Termination proof of /tmp/tmpIu1b-8/22.itrs

Termination of the given ITRSProblem could successfully be proven:

↳ ITRS

  ↳ ITRStoQTRSProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

Cond_eval_1(TRUE, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(&&(>@z(x, 0@z), >=@z(0@z, y)), x, y)
eval_2(x, y) → Cond_eval_2(&&(>@z(x, 0@z), >@z(y, 0@z)), x, y)
Cond_eval_21(TRUE, x, y) → eval_1(-@z(x, 1@z), y)
Cond_eval_2(TRUE, x, y) → eval_2(x, -@z(y, 1@z))
eval_1(x, y) → Cond_eval_1(>@z(x, 0@z), x, y)

The set Q consists of the following terms:

Cond_eval_1(TRUE, x0, x1)
eval_2(x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
eval_1(x0, x1)

Represented integers and predefined function symbols by Terms

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

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

Cond_eval_1(true, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
eval_2(x, y) → Cond_eval_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
Cond_eval_21(true, x, y) → eval_1(minus_int(x, pos(s(0))), y)
Cond_eval_2(true, x, y) → eval_2(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(greater_int(x, pos(0)), x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
EVAL_2(x, y) → AND(greater_int(x, pos(0)), greatereq_int(pos(0), y))
EVAL_2(x, y) → GREATER_INT(x, pos(0))
EVAL_2(x, y) → GREATEREQ_INT(pos(0), y)
EVAL_2(x, y) → COND_EVAL_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
EVAL_2(x, y) → AND(greater_int(x, pos(0)), greater_int(y, pos(0)))
EVAL_2(x, y) → GREATER_INT(y, pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
COND_EVAL_21(true, x, y) → MINUS_INT(x, pos(s(0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
COND_EVAL_2(true, x, y) → MINUS_INT(y, pos(s(0)))
EVAL_1(x, y) → COND_EVAL_1(greater_int(x, pos(0)), x, y)
EVAL_1(x, y) → GREATER_INT(x, pos(0))
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))
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))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_eval_1(true, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
eval_2(x, y) → Cond_eval_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
Cond_eval_21(true, x, y) → eval_1(minus_int(x, pos(s(0))), y)
Cond_eval_2(true, x, y) → eval_2(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(greater_int(x, pos(0)), x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
EVAL_2(x, y) → AND(greater_int(x, pos(0)), greatereq_int(pos(0), y))
EVAL_2(x, y) → GREATER_INT(x, pos(0))
EVAL_2(x, y) → GREATEREQ_INT(pos(0), y)
EVAL_2(x, y) → COND_EVAL_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
EVAL_2(x, y) → AND(greater_int(x, pos(0)), greater_int(y, pos(0)))
EVAL_2(x, y) → GREATER_INT(y, pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
COND_EVAL_21(true, x, y) → MINUS_INT(x, pos(s(0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
COND_EVAL_2(true, x, y) → MINUS_INT(y, pos(s(0)))
EVAL_1(x, y) → COND_EVAL_1(greater_int(x, pos(0)), x, y)
EVAL_1(x, y) → GREATER_INT(x, pos(0))
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))
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))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_eval_1(true, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
eval_2(x, y) → Cond_eval_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
Cond_eval_21(true, x, y) → eval_1(minus_int(x, pos(s(0))), y)
Cond_eval_2(true, x, y) → eval_2(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(greater_int(x, pos(0)), x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 12 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:

Cond_eval_1(true, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
eval_2(x, y) → Cond_eval_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
Cond_eval_21(true, x, y) → eval_1(minus_int(x, pos(s(0))), y)
Cond_eval_2(true, x, y) → eval_2(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(greater_int(x, pos(0)), x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

Cond_eval_1(true, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
eval_2(x, y) → Cond_eval_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
Cond_eval_21(true, x, y) → eval_1(minus_int(x, pos(s(0))), y)
Cond_eval_2(true, x, y) → eval_2(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(greater_int(x, pos(0)), x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

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

The TRS R consists of the following rules:

Cond_eval_1(true, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
eval_2(x, y) → Cond_eval_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
Cond_eval_21(true, x, y) → eval_1(minus_int(x, pos(s(0))), y)
Cond_eval_2(true, x, y) → eval_2(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(greater_int(x, pos(0)), x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

The following dependency pairs can be deleted:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

Cond_eval_1(true, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
eval_2(x, y) → Cond_eval_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
Cond_eval_21(true, x, y) → eval_1(minus_int(x, pos(s(0))), y)
Cond_eval_2(true, x, y) → eval_2(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(greater_int(x, pos(0)), x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

Cond_eval_1(true, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
eval_2(x, y) → Cond_eval_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
Cond_eval_21(true, x, y) → eval_1(minus_int(x, pos(s(0))), y)
Cond_eval_2(true, x, y) → eval_2(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(greater_int(x, pos(0)), x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

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:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

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

              ↳ 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:

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

The TRS R consists of the following rules:

Cond_eval_1(true, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
eval_2(x, y) → Cond_eval_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
Cond_eval_21(true, x, y) → eval_1(minus_int(x, pos(s(0))), y)
Cond_eval_2(true, x, y) → eval_2(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(greater_int(x, pos(0)), x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

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:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

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

              ↳ 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:

EVAL_2(x, y) → COND_EVAL_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(x, y) → COND_EVAL_1(greater_int(x, pos(0)), x, y)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))

The TRS R consists of the following rules:

Cond_eval_1(true, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
eval_2(x, y) → Cond_eval_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
Cond_eval_21(true, x, y) → eval_1(minus_int(x, pos(s(0))), y)
Cond_eval_2(true, x, y) → eval_2(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(greater_int(x, pos(0)), x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ 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:

EVAL_2(x, y) → COND_EVAL_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(x, y) → COND_EVAL_1(greater_int(x, pos(0)), x, y)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

Cond_eval_1(true, x0, x1)
eval_2(x0, x1)
Cond_eval_21(true, x0, x1)
Cond_eval_2(true, x0, x1)
eval_1(x0, x1)

↳ 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:

EVAL_2(x, y) → COND_EVAL_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(x, y) → COND_EVAL_1(greater_int(x, pos(0)), x, y)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(x, y) → COND_EVAL_21(and(greater_int(x, pos(0)), greatereq_int(pos(0), y)), x, y) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(y0, pos(0)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, pos(0))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(s(x0)), y1)
EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, 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_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(x, y) → COND_EVAL_1(greater_int(x, pos(0)), x, y)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, pos(0)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, pos(0))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(s(x0)), y1)
EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_1(x, y) → COND_EVAL_1(greater_int(x, pos(0)), x, y) at position [0] we obtained the following new rules [LPAR04]:

EVAL_1(neg(s(x0)), y1) → COND_EVAL_1(false, neg(s(x0)), y1)
EVAL_1(pos(0), y1) → COND_EVAL_1(false, pos(0), y1)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_1(neg(0), y1) → COND_EVAL_1(false, neg(0), 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_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, pos(0)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, pos(0))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(s(x0)), y1)
EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_1(neg(s(x0)), y1) → COND_EVAL_1(false, neg(s(x0)), y1)
EVAL_1(pos(0), y1) → COND_EVAL_1(false, pos(0), y1)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_1(neg(0), y1) → COND_EVAL_1(false, neg(0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, pos(0)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_2(neg(s(x0)), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(s(x0)), y1)
EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(x, y) → COND_EVAL_2(and(greater_int(x, pos(0)), greater_int(y, pos(0))), x, y) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, pos(0)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_2(neg(s(x0)), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(s(x0)), y1)
EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(y0, pos(0)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, pos(0)) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(s(x0)), pos(0)) → COND_EVAL_21(and(false, true), neg(s(x0)), pos(0))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(and(true, true), pos(s(x0)), pos(0))
EVAL_2(neg(0), pos(0)) → COND_EVAL_21(and(false, true), neg(0), pos(0))
EVAL_2(pos(0), pos(0)) → COND_EVAL_21(and(false, true), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_2(neg(s(x0)), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(s(x0)), y1)
EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(neg(s(x0)), pos(0)) → COND_EVAL_21(and(false, true), neg(s(x0)), pos(0))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(and(true, true), pos(s(x0)), pos(0))
EVAL_2(neg(0), pos(0)) → COND_EVAL_21(and(false, true), neg(0), pos(0))
EVAL_2(pos(0), pos(0)) → COND_EVAL_21(and(false, true), pos(0), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

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

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(neg(s(x0)), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(s(x0)), y1)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(and(true, true), pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(and(true, true), pos(s(x0)), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

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

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(neg(s(x0)), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(s(x0)), y1)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(neg(s(x0)), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(s(y0)), pos(0)) → COND_EVAL_21(and(false, true), neg(s(y0)), pos(0))
EVAL_2(neg(s(y0)), pos(s(x0))) → COND_EVAL_21(and(false, false), neg(s(y0)), pos(s(x0)))
EVAL_2(neg(s(y0)), neg(x1)) → COND_EVAL_21(and(false, true), neg(s(y0)), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

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

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(neg(s(y0)), pos(0)) → COND_EVAL_21(and(false, true), neg(s(y0)), pos(0))
EVAL_2(neg(s(y0)), pos(s(x0))) → COND_EVAL_21(and(false, false), neg(s(y0)), pos(s(x0)))
EVAL_2(neg(s(y0)), neg(x1)) → COND_EVAL_21(and(false, true), neg(s(y0)), neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

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

EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(neg(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), neg(0), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(0), pos(s(x0))) → COND_EVAL_21(and(false, false), neg(0), pos(s(x0)))
EVAL_2(neg(0), neg(x1)) → COND_EVAL_21(and(false, true), neg(0), neg(x1))
EVAL_2(neg(0), pos(0)) → COND_EVAL_21(and(false, true), 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

                                    ↳ 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_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(neg(0), pos(s(x0))) → COND_EVAL_21(and(false, false), neg(0), pos(s(x0)))
EVAL_2(neg(0), neg(x1)) → COND_EVAL_21(and(false, true), neg(0), neg(x1))
EVAL_2(neg(0), pos(0)) → COND_EVAL_21(and(false, true), neg(0), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ 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:

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(pos(s(x0)), y1) → COND_EVAL_21(and(true, greatereq_int(pos(0), y1)), pos(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(y0)), pos(0)) → COND_EVAL_21(and(true, true), pos(s(y0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(and(true, true), pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_21(and(true, false), pos(s(y0)), pos(s(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

                                    ↳ Narrowing

                                      ↳ 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:

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), pos(0)) → COND_EVAL_21(and(true, true), pos(s(y0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(and(true, true), pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_21(and(true, false), pos(s(y0)), pos(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ 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_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), pos(0)) → COND_EVAL_21(and(true, true), pos(s(y0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(and(true, true), pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL_2(pos(s(y0)), pos(0)) → COND_EVAL_21(and(true, true), pos(s(y0)), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(y0)), pos(0)) → COND_EVAL_21(true, pos(s(y0)), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

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

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(and(true, true), pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(and(true, true), pos(s(y0)), neg(x1)) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

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

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(pos(0), y1) → COND_EVAL_21(and(false, greatereq_int(pos(0), y1)), pos(0), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(0), neg(x1)) → COND_EVAL_21(and(false, true), pos(0), neg(x1))
EVAL_2(pos(0), pos(s(x0))) → COND_EVAL_21(and(false, false), pos(0), pos(s(x0)))
EVAL_2(pos(0), pos(0)) → COND_EVAL_21(and(false, true), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

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

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(0), neg(x1)) → COND_EVAL_21(and(false, true), pos(0), neg(x1))
EVAL_2(pos(0), pos(s(x0))) → COND_EVAL_21(and(false, false), pos(0), pos(s(x0)))
EVAL_2(pos(0), pos(0)) → COND_EVAL_21(and(false, true), pos(0), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

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

EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

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

EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

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

EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0)))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(y0, pos(s(x0))) → COND_EVAL_21(and(greater_int(y0, pos(0)), false), y0, pos(s(x0))) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(0), pos(s(y1))) → COND_EVAL_21(and(false, false), neg(0), pos(s(y1)))
EVAL_2(neg(s(x0)), pos(s(y1))) → COND_EVAL_21(and(false, false), neg(s(x0)), pos(s(y1)))
EVAL_2(pos(0), pos(s(y1))) → COND_EVAL_21(and(false, false), pos(0), pos(s(y1)))
EVAL_2(pos(s(x0)), pos(s(y1))) → COND_EVAL_21(and(true, false), pos(s(x0)), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

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

COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(neg(0), pos(s(y1))) → COND_EVAL_21(and(false, false), neg(0), pos(s(y1)))
EVAL_2(neg(s(x0)), pos(s(y1))) → COND_EVAL_21(and(false, false), neg(s(x0)), pos(s(y1)))
EVAL_2(pos(0), pos(s(y1))) → COND_EVAL_21(and(false, false), pos(0), pos(s(y1)))
EVAL_2(pos(s(x0)), pos(s(y1))) → COND_EVAL_21(and(true, false), pos(s(x0)), pos(s(y1)))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(y0, neg(x1)) → COND_EVAL_21(and(greater_int(y0, pos(0)), true), y0, neg(x1)) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(0), neg(y1)) → COND_EVAL_21(and(false, true), pos(0), neg(y1))
EVAL_2(neg(s(x0)), neg(y1)) → COND_EVAL_21(and(false, true), neg(s(x0)), neg(y1))
EVAL_2(neg(0), neg(y1)) → COND_EVAL_21(and(false, true), neg(0), neg(y1))
EVAL_2(pos(s(x0)), neg(y1)) → COND_EVAL_21(and(true, true), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(0), neg(y1)) → COND_EVAL_21(and(false, true), pos(0), neg(y1))
EVAL_2(neg(s(x0)), neg(y1)) → COND_EVAL_21(and(false, true), neg(s(x0)), neg(y1))
EVAL_2(neg(0), neg(y1)) → COND_EVAL_21(and(false, true), neg(0), neg(y1))
EVAL_2(pos(s(x0)), neg(y1)) → COND_EVAL_21(and(true, true), pos(s(x0)), neg(y1))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

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

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(x0)), neg(y1)) → COND_EVAL_21(and(true, true), pos(s(x0)), neg(y1))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL_2(pos(s(x0)), neg(y1)) → COND_EVAL_21(and(true, true), pos(s(x0)), neg(y1)) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(x0)), neg(y1)) → COND_EVAL_21(true, 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

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

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(neg(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(0), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(0), neg(0)) → COND_EVAL_2(and(false, false), neg(0), neg(0))
EVAL_2(neg(0), pos(0)) → COND_EVAL_2(and(false, false), neg(0), pos(0))
EVAL_2(neg(0), pos(s(x0))) → COND_EVAL_2(and(false, true), neg(0), pos(s(x0)))
EVAL_2(neg(0), neg(s(x0))) → COND_EVAL_2(and(false, false), neg(0), neg(s(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ 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:

COND_EVAL_1(true, x, y) → EVAL_2(x, y)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(neg(0), neg(0)) → COND_EVAL_2(and(false, false), neg(0), neg(0))
EVAL_2(neg(0), pos(0)) → COND_EVAL_2(and(false, false), neg(0), pos(0))
EVAL_2(neg(0), pos(s(x0))) → COND_EVAL_2(and(false, true), neg(0), pos(s(x0)))
EVAL_2(neg(0), neg(s(x0))) → COND_EVAL_2(and(false, false), neg(0), neg(s(x0)))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

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

EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(neg(s(x0)), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), neg(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(s(y0)), pos(0)) → COND_EVAL_2(and(false, false), neg(s(y0)), pos(0))
EVAL_2(neg(s(y0)), pos(s(x0))) → COND_EVAL_2(and(false, true), neg(s(y0)), pos(s(x0)))
EVAL_2(neg(s(y0)), neg(s(x0))) → COND_EVAL_2(and(false, false), neg(s(y0)), neg(s(x0)))
EVAL_2(neg(s(y0)), neg(0)) → COND_EVAL_2(and(false, false), neg(s(y0)), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(neg(s(y0)), pos(0)) → COND_EVAL_2(and(false, false), neg(s(y0)), pos(0))
EVAL_2(neg(s(y0)), pos(s(x0))) → COND_EVAL_2(and(false, true), neg(s(y0)), pos(s(x0)))
EVAL_2(neg(s(y0)), neg(s(x0))) → COND_EVAL_2(and(false, false), neg(s(y0)), neg(s(x0)))
EVAL_2(neg(s(y0)), neg(0)) → COND_EVAL_2(and(false, false), neg(s(y0)), neg(0))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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:

EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(y0, neg(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(0)) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(0), neg(0)) → COND_EVAL_2(and(false, false), neg(0), neg(0))
EVAL_2(pos(0), neg(0)) → COND_EVAL_2(and(false, false), pos(0), neg(0))
EVAL_2(pos(s(x0)), neg(0)) → COND_EVAL_2(and(true, false), pos(s(x0)), neg(0))
EVAL_2(neg(s(x0)), neg(0)) → COND_EVAL_2(and(false, false), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(neg(0), neg(0)) → COND_EVAL_2(and(false, false), neg(0), neg(0))
EVAL_2(pos(0), neg(0)) → COND_EVAL_2(and(false, false), pos(0), neg(0))
EVAL_2(pos(s(x0)), neg(0)) → COND_EVAL_2(and(true, false), pos(s(x0)), neg(0))
EVAL_2(neg(s(x0)), neg(0)) → COND_EVAL_2(and(false, false), neg(s(x0)), neg(0))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(pos(s(x0)), y1) → COND_EVAL_2(and(true, greater_int(y1, pos(0))), pos(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(y0)), neg(s(x0))) → COND_EVAL_2(and(true, false), pos(s(y0)), neg(s(x0)))
EVAL_2(pos(s(y0)), pos(0)) → COND_EVAL_2(and(true, false), pos(s(y0)), pos(0))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(and(true, true), pos(s(y0)), pos(s(x0)))
EVAL_2(pos(s(y0)), neg(0)) → COND_EVAL_2(and(true, false), pos(s(y0)), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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:

COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), neg(s(x0))) → COND_EVAL_2(and(true, false), pos(s(y0)), neg(s(x0)))
EVAL_2(pos(s(y0)), pos(0)) → COND_EVAL_2(and(true, false), pos(s(y0)), pos(0))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(and(true, true), pos(s(y0)), pos(s(x0)))
EVAL_2(pos(s(y0)), neg(0)) → COND_EVAL_2(and(true, false), pos(s(y0)), neg(0))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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:

EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(and(true, true), pos(s(y0)), pos(s(x0)))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(and(true, true), pos(s(y0)), pos(s(x0))) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

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

EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1)
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(pos(0), y1) → COND_EVAL_2(and(false, greater_int(y1, pos(0))), pos(0), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(0), neg(s(x0))) → COND_EVAL_2(and(false, false), pos(0), neg(s(x0)))
EVAL_2(pos(0), pos(s(x0))) → COND_EVAL_2(and(false, true), pos(0), pos(s(x0)))
EVAL_2(pos(0), pos(0)) → COND_EVAL_2(and(false, false), pos(0), pos(0))
EVAL_2(pos(0), neg(0)) → COND_EVAL_2(and(false, false), pos(0), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

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

COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
EVAL_2(pos(0), neg(s(x0))) → COND_EVAL_2(and(false, false), pos(0), neg(s(x0)))
EVAL_2(pos(0), pos(s(x0))) → COND_EVAL_2(and(false, true), pos(0), pos(s(x0)))
EVAL_2(pos(0), pos(0)) → COND_EVAL_2(and(false, false), pos(0), pos(0))
EVAL_2(pos(0), neg(0)) → COND_EVAL_2(and(false, false), pos(0), neg(0))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

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

EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(y0, pos(0)) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, pos(0)) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(s(x0)), pos(0)) → COND_EVAL_2(and(false, false), neg(s(x0)), pos(0))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_2(and(true, false), pos(s(x0)), pos(0))
EVAL_2(pos(0), pos(0)) → COND_EVAL_2(and(false, false), pos(0), pos(0))
EVAL_2(neg(0), pos(0)) → COND_EVAL_2(and(false, false), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

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

COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
EVAL_2(neg(s(x0)), pos(0)) → COND_EVAL_2(and(false, false), neg(s(x0)), pos(0))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_2(and(true, false), pos(s(x0)), pos(0))
EVAL_2(pos(0), pos(0)) → COND_EVAL_2(and(false, false), pos(0), pos(0))
EVAL_2(neg(0), pos(0)) → COND_EVAL_2(and(false, false), neg(0), pos(0))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

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

EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(y0, pos(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), true), y0, pos(s(x0))) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(s(x0)), pos(s(y1))) → COND_EVAL_2(and(false, true), neg(s(x0)), pos(s(y1)))
EVAL_2(pos(0), pos(s(y1))) → COND_EVAL_2(and(false, true), pos(0), pos(s(y1)))
EVAL_2(pos(s(x0)), pos(s(y1))) → COND_EVAL_2(and(true, true), pos(s(x0)), pos(s(y1)))
EVAL_2(neg(0), pos(s(y1))) → COND_EVAL_2(and(false, true), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

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

COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
EVAL_2(neg(s(x0)), pos(s(y1))) → COND_EVAL_2(and(false, true), neg(s(x0)), pos(s(y1)))
EVAL_2(pos(0), pos(s(y1))) → COND_EVAL_2(and(false, true), pos(0), pos(s(y1)))
EVAL_2(pos(s(x0)), pos(s(y1))) → COND_EVAL_2(and(true, true), pos(s(x0)), pos(s(y1)))
EVAL_2(neg(0), pos(s(y1))) → COND_EVAL_2(and(false, true), neg(0), pos(s(y1)))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

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

EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
EVAL_2(pos(s(x0)), pos(s(y1))) → COND_EVAL_2(and(true, true), pos(s(x0)), pos(s(y1)))

The TRS R consists of the following rules:

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
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

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

EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
EVAL_2(pos(s(x0)), pos(s(y1))) → COND_EVAL_2(and(true, true), pos(s(x0)), pos(s(y1)))

The TRS R consists of the following rules:

and(true, true) → true
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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(true, false) → false

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL_2(pos(s(x0)), pos(s(y1))) → COND_EVAL_2(and(true, true), pos(s(x0)), pos(s(y1))) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(x0)), pos(s(y1))) → COND_EVAL_2(true, pos(s(x0)), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

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

EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))

The TRS R consists of the following rules:

and(true, true) → true
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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(true, false) → false

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

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

EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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(true, false) → false

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(y0, neg(s(x0))) → COND_EVAL_2(and(greater_int(y0, pos(0)), false), y0, neg(s(x0))) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(pos(0), neg(s(y1))) → COND_EVAL_2(and(false, false), pos(0), neg(s(y1)))
EVAL_2(pos(s(x0)), neg(s(y1))) → COND_EVAL_2(and(true, false), pos(s(x0)), neg(s(y1)))
EVAL_2(neg(s(x0)), neg(s(y1))) → COND_EVAL_2(and(false, false), neg(s(x0)), neg(s(y1)))
EVAL_2(neg(0), neg(s(y1))) → COND_EVAL_2(and(false, false), neg(0), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

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

COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
EVAL_2(pos(0), neg(s(y1))) → COND_EVAL_2(and(false, false), pos(0), neg(s(y1)))
EVAL_2(pos(s(x0)), neg(s(y1))) → COND_EVAL_2(and(true, false), pos(s(x0)), neg(s(y1)))
EVAL_2(neg(s(x0)), neg(s(y1))) → COND_EVAL_2(and(false, false), neg(s(x0)), neg(s(y1)))
EVAL_2(neg(0), neg(s(y1))) → COND_EVAL_2(and(false, false), neg(0), neg(s(y1)))

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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(true, false) → false

The set Q consists of the following terms:

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

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

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

EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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(true, false) → false

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

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

EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

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].

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

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

EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL_21(true, x, y) → EVAL_1(minus_int(x, pos(s(0))), y) at position [0] we obtained the following new rules [LPAR04]:

COND_EVAL_21(true, neg(x0), y1) → EVAL_1(neg(plus_nat(x0, s(0))), y1)
COND_EVAL_21(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

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

EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))
COND_EVAL_21(true, neg(x0), y1) → EVAL_1(neg(plus_nat(x0, s(0))), y1)
COND_EVAL_21(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

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

COND_EVAL_21(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL_2(true, x, y) → EVAL_2(x, minus_int(y, pos(s(0)))) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL_2(true, y0, neg(x0)) → EVAL_2(y0, neg(plus_nat(x0, s(0))))
COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

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

COND_EVAL_21(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, y0, neg(x0)) → EVAL_2(y0, neg(plus_nat(x0, s(0))))
COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(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))

The set Q consists of the following terms:

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

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

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(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))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL_1(true, x, y) → EVAL_2(x, y) we obtained the following new rules [LPAR04]:

COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(0)))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), 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))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL_21(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1) we obtained the following new rules [LPAR04]:

COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(s(z0), s(0)), pos(0))
COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(s(z0), s(0)), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(0)))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(s(z0), s(0)), pos(0))
COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(s(z0), s(0)), neg(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))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(s(z0), s(0)), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(z0, 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(0)))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(s(z0), s(0)), neg(z1))
COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(z0, 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))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(s(z0), s(0)), neg(z1)) at position [0] we obtained the following new rules [LPAR04]:

COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(z0, 0), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

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

EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(0)))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(z0, 0), pos(0))
COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(z0, 0), neg(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))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule EVAL_1(pos(s(x0)), y1) → COND_EVAL_1(true, pos(s(x0)), y1) we obtained the following new rules [LPAR04]:

EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

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

EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(0)))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(z0, 0), pos(0))
COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(z0, 0), neg(z1))
EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(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))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL_2(true, y0, pos(x0)) → EVAL_2(y0, minus_nat(x0, s(0))) we obtained the following new rules [LPAR04]:

COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(s(z1), s(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

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

EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(z0, 0), pos(0))
COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(z0, 0), neg(z1))
EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(s(z1), s(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))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(s(z1), s(0))) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(z1, 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

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

EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(z0, 0), pos(0))
COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(z0, 0), neg(z1))
EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(z1, 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))

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

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

EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(z0, 0), pos(0))
COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(z0, 0), neg(z1))
EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(z1, 0))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL_21(true, pos(s(z0)), pos(0)) → EVAL_1(minus_nat(z0, 0), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

COND_EVAL_21(true, pos(s(0)), pos(0)) → EVAL_1(pos(0), pos(0))
COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

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

EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(z0, 0), neg(z1))
EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(z1, 0))
COND_EVAL_21(true, pos(s(0)), pos(0)) → EVAL_1(pos(0), pos(0))
COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

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

COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(z0, 0), neg(z1))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(z1, 0))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL_21(true, pos(s(z0)), neg(z1)) → EVAL_1(minus_nat(z0, 0), neg(z1)) at position [0] we obtained the following new rules [LPAR04]:

COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
COND_EVAL_21(true, pos(s(0)), neg(y1)) → EVAL_1(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

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

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

COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(z1, 0))
COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
COND_EVAL_21(true, pos(s(0)), neg(y1)) → EVAL_1(pos(0), neg(y1))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

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

EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(z1, 0))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL_2(true, pos(s(z0)), pos(s(z1))) → EVAL_2(pos(s(z0)), minus_nat(z1, 0)) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL_2(true, pos(s(y0)), pos(s(s(x0)))) → EVAL_2(pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(y0)), pos(s(0))) → EVAL_2(pos(s(y0)), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

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

EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(y0)), pos(s(s(x0)))) → EVAL_2(pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(y0)), pos(s(0))) → EVAL_2(pos(s(y0)), pos(0))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

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

EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(y0)), pos(s(s(x0)))) → EVAL_2(pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(y0)), pos(s(0))) → EVAL_2(pos(s(y0)), pos(0))

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

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

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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

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

EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1)
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(y0)), pos(s(s(x0)))) → EVAL_2(pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(y0)), pos(s(0))) → EVAL_2(pos(s(y0)), pos(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL_1(true, pos(s(z0)), z1) → EVAL_2(pos(s(z0)), z1) we obtained the following new rules [LPAR04]:

COND_EVAL_1(true, pos(s(z0)), neg(z1)) → EVAL_2(pos(s(z0)), neg(z1))
COND_EVAL_1(true, pos(s(z0)), pos(0)) → EVAL_2(pos(s(z0)), 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

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

EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))
COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(y0)), pos(s(s(x0)))) → EVAL_2(pos(s(y0)), pos(s(x0)))
COND_EVAL_2(true, pos(s(y0)), pos(s(0))) → EVAL_2(pos(s(y0)), pos(0))
COND_EVAL_1(true, pos(s(z0)), neg(z1)) → EVAL_2(pos(s(z0)), neg(z1))
COND_EVAL_1(true, pos(s(z0)), pos(0)) → EVAL_2(pos(s(z0)), pos(0))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

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

COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
COND_EVAL_1(true, pos(s(z0)), neg(z1)) → EVAL_2(pos(s(z0)), neg(z1))
EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule EVAL_2(pos(s(y0)), neg(x1)) → COND_EVAL_21(true, pos(s(y0)), neg(x1)) we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(s(y_0))), neg(x1)) → COND_EVAL_21(true, pos(s(s(y_0))), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

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

COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
COND_EVAL_1(true, pos(s(z0)), neg(z1)) → EVAL_2(pos(s(z0)), neg(z1))
EVAL_2(pos(s(s(y_0))), neg(x1)) → COND_EVAL_21(true, pos(s(s(y_0))), neg(x1))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND_EVAL_1(true, pos(s(z0)), neg(z1)) → EVAL_2(pos(s(z0)), neg(z1)) we obtained the following new rules [LPAR04]:

COND_EVAL_1(true, pos(s(s(y_0))), neg(x1)) → EVAL_2(pos(s(s(y_0))), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

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

COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1))
EVAL_2(pos(s(s(y_0))), neg(x1)) → COND_EVAL_21(true, pos(s(s(y_0))), neg(x1))
COND_EVAL_1(true, pos(s(s(y_0))), neg(x1)) → EVAL_2(pos(s(s(y_0))), neg(x1))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule EVAL_1(pos(s(x0)), neg(z1)) → COND_EVAL_1(true, pos(s(x0)), neg(z1)) we obtained the following new rules [LPAR04]:

EVAL_1(pos(s(s(y_0))), neg(x1)) → COND_EVAL_1(true, pos(s(s(y_0))), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

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

COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1))
EVAL_2(pos(s(s(y_0))), neg(x1)) → COND_EVAL_21(true, pos(s(s(y_0))), neg(x1))
COND_EVAL_1(true, pos(s(s(y_0))), neg(x1)) → EVAL_2(pos(s(s(y_0))), neg(x1))
EVAL_1(pos(s(s(y_0))), neg(x1)) → COND_EVAL_1(true, pos(s(s(y_0))), neg(x1))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND_EVAL_21(true, pos(s(s(x0))), neg(y1)) → EVAL_1(pos(s(x0)), neg(y1)) we obtained the following new rules [LPAR04]:

COND_EVAL_21(true, pos(s(s(s(y_0)))), neg(x1)) → EVAL_1(pos(s(s(y_0))), neg(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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                                              ↳ MRRProof

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

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

EVAL_2(pos(s(s(y_0))), neg(x1)) → COND_EVAL_21(true, pos(s(s(y_0))), neg(x1))
COND_EVAL_1(true, pos(s(s(y_0))), neg(x1)) → EVAL_2(pos(s(s(y_0))), neg(x1))
EVAL_1(pos(s(s(y_0))), neg(x1)) → COND_EVAL_1(true, pos(s(s(y_0))), neg(x1))
COND_EVAL_21(true, pos(s(s(s(y_0)))), neg(x1)) → EVAL_1(pos(s(s(y_0))), neg(x1))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

EVAL_2(pos(s(s(y_0))), neg(x1)) → COND_EVAL_21(true, pos(s(s(y_0))), neg(x1))
COND_EVAL_1(true, pos(s(s(y_0))), neg(x1)) → EVAL_2(pos(s(s(y_0))), neg(x1))
EVAL_1(pos(s(s(y_0))), neg(x1)) → COND_EVAL_1(true, pos(s(s(y_0))), neg(x1))
COND_EVAL_21(true, pos(s(s(s(y_0)))), neg(x1)) → EVAL_1(pos(s(s(y_0))), neg(x1))

Used ordering: Polynomial interpretation [POLO]:

POL(COND_EVAL_1(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(COND_EVAL_21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(EVAL_1(x₁, x₂)) = 3 + x₁ + x₂
POL(EVAL_2(x₁, x₂)) = 1 + x₁ + x₂
POL(neg(x₁)) = x₁
POL(pos(x₁)) = x₁
POL(s(x₁)) = 4 + x₁
POL(true) = 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                                              ↳ MRRProof

                                                                                                                                                                                                                                                                                                                                ↳ 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

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                            ↳ QDP

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

COND_EVAL_1(true, pos(s(z0)), pos(0)) → EVAL_2(pos(s(z0)), pos(0))
EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0))
COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule EVAL_2(pos(s(x0)), pos(0)) → COND_EVAL_21(true, pos(s(x0)), pos(0)) we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(s(y_0))), pos(0)) → COND_EVAL_21(true, pos(s(s(y_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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                            ↳ QDP

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

COND_EVAL_1(true, pos(s(z0)), pos(0)) → EVAL_2(pos(s(z0)), pos(0))
COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(s(y_0))), pos(0)) → COND_EVAL_21(true, pos(s(s(y_0))), pos(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND_EVAL_1(true, pos(s(z0)), pos(0)) → EVAL_2(pos(s(z0)), pos(0)) we obtained the following new rules [LPAR04]:

COND_EVAL_1(true, pos(s(s(y_0))), pos(0)) → EVAL_2(pos(s(s(y_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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                            ↳ QDP

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

COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0))
EVAL_2(pos(s(s(y_0))), pos(0)) → COND_EVAL_21(true, pos(s(s(y_0))), pos(0))
COND_EVAL_1(true, pos(s(s(y_0))), pos(0)) → EVAL_2(pos(s(s(y_0))), pos(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule EVAL_1(pos(s(x0)), pos(0)) → COND_EVAL_1(true, pos(s(x0)), pos(0)) we obtained the following new rules [LPAR04]:

EVAL_1(pos(s(s(y_0))), pos(0)) → COND_EVAL_1(true, pos(s(s(y_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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                            ↳ QDP

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

COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0))
EVAL_2(pos(s(s(y_0))), pos(0)) → COND_EVAL_21(true, pos(s(s(y_0))), pos(0))
COND_EVAL_1(true, pos(s(s(y_0))), pos(0)) → EVAL_2(pos(s(s(y_0))), pos(0))
EVAL_1(pos(s(s(y_0))), pos(0)) → COND_EVAL_1(true, pos(s(s(y_0))), pos(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND_EVAL_21(true, pos(s(s(x0))), pos(0)) → EVAL_1(pos(s(x0)), pos(0)) we obtained the following new rules [LPAR04]:

COND_EVAL_21(true, pos(s(s(s(y_0)))), pos(0)) → EVAL_1(pos(s(s(y_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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                                              ↳ MRRProof

                                                                                                                                                                                                                                                                                                            ↳ QDP

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

EVAL_2(pos(s(s(y_0))), pos(0)) → COND_EVAL_21(true, pos(s(s(y_0))), pos(0))
COND_EVAL_1(true, pos(s(s(y_0))), pos(0)) → EVAL_2(pos(s(s(y_0))), pos(0))
EVAL_1(pos(s(s(y_0))), pos(0)) → COND_EVAL_1(true, pos(s(s(y_0))), pos(0))
COND_EVAL_21(true, pos(s(s(s(y_0)))), pos(0)) → EVAL_1(pos(s(s(y_0))), pos(0))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND_EVAL_1(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(COND_EVAL_21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(EVAL_1(x₁, x₂)) = 3 + x₁ + x₂
POL(EVAL_2(x₁, x₂)) = 1 + x₁ + x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 4 + x₁
POL(true) = 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                                              ↳ MRRProof

                                                                                                                                                                                                                                                                                                                                ↳ 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

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

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

COND_EVAL_2(true, pos(s(y0)), pos(s(s(x0)))) → EVAL_2(pos(s(y0)), pos(s(x0)))
EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule EVAL_2(pos(s(y0)), pos(s(x0))) → COND_EVAL_2(true, pos(s(y0)), pos(s(x0))) we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(x0)), pos(s(s(y_1)))) → COND_EVAL_2(true, pos(s(x0)), pos(s(s(y_1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

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

COND_EVAL_2(true, pos(s(y0)), pos(s(s(x0)))) → EVAL_2(pos(s(y0)), pos(s(x0)))
EVAL_2(pos(s(x0)), pos(s(s(y_1)))) → COND_EVAL_2(true, pos(s(x0)), pos(s(s(y_1))))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND_EVAL_2(true, pos(s(y0)), pos(s(s(x0)))) → EVAL_2(pos(s(y0)), pos(s(x0))) we obtained the following new rules [LPAR04]:

COND_EVAL_2(true, pos(s(x0)), pos(s(s(s(y_1))))) → EVAL_2(pos(s(x0)), pos(s(s(y_1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ MRRProof

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

EVAL_2(pos(s(x0)), pos(s(s(y_1)))) → COND_EVAL_2(true, pos(s(x0)), pos(s(s(y_1))))
COND_EVAL_2(true, pos(s(x0)), pos(s(s(s(y_1))))) → EVAL_2(pos(s(x0)), pos(s(s(y_1))))

Used ordering: Polynomial interpretation [POLO]:

POL(COND_EVAL_2(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(EVAL_2(x₁, x₂)) = 1 + x₁ + x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 2 + x₁
POL(true) = 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

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ 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

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                          ↳ AND

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ MRRProof

                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                          ↳ PisEmptyProof

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.