Termination proof of /tmp/tmp4dDzWW/csharp3.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_b14(TRUE, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, -@z(sv23_37, sv14_14), +@z(sv24_38, 1@z))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(&&(>=@z(sv23_37, sv14_14), <@z(1@z, sv14_14)), sv14_14, sv23_37, sv24_38)

The set Q consists of the following terms:

Cond_b14(TRUE, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)

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_b14(true, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
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))
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)
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))
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))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B15(sv14_14, sv23_37, sv24_38) → MINUS_INT(sv23_37, sv14_14)
B15(sv14_14, sv23_37, sv24_38) → PLUS_INT(pos(s(0)), sv24_38)
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(sv14_14, sv23_37, sv24_38) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
B14(sv14_14, sv23_37, sv24_38) → AND(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14))
B14(sv14_14, sv23_37, sv24_38) → GREATEREQ_INT(sv23_37, sv14_14)
B14(sv14_14, sv23_37, sv24_38) → LESS_INT(pos(s(0)), sv14_14)
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)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(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)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_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))
LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))
LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))

The TRS R consists of the following rules:

Cond_b14(true, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
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))
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)
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))
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))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B15(sv14_14, sv23_37, sv24_38) → MINUS_INT(sv23_37, sv14_14)
B15(sv14_14, sv23_37, sv24_38) → PLUS_INT(pos(s(0)), sv24_38)
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(sv14_14, sv23_37, sv24_38) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
B14(sv14_14, sv23_37, sv24_38) → AND(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14))
B14(sv14_14, sv23_37, sv24_38) → GREATEREQ_INT(sv23_37, sv14_14)
B14(sv14_14, sv23_37, sv24_38) → LESS_INT(pos(s(0)), sv14_14)
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)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(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)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_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))
LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))
LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))

The TRS R consists of the following rules:

Cond_b14(true, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
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))
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)
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))
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))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 13 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:

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

The TRS R consists of the following rules:

Cond_b14(true, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
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))
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)
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))
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))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

LESS_INT(neg(s(x)), neg(s(y))) → LESS_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:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESS_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:

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

The TRS R consists of the following rules:

Cond_b14(true, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
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))
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)
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))
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))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESS_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_b14(true, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
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))
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)
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))
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))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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_b14(true, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
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))
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)
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))
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))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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_b14(true, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
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))
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)
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))
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))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

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

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

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_b14(true, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
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))
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)
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))
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))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

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

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(sv14_14, sv23_37, sv24_38) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)

The TRS R consists of the following rules:

Cond_b14(true, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)
b14(sv14_14, sv23_37, sv24_38) → Cond_b14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
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))
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)
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))
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))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(sv14_14, sv23_37, sv24_38) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(sv14_14, sv23_37, sv24_38) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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: B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
Positions in right side of the pair:

[2,0]

Pair: B14(sv14_14, sv23_37, sv24_38) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
Positions in right side of the pair:

[0,1,0]

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

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38, x_removed) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(x_removed, sv24_38), x_removed)
B10(sv14_14, sv23_37, sv24_38, x_removed) → B14(sv14_14, sv23_37, sv24_38, x_removed)
B14(sv14_14, sv23_37, sv24_38, x_removed) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(x_removed, sv14_14)), sv14_14, sv23_37, sv24_38, x_removed)
COND_B14(true, sv14_14, sv23_37, sv24_38, x_removed) → B15(sv14_14, sv23_37, sv24_38, x_removed)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

[2,0]

Pair: B14(sv14_14, sv23_37, sv24_38) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
Positions in right side of the pair:

[0,1,0]

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

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38, x_removed) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(x_removed, sv24_38), x_removed)
B10(sv14_14, sv23_37, sv24_38, x_removed) → B14(sv14_14, sv23_37, sv24_38, x_removed)
B14(sv14_14, sv23_37, sv24_38, x_removed) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(x_removed, sv14_14)), sv14_14, sv23_37, sv24_38, x_removed)
COND_B14(true, sv14_14, sv23_37, sv24_38, x_removed) → B15(sv14_14, sv23_37, sv24_38, x_removed)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule B14(sv14_14, sv23_37, sv24_38) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38) at position [0] we obtained the following new rules [LPAR04]:

B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x0)))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), less_int(pos(s(0)), neg(s(x1)))), neg(s(x1)), neg(s(x0)), y2)
B14(pos(0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), neg(0), y2)
B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
B14(pos(0), neg(s(x0)), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(0))), pos(0), neg(s(x0)), y2)
B14(pos(0), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), pos(x0), y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(0), neg(s(x0)), y2) → COND_B14(and(false, less_int(pos(s(0)), neg(0))), neg(0), neg(s(x0)), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x0)))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), less_int(pos(s(0)), neg(s(x1)))), neg(s(x1)), neg(s(x0)), y2)
B14(pos(0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), neg(0), y2)
B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
B14(pos(0), neg(s(x0)), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(0))), pos(0), neg(s(x0)), y2)
B14(pos(0), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), pos(x0), y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(0), neg(s(x0)), y2) → COND_B14(and(false, less_int(pos(s(0)), neg(0))), neg(0), neg(s(x0)), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x0)))), pos(s(x0)), pos(0), y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), less_int(pos(s(0)), neg(s(x1)))), neg(s(x1)), neg(s(x0)), y2)
B14(pos(0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), neg(0), y2)
B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
B14(pos(0), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), pos(x0), y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x0)))), pos(s(x0)), pos(0), y2) at position [0,1] we obtained the following new rules [LPAR04]:

B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), less_int(pos(s(0)), neg(s(x1)))), neg(s(x1)), neg(s(x0)), y2)
B14(pos(0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), neg(0), y2)
B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
B14(pos(0), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), pos(x0), y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), less_int(pos(s(0)), neg(s(x1)))), neg(s(x1)), neg(s(x0)), y2) at position [0,1] we obtained the following new rules [LPAR04]:

B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B14(pos(0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), neg(0), y2)
B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
B14(pos(0), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), pos(x0), y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule B14(pos(0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), neg(0), y2) at position [0,1] we obtained the following new rules [LPAR04]:

B14(pos(0), neg(0), y2) → COND_B14(and(true, false), pos(0), neg(0), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
B14(pos(0), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), pos(x0), y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(0), neg(0), y2) → COND_B14(and(true, false), pos(0), neg(0), y2)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                ↳ UsableRulesProof

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

B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(0), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), pos(x0), y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule B14(pos(0), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), pos(0))), pos(0), pos(x0), y2) at position [0,1] we obtained the following new rules [LPAR04]:

B14(pos(0), pos(x0), y2) → COND_B14(and(true, false), pos(0), pos(x0), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(0), pos(x0), y2) → COND_B14(and(true, false), pos(0), pos(x0), y2)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                ↳ UsableRulesProof

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

COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), neg(x0), y2) at position [0,1] we obtained the following new rules [LPAR04]:

B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                ↳ UsableRulesProof

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

COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(s(0)), pos(s(x1)))), pos(s(x1)), pos(s(x0)), y2) at position [0,1] we obtained the following new rules [LPAR04]:

B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                ↳ UsableRulesProof

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

COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                ↳ UsableRulesProof

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

COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule B14(pos(0), y1, y2) → COND_B14(and(greatereq_int(y1, pos(0)), false), pos(0), y1, y2) at position [0] we obtained the following new rules [LPAR04]:

B14(pos(0), neg(0), y1) → COND_B14(and(true, false), pos(0), neg(0), y1)
B14(pos(0), neg(s(x0)), y1) → COND_B14(and(false, false), pos(0), neg(s(x0)), y1)
B14(pos(0), pos(x0), y1) → COND_B14(and(true, false), pos(0), pos(x0), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
B14(pos(0), neg(0), y1) → COND_B14(and(true, false), pos(0), neg(0), y1)
B14(pos(0), neg(s(x0)), y1) → COND_B14(and(false, false), pos(0), neg(s(x0)), y1)
B14(pos(0), pos(x0), y1) → COND_B14(and(true, false), pos(0), pos(x0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule B14(neg(0), y1, y2) → COND_B14(and(greatereq_int(y1, neg(0)), false), neg(0), y1, y2) at position [0] we obtained the following new rules [LPAR04]:

B14(neg(0), neg(s(x0)), y1) → COND_B14(and(false, false), neg(0), neg(s(x0)), y1)
B14(neg(0), pos(x0), y1) → COND_B14(and(true, false), neg(0), pos(x0), y1)
B14(neg(0), neg(0), y1) → COND_B14(and(true, false), neg(0), neg(0), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
B14(neg(0), neg(s(x0)), y1) → COND_B14(and(false, false), neg(0), neg(s(x0)), y1)
B14(neg(0), pos(x0), y1) → COND_B14(and(true, false), neg(0), pos(x0), y1)
B14(neg(0), neg(0), y1) → COND_B14(and(true, false), neg(0), neg(0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule B14(neg(x1), pos(x0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x1))), neg(x1), pos(x0), y2) at position [0] we obtained the following new rules [LPAR04]:

B14(neg(0), pos(y1), y2) → COND_B14(and(true, false), neg(0), pos(y1), y2)
B14(neg(s(x1)), pos(y1), y2) → COND_B14(and(true, false), neg(s(x1)), pos(y1), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
B14(neg(0), pos(y1), y2) → COND_B14(and(true, false), neg(0), pos(y1), y2)
B14(neg(s(x1)), pos(y1), y2) → COND_B14(and(true, false), neg(s(x1)), pos(y1), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                ↳ UsableRulesProof

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

B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule B14(neg(x0), neg(0), y2) → COND_B14(and(true, less_int(pos(s(0)), neg(x0))), neg(x0), neg(0), y2) at position [0] we obtained the following new rules [LPAR04]:

B14(neg(0), neg(0), y1) → COND_B14(and(true, false), neg(0), neg(0), y1)
B14(neg(s(x1)), neg(0), y1) → COND_B14(and(true, false), neg(s(x1)), neg(0), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
B14(neg(0), neg(0), y1) → COND_B14(and(true, false), neg(0), neg(0), y1)
B14(neg(s(x1)), neg(0), y1) → COND_B14(and(true, false), neg(s(x1)), neg(0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule B14(neg(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, neg(s(x1))), false), neg(s(x1)), y1, y2) at position [0] we obtained the following new rules [LPAR04]:

B14(neg(s(y0)), neg(0), y2) → COND_B14(and(true, false), neg(s(y0)), neg(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(neg(s(y0)), pos(x0), y2) → COND_B14(and(true, false), neg(s(y0)), pos(x0), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
B14(neg(s(y0)), neg(0), y2) → COND_B14(and(true, false), neg(s(y0)), neg(0), y2)
B14(neg(s(y0)), pos(x0), y2) → COND_B14(and(true, false), neg(s(y0)), pos(x0), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule B14(pos(s(x0)), pos(0), y2) → COND_B14(and(false, less_int(pos(0), pos(x0))), pos(s(x0)), pos(0), y2) at position [0] we obtained the following new rules [LPAR04]:

B14(pos(s(s(x0))), pos(0), y1) → COND_B14(and(false, true), pos(s(s(x0))), pos(0), y1)
B14(pos(s(0)), pos(0), y1) → COND_B14(and(false, false), 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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
B14(pos(s(s(x0))), pos(0), y1) → COND_B14(and(false, true), pos(s(s(x0))), pos(0), y1)
B14(pos(s(0)), pos(0), y1) → COND_B14(and(false, false), pos(s(0)), pos(0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                ↳ UsableRulesProof

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

B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule B14(pos(s(x1)), neg(x0), y2) → COND_B14(and(false, less_int(pos(0), pos(x1))), pos(s(x1)), neg(x0), y2) at position [0] we obtained the following new rules [LPAR04]:

B14(pos(s(0)), neg(y1), y2) → COND_B14(and(false, false), pos(s(0)), neg(y1), y2)
B14(pos(s(s(x0))), neg(y1), y2) → COND_B14(and(false, true), pos(s(s(x0))), neg(y1), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
B14(pos(s(0)), neg(y1), y2) → COND_B14(and(false, false), pos(s(0)), neg(y1), y2)
B14(pos(s(s(x0))), neg(y1), y2) → COND_B14(and(false, true), pos(s(s(x0))), neg(y1), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                ↳ UsableRulesProof

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

COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38) we obtained the following new rules [LPAR04]:

COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                ↳ UsableRulesProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38)) we obtained the following new rules [LPAR04]:

B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_int(neg(s(z1)), neg(s(z0))), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_int(pos(s(z1)), pos(s(z0))), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_int(neg(s(z1)), neg(s(z0))), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_int(pos(s(z1)), pos(s(z0))), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_int(neg(s(z1)), neg(s(z0))), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_int(pos(s(z1)), pos(s(z0))), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_int(neg(s(z1)), neg(s(z0))), plus_int(pos(s(0)), z2)) at position [1] we obtained the following new rules [LPAR04]:

B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(s(z0), s(z1)), plus_int(pos(s(0)), z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_int(pos(s(z1)), pos(s(z0))), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(s(z0), s(z1)), plus_int(pos(s(0)), z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_int(pos(s(z1)), pos(s(z0))), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(s(z0), s(z1)), plus_int(pos(s(0)), z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_int(pos(s(z1)), pos(s(z0))), plus_int(pos(s(0)), z2)) at position [1] we obtained the following new rules [LPAR04]:

B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_nat(s(z1), s(z0)), plus_int(pos(s(0)), z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(s(z0), s(z1)), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_nat(s(z1), s(z0)), plus_int(pos(s(0)), z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(s(z0), s(z1)), plus_int(pos(s(0)), z2)) at position [1] we obtained the following new rules [LPAR04]:

B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(z0, z1), plus_int(pos(s(0)), z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_nat(s(z1), s(z0)), plus_int(pos(s(0)), z2))
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(z0, z1), plus_int(pos(s(0)), z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_nat(s(z1), s(z0)), plus_int(pos(s(0)), z2)) at position [1] we obtained the following new rules [LPAR04]:

B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_nat(z1, z0), plus_int(pos(s(0)), z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                ↳ UsableRulesProof

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

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(z0, z1), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_nat(z1, z0), plus_int(pos(s(0)), z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38) we obtained the following new rules [LPAR04]:

B10(pos(s(z0)), y_0, y_1) → B14(pos(s(z0)), y_0, y_1)
B10(neg(s(z0)), y_0, y_1) → B14(neg(s(z0)), y_0, y_1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(z0, z1), plus_int(pos(s(0)), z2))
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_nat(z1, z0), plus_int(pos(s(0)), z2))
B10(pos(s(z0)), y_0, y_1) → B14(pos(s(z0)), y_0, y_1)
B10(neg(s(z0)), y_0, y_1) → B14(neg(s(z0)), y_0, y_1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(z0, z1), plus_int(pos(s(0)), z2))
B10(neg(s(z0)), y_0, y_1) → B14(neg(s(z0)), y_0, y_1)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(z0, z1), plus_int(pos(s(0)), z2))
B10(neg(s(z0)), y_0, y_1) → B14(neg(s(z0)), y_0, y_1)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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].

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(z0, z1), plus_int(pos(s(0)), z2))
B10(neg(s(z0)), y_0, y_1) → B14(neg(s(z0)), y_0, y_1)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)

The TRS R consists of the following rules:

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)
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))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → 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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule B10(neg(s(z0)), y_0, y_1) → B14(neg(s(z0)), y_0, y_1) we obtained the following new rules [LPAR04]:

B10(neg(s(x0)), neg(s(y_1)), x2) → B14(neg(s(x0)), neg(s(y_1)), x2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(z0, z1), plus_int(pos(s(0)), z2))
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B10(neg(s(x0)), neg(s(y_1)), x2) → B14(neg(s(x0)), neg(s(y_1)), x2)

The TRS R consists of the following rules:

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)
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))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → 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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule B15(neg(s(z0)), neg(s(z1)), z2) → B10(neg(s(z0)), minus_nat(z0, z1), plus_int(pos(s(0)), z2)) at position [1] we obtained the following new rules [LPAR04]:

B15(neg(s(s(x0))), neg(s(0)), y2) → B10(neg(s(s(x0))), pos(s(x0)), plus_int(pos(s(0)), y2))
B15(neg(s(0)), neg(s(s(x0))), y2) → B10(neg(s(0)), neg(s(x0)), plus_int(pos(s(0)), y2))
B15(neg(s(0)), neg(s(0)), y2) → B10(neg(s(0)), pos(0), plus_int(pos(s(0)), y2))
B15(neg(s(s(x0))), neg(s(s(x1))), y2) → B10(neg(s(s(x0))), minus_nat(x0, x1), plus_int(pos(s(0)), y2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
B10(neg(s(x0)), neg(s(y_1)), x2) → B14(neg(s(x0)), neg(s(y_1)), x2)
B15(neg(s(s(x0))), neg(s(0)), y2) → B10(neg(s(s(x0))), pos(s(x0)), plus_int(pos(s(0)), y2))
B15(neg(s(0)), neg(s(s(x0))), y2) → B10(neg(s(0)), neg(s(x0)), plus_int(pos(s(0)), y2))
B15(neg(s(0)), neg(s(0)), y2) → B10(neg(s(0)), pos(0), plus_int(pos(s(0)), y2))
B15(neg(s(s(x0))), neg(s(s(x1))), y2) → B10(neg(s(s(x0))), minus_nat(x0, x1), plus_int(pos(s(0)), y2))

The TRS R consists of the following rules:

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)
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))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → 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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

B15(neg(s(0)), neg(s(s(x0))), y2) → B10(neg(s(0)), neg(s(x0)), plus_int(pos(s(0)), y2))
B10(neg(s(x0)), neg(s(y_1)), x2) → B14(neg(s(x0)), neg(s(y_1)), x2)
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
B15(neg(s(s(x0))), neg(s(s(x1))), y2) → B10(neg(s(s(x0))), minus_nat(x0, x1), plus_int(pos(s(0)), y2))

The TRS R consists of the following rules:

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)
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))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → 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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule B10(neg(s(x0)), neg(s(y_1)), x2) → B14(neg(s(x0)), neg(s(y_1)), x2) we obtained the following new rules [LPAR04]:

B10(neg(s(0)), neg(s(z0)), y_0) → B14(neg(s(0)), neg(s(z0)), y_0)
B10(neg(s(s(z0))), neg(s(x1)), y_1) → B14(neg(s(s(z0))), neg(s(x1)), y_1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

B15(neg(s(0)), neg(s(s(x0))), y2) → B10(neg(s(0)), neg(s(x0)), plus_int(pos(s(0)), y2))
B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
B15(neg(s(s(x0))), neg(s(s(x1))), y2) → B10(neg(s(s(x0))), minus_nat(x0, x1), plus_int(pos(s(0)), y2))
B10(neg(s(0)), neg(s(z0)), y_0) → B14(neg(s(0)), neg(s(z0)), y_0)
B10(neg(s(s(z0))), neg(s(x1)), y_1) → B14(neg(s(s(z0))), neg(s(x1)), y_1)

The TRS R consists of the following rules:

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)
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))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → 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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule B14(neg(s(x1)), neg(s(x0)), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)), y2) at position [0] we obtained the following new rules [LPAR04]:

B14(neg(s(0)), neg(s(s(x0))), y2) → COND_B14(and(false, false), neg(s(0)), neg(s(s(x0))), y2)
B14(neg(s(s(x1))), neg(s(s(x0))), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(s(x1))), neg(s(s(x0))), y2)
B14(neg(s(x0)), neg(s(0)), y2) → COND_B14(and(true, false), neg(s(x0)), neg(s(0)), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

B15(neg(s(0)), neg(s(s(x0))), y2) → B10(neg(s(0)), neg(s(x0)), plus_int(pos(s(0)), y2))
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)
B15(neg(s(s(x0))), neg(s(s(x1))), y2) → B10(neg(s(s(x0))), minus_nat(x0, x1), plus_int(pos(s(0)), y2))
B10(neg(s(0)), neg(s(z0)), y_0) → B14(neg(s(0)), neg(s(z0)), y_0)
B10(neg(s(s(z0))), neg(s(x1)), y_1) → B14(neg(s(s(z0))), neg(s(x1)), y_1)
B14(neg(s(0)), neg(s(s(x0))), y2) → COND_B14(and(false, false), neg(s(0)), neg(s(s(x0))), y2)
B14(neg(s(s(x1))), neg(s(s(x0))), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(s(x1))), neg(s(s(x0))), y2)
B14(neg(s(x0)), neg(s(0)), y2) → COND_B14(and(true, false), neg(s(x0)), neg(s(0)), y2)

The TRS R consists of the following rules:

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)
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))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → 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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

B15(neg(s(s(x0))), neg(s(s(x1))), y2) → B10(neg(s(s(x0))), minus_nat(x0, x1), plus_int(pos(s(0)), y2))
B10(neg(s(s(z0))), neg(s(x1)), y_1) → B14(neg(s(s(z0))), neg(s(x1)), y_1)
B14(neg(s(s(x1))), neg(s(s(x0))), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(s(x1))), neg(s(s(x0))), y2)
COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2)

The TRS R consists of the following rules:

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)
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))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → 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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_B14(true, neg(s(z0)), neg(s(z1)), z2) → B15(neg(s(z0)), neg(s(z1)), z2) we obtained the following new rules [LPAR04]:

COND_B14(true, neg(s(s(z0))), neg(s(s(z1))), z2) → B15(neg(s(s(z0))), neg(s(s(z1))), z2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

B15(neg(s(s(x0))), neg(s(s(x1))), y2) → B10(neg(s(s(x0))), minus_nat(x0, x1), plus_int(pos(s(0)), y2))
B10(neg(s(s(z0))), neg(s(x1)), y_1) → B14(neg(s(s(z0))), neg(s(x1)), y_1)
B14(neg(s(s(x1))), neg(s(s(x0))), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(s(x1))), neg(s(s(x0))), y2)
COND_B14(true, neg(s(s(z0))), neg(s(s(z1))), z2) → B15(neg(s(s(z0))), neg(s(s(z1))), z2)

The TRS R consists of the following rules:

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)
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))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → 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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

The following pairs can be oriented strictly and are deleted.

COND_B14(true, neg(s(s(z0))), neg(s(s(z1))), z2) → B15(neg(s(s(z0))), neg(s(s(z1))), z2)

The remaining pairs can at least be oriented weakly.

B15(neg(s(s(x0))), neg(s(s(x1))), y2) → B10(neg(s(s(x0))), minus_nat(x0, x1), plus_int(pos(s(0)), y2))
B10(neg(s(s(z0))), neg(s(x1)), y_1) → B14(neg(s(s(z0))), neg(s(x1)), y_1)
B14(neg(s(s(x1))), neg(s(s(x0))), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(s(x1))), neg(s(s(x0))), y2)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(B10(x₁, x₂, x₃)) = 1
POL(B14(x₁, x₂, x₃)) = 1
POL(B15(x₁, x₂, x₃)) = 1
POL(COND_B14(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(and(x₁, x₂)) = 0
POL(false) = 0
POL(greatereq_int(x₁, x₂)) = 0
POL(minus_nat(x₁, x₂)) = 0
POL(neg(x₁)) = 0
POL(plus_int(x₁, x₂)) = 0
POL(plus_nat(x₁, x₂)) = 0
POL(pos(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = 1

The following usable rules [FROCOS05] were oriented:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

B15(neg(s(s(x0))), neg(s(s(x1))), y2) → B10(neg(s(s(x0))), minus_nat(x0, x1), plus_int(pos(s(0)), y2))
B10(neg(s(s(z0))), neg(s(x1)), y_1) → B14(neg(s(s(z0))), neg(s(x1)), y_1)
B14(neg(s(s(x1))), neg(s(s(x0))), y2) → COND_B14(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(s(x1))), neg(s(s(x0))), y2)

The TRS R consists of the following rules:

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)
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))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → 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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                ↳ UsableRulesProof

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

COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
B10(pos(s(z0)), y_0, y_1) → B14(pos(s(z0)), y_0, y_1)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_nat(z1, z0), plus_int(pos(s(0)), z2))
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QDPOrderProof

                ↳ UsableRulesProof

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

COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
B10(pos(s(z0)), y_0, y_1) → B14(pos(s(z0)), y_0, y_1)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_nat(z1, z0), plus_int(pos(s(0)), z2))
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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.

B15(pos(s(z0)), pos(s(z1)), z2) → B10(pos(s(z0)), minus_nat(z1, z0), plus_int(pos(s(0)), z2))

The remaining pairs can at least be oriented weakly.

COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
B10(pos(s(z0)), y_0, y_1) → B14(pos(s(z0)), y_0, y_1)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(B10(x₁, x₂, x₃)) = x₂
POL(B14(x₁, x₂, x₃)) = x₂
POL(B15(x₁, x₂, x₃)) = x₂
POL(COND_B14(x₁, x₂, x₃, x₄)) = x₃
POL(and(x₁, x₂)) = 0
POL(false) = 0
POL(greatereq_int(x₁, x₂)) = 0
POL(less_int(x₁, x₂)) = 0
POL(minus_int(x₁, x₂)) = x₁
POL(minus_nat(x₁, x₂)) = x₁
POL(neg(x₁)) = 0
POL(plus_int(x₁, x₂)) = 0
POL(plus_nat(x₁, x₂)) = 0
POL(pos(x₁)) = x₁
POL(s(x₁)) = 1 + x₁
POL(true) = 0

The following usable rules [FROCOS05] were oriented:

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_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_int(pos(x), pos(y)) → minus_nat(x, y)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ RemovalProof

                                                                                                                                                                                          ↳ RemovalProof

                ↳ UsableRulesProof

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

COND_B14(true, pos(s(z0)), pos(s(z1)), z2) → B15(pos(s(z0)), pos(s(z1)), z2)
B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
B10(pos(s(z0)), y_0, y_1) → B14(pos(s(z0)), y_0, y_1)
B14(pos(s(x1)), y1, y2) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2)
COND_B14(true, pos(s(z0)), z1, z2) → B15(pos(s(z0)), z1, z2)
B14(pos(s(x1)), pos(s(x0)), y2) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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: B15(pos(s(z0)), z1, z2) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(pos(s(0)), z2))
Positions in right side of the pair:

[2,0]

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ RemovalProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                          ↳ RemovalProof

                ↳ UsableRulesProof

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

COND_B14(true, pos(s(z0)), pos(s(z1)), z2, x_removed) → B15(pos(s(z0)), pos(s(z1)), z2, x_removed)
B15(pos(s(z0)), z1, z2, x_removed) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(x_removed, z2), x_removed)
B10(pos(s(z0)), y_0, y_1, x_removed) → B14(pos(s(z0)), y_0, y_1, x_removed)
B14(pos(s(x1)), y1, y2, x_removed) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2, x_removed)
B14(pos(s(x1)), pos(s(x0)), y2, x_removed) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2, x_removed)
COND_B14(true, pos(s(z0)), z1, z2, x_removed) → B15(pos(s(z0)), z1, z2, x_removed)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

[2,0]

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                              ↳ AND

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ RemovalProof

                                                                                                                                                                                          ↳ RemovalProof

                                                                                                                                                                                            ↳ QDP

                ↳ UsableRulesProof

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

COND_B14(true, pos(s(z0)), pos(s(z1)), z2, x_removed) → B15(pos(s(z0)), pos(s(z1)), z2, x_removed)
B15(pos(s(z0)), z1, z2, x_removed) → B10(pos(s(z0)), minus_int(z1, pos(s(z0))), plus_int(x_removed, z2), x_removed)
B10(pos(s(z0)), y_0, y_1, x_removed) → B14(pos(s(z0)), y_0, y_1, x_removed)
B14(pos(s(x1)), y1, y2, x_removed) → COND_B14(and(greatereq_int(y1, pos(s(x1))), less_int(pos(0), pos(x1))), pos(s(x1)), y1, y2, x_removed)
B14(pos(s(x1)), pos(s(x0)), y2, x_removed) → COND_B14(and(greatereq_int(pos(x0), pos(x1)), less_int(pos(0), pos(x1))), pos(s(x1)), pos(s(x0)), y2, x_removed)
COND_B14(true, pos(s(z0)), z1, z2, x_removed) → B15(pos(s(z0)), z1, z2, x_removed)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(sv14_14, sv23_37, sv24_38) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

Cond_b14(true, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)
b14(x0, x1, x2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

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

B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, minus_int(sv23_37, sv14_14), plus_int(pos(s(0)), sv24_38))
B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
B14(sv14_14, sv23_37, sv24_38) → COND_B14(and(greatereq_int(sv23_37, sv14_14), less_int(pos(s(0)), sv14_14)), sv14_14, sv23_37, sv24_38)
COND_B14(true, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)

The TRS R consists of the following rules:

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))
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(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))
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), neg(0)) → false
less_int(pos(s(x)), neg(s(y))) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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