → F(pos(z0), pos(plus_nat(s(0), z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(plus_nat(s(0), s(z0))))
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(plus_nat(s(0), 0)))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(plus_nat(0, z0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(plus_nat(s(0), s(z0))))
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(plus_nat(s(0), 0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(plus_nat(0, z0))))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(plus_nat(0, s(z0)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(plus_nat(s(0), 0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(plus_nat(0, z0))))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(plus_nat(0, s(z0)))))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, pos(0), z0) → F(pos(0), pos(s(plus_nat(0, 0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(plus_nat(0, z0))))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(plus_nat(0, s(z0)))))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(plus_nat(0, 0))))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(plus_nat(0, s(z0)))))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(plus_nat(0, 0))))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(plus_nat(0, 0))))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(0), y1) → COND_F(and(false, equal_int(y1, neg(s(0)))), pos(0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

F(pos(0), neg(s(x0))) → COND_F(and(false, equal_int(neg(x0), neg(0))), pos(0), neg(s(x0)))
F(pos(0), pos(0)) → COND_F(and(false, false), pos(0), pos(0))
F(pos(0), pos(s(x0))) → COND_F(and(false, false), pos(0), pos(s(x0)))
F(pos(0), neg(0)) → COND_F(and(false, false), pos(0), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, x0, y1) → F(x0, x0)
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
F(pos(0), neg(s(x0))) → COND_F(and(false, equal_int(neg(x0), neg(0))), pos(0), neg(s(x0)))
F(pos(0), pos(0)) → COND_F(and(false, false), pos(0), pos(0))
F(pos(0), pos(s(x0))) → COND_F(and(false, false), pos(0), pos(s(x0)))
F(pos(0), neg(0)) → COND_F(and(false, false), pos(0), neg(0))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

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

COND_F(true, x0, y1) → F(x0, x0)
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, x0, y1) → F(x0, x0) we obtained the following new rules [LPAR04]:

COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, neg(z0), z1) → F(neg(z0), neg(z0))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

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

F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, neg(z0), z1) → F(neg(z0), neg(z0))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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 2 SCCs.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                            ↳ QDP

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

COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
COND_F(true, neg(z0), z1) → F(neg(z0), neg(z0))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                            ↳ QDP

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

COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
COND_F(true, neg(z0), z1) → F(neg(z0), neg(z0))

The TRS R consists of the following rules:

minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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].

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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

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

COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)
COND_F(true, neg(z0), z1) → F(neg(z0), neg(z0))

The TRS R consists of the following rules:

minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

COND_F(true, neg(z0), z1) → F(neg(z0), minus_nat(s(0), z0))
COND_F(true, neg(z0), z1) → F(neg(z0), neg(z0))

The remaining pairs can at least be oriented weakly.

F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND_F(x₁, x₂, x₃)) = x₁
POL(F(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 0
POL(equal_int(x₁, x₂)) = 0
POL(false) = 0
POL(minus_nat(x₁, x₂)) = 0
POL(neg(x₁)) = 0
POL(plus_nat(x₁, x₂)) = 1 + x₁ + x₂
POL(pos(x₁)) = 0
POL(s(x₁)) = 1
POL(true) = 1

The following usable rules [FROCOS05] were oriented:

and(false, true) → false
and(false, false) → false

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

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

F(neg(x0), y1) → COND_F(and(false, equal_int(y1, neg(plus_nat(x0, s(0))))), neg(x0), y1)

The TRS R consists of the following rules:

minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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 0 SCCs with 1 less node.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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].

plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule F(pos(x0), y1) → COND_F(and(greatereq_int(pos(x0), pos(s(0))), equal_int(y1, minus_nat(x0, s(0)))), pos(x0), y1) we obtained the following new rules [LPAR04]:

F(pos(0), pos(s(0))) → COND_F(and(greatereq_int(pos(0), pos(s(0))), equal_int(pos(s(0)), minus_nat(0, s(0)))), pos(0), pos(s(0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(greatereq_int(pos(s(z0)), pos(s(0))), equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(greatereq_int(pos(s(z0)), pos(s(0))), equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
F(pos(0), pos(s(0))) → COND_F(and(greatereq_int(pos(0), pos(s(0))), equal_int(pos(s(0)), minus_nat(0, s(0)))), pos(0), pos(s(0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(greatereq_int(pos(s(z0)), pos(s(0))), equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(greatereq_int(pos(s(z0)), pos(s(0))), equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

F(pos(0), pos(s(0))) → COND_F(and(false, equal_int(pos(s(0)), minus_nat(0, s(0)))), pos(0), pos(s(0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(greatereq_int(pos(s(z0)), pos(s(0))), equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(greatereq_int(pos(s(z0)), pos(s(0))), equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
F(pos(0), pos(s(0))) → COND_F(and(false, equal_int(pos(s(0)), minus_nat(0, s(0)))), pos(0), pos(s(0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(greatereq_int(pos(z0), pos(0)), equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(greatereq_int(pos(s(z0)), pos(s(0))), equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
F(pos(0), pos(s(0))) → COND_F(and(false, equal_int(pos(s(0)), minus_nat(0, s(0)))), pos(0), pos(s(0)))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(greatereq_int(pos(z0), pos(0)), equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

F(pos(s(z0)), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(0)), equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
F(pos(0), pos(s(0))) → COND_F(and(false, equal_int(pos(s(0)), minus_nat(0, s(0)))), pos(0), pos(s(0)))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(greatereq_int(pos(z0), pos(0)), equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(0)), equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(pos(0), pos(s(0))) → COND_F(and(false, equal_int(pos(s(0)), minus_nat(0, s(0)))), pos(0), pos(s(0))) at position [0,1,1] we obtained the following new rules [LPAR04]:

F(pos(0), pos(s(0))) → COND_F(and(false, equal_int(pos(s(0)), neg(s(0)))), pos(0), pos(s(0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(greatereq_int(pos(z0), pos(0)), equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(0)), equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))
F(pos(0), pos(s(0))) → COND_F(and(false, equal_int(pos(s(0)), neg(s(0)))), pos(0), pos(s(0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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 1 less node.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

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

F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(greatereq_int(pos(z0), pos(0)), equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(0)), equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

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

F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(0)), equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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

F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

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

F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(s(z0)))) at position [0,1,1] we obtained the following new rules [LPAR04]:

F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

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

F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0)))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(s(z0), s(0)))), pos(s(z0)), pos(s(z0))) at position [0,1,1] we obtained the following new rules [LPAR04]:

F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Instantiation

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

F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1)
COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule F(pos(s(x0)), y1) → COND_F(and(true, equal_int(y1, minus_nat(x0, 0))), pos(s(x0)), y1) we obtained the following new rules [LPAR04]:

F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)))
F(pos(s(x0)), pos(s(s(x0)))) → COND_F(and(true, equal_int(pos(s(s(x0))), minus_nat(x0, 0))), pos(s(x0)), pos(s(s(x0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ UsableRulesProof

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)))

The TRS R consists of the following rules:

greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greatereq_int(pos(x), pos(0)) → true

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Instantiation

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

COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0)))
F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(x), pos(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, pos(z0), z1) → F(pos(z0), pos(s(z0))) we obtained the following new rules [LPAR04]:

COND_F(true, pos(z0), pos(z0)) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), pos(s(z0))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), pos(s(s(z0)))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(z0), pos(s(z0))) → F(pos(z0), pos(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

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

F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), pos(z0)) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), pos(s(z0))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), pos(s(s(z0)))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(z0), pos(s(z0))) → F(pos(z0), pos(s(z0)))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(x), pos(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(s(z0)))) we obtained the following new rules [LPAR04]:

COND_F(true, pos(s(x0)), pos(s(x0))) → F(pos(s(x0)), pos(s(s(x0))))
COND_F(true, pos(s(x0)), pos(s(s(x0)))) → F(pos(s(x0)), pos(s(s(x0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Instantiation

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

F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), pos(z0)) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), pos(s(z0))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), pos(s(s(z0)))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(z0), pos(s(z0))) → F(pos(z0), pos(s(z0)))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(x), pos(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, pos(s(z0)), z1) → F(pos(s(z0)), pos(s(z0))) we obtained the following new rules [LPAR04]:

COND_F(true, pos(s(x0)), pos(s(x0))) → F(pos(s(x0)), pos(s(x0)))
COND_F(true, pos(s(x0)), pos(s(s(x0)))) → F(pos(s(x0)), 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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Instantiation

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

F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(0), z0) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), pos(z0)) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), pos(s(z0))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), pos(s(s(z0)))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(z0), pos(s(z0))) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(x0)), pos(s(x0))) → F(pos(s(x0)), pos(s(x0)))
COND_F(true, pos(s(x0)), pos(s(s(x0)))) → F(pos(s(x0)), pos(s(x0)))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(x), pos(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, pos(0), z0) → F(pos(0), pos(s(0))) we obtained the following new rules [LPAR04]:

COND_F(true, pos(0), pos(s(0))) → F(pos(0), pos(s(0)))
COND_F(true, pos(0), pos(0)) → F(pos(0), pos(s(0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Instantiation

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

F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), pos(z0)) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), pos(s(z0))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), pos(s(s(z0)))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(z0), pos(s(z0))) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(x0)), pos(s(x0))) → F(pos(s(x0)), pos(s(x0)))
COND_F(true, pos(s(x0)), pos(s(s(x0)))) → F(pos(s(x0)), pos(s(x0)))
COND_F(true, pos(0), pos(s(0))) → F(pos(0), pos(s(0)))
COND_F(true, pos(0), pos(0)) → F(pos(0), pos(s(0)))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(x), pos(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, pos(z0), z1) → F(pos(z0), pos(z0)) we obtained the following new rules [LPAR04]:

COND_F(true, pos(s(z0)), pos(s(z0))) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), pos(s(z0))) → F(pos(z0), pos(z0))
COND_F(true, pos(s(z0)), pos(s(s(z0)))) → F(pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), pos(z0)) → F(pos(z0), pos(z0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ RemovalProof

                                                                                                                                                                                                                                          ↳ RemovalProof

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

F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
F(pos(s(z0)), pos(s(s(z0)))) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))))
F(pos(s(z0)), pos(s(z0))) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)))
COND_F(true, pos(z0), pos(z0)) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(z0)), pos(s(z0))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(s(z0)), pos(s(s(z0)))) → F(pos(s(z0)), pos(s(s(z0))))
COND_F(true, pos(z0), pos(s(z0))) → F(pos(z0), pos(s(z0)))
COND_F(true, pos(s(x0)), pos(s(x0))) → F(pos(s(x0)), pos(s(x0)))
COND_F(true, pos(s(x0)), pos(s(s(x0)))) → F(pos(s(x0)), pos(s(x0)))
COND_F(true, pos(0), pos(s(0))) → F(pos(0), pos(s(0)))
COND_F(true, pos(0), pos(0)) → F(pos(0), pos(s(0)))
COND_F(true, pos(z0), pos(s(z0))) → F(pos(z0), pos(z0))
COND_F(true, pos(z0), pos(z0)) → F(pos(z0), pos(z0))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(x), pos(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_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: F(pos(z0), pos(s(z0))) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)))
Positions in right side of the pair:

[0,0,1]

Pair: F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
Positions in right side of the pair:

[0,0,1]

Pair: COND_F(true, pos(0), pos(s(0))) → F(pos(0), pos(s(0)))
Positions in right side of the pair:

Pair: COND_F(true, pos(0), pos(0)) → F(pos(0), pos(s(0)))
Positions in right side of the pair:

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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ RemovalProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                          ↳ RemovalProof

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

F(pos(z0), pos(s(z0)), x_removed) → COND_F(and(greatereq_int(pos(z0), x_removed), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)), x_removed)
COND_F(true, pos(s(z0)), pos(s(s(z0))), x_removed) → F(pos(s(z0)), pos(s(s(z0))), x_removed)
COND_F(true, pos(z0), pos(s(z0)), x_removed) → F(pos(z0), pos(s(z0)), x_removed)
COND_F(true, pos(s(x0)), pos(s(s(x0))), x_removed) → F(pos(s(x0)), pos(s(x0)), x_removed)
COND_F(true, pos(0), pos(s(0)), x_removed) → F(pos(0), x_removed, x_removed)
COND_F(true, pos(z0), pos(s(z0)), x_removed) → F(pos(z0), pos(z0), x_removed)
F(pos(z0), pos(z0), x_removed) → COND_F(and(greatereq_int(pos(z0), x_removed), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0), x_removed)
COND_F(true, pos(z0), pos(z0), x_removed) → F(pos(z0), pos(s(z0)), x_removed)
COND_F(true, pos(s(z0)), pos(s(z0)), x_removed) → F(pos(s(z0)), pos(s(s(z0))), x_removed)
COND_F(true, pos(s(x0)), pos(s(x0)), x_removed) → F(pos(s(x0)), pos(s(x0)), x_removed)
COND_F(true, pos(0), pos(0), x_removed) → F(pos(0), x_removed, x_removed)
COND_F(true, pos(z0), pos(z0), x_removed) → F(pos(z0), pos(z0), x_removed)
F(pos(s(z0)), pos(s(s(z0))), x_removed) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))), x_removed)
F(pos(s(z0)), pos(s(z0)), x_removed) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)), x_removed)

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(x), pos(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

[0,0,1]

Pair: F(pos(z0), pos(z0)) → COND_F(and(greatereq_int(pos(z0), pos(s(0))), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0))
Positions in right side of the pair:

[0,0,1]

Pair: COND_F(true, pos(0), pos(s(0))) → F(pos(0), pos(s(0)))
Positions in right side of the pair:

Pair: COND_F(true, pos(0), pos(0)) → F(pos(0), pos(s(0)))
Positions in right side of the pair:

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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ UsableRulesProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QReductionProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Narrowing

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                          ↳ AND

                                                                                                                                                            ↳ QDP

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ RemovalProof

                                                                                                                                                                                                                                          ↳ RemovalProof

                                                                                                                                                                                                                                            ↳ QDP

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

F(pos(z0), pos(s(z0)), x_removed) → COND_F(and(greatereq_int(pos(z0), x_removed), equal_int(pos(s(z0)), minus_nat(z0, s(0)))), pos(z0), pos(s(z0)), x_removed)
COND_F(true, pos(s(z0)), pos(s(s(z0))), x_removed) → F(pos(s(z0)), pos(s(s(z0))), x_removed)
COND_F(true, pos(z0), pos(s(z0)), x_removed) → F(pos(z0), pos(s(z0)), x_removed)
COND_F(true, pos(s(x0)), pos(s(s(x0))), x_removed) → F(pos(s(x0)), pos(s(x0)), x_removed)
COND_F(true, pos(0), pos(s(0)), x_removed) → F(pos(0), x_removed, x_removed)
COND_F(true, pos(z0), pos(s(z0)), x_removed) → F(pos(z0), pos(z0), x_removed)
F(pos(z0), pos(z0), x_removed) → COND_F(and(greatereq_int(pos(z0), x_removed), equal_int(pos(z0), minus_nat(z0, s(0)))), pos(z0), pos(z0), x_removed)
COND_F(true, pos(z0), pos(z0), x_removed) → F(pos(z0), pos(s(z0)), x_removed)
COND_F(true, pos(s(z0)), pos(s(z0)), x_removed) → F(pos(s(z0)), pos(s(s(z0))), x_removed)
COND_F(true, pos(s(x0)), pos(s(x0)), x_removed) → F(pos(s(x0)), pos(s(x0)), x_removed)
COND_F(true, pos(0), pos(0), x_removed) → F(pos(0), x_removed, x_removed)
COND_F(true, pos(z0), pos(z0), x_removed) → F(pos(z0), pos(z0), x_removed)
F(pos(s(z0)), pos(s(s(z0))), x_removed) → COND_F(and(true, equal_int(pos(s(s(z0))), minus_nat(z0, 0))), pos(s(z0)), pos(s(s(z0))), x_removed)
F(pos(s(z0)), pos(s(z0)), x_removed) → COND_F(and(true, equal_int(pos(s(z0)), minus_nat(z0, 0))), pos(s(z0)), pos(s(z0)), x_removed)

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
and(false, false) → false
and(false, true) → false
greatereq_int(pos(x), pos(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false

The set Q consists of the following terms:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))

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