Termination proof of /tmp/tmpk9emVZ/a.09.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:

eval(x, y, z) → Cond_eval(&&(>=@z(x, z), >@z(y, 0@z)), x, y, z)
Cond_eval(TRUE, x, y, z) → eval(x, y, +@z(z, y))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(TRUE, 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:

eval(x, y, z) → Cond_eval(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(x, y, plus_int(z, 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

EVAL(x, y, z) → COND_EVAL(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
EVAL(x, y, z) → AND(greatereq_int(x, z), greater_int(y, pos(0)))
EVAL(x, y, z) → GREATEREQ_INT(x, z)
EVAL(x, y, z) → GREATER_INT(y, pos(0))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
COND_EVAL(true, x, y, z) → PLUS_INT(z, 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))
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(x, y, plus_int(z, 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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:

EVAL(x, y, z) → COND_EVAL(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
EVAL(x, y, z) → AND(greatereq_int(x, z), greater_int(y, pos(0)))
EVAL(x, y, z) → GREATEREQ_INT(x, z)
EVAL(x, y, z) → GREATER_INT(y, pos(0))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
COND_EVAL(true, x, y, z) → PLUS_INT(z, 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))
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(x, y, plus_int(z, 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 8 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:

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

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(x, y, plus_int(z, 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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:

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

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

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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].

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ 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

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(x, y, plus_int(z, 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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].

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ 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

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(x, y, plus_int(z, 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

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

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

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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].

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_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:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATER_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:

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

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(x, y, plus_int(z, 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

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

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

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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].

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATER_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:

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

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(x, y, plus_int(z, 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

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:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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].

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ 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

              ↳ QDP

                ↳ UsableRulesProof

              ↳ 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:

eval(x, y, z) → Cond_eval(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(x, y, plus_int(z, 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

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:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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].

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(x, y, z) → COND_EVAL(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(x, y, plus_int(z, 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(x, y, z) → COND_EVAL(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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].

eval(x0, x1, x2)
Cond_eval(true, x0, x1, x2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(x, y, z) → COND_EVAL(and(greatereq_int(x, z), greater_int(y, pos(0))), x, y, z)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(neg(0), y1, neg(x0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, neg(x0))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(neg(s(x0)), y1, pos(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, pos(0))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(pos(x0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, pos(0))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(neg(0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, pos(0))
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(0), y1, neg(x0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, neg(x0))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(neg(s(x0)), y1, pos(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, pos(0))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(pos(x0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, pos(0))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(neg(0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, pos(0))
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(neg(0), neg(s(x0)), neg(y1)) → COND_EVAL(and(true, false), neg(0), neg(s(x0)), neg(y1))
EVAL(neg(0), neg(0), neg(y1)) → COND_EVAL(and(true, false), neg(0), neg(0), neg(y1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(and(true, true), neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(0), neg(y1)) → COND_EVAL(and(true, false), neg(0), pos(0), neg(y1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(neg(s(x0)), y1, pos(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, pos(0))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(pos(x0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, pos(0))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(neg(0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, pos(0))
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), neg(s(x0)), neg(y1)) → COND_EVAL(and(true, false), neg(0), neg(s(x0)), neg(y1))
EVAL(neg(0), neg(0), neg(y1)) → COND_EVAL(and(true, false), neg(0), neg(0), neg(y1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(and(true, true), neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(0), neg(y1)) → COND_EVAL(and(true, false), neg(0), pos(0), neg(y1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, pos(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, pos(0))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(pos(x0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, pos(0))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(neg(0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, pos(0))
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(and(true, true), neg(0), pos(s(x0)), neg(y1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, pos(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, pos(0))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(pos(x0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, pos(0))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(neg(0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, pos(0))
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(neg(s(y0)), neg(0), pos(0)) → COND_EVAL(and(false, false), neg(s(y0)), neg(0), pos(0))
EVAL(neg(s(y0)), pos(s(x0)), pos(0)) → COND_EVAL(and(false, true), neg(s(y0)), pos(s(x0)), pos(0))
EVAL(neg(s(y0)), pos(0), pos(0)) → COND_EVAL(and(false, false), neg(s(y0)), pos(0), pos(0))
EVAL(neg(s(y0)), neg(s(x0)), pos(0)) → COND_EVAL(and(false, false), neg(s(y0)), neg(s(x0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(pos(x0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, pos(0))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(neg(0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, pos(0))
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(s(y0)), neg(0), pos(0)) → COND_EVAL(and(false, false), neg(s(y0)), neg(0), pos(0))
EVAL(neg(s(y0)), pos(s(x0)), pos(0)) → COND_EVAL(and(false, true), neg(s(y0)), pos(s(x0)), pos(0))
EVAL(neg(s(y0)), pos(0), pos(0)) → COND_EVAL(and(false, false), neg(s(y0)), pos(0), pos(0))
EVAL(neg(s(y0)), neg(s(x0)), pos(0)) → COND_EVAL(and(false, false), neg(s(y0)), neg(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(pos(x0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, pos(0))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(neg(0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, pos(0))
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(and(true, true), pos(y0), pos(s(x0)), pos(0))
EVAL(pos(y0), neg(s(x0)), pos(0)) → COND_EVAL(and(true, false), pos(y0), neg(s(x0)), pos(0))
EVAL(pos(y0), neg(0), pos(0)) → COND_EVAL(and(true, false), pos(y0), neg(0), pos(0))
EVAL(pos(y0), pos(0), pos(0)) → COND_EVAL(and(true, false), pos(y0), pos(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(neg(0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, pos(0))
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(and(true, true), pos(y0), pos(s(x0)), pos(0))
EVAL(pos(y0), neg(s(x0)), pos(0)) → COND_EVAL(and(true, false), pos(y0), neg(s(x0)), pos(0))
EVAL(pos(y0), neg(0), pos(0)) → COND_EVAL(and(true, false), pos(y0), neg(0), pos(0))
EVAL(pos(y0), pos(0), pos(0)) → COND_EVAL(and(true, false), pos(y0), pos(0), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(neg(0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, pos(0))
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(and(true, true), pos(y0), pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(and(true, true), pos(y0), pos(s(x0)), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(neg(0), y1, pos(0)) → COND_EVAL(and(true, greater_int(y1, pos(0))), neg(0), y1, pos(0))
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(and(true, true), neg(0), pos(s(x0)), pos(0))
EVAL(neg(0), neg(0), pos(0)) → COND_EVAL(and(true, false), neg(0), neg(0), pos(0))
EVAL(neg(0), pos(0), pos(0)) → COND_EVAL(and(true, false), neg(0), pos(0), pos(0))
EVAL(neg(0), neg(s(x0)), pos(0)) → COND_EVAL(and(true, false), neg(0), neg(s(x0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(and(true, true), neg(0), pos(s(x0)), pos(0))
EVAL(neg(0), neg(0), pos(0)) → COND_EVAL(and(true, false), neg(0), neg(0), pos(0))
EVAL(neg(0), pos(0), pos(0)) → COND_EVAL(and(true, false), neg(0), pos(0), pos(0))
EVAL(neg(0), neg(s(x0)), pos(0)) → COND_EVAL(and(true, false), neg(0), neg(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(and(true, true), neg(0), pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(pos(0), y1, pos(s(x0))) → COND_EVAL(and(false, greater_int(y1, pos(0))), pos(0), y1, pos(s(x0)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(pos(0), neg(s(x0)), pos(s(y1))) → COND_EVAL(and(false, false), pos(0), neg(s(x0)), pos(s(y1)))
EVAL(pos(0), neg(0), pos(s(y1))) → COND_EVAL(and(false, false), pos(0), neg(0), pos(s(y1)))
EVAL(pos(0), pos(s(x0)), pos(s(y1))) → COND_EVAL(and(false, true), pos(0), pos(s(x0)), pos(s(y1)))
EVAL(pos(0), pos(0), pos(s(y1))) → COND_EVAL(and(false, false), pos(0), pos(0), pos(s(y1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(0), neg(s(x0)), pos(s(y1))) → COND_EVAL(and(false, false), pos(0), neg(s(x0)), pos(s(y1)))
EVAL(pos(0), neg(0), pos(s(y1))) → COND_EVAL(and(false, false), pos(0), neg(0), pos(s(y1)))
EVAL(pos(0), pos(s(x0)), pos(s(y1))) → COND_EVAL(and(false, true), pos(0), pos(s(x0)), pos(s(y1)))
EVAL(pos(0), pos(0), pos(s(y1))) → COND_EVAL(and(false, false), pos(0), pos(0), pos(s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(s(x0)), y1, neg(0)) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1, neg(0))
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(neg(s(y0)), neg(0), neg(0)) → COND_EVAL(and(false, false), neg(s(y0)), neg(0), neg(0))
EVAL(neg(s(y0)), pos(0), neg(0)) → COND_EVAL(and(false, false), neg(s(y0)), pos(0), neg(0))
EVAL(neg(s(y0)), pos(s(x0)), neg(0)) → COND_EVAL(and(false, true), neg(s(y0)), pos(s(x0)), neg(0))
EVAL(neg(s(y0)), neg(s(x0)), neg(0)) → COND_EVAL(and(false, false), neg(s(y0)), neg(s(x0)), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(neg(s(y0)), neg(0), neg(0)) → COND_EVAL(and(false, false), neg(s(y0)), neg(0), neg(0))
EVAL(neg(s(y0)), pos(0), neg(0)) → COND_EVAL(and(false, false), neg(s(y0)), pos(0), neg(0))
EVAL(neg(s(y0)), pos(s(x0)), neg(0)) → COND_EVAL(and(false, true), neg(s(y0)), pos(s(x0)), neg(0))
EVAL(neg(s(y0)), neg(s(x0)), neg(0)) → COND_EVAL(and(false, false), neg(s(y0)), neg(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1)))
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(neg(x0), y1, pos(s(x1))) → COND_EVAL(and(false, greater_int(y1, pos(0))), neg(x0), y1, pos(s(x1))) at position [0] we obtained the following new rules [LPAR04]:

EVAL(neg(y0), neg(s(x0)), pos(s(y2))) → COND_EVAL(and(false, false), neg(y0), neg(s(x0)), pos(s(y2)))
EVAL(neg(y0), neg(0), pos(s(y2))) → COND_EVAL(and(false, false), neg(y0), neg(0), pos(s(y2)))
EVAL(neg(y0), pos(0), pos(s(y2))) → COND_EVAL(and(false, false), neg(y0), pos(0), pos(s(y2)))
EVAL(neg(y0), pos(s(x0)), pos(s(y2))) → COND_EVAL(and(false, true), neg(y0), pos(s(x0)), pos(s(y2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(neg(y0), neg(s(x0)), pos(s(y2))) → COND_EVAL(and(false, false), neg(y0), neg(s(x0)), pos(s(y2)))
EVAL(neg(y0), neg(0), pos(s(y2))) → COND_EVAL(and(false, false), neg(y0), neg(0), pos(s(y2)))
EVAL(neg(y0), pos(0), pos(s(y2))) → COND_EVAL(and(false, false), neg(y0), pos(0), pos(s(y2)))
EVAL(neg(y0), pos(s(x0)), pos(s(y2))) → COND_EVAL(and(false, true), neg(y0), pos(s(x0)), pos(s(y2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(pos(x0), y1, neg(x1)) → COND_EVAL(and(true, greater_int(y1, pos(0))), pos(x0), y1, neg(x1)) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(y0), pos(0), neg(y2)) → COND_EVAL(and(true, false), pos(y0), pos(0), neg(y2))
EVAL(pos(y0), neg(0), neg(y2)) → COND_EVAL(and(true, false), pos(y0), neg(0), neg(y2))
EVAL(pos(y0), neg(s(x0)), neg(y2)) → COND_EVAL(and(true, false), pos(y0), neg(s(x0)), neg(y2))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(and(true, true), pos(y0), pos(s(x0)), neg(y2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(0), neg(y2)) → COND_EVAL(and(true, false), pos(y0), pos(0), neg(y2))
EVAL(pos(y0), neg(0), neg(y2)) → COND_EVAL(and(true, false), pos(y0), neg(0), neg(y2))
EVAL(pos(y0), neg(s(x0)), neg(y2)) → COND_EVAL(and(true, false), pos(y0), neg(s(x0)), neg(y2))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(and(true, true), pos(y0), pos(s(x0)), neg(y2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(and(true, true), pos(y0), pos(s(x0)), neg(y2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(and(true, true), pos(y0), pos(s(x0)), neg(y2)) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

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

COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y))
EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL(true, x, y, z) → EVAL(x, y, plus_int(z, y)) we obtained the following new rules [LPAR04]:

COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), plus_int(pos(0), pos(s(z1))))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(s(z0)), z1, pos(s(z2))) → EVAL(pos(s(z0)), z1, plus_int(pos(s(z2)), z1))
COND_EVAL(true, neg(s(z0)), z1, neg(s(z2))) → EVAL(neg(s(z0)), z1, plus_int(neg(s(z2)), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), plus_int(neg(z2), pos(s(z1))))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), plus_int(pos(0), pos(s(z0))))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), plus_int(neg(z1), pos(s(z0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), plus_int(pos(0), pos(s(z1))))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(s(z0)), z1, pos(s(z2))) → EVAL(pos(s(z0)), z1, plus_int(pos(s(z2)), z1))
COND_EVAL(true, neg(s(z0)), z1, neg(s(z2))) → EVAL(neg(s(z0)), z1, plus_int(neg(s(z2)), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), plus_int(neg(z2), pos(s(z1))))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), plus_int(pos(0), pos(s(z0))))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), plus_int(neg(z1), pos(s(z0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), plus_int(pos(0), pos(s(z1)))) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(plus_nat(0, s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(s(z0)), z1, pos(s(z2))) → EVAL(pos(s(z0)), z1, plus_int(pos(s(z2)), z1))
COND_EVAL(true, neg(s(z0)), z1, neg(s(z2))) → EVAL(neg(s(z0)), z1, plus_int(neg(s(z2)), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), plus_int(neg(z2), pos(s(z1))))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), plus_int(pos(0), pos(s(z0))))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), plus_int(neg(z1), pos(s(z0))))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(plus_nat(0, s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), plus_int(neg(z2), pos(s(z1)))) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(s(z0)), z1, pos(s(z2))) → EVAL(pos(s(z0)), z1, plus_int(pos(s(z2)), z1))
COND_EVAL(true, neg(s(z0)), z1, neg(s(z2))) → EVAL(neg(s(z0)), z1, plus_int(neg(s(z2)), z1))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), plus_int(pos(0), pos(s(z0))))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), plus_int(neg(z1), pos(s(z0))))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(plus_nat(0, s(z1))))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), plus_int(pos(0), pos(s(z0)))) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(plus_nat(0, s(z0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(s(z0)), z1, pos(s(z2))) → EVAL(pos(s(z0)), z1, plus_int(pos(s(z2)), z1))
COND_EVAL(true, neg(s(z0)), z1, neg(s(z2))) → EVAL(neg(s(z0)), z1, plus_int(neg(s(z2)), z1))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), plus_int(neg(z1), pos(s(z0))))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(plus_nat(0, s(z1))))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(plus_nat(0, s(z0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), plus_int(neg(z1), pos(s(z0)))) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(s(z0)), z1, pos(s(z2))) → EVAL(pos(s(z0)), z1, plus_int(pos(s(z2)), z1))
COND_EVAL(true, neg(s(z0)), z1, neg(s(z2))) → EVAL(neg(s(z0)), z1, plus_int(neg(s(z2)), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(plus_nat(0, s(z1))))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(plus_nat(0, s(z0))))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(plus_nat(0, s(z1)))) at position [2,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(s(z0)), z1, pos(s(z2))) → EVAL(pos(s(z0)), z1, plus_int(pos(s(z2)), z1))
COND_EVAL(true, neg(s(z0)), z1, neg(s(z2))) → EVAL(neg(s(z0)), z1, plus_int(neg(s(z2)), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(plus_nat(0, s(z0))))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(plus_nat(0, s(z0)))) at position [2,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(s(z0)), z1, pos(s(z2))) → EVAL(pos(s(z0)), z1, plus_int(pos(s(z2)), z1))
COND_EVAL(true, neg(s(z0)), z1, neg(s(z2))) → EVAL(neg(s(z0)), z1, plus_int(neg(s(z2)), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(plus_nat(s(y2), x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, neg(s(z0)), z1, neg(s(z2))) → EVAL(neg(s(z0)), z1, plus_int(neg(s(z2)), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(plus_nat(s(y2), x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(plus_nat(s(y2), x1))) at position [2,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, neg(s(z0)), z1, neg(s(z2))) → EVAL(neg(s(z0)), z1, plus_int(neg(s(z2)), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(plus_nat(s(y2), x1)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(plus_nat(s(y2), x1)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(plus_nat(s(y2), x1))) at position [2,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

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

EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule EVAL(neg(s(x0)), y1, neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), greater_int(y1, pos(0))), neg(s(x0)), y1, neg(s(x1))) we obtained the following new rules [LPAR04]:

EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(pos(s(z1)), pos(0))), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(neg(s(z1)), pos(0))), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(pos(0), pos(0))), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(neg(0), pos(0))), neg(s(x0)), neg(0), neg(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

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

EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(pos(s(z1)), pos(0))), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(neg(s(z1)), pos(0))), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(pos(0), pos(0))), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(neg(0), pos(0))), neg(s(x0)), neg(0), neg(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(pos(s(z1)), pos(0))), neg(s(x0)), pos(s(z1)), neg(s(x2))) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

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

EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(neg(s(z1)), pos(0))), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(pos(0), pos(0))), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(neg(0), pos(0))), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(neg(s(z1)), pos(0))), neg(s(x0)), neg(s(z1)), neg(s(x2))) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

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

EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(pos(0), pos(0))), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(neg(0), pos(0))), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(pos(0), pos(0))), neg(s(x0)), pos(0), neg(s(x2))) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

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

EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(neg(0), pos(0))), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), greater_int(neg(0), pos(0))), neg(s(x0)), neg(0), neg(s(x2))) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

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

EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1)))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule EVAL(pos(s(x0)), y1, pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), greater_int(y1, pos(0))), pos(s(x0)), y1, pos(s(x1))) we obtained the following new rules [LPAR04]:

EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(0), pos(0))), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(0), pos(0))), pos(s(x0)), pos(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(x0)), neg(s(z1)), pos(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

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

EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(0), pos(0))), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(0), pos(0))), pos(s(x0)), pos(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(x0)), neg(s(z1)), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(0), pos(0))), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(x0)), neg(s(z1)), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(0), pos(0))), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(x0)), neg(s(z1)), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(0), pos(0))), pos(s(x0)), neg(0), pos(s(x2))) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(x0)), neg(s(z1)), pos(s(x2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(x0)), neg(s(z1)), pos(s(x2))) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2)
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(y0, neg(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(s(x0)), y2) at position [0] we obtained the following new rules [LPAR04]:

EVAL(neg(0), neg(s(y1)), neg(x0)) → COND_EVAL(and(true, false), neg(0), neg(s(y1)), neg(x0))
EVAL(neg(s(x0)), neg(s(y1)), neg(0)) → COND_EVAL(and(false, false), neg(s(x0)), neg(s(y1)), neg(0))
EVAL(pos(s(x0)), neg(s(y1)), pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), false), pos(s(x0)), neg(s(y1)), pos(s(x1)))
EVAL(neg(x0), neg(s(y1)), pos(s(x1))) → COND_EVAL(and(false, false), neg(x0), neg(s(y1)), pos(s(x1)))
EVAL(pos(0), neg(s(y1)), pos(s(x0))) → COND_EVAL(and(false, false), pos(0), neg(s(y1)), pos(s(x0)))
EVAL(pos(x0), neg(s(y1)), neg(x1)) → COND_EVAL(and(true, false), pos(x0), neg(s(y1)), neg(x1))
EVAL(neg(s(x0)), neg(s(y1)), neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(y1)), neg(s(x1)))
EVAL(pos(x0), neg(s(y1)), pos(0)) → COND_EVAL(and(true, false), pos(x0), neg(s(y1)), pos(0))
EVAL(neg(s(x0)), neg(s(y1)), pos(0)) → COND_EVAL(and(false, false), neg(s(x0)), neg(s(y1)), pos(0))
EVAL(neg(0), neg(s(y1)), pos(0)) → COND_EVAL(and(true, false), neg(0), neg(s(y1)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
EVAL(neg(0), neg(s(y1)), neg(x0)) → COND_EVAL(and(true, false), neg(0), neg(s(y1)), neg(x0))
EVAL(neg(s(x0)), neg(s(y1)), neg(0)) → COND_EVAL(and(false, false), neg(s(x0)), neg(s(y1)), neg(0))
EVAL(neg(x0), neg(s(y1)), pos(s(x1))) → COND_EVAL(and(false, false), neg(x0), neg(s(y1)), pos(s(x1)))
EVAL(pos(0), neg(s(y1)), pos(s(x0))) → COND_EVAL(and(false, false), pos(0), neg(s(y1)), pos(s(x0)))
EVAL(pos(x0), neg(s(y1)), neg(x1)) → COND_EVAL(and(true, false), pos(x0), neg(s(y1)), neg(x1))
EVAL(pos(x0), neg(s(y1)), pos(0)) → COND_EVAL(and(true, false), pos(x0), neg(s(y1)), pos(0))
EVAL(neg(s(x0)), neg(s(y1)), pos(0)) → COND_EVAL(and(false, false), neg(s(x0)), neg(s(y1)), pos(0))
EVAL(neg(0), neg(s(y1)), pos(0)) → COND_EVAL(and(true, false), neg(0), neg(s(y1)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                    ↳ QDP

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

EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(z1)), plus_int(z2, neg(s(z1)))) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, y0, neg(s(y1)), neg(x0)) → EVAL(y0, neg(s(y1)), neg(plus_nat(x0, s(y1))))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                    ↳ QDP

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

EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2)
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), neg(x0)) → EVAL(y0, neg(s(y1)), neg(plus_nat(x0, s(y1))))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(y0, neg(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, neg(0), y2) at position [0] we obtained the following new rules [LPAR04]:

EVAL(neg(s(x0)), neg(0), neg(0)) → COND_EVAL(and(false, false), neg(s(x0)), neg(0), neg(0))
EVAL(neg(x0), neg(0), pos(s(x1))) → COND_EVAL(and(false, false), neg(x0), neg(0), pos(s(x1)))
EVAL(pos(x0), neg(0), neg(x1)) → COND_EVAL(and(true, false), pos(x0), neg(0), neg(x1))
EVAL(pos(s(x0)), neg(0), pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), false), pos(s(x0)), neg(0), pos(s(x1)))
EVAL(neg(s(x0)), neg(0), pos(0)) → COND_EVAL(and(false, false), neg(s(x0)), neg(0), pos(0))
EVAL(neg(0), neg(0), neg(x0)) → COND_EVAL(and(true, false), neg(0), neg(0), neg(x0))
EVAL(neg(s(x0)), neg(0), neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(0), neg(s(x1)))
EVAL(pos(0), neg(0), pos(s(x0))) → COND_EVAL(and(false, false), pos(0), neg(0), pos(s(x0)))
EVAL(pos(x0), neg(0), pos(0)) → COND_EVAL(and(true, false), pos(x0), neg(0), pos(0))
EVAL(neg(0), neg(0), pos(0)) → COND_EVAL(and(true, false), neg(0), neg(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

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

EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), neg(x0)) → EVAL(y0, neg(s(y1)), neg(plus_nat(x0, s(y1))))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))
EVAL(neg(s(x0)), neg(0), neg(0)) → COND_EVAL(and(false, false), neg(s(x0)), neg(0), neg(0))
EVAL(neg(x0), neg(0), pos(s(x1))) → COND_EVAL(and(false, false), neg(x0), neg(0), pos(s(x1)))
EVAL(pos(x0), neg(0), neg(x1)) → COND_EVAL(and(true, false), pos(x0), neg(0), neg(x1))
EVAL(neg(s(x0)), neg(0), pos(0)) → COND_EVAL(and(false, false), neg(s(x0)), neg(0), pos(0))
EVAL(neg(0), neg(0), neg(x0)) → COND_EVAL(and(true, false), neg(0), neg(0), neg(x0))
EVAL(pos(0), neg(0), pos(s(x0))) → COND_EVAL(and(false, false), pos(0), neg(0), pos(s(x0)))
EVAL(pos(x0), neg(0), pos(0)) → COND_EVAL(and(true, false), pos(x0), neg(0), pos(0))
EVAL(neg(0), neg(0), pos(0)) → COND_EVAL(and(true, false), neg(0), neg(0), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), neg(x0)) → EVAL(y0, neg(s(y1)), neg(plus_nat(x0, s(y1))))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0)))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), neg(x0)) → EVAL(y0, neg(s(y1)), neg(plus_nat(x0, s(y1))))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, z0, neg(0), z1) → EVAL(z0, neg(0), plus_int(z1, neg(0))) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, y0, neg(0), neg(x0)) → EVAL(y0, neg(0), neg(plus_nat(x0, 0)))
COND_EVAL(true, y0, neg(0), pos(x0)) → EVAL(y0, neg(0), minus_nat(x0, 0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

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

EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), neg(x0)) → EVAL(y0, neg(s(y1)), neg(plus_nat(x0, s(y1))))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))
COND_EVAL(true, y0, neg(0), neg(x0)) → EVAL(y0, neg(0), neg(plus_nat(x0, 0)))
COND_EVAL(true, y0, neg(0), pos(x0)) → EVAL(y0, neg(0), minus_nat(x0, 0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, y0, neg(s(y1)), neg(x0)) → EVAL(y0, neg(s(y1)), neg(plus_nat(x0, s(y1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, y0, neg(0), neg(x0)) → EVAL(y0, neg(0), neg(plus_nat(x0, 0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, y0, neg(s(y1)), neg(x0)) → EVAL(y0, neg(s(y1)), neg(plus_nat(x0, s(y1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, y0, neg(0), neg(x0)) → EVAL(y0, neg(0), neg(plus_nat(x0, 0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
COND_EVAL(true, y0, neg(s(y1)), neg(x0)) → EVAL(y0, neg(s(y1)), neg(plus_nat(x0, s(y1))))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, y0, neg(0), neg(x0)) → EVAL(y0, neg(0), neg(plus_nat(x0, 0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL(true, y0, neg(s(y1)), neg(x0)) → EVAL(y0, neg(s(y1)), neg(plus_nat(x0, s(y1)))) we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(s(z0)), neg(s(x1)), neg(s(z2))) → EVAL(neg(s(z0)), neg(s(x1)), neg(plus_nat(s(z2), s(x1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, y0, neg(0), neg(x0)) → EVAL(y0, neg(0), neg(plus_nat(x0, 0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
COND_EVAL(true, neg(s(z0)), neg(s(x1)), neg(s(z2))) → EVAL(neg(s(z0)), neg(s(x1)), neg(plus_nat(s(z2), s(x1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(s(z0)), neg(s(x1)), neg(s(z2))) → EVAL(neg(s(z0)), neg(s(x1)), neg(plus_nat(s(z2), s(x1)))) at position [2,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(s(z0)), neg(s(x1)), neg(s(z2))) → EVAL(neg(s(z0)), neg(s(x1)), neg(s(plus_nat(z2, s(x1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
COND_EVAL(true, y0, neg(0), neg(x0)) → EVAL(y0, neg(0), neg(plus_nat(x0, 0)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
COND_EVAL(true, neg(s(z0)), neg(s(x1)), neg(s(z2))) → EVAL(neg(s(z0)), neg(s(x1)), neg(s(plus_nat(z2, s(x1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL(true, y0, neg(0), neg(x0)) → EVAL(y0, neg(0), neg(plus_nat(x0, 0))) we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(s(z0)), neg(0), neg(s(z2))) → EVAL(neg(s(z0)), neg(0), neg(plus_nat(s(z2), 0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
COND_EVAL(true, neg(s(z0)), neg(s(x1)), neg(s(z2))) → EVAL(neg(s(z0)), neg(s(x1)), neg(s(plus_nat(z2, s(x1)))))
COND_EVAL(true, neg(s(z0)), neg(0), neg(s(z2))) → EVAL(neg(s(z0)), neg(0), neg(plus_nat(s(z2), 0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(s(z0)), neg(0), neg(s(z2))) → EVAL(neg(s(z0)), neg(0), neg(plus_nat(s(z2), 0))) at position [2,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(s(z0)), neg(0), neg(s(z2))) → EVAL(neg(s(z0)), neg(0), neg(s(plus_nat(z2, 0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(x1), neg(s(y2))) → EVAL(neg(s(y0)), neg(x1), neg(s(plus_nat(y2, x1))))
EVAL(neg(s(z0)), neg(z1), neg(s(y_0))) → COND_EVAL(and(greatereq_int(neg(z0), neg(y_0)), greater_int(neg(z1), pos(0))), neg(s(z0)), neg(z1), neg(s(y_0)))
EVAL(neg(s(x0)), neg(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), neg(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), neg(0), neg(s(x2)))
COND_EVAL(true, neg(s(z0)), neg(s(x1)), neg(s(z2))) → EVAL(neg(s(z0)), neg(s(x1)), neg(s(plus_nat(z2, s(x1)))))
COND_EVAL(true, neg(s(z0)), neg(0), neg(s(z2))) → EVAL(neg(s(z0)), neg(0), neg(s(plus_nat(z2, 0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

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

EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
COND_EVAL(true, y0, neg(0), pos(x0)) → EVAL(y0, neg(0), minus_nat(x0, 0))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                    ↳ QDP

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

EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
COND_EVAL(true, y0, neg(0), pos(x0)) → EVAL(y0, neg(0), minus_nat(x0, 0))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                    ↳ QDP

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

EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
COND_EVAL(true, y0, neg(0), pos(x0)) → EVAL(y0, neg(0), minus_nat(x0, 0))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL(true, y0, neg(0), pos(x0)) → EVAL(y0, neg(0), minus_nat(x0, 0)) we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), minus_nat(s(z2), 0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

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

EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))
COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), minus_nat(s(z2), 0))

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), minus_nat(s(z2), 0)) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                    ↳ QDP

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

EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1)))
COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(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))
greater_int(neg(0), pos(0)) → false
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL(true, y0, neg(s(y1)), pos(x0)) → EVAL(y0, neg(s(y1)), minus_nat(x0, s(y1))) we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(s(z2), s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

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

EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))
COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(s(z2), s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(s(z2), s(x1))) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(z2, x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                    ↳ QDP

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

EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))
COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(z2, x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, pos(s(y0)), neg(x1), pos(s(y2))) → EVAL(pos(s(y0)), neg(x1), minus_nat(s(y2), x1)) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(y0)), neg(0), pos(s(x0))) → EVAL(pos(s(y0)), neg(0), pos(s(x0)))
COND_EVAL(true, pos(s(y0)), neg(s(x1)), pos(s(x0))) → EVAL(pos(s(y0)), neg(s(x1)), minus_nat(x0, x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                    ↳ QDP

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

EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))
COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(z2, x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule EVAL(pos(s(z0)), neg(z1), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(z1), pos(0))), pos(s(z0)), neg(z1), pos(s(x2))) we obtained the following new rules [LPAR04]:

EVAL(pos(s(z0)), neg(0), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(z0), pos(z1)), greater_int(neg(0), pos(0))), pos(s(z0)), neg(0), pos(s(z1)))
EVAL(pos(s(z0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(z0)), neg(s(z1)), pos(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

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

EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))
COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(z2, x1))
EVAL(pos(s(z0)), neg(0), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(z0), pos(z1)), greater_int(neg(0), pos(0))), pos(s(z0)), neg(0), pos(s(z1)))
EVAL(pos(s(z0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(z0)), neg(s(z1)), pos(s(x2)))

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

The set Q consists of the following terms:

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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(z2, x1))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
EVAL(pos(s(z0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(z0)), neg(s(z1)), pos(s(x2)))

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(z2, x1))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))
EVAL(pos(s(z0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(z0)), neg(s(z1)), pos(s(x2)))

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(pos(s(z0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), greater_int(neg(s(z1)), pos(0))), pos(s(z0)), neg(s(z1)), pos(s(x2))) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(pos(s(z0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(z0), pos(x2)), false), pos(s(z0)), neg(s(z1)), pos(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(z2, x1))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(z2, x1))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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].

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ QReductionProof

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(s(x1)), pos(s(z2))) → EVAL(pos(s(z0)), neg(s(x1)), minus_nat(z2, x1))
EVAL(pos(s(x0)), neg(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(s(z1)), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(z0)), neg(0), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(z0), pos(z1)), greater_int(neg(0), pos(0))), pos(s(z0)), neg(0), pos(s(z1)))

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(z0)), neg(0), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(z0), pos(z1)), greater_int(neg(0), pos(0))), pos(s(z0)), neg(0), pos(s(z1)))

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))
greater_int(neg(0), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true

The set Q consists of the following terms:

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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), 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_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))
EVAL(pos(s(z0)), neg(0), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(z0), pos(z1)), greater_int(neg(0), pos(0))), pos(s(z0)), neg(0), pos(s(z1)))

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))
greater_int(neg(0), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true

The set Q consists of the following terms:

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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

EVAL(pos(s(z0)), neg(0), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(z0), pos(z1)), false), pos(s(z0)), neg(0), pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))

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))
greater_int(neg(0), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true

The set Q consists of the following terms:

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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ QReductionProof

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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].

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Narrowing

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                                ↳ AND

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ QReductionProof

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                    ↳ QDP

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

COND_EVAL(true, pos(s(z0)), neg(0), pos(s(z2))) → EVAL(pos(s(z0)), neg(0), pos(s(z2)))
EVAL(pos(s(x0)), neg(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), neg(0), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

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

COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(0), pos(0))), pos(s(x0)), pos(0), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

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

COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(0), pos(0))), pos(s(x0)), pos(0), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(z1))) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

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

COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(0), pos(0))), pos(s(x0)), pos(0), pos(s(x2)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(s(z1)), pos(0))), pos(s(x0)), pos(s(z1)), pos(s(x2))) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

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

COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(0), pos(0))), pos(s(x0)), pos(0), pos(s(x2)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), greater_int(pos(0), pos(0))), pos(s(x0)), pos(0), pos(s(x2))) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

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

COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2)
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(y0, pos(0), y2) → COND_EVAL(and(greatereq_int(y0, y2), false), y0, pos(0), y2) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(x0), pos(0), neg(x1)) → COND_EVAL(and(true, false), pos(x0), pos(0), neg(x1))
EVAL(neg(s(x0)), pos(0), neg(s(x1))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), pos(0), neg(s(x1)))
EVAL(neg(0), pos(0), pos(0)) → COND_EVAL(and(true, false), neg(0), pos(0), pos(0))
EVAL(pos(s(x0)), pos(0), pos(s(x1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x1)), false), pos(s(x0)), pos(0), pos(s(x1)))
EVAL(neg(0), pos(0), neg(x0)) → COND_EVAL(and(true, false), neg(0), pos(0), neg(x0))
EVAL(pos(0), pos(0), pos(s(x0))) → COND_EVAL(and(false, false), pos(0), pos(0), pos(s(x0)))
EVAL(neg(s(x0)), pos(0), neg(0)) → COND_EVAL(and(false, false), neg(s(x0)), pos(0), neg(0))
EVAL(neg(s(x0)), pos(0), pos(0)) → COND_EVAL(and(false, false), neg(s(x0)), pos(0), pos(0))
EVAL(neg(x0), pos(0), pos(s(x1))) → COND_EVAL(and(false, false), neg(x0), pos(0), pos(s(x1)))
EVAL(pos(x0), pos(0), pos(0)) → COND_EVAL(and(true, false), pos(x0), pos(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

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

COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))
EVAL(pos(x0), pos(0), neg(x1)) → COND_EVAL(and(true, false), pos(x0), pos(0), neg(x1))
EVAL(neg(0), pos(0), pos(0)) → COND_EVAL(and(true, false), neg(0), pos(0), pos(0))
EVAL(neg(0), pos(0), neg(x0)) → COND_EVAL(and(true, false), neg(0), pos(0), neg(x0))
EVAL(pos(0), pos(0), pos(s(x0))) → COND_EVAL(and(false, false), pos(0), pos(0), pos(s(x0)))
EVAL(neg(s(x0)), pos(0), neg(0)) → COND_EVAL(and(false, false), neg(s(x0)), pos(0), neg(0))
EVAL(neg(s(x0)), pos(0), pos(0)) → COND_EVAL(and(false, false), neg(s(x0)), pos(0), pos(0))
EVAL(neg(x0), pos(0), pos(s(x1))) → COND_EVAL(and(false, false), neg(x0), pos(0), pos(s(x1)))
EVAL(pos(x0), pos(0), pos(0)) → COND_EVAL(and(true, false), pos(x0), pos(0), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, z0, pos(0), z1) → EVAL(z0, pos(0), plus_int(z1, pos(0))) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, y0, pos(0), pos(x0)) → EVAL(y0, pos(0), pos(plus_nat(x0, 0)))
COND_EVAL(true, y0, pos(0), neg(x0)) → EVAL(y0, pos(0), minus_nat(0, x0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, y0, pos(0), pos(x0)) → EVAL(y0, pos(0), pos(plus_nat(x0, 0)))
COND_EVAL(true, y0, pos(0), neg(x0)) → EVAL(y0, pos(0), minus_nat(0, x0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(plus_nat(s(z2), 0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, y0, pos(0), neg(x0)) → EVAL(y0, pos(0), minus_nat(0, x0))
COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(plus_nat(s(z2), 0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(plus_nat(s(z2), 0))) at position [2,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, y0, pos(0), neg(x0)) → EVAL(y0, pos(0), minus_nat(0, x0))
COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL(true, y0, pos(0), neg(x0)) → EVAL(y0, pos(0), minus_nat(0, x0)) we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), minus_nat(0, s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Rewriting

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), minus_nat(0, s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), minus_nat(0, s(z2))) at position [2] we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ QDPOrderProof

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))
COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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_EVAL(true, pos(z0), pos(s(z1)), pos(0)) → EVAL(pos(z0), pos(s(z1)), pos(s(z1)))
COND_EVAL(true, neg(0), pos(s(z0)), pos(0)) → EVAL(neg(0), pos(s(z0)), pos(s(z0)))

The remaining pairs can at least be oriented weakly.

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(COND_EVAL(x₁, x₂, x₃, x₄)) = x₄
POL(EVAL(x₁, x₂, x₃)) = x₃
POL(and(x₁, x₂)) = 0
POL(false) = 0
POL(greater_int(x₁, x₂)) = 0
POL(greatereq_int(x₁, x₂)) = 1
POL(minus_nat(x₁, x₂)) = 1
POL(neg(x₁)) = 1
POL(plus_int(x₁, x₂)) = x₁ + x₂
POL(plus_nat(x₁, x₂)) = x₁ + x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 0
POL(true) = 0

The following usable rules [FROCOS05] were oriented:

minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
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))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ QDPOrderProof

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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.

EVAL(neg(s(x0)), pos(0), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), false), neg(s(x0)), pos(0), neg(s(x2)))
EVAL(pos(s(x0)), pos(0), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), false), pos(s(x0)), pos(0), pos(s(x2)))

The remaining pairs can at least be oriented weakly.

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(COND_EVAL(x₁, x₂, x₃, x₄)) = x₁ + x₂
POL(EVAL(x₁, x₂, x₃)) = 1 + x₁ + x₃
POL(and(x₁, x₂)) = x₂
POL(false) = 0
POL(greater_int(x₁, x₂)) = 1
POL(greatereq_int(x₁, x₂)) = 0
POL(minus_nat(x₁, x₂)) = 0
POL(neg(x₁)) = x₁
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:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ QDPOrderProof

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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.

EVAL(pos(y0), pos(s(x0)), pos(0)) → COND_EVAL(true, pos(y0), pos(s(x0)), pos(0))

The remaining pairs can at least be oriented weakly.

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(EVAL(x₁, x₂, x₃)) =
/ 0A -I \

\ 0A -I /

· x₁ +
/ 0A \

\ 0A /

+
/ -I -I \

\ 0A -I /

· x₂ +
/ 0A 0A \

\ 0A 0A /

· x₃

POL(neg(x₁)) =
/ 0A 1A \

\ 0A 0A /

· x₁ +
/ -I \

\ 0A /

POL(s(x₁)) =
/ 0A 1A \

\ -I -I /

· x₁ +
/ -I \

\ -I /

POL(pos(x₁)) =
/ -I -I \

\ -I 1A /

· x₁ +
/ 0A \

\ 0A /

POL(COND_EVAL(x₁, x₂, x₃, x₄)) =
/ -I 0A \

\ -I 0A /

· x₁ +
/ -I \

\ -I /

+
/ 0A -I \

\ 0A -I /

· x₂ +
/ -I -I \

\ -I -I /

· x₃ +
/ 0A -I \

\ 0A -I /

· x₄

POL(and(x₁, x₂)) =
/ 0A 0A \

\ 0A -I /

· x₁ +
/ -I \

\ -I /

+
/ 0A 0A \

\ -I -I /

· x₂

POL(greatereq_int(x₁, x₂)) =
/ 0A -I \

\ 0A -I /

· x₁ +
/ -I \

\ 0A /

+
/ 0A -I \

\ 0A -I /

· x₂

POL(greater_int(x₁, x₂)) =
/ 0A 0A \

\ 0A 0A /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A -I \

\ 0A -I /

· x₂

POL(0) =
/ 0A \

\ 0A /

POL(true) =
/ 0A \

\ 0A /

POL(plus_int(x₁, x₂)) =
/ 0A -I \

\ 0A -I /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A 0A \

\ 0A 0A /

· x₂

POL(minus_nat(x₁, x₂)) =
/ -I -I \

\ -I -I /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A 0A \

\ 0A 1A /

· x₂

POL(plus_nat(x₁, x₂)) =
/ 0A -I \

\ -I -I /

· x₁ +
/ 0A \

\ -I /

+
/ 0A 1A \

\ -I 0A /

· x₂

POL(false) =
/ 0A \

\ 0A /

The following usable rules [FROCOS05] were oriented:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ QDPOrderProof

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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_EVAL(true, pos(s(z0)), pos(0), pos(s(z2))) → EVAL(pos(s(z0)), pos(0), pos(s(plus_nat(z2, 0))))

The remaining pairs can at least be oriented weakly.

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(EVAL(x₁, x₂, x₃)) =
/ -I -I \

\ -I -I /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A 0A \

\ 1A 0A /

· x₂ +
/ -I -I \

\ 0A 0A /

· x₃

POL(neg(x₁)) =
/ 0A 1A \

\ 0A 0A /

· x₁ +
/ 0A \

\ 0A /

POL(s(x₁)) =
/ 0A -I \

\ 0A 0A /

· x₁ +
/ 0A \

\ 1A /

POL(pos(x₁)) =
/ 0A 0A \

\ 0A 0A /

· x₁ +
/ -I \

\ -I /

POL(COND_EVAL(x₁, x₂, x₃, x₄)) =
/ 0A 0A \

\ 1A -I /

· x₁ +
/ 0A \

\ 0A /

+
/ -I -I \

\ -I -I /

· x₂ +
/ 0A -I \

\ 1A 0A /

· x₃ +
/ -I -I \

\ 0A 0A /

· x₄

POL(and(x₁, x₂)) =
/ -I 0A \

\ -I 0A /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A 0A \

\ -I 0A /

· x₂

POL(greatereq_int(x₁, x₂)) =
/ -I 0A \

\ -I -I /

· x₁ +
/ 1A \

\ 0A /

+
/ 0A 0A \

\ -I -I /

· x₂

POL(greater_int(x₁, x₂)) =
/ -I 0A \

\ 0A -I /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A -I \

\ -I 0A /

· x₂

POL(0) =
/ 0A \

\ -I /

POL(true) =
/ 1A \

\ 0A /

POL(plus_int(x₁, x₂)) =
/ 0A 0A \

\ 0A 0A /

· x₁ +
/ 1A \

\ 1A /

+
/ 0A 0A \

\ 1A 0A /

· x₂

POL(minus_nat(x₁, x₂)) =
/ 0A 0A \

\ 0A 0A /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A 1A \

\ 0A 0A /

· x₂

POL(plus_nat(x₁, x₂)) =
/ -I -I \

\ -I -I /

· x₁ +
/ 0A \

\ 1A /

+
/ 0A 0A \

\ 0A 0A /

· x₂

POL(false) =
/ 0A \

\ 0A /

The following usable rules [FROCOS05] were oriented:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ QDPOrderProof

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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.

EVAL(neg(0), pos(s(x0)), pos(0)) → COND_EVAL(true, neg(0), pos(s(x0)), pos(0))

The remaining pairs can at least be oriented weakly.

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(EVAL(x₁, x₂, x₃)) =
/ -I 0A \

\ 0A 0A /

· x₁ +
/ -I \

\ 1A /

+
/ -I -I \

\ -I -I /

· x₂ +
/ 0A 0A \

\ 1A 1A /

· x₃

POL(neg(x₁)) =
/ 0A 0A \

\ -I 0A /

· x₁ +
/ 0A \

\ -I /

POL(s(x₁)) =
/ -I -I \

\ 0A 0A /

· x₁ +
/ 0A \

\ 0A /

POL(pos(x₁)) =
/ -I -I \

\ 0A -I /

· x₁ +
/ 0A \

\ 0A /

POL(COND_EVAL(x₁, x₂, x₃, x₄)) =
/ 0A -I \

\ 0A 0A /

· x₁ +
/ -I \

\ 1A /

+
/ -I 0A \

\ 0A 0A /

· x₂ +
/ -I -I \

\ 0A -I /

· x₃ +
/ 0A -I \

\ 1A 0A /

· x₄

POL(and(x₁, x₂)) =
/ 0A -I \

\ -I 0A /

· x₁ +
/ -I \

\ 0A /

+
/ -I -I \

\ -I -I /

· x₂

POL(greatereq_int(x₁, x₂)) =
/ -I -I \

\ -I -I /

· x₁ +
/ -I \

\ 0A /

+
/ 0A -I \

\ 1A -I /

· x₂

POL(greater_int(x₁, x₂)) =
/ 0A 0A \

\ 0A 0A /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A -I \

\ 0A -I /

· x₂

POL(0) =
/ 1A \

\ -I /

POL(true) =
/ 0A \

\ 0A /

POL(plus_int(x₁, x₂)) =
/ 0A -I \

\ 0A -I /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A 0A \

\ 0A 0A /

· x₂

POL(minus_nat(x₁, x₂)) =
/ -I -I \

\ -I -I /

· x₁ +
/ -I \

\ -I /

+
/ 0A 0A \

\ 0A 0A /

· x₂

POL(plus_nat(x₁, x₂)) =
/ -I -I \

\ 0A 0A /

· x₁ +
/ 0A \

\ -I /

+
/ 0A -I \

\ 0A 0A /

· x₂

POL(false) =
/ 0A \

\ 0A /

The following usable rules [FROCOS05] were oriented:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ QDPOrderProof

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))
COND_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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_EVAL(true, neg(s(z0)), pos(0), neg(s(z2))) → EVAL(neg(s(z0)), pos(0), neg(s(z2)))

The remaining pairs can at least be oriented weakly.

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))

Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(EVAL(x₁, x₂, x₃)) =
/ 0A 0A \

\ 0A 0A /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A 0A \

\ 0A 0A /

· x₂ +
/ 0A 0A \

\ 0A 0A /

· x₃

POL(neg(x₁)) =
/ -I -I \

\ -I 0A /

· x₁ +
/ 0A \

\ 0A /

POL(s(x₁)) =
/ 0A -I \

\ -I -I /

· x₁ +
/ 1A \

\ 0A /

POL(pos(x₁)) =
/ 0A -I \

\ -I -I /

· x₁ +
/ 0A \

\ 0A /

POL(COND_EVAL(x₁, x₂, x₃, x₄)) =
/ 0A 0A \

\ 0A 0A /

· x₁ +
/ -I \

\ 0A /

+
/ 0A 0A \

\ 0A 0A /

· x₂ +
/ 0A 0A \

\ 0A 0A /

· x₃ +
/ 0A 0A \

\ 0A 0A /

· x₄

POL(and(x₁, x₂)) =
/ 0A -I \

\ 0A -I /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A -I \

\ 0A 0A /

· x₂

POL(greatereq_int(x₁, x₂)) =
/ 0A -I \

\ 0A -I /

· x₁ +
/ 0A \

\ 1A /

+
/ 0A -I \

\ 0A -I /

· x₂

POL(greater_int(x₁, x₂)) =
/ 0A 0A \

\ 0A 0A /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A -I \

\ -I -I /

· x₂

POL(0) =
/ 0A \

\ 0A /

POL(true) =
/ 0A \

\ 1A /

POL(plus_int(x₁, x₂)) =
/ 0A 0A \

\ 0A 0A /

· x₁ +
/ 0A \

\ 0A /

+
/ 0A 1A \

\ -I 0A /

· x₂

POL(minus_nat(x₁, x₂)) =
/ 0A -I \

\ -I -I /

· x₁ +
/ 0A \

\ 0A /

+
/ -I -I \

\ -I -I /

· x₂

POL(plus_nat(x₁, x₂)) =
/ 0A -I \

\ 0A 0A /

· x₁ +
/ -I \

\ 0A /

+
/ 0A -I \

\ 0A 0A /

· x₂

POL(false) =
/ 0A \

\ 0A /

The following usable rules [FROCOS05] were oriented:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                                                                        ↳ QDP

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

EVAL(neg(s(z0)), pos(z1), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(z0), neg(x2)), greater_int(pos(z1), pos(0))), neg(s(z0)), pos(z1), neg(s(x2)))
COND_EVAL(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), plus_int(z2, pos(s(z1))))
EVAL(y0, pos(s(x0)), y2) → COND_EVAL(and(greatereq_int(y0, y2), true), y0, pos(s(x0)), y2)
COND_EVAL(true, pos(z0), pos(s(z1)), neg(z2)) → EVAL(pos(z0), pos(s(z1)), minus_nat(s(z1), z2))
EVAL(pos(s(z0)), pos(z1), pos(s(y_0))) → COND_EVAL(and(greatereq_int(pos(z0), pos(y_0)), greater_int(pos(z1), pos(0))), pos(s(z0)), pos(z1), pos(s(y_0)))
COND_EVAL(true, neg(s(y0)), pos(x1), neg(s(y2))) → EVAL(neg(s(y0)), pos(x1), minus_nat(x1, s(y2)))
EVAL(neg(s(x0)), pos(s(z1)), neg(s(x2))) → COND_EVAL(and(greatereq_int(neg(x0), neg(x2)), true), neg(s(x0)), pos(s(z1)), neg(s(x2)))
COND_EVAL(true, pos(s(y0)), pos(x1), pos(s(y2))) → EVAL(pos(s(y0)), pos(x1), pos(s(plus_nat(y2, x1))))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(z1))) → COND_EVAL(and(greatereq_int(pos(x0), pos(z1)), true), pos(s(x0)), pos(s(z1)), pos(s(z1)))
EVAL(pos(s(x0)), pos(s(z1)), pos(s(x2))) → COND_EVAL(and(greatereq_int(pos(x0), pos(x2)), true), pos(s(x0)), pos(s(z1)), pos(s(x2)))
EVAL(pos(y0), pos(s(x0)), neg(y2)) → COND_EVAL(true, pos(y0), pos(s(x0)), neg(y2))
COND_EVAL(true, neg(0), pos(s(z0)), neg(z1)) → EVAL(neg(0), pos(s(z0)), minus_nat(s(z0), z1))
EVAL(neg(0), pos(s(x0)), neg(y1)) → COND_EVAL(true, neg(0), pos(s(x0)), neg(y1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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